UC-NRI vV^cVVVX> >>-** v $wgss$$&f$ v^.V^V^.N-v. gggMp|;: :;<;? : ifSi:S;: '--}' Jv*i^v*S?"^S ;:v^;];;^x : ;;; ; ;;, ;? : 'Sf : KW ps : oSj;;;$> $1 1 > E O i> -n i-n\o >-o t un o " ' " * L - ^- CO T^ LO ON CO I CO ON ON I co a 00 I >. 3 ]5> *c i 01 3 "3 "d o [ PQ ^ I 1 1 R I I I I I ! ! ! ^288828 f 88^ tfff4i r)- ^F o" co to : a; >eneion of tolanet for - of UfV d Drag. RbW.o* ai J>rev*c( ty a Ce<* Jwirf usW m cooju^chot? * O f FIG. 28. N.P.L. Aerodynamical Balance. 44 AEROPLANE DESIGN angles, each arm being counterbalanced. The centre lines of these arms meet in a point at which a steel centre is fixed. The weight of the balance is taken on this point, which rests in a hollow cone in a column fixed to the floor of the room. Three degrees of freedom are thus allowed, permitting measure- ments to be made of the moments about the centre lines of the three arms, which constitute a system of rectangular axes of reference. The vertical arm of the balance passes through the floor of the tunnel and supports the model under test. It can be rotated from the outside of the tunnel, and this rotation provides one of the two angle settings which have been shown to be necessary for general work. The two horizontal arms are set along and at right angles to the wind direction, and are used for determinations of lift and drag, or lateral force and drag, as may be required. The forces on the model are counter- balanced by dead weights hung from the ends of the horizontal arms, fine adjustment being provided by the movement of a jockey weight along the arm. The rotation of the balance about the vertical axis is prevented by a torsion wire, the twist in which is measured on a torsion head, and thus the moment about a vertical axis is determined. The force along the vertical axis is measured on a horizontal weighing lever, the force on the model being transmitted to the lever through a vertical rod which slides freely inside the vertical arm of the balance. Two moments are also measured on this latter weighing lever. The model is held to the vertical balance arm by a special attachment, which allows rotation to occur about a horizontal axis in any desired direction, this axis being much nearer to the model than the axis of rotation of the main balance arms. The rotation of this special attachment is controlled by connecting it by a short arm to the top of the vertical sliding-rod. The immediate attachment to the model allows an alteration of angle to be made about a horizontal axis, which is fixed relative to the model. This adjustment can only be made from inside the tunnel. Of the six force and couple measurements necessary for the examination of an unsymmetrically situated model, it is possible to make four simultaneously. Except in special circumstances, however, it is not desirable to make so many observations at the same time, and locking arrangements are therefore provided to reduce the number of degrees of freedom. A simple locking device also holds the balance from movement in any direction when not in use. Rapid oscillations are damped out by means of dash-pots. On the lower part of the vertical arm, weights can be attached THE PROPERTIES OF AEROFOILS 45 which allow changes to be made in the sensitivity of the balance, so that models of greatly varying size can be readily tested. LIFT AND DRAG MEASUREMENTS. For this purpose the balance is supported at the point O only, and a locking device prevents rotation about a vertical axis. The balance is then free to rotate about two horizontal axes only. VERTICAL FORCE MEASUREMENT. The vertical rod in the upper portion of the balance is guided by four rollers, so that it can slide in a vertical direction but not twist. The rod will move along its axis under a force of O'oooi Ib. COUPLE ABOUT A VERTICAL Axis. For this purpose the centre H is held in a conical cup by the spring K, which is not sufficiently powerful to lift the upper centre O off its seating. The couples are therefore measured about an axis through O and II, and special precautions have been taken in the balance to ensure that the axis OH is in the vertical position. The rotation about this axis is controlled by the torsion wire T, the twist being measured on the torsion head TH by the amount of rotation necessary to bring a crosswire attached to the balance into alignment with a crosswire in the microscope M, which is fixed to the balance support. MEASUREMENT OF DRAG ALONE. For this purpose the support for the centre O is lowered until the balance rests on two points on either side, the centre point then being out of use. This measurement is used in those cases where there is no appreciable lift. The Eiffel Laboratory.* Eiffel's early wind-channel ex- periments were conducted in a laboratory erected in the Champ de Mars at Paris. These experiments were carried out to determine the force exerted upon a flat plate, and were made in conjunction with the method of dropping flat plates from the Eiffel Tower in Paris for a similar purpose. Much useful work was carried out in this early tunnel, but in order to be able to experiment at speeds more nearly approaching those of an aeroplane in flight there was built at Auteuil in 1912 a new laboratory and wind tunnel, of which illustrations are shown in Figs. 29-33. The experimental chamber (see Figs. 30 and 32) is an air- tight room. Leading to this room are a pair of funnel-shaped collectors (see Fig. 31, p. 40) through which the air is drawn from the hangar outside (see Fig. 30, p. 46). In the new channel the * From information communicated bv Mons. G. Eiffel. 4 6 AEROPLANE DESIGN outer and inner diameters of the larger collector are 13 feet and 6J feet respectively, and it is 1 1 feet in length. The effect of so reducing the cross-sectional area is to raise the velocity of the t FIG. 29. Sectional Elevation of Wind Tunnel in Eiffel Aerodynamical Laboratory. air stream and diminish its pressure correspondingly. Conse- quently the model is under investigation in a region of high velocity and low pressure (see Fig. 33). By measuring the FIG. 30. Sectional Plan of Wind Channel in Eiffel Aerodynamical Laboratory. difference in pressure between the experimental chamber and the air in the hangar outside, the velocity of the air stream can be deduced from the formula #2_ 2gh Formula 10 where h is the difference in pressure observed. Massing across the experimental chamber the air stream enters the discharger, which is an expanding chamber 30 feet long, leading to the air- screw. This airscrew is actuated by a 50 h.p. electric motor. This discharger serves to lower the velocity and raise the pressure of the air stream, thus reducing the power required THE PROPERTIES OF AEROFOILS 47 to drive the airscrew and returning the air to the hangar without setting up pulsations. A maximum speed of TOO feet per second can be attained in the experimental chamber. The observers are situated on a platform above the air stream and model as shown in Fig. 32, a position which is very convenient for experimental purposes. The investigations carried out by Eiffel include the deter- mination of forces and moments upon flat plates and aerofoils, the resistances of wing structures, scale model tests, the appli- cation of results to full-sized machines, and the performance of model airscrews. The Flat Plate. The force exerted upon a flat plate suspended normally to a current of air is at the basis of ex- perimental aeronautics. A very thorough investigation of this fundamental problem was carried out by Dr. Stanton at the N.P.L. As a result he found that the force (F) varied directly as the area of the plate (A), the square of the velocity of the plate relative to the wind (V), and that the relationship could be expressed by means of the formula F = K AV 2 Formula n g where K is a coefficient depending upon the units used and the size of the plate under investigation. From Formula 1 1 we see that the pressure per unit area = ~ = K^V 2 . Formula n (a) . A S These relationships are independent of the system of units used ; if the units are the foot, pound, and second, the value of K increases from O'52 for a plate 2 inches square to a value of 0*62 for plates between 5 and 10 feet square. Eiffel obtained very similar results at his laboratory, and enlarged the scope of the investigation to include plates of varying aspect ratio, that is, ratio of span to chord. He found that the effect of increasing the span relative to the chord was to increase the normal pressure on the plate. His results are embodied in Table XIV. TABLE XIV. INFLUENCE OF ASPECT RATIO ON THE NORMAL PRESSURE OF A FLAT PLATE. (EIFFEL.) Aspect ratio ... 11-53 6 10 14-6 20 30 41-5 50 Ratio of pressures Rectangle Square I ' r 4 I>07 ri I145 T ' 25 I>34 *' 4 I>435 I>47> 48 AEROPLANE DESIGN It will be seen from the above table that the normal pressure on a plate of aspect ratio 6 is 10%, and of aspect ratio 14-6 is 2 5% greater than that on a square plate of the same area at the same speed. This effect is due to the lateral escape of the air towards the ends of the plate, and will be more fully considered in relation to aerofoil sections. The Inclined Flat Plate. The next step was to investigate the effect of inclining the plate to the direction of the air stream, and this was undertaken both by the American experimenter, 20 30 -fO* 50 60 Angle of hade nee O do" 90 FIG. 34. Effect of Aspect Ratio upon Pressure on Inclined Plane. Langley, and Eiffel, from the latter of whom most of our infor- mation on this problem is derived. It was found that for small angles of incidence of the plate to the direction of the air stream the resultant force on the plate is given by the expression : Force = F = C ^ A V 2 8 Formula 12 and the pressure per unit area : = A = C | Formula 12 (a) Reproduced by courtesy oj M. Eiffel, FIG. 32. Experimental Chamber in Eiffel Laboratory. Reproduced by courtesy ofM. Eiffel. FIG. 33. Model under Test in Eiffel Laboratory. Facing fage 48. THE PROPERTIES OF AEROFOILS 49 that is, in this case the force is proportional to the angle of incidence of the plate. As the angle of the plate relative to the air stream increases, Formula 12 ceases to hold good, and the force tends towards the value given by Formula n. Fig. 34 shows that the pressure on a square plate between the angles of 25 and 90 is greater than that at 90 that is, when the plate is normal to the wind direction. The effect of aspect ratio upon an inclined flat plate is very clearly exhibited by the graphs shown in Fig. 34. The series of curves there drawn are due to results obtained by Eiffel, and give the ratio between the pressure at any angle 6 and the normal pressure, for all angles from o to 90. It will be seen that increase of aspect ratio produces a smaller maximum normal pressure, but that for small angles of incidence the normal pressure is greatest for the largest aspect ratio. The resultant force (F) on an inclined flat plate can be FIG. 35. resolved into two particular components of great use in aero- nautical problems. The first of these components is that per- pendicular to the direction of the air stream, and is known as the Lift ; while the second is the component in the direction of the air stream, and is known as the Drag. These components are illustrated in Fig. 35. It is customary to express these components in the manner shown by the relationships in Formulae 13 and 14. Lift = K y ^AV 2 Formula 13 o Drag = K X ^AV 2 Formula 14 o where K y and K x> known as the Lift and Drag absolute co- E 50 AEROPLANE DESIGN efficients respectively, are dependent upon the angle of incidence. Formulae 13 and 14 may be regarded as the two fundamental r LIFT . equations of aerodynamics, and the ratio is a measure of the efficiency of the surface under test. The determination of the Lift and Drag coefficients for surfaces of various shapes is a function that has been admirably performed by the wind tunnel. Flat Plate moving Edgewise. The investigation of the forces in this case was carried out by Zahm, who expressed the results obtained in the relationship F = K A-93 V 1 ' 86 Formula 15 We shall consider this question further when dealing with the subject of skin friction. These fundamental data, while not directly applicable to practical aeronautical design work, provide an essential founda- tion for reference in the ever-growing field of aeronautical knowledge, and enable the true significance of the co-efficients for objects of special shapes, such as aerofoils and stream-line sections, to be more fully understood. The Aerofoil. Lilienthal was one of the first to investigate by means of scale models the properties of the cambered aero- foil, and to point out its much superior efficiency over that of the flat plate. To-day, the analysis of a wing section enables the values of the lift and drag coefficients to be determined 'over a large range of angles and also provides information concerning the pressure distribution over the upper and lower surfaces. These results are obtained from experiments carried out in wind tunnels upon carefully prepared scale models. The extreme accuracy with which the forces can be measured and the conditions of flight approximated, make wind-tunnel experi- ments of increasing importance and value. To-day, when a new type of t machine is being designed, an accurate model is made and tested, and from the results information may be gathered leading to an increased efficiency in design. From the point of view of aeroplane design, the determina- tion of the lift and drag of an aerofoil for various angles of incidence is the most important measurement required, and it will therefore be useful to consider briefly the most general method of recording these characteristics of an aerofoil and THE PROPERTIES OF AEROFOILS 51 their common features. Table XV. gives the results of tests in the wind tunnel made at the N.P.L. upon an aerofoil section known as the R'.A.F. 6. It will be noted that the lift, drag, and Lift/Drag coefficients are given in absolute units. This is the method now adopted in England in giving the results of tests upon modern aerofoils, and expresses the values of K y and K x in Formulae 13 and 14. Lift/Drag. 4*5 10*9 14-3 14*1 12*9 J1 '4 10*4 9-3 6'9 4*1 3*0 2*6 2*3 The curves obtained from the above results are shown plotted in Figs. 36, 37, and 38, and may be regarded as typical of the curves obtained from tests upon model aerofoils possessing no freak characteristics. It will be observed that the point of no lift occurs at a small negative angle of incidence : that is ? when the leading edge of the aerofoil is inclined downwards to the direction of the wind stream. The actual value of the point of no lift is of importance when considering questions of stability and control. The slope of the lift curve remains practically constant up to a point shown by c in Fig. 36, and is of importance in considering stability. The angle of incidence corresponding to this point is known as the critical angle. The value of the lift corresponding to the maximum Lift/Drag ratio is indicated by the point B (Figs. 38 and 36). The angle of incidence corresponding to this point will approximate very closely to that chosen for the most efficient flying position. Moreover, the value of the lift at this point should be high in order that the area of the planes may not be excessive. On the other hand, it should not approach the point of maximum lift C too closely, or there will be in- TABLE XV. R.A.F. 6 COEFFICIENTS. Angle of Lift coefficient Drag coefficient incidence. absolute. absolute. - 2 0*003 O*02OI 0-074 0*0165 2 0*173 0*0159 4 0*275 0*0193 6 0-354 0*0252 8 0*423 0*0329 10 0*496 0*0433 12 0*564 0*0545 14 o*593 0*0640 16 0-605 0*0875 18 '55 0*1336 20 0*500 0*1665 22 0*476 0*1845 24 0*456 0*2015 52 AEROPLANE DESIGN sufficient latitude for manoeuvring. Upon the value of the critical angle depends the landing speed of the machine ; and for 16' 30' 24* of INCIDENCE FIG. 36. Lift Curve. 7 8- OF INCIDENCE FIG. 37. Drag Curve. a given wing area the aerofoil having the maximum lift co- efficient will give the slowest landing speed. The critical angle THE PROPERTIES OF AEROFOILS 53 is influenced greatly by the shape of the aerofoil and slightly by the aspect ratio. . e IT -2* 0* 16* 30' ANGUE OF INCIDENCC FIG. 38. Typical Lift/Drag Curve for Aerofoil Section. I T e- Angle of Incidence FIG. 39. Variation of Lift/ Drag Ratio with Increase of Speed. After passing the critical angle, the lift diminishes sometimes slowly and sometimes rapidly, there being a corresponding 54 AEROPLANE DESIGN increase in the drag. When testing model aerofoils at low speeds there is occasionally a rapid drop in the lift just after the critical angle, and then a second rise in the value of the lift Profile, of R.AF6 o 8 12 ANGLE OF INOOENCE. 16 FIG. 40. Typical Curves for an Aerofoil Section. Combination of Figs. 36, 37, 38. coefficient to approximately the same value as that at first obtained. On increasing the speed of the air current, however, this temporary depression disappears. THE PROPERTIES OF AEROFOILS 55 Fig. 37 shows the drag curve, from which it will be seen that the drag diminishes to a minimum value between o and 2, and that it remains fairly constant in this neighbourhood, and then follows approximately a parabolic law up to the critical angle, after passing which point there is a very rapid increase. Fig. 38 shows the Lift/Drag curve for the aerofoil whose curves of lift and drag are given in Figs. 36 and 37, and is plotted from the calculated results shown in Table XV. Fig. 39 shows the effect upon the Lift/Drag curve of increasing the speed of the air current in the wind tunnel for the same aerofoil. For aerofoils in general use the critical angle is usually about 1 6, the corresponding lift coefficient varying from '45 to '70. The maximum Lift/Drag ratio occurs at about 4 angle of incidence and varies in value between 15 and 18. The minimum drag so far obtained is about '006. It is usual to incorporate all these three curves on one chart, as shown in Fig. 40. Pressure Distribution over an Aerofoil. Having con- sidered the characteristic points of an aerofoil, it is desirable to investigate the nature of the airflow producing these charac- teristics, and to examine the effect upon this flow of changes in the shape of the aerofoil. To establish the principles underlying the remarkable efficiency of a good aerofoil section as compared with an inclined flat plate, the N.P.L. investigated the distribution of pressure over the surface of an aerofoil. The following infor- mation is taken from the Reports for the years 1911-1912-1913. In order to make a thorough analysis of the pressure distribution over a large range of angles of incidence, it was found advisable to limit the scope of the experiments to three different shapes, namely i. A flat plate. ii. An aerofoil with both surfaces cambered. iii. An aerofoil with the top surface only cambered. The models used are illustrated in Fig. 41, the flat plate being made of thin steel "02" thick, 12" long, and 2j" wide, while the other two models were moulded with wax upon thin brass sheet curved to the desired shape, 12" long by 2|" wide, the upper surfaces of these two models being exactly similar. The pressure was observed at eight different points along the median section, the position of the holes being indicated in Fig. 41. These holes were 1/64" in diameter and each com- municated when under observation with a manometer by means of a length of tubing. All the holes except the one under test 56 AEROPLANE DESIGN were plugged with plasticine, and the whole apparatus was designed to interfere as little as possible with the flow of the air around the aerofoil. The speed of the wind stream was measured in the usual way by observing the pressure difference shown by the Pitot tube, and was found to be about 17 feet per second. N? 1 .' ? 3 + 5 67 N2 , 2 N95 FIG. 41. Aerofoil Sections. The pressure diagrams obtained for these three model aero- foils are shown in Fig. 42. Ordinates below the datum line indicate negative pressure or suction, while those above indicate positive pressure. It will be seen that for ordinary flight angles both the negative pressure over the top surface and the positive pressure over the bottom surface reach a maximum very near to the leading edge and fall away almost to zero at the trailing edge, and for certain angles of incidence they even change sign. It is this phenomenon which accounts for the position of the centre of pressure the point on the chord at which the resultant force acts being much ahead of the centre of the chord for flight angles, and which points to the necessity for making the leading edge of an aerofoil very much stronger than the trailing edge. Applying these results to full-size wings, the force per square foot, at an angle of incidence of 10 and a speed of 60 miles per hour, is about 35 Ibs. at the leading edge and only 2 Ibs. at the trailing edge. The observations show that for each aerofoil there is a critical angle above which the pressure over the upper surface, after passing through a period of extreme unsteadiness, THE PROPERTIES OF AEROFOILS 57 A. 1 B 5 I -I FIG. 42. Distribution of Pressure on Median Section of Aerofoils Nos. i and 3. 58 AEROPLANE DESIGN becomes uniform. For angles below the critical angle the pressure over both surfaces varies with the angle of incidence according to definite laws, but after the unsteady region is passed the distribution over the upper surface becomes uniform, while pressure on the lower surface falls off to an extent sufficient to cause a change of sign near the trailing edge. A determination of the lift and drag on these aerofoils was also carried out, and the results are shown plotted in Fig. 43 (a) and (b). From these curves it appears that the critical angle, above which the pressure distribution becomes unsteady, corresponds to the critical angle of the lift curve at which there is a falling off in the lift and a large increase in the drag. This indicates that these phenomena are due to the sudden alteration in the Aerofoil No.1. Aerofoil No. 2. Aerofoil No. 3. o-ao 0-10 r o o-oo 0* 5* 10* 15 0' 25* Angle of Incidence. -o-io Aerofoil No.3: AeropoilNol^ r (b) o* 5 to* /5 eo* Angle op Incidence. FIG. 43. Lift and Drag Curves for three Aerofoils. pressure distribution over the upper surface, owing to a break- down in the character of the fluid flow in the neighbourhood of this angle. The value of the critical angle and the amount of change that occurs at this point is largely influenced by changing the position of the maximum ordinate of the aerofoil section, as will be seen shortly. A striking peculiarity illus- trated by these pressure distribution curves is that it is possible to have a very high negative pressure or suction near the leading edge when the angle of incidence is such that a positive pressure would have been anticipated. The principle under- lying this departure from expected conditions is known as the ' Phenomenon of the Dipping Front Edge,' the explanation being that the stream-lines approaching the leading edge are deflected upwards before reaching it, and consequently, although the local angle of incidence with the general wind direction may be positive, the actual angle made with the local wind is THE PROPERTIES OF AEROFOILS 59 negative. This upward deflection of the stream-lines is accom- panied by the formation of a general low-pressure region above FIG. 44. Flow past an Aerofoil Section, showing Development of Eddy Motion with Increase of Angle of Incidence. and a high-pressure region below the aerofoil. The photographs reproduced in Fig. 44 show the effect of the disturbance for different angles of incidence. Aerofoil Efficiency. For an aerofoil to be of practical value it is essential that at some angle of incidence there should exist a high value of the ratio of L/D, accompanied by a high value of the lift coefficient. In the case of the flat plate, although the maximum ratio of L/D may be high at ordinary angles, the corresponding value of the lift, as shown by Fig. 43 (a), is much too low for practical purposes. The total lift on the aerofoil is seen from the same figure to be much greater than that for the flat plate, and there is also a much greater 60 AEROPLANE DESIGN range between the angle of no lift and the critical angle, thus allowing much more latitude for adjustment during flight. The most important consideration leading to the greater efficiency of the aerofoil is as follows : Whereas the resultant force on a flat plate can never act forwards of a normal to itself, a good aero- foil section, on account of the upward deflection of the stream- lines shown in Fig. 44, and the consequent pressure distribution over the front portion of the aerofoil, can and usually does have a resultant force upon it acting in a direction well forward of the normal to the chord. These cases are illustrated in Figs. 35 and 45. If the surface of the flat plate offered no resistance to FIG. 45. the airflow, which corresponds to a condition of maximum efficiency, the resultant would be exactly perpendicular to the plate. The effect of skin friction, however, is such that the resultant acts behind the normal to the chord. For the aerofoil the pressure distribution is such that the resultant acts forward of the normal to the chord. Resolving normally and along the chord, we therefore have a component acting along the chord practically in the opposite direction to the drag force, thus reducing the value of the total drag, and thereby increasing the value of the L/D ratio. The increased efficiency of an aerofoil is principally dependent upon the production of this component acting in opposition to the drag. Reference to the curves in Fig 42 shows that this is due to the uneven pressure distribution over the upper surface. If the pressure distribution were uniform, this opposing component would disappear entirely and the drag would be greatly increased, and this is actually what occurs after the critical angle is passed. The more pronounced this uneven pressure distribution effect can be made without causing a breakdown in the airflow, the more efficient the aerofoil becomes. THE PROPERTIES OF AEROFOILS 61 Pressure Distribution over the Entire Surface of an Aerofoil. The experiments just described relating to the pressure distribution over the median section of a model aero- 7. 6, 5. 4. 3, e. t 3" { A- 6 c ft- 1 SL. ,, 18 FIG. 46. Plan and Section of Aerofoil, showing Observation Points. foil were subsequently extended to cover the entire surface, the observations being made at four other sections, all compara- tively near to the wing-tips, as well as at the median section. Ah SccHon A . o -j i-o is a-o Scale of Absolul-c Pressures. Ar SecMon E Incidence. 4 Incidence. 12* Incidence. FIG. 47. Curves showing Pressure Distribution over Aerofoil at Median and End Sections. The positions of these observation points are indicated in Fig. 46, and the results obtained are shown in Fig. 47. Positive pressures are denoted by normals drawn downwards from the 62 AEROPLANE DESIGN upper or lower surface, and negative pressures by normals drawn upwards. These pressures are given in absolute units. To convert to pounds per square foot at V miles per hour, multiply by -00510 V 2 . The pressure distribution is shown for three angles of incidence, o,4,and 12, for the median section and the extreme end section, side by side in order to give a clearer conception of the very different airflow existing at these two sections. It is found that the points of highest pressure on the aerofoil gradually recede from the leading edge, until in the neighbour- hood of the wing-tip the maximum 'pressures occur close to the trailing edge and are due to suction entirely. As a result of this the direction of the resultant lift force instead of being inclined toward the direction of motion, is inclined in the opposite way, and hence its component in the direction of motion increases the drag force. The value of the drift is a minimum at the central section and increases gradually towards the wing- tips and then rises very rapidly at the extreme ends. The lift coefficient falls off considerably near the tips, its value only being about one-half that at the central section. This is due to the lateral escape of the air on the under side of the wing and the influx of air above the wing. The result of this variation in the characteristics of the aerofoil section at the wing-tips is a reduction in the L/D ratio of the \\ing as a whole ; that is, the efficiency of the supporting surface is diminished owing to this effect, which is often called the * End Effect.' Full-scale Pressure Distribution Experiments.* In a paper read before the Aeronautical Society, Captain Farren gave an account of the investigation of the distribution of pressure over the wings of a full size machine when in flight. The method adopted was very similar to that used for model aerofoils. A number of small tubes were run through the wings,, with the outer ends open and fixed at the point in the surface of the wing at which it was desired to measure the pressure. The inner ends of the tubes were connected to manometer tubes so arranged that pressure differences could be recorded photographically. A diagrammatic sketch of the arrangement is shown in Fig. 48. As in the model experiments, all the holes except the one under observation at the moment were sealed up, and great difficulty was encountered in ensuring that there were no leaks in the tubes. Difficulty was experienced in comparing the results with those obtained in model aerofoil experiments, as it was not possible to determine the attitude * Aeronautical Journal, February, 1919. THE PROPERTIES OF AEROFOILS of the machine exactly, but by installing a yawmeter in a vertical plane, it may be possible to record the correct angle of incidence on the photographic record. Rb. FIG. 48. Arrangement of Manometer Tubes for Investigation of Pressure Distribution in Full-scale Machines. Fig. 49 shows a comparison of the pressures obtained in a test upon a model biplane in the wind tunnel, and corresponding full-scale machine tested in the manner indicated above. Aspect Ratio. The pressure distribution diagrams given* in Fig. 47 lead one to expect that the efficiency of a wing surface will be increased by an increase in aspect ratio. Table XVI. shows that this is precisely what occurs. Aspect ratio. TABLE XVI. INFLUENCE OF ASPECT RATIO. Angles of Incidence. 3 6 Lift L 4P Lift L/D Lift 60 '47 ... .62 *54 '60 70 58 73 64 ... 78 84 73 ... "85 '77 ... -90 *94 86 '95 90 ... ' 9 6 I '00 I'OO I'OO I'OO 1*00 1-05 i '06 i "04 no ... 1-04 i -08 i '09 ... i'o8 1-16 ... i '08 9 L/D '55 72 'S3- 92 I'OO I'OO/ 1-15 6 4 AEROPLANE DESIGN An aspect ratio of 6 has been taken as a standard of reference and the lift, and L/D of other aspect ratios expressed (V Q. 0-8 04 O 0-4 O -04 -0* -1-2 -1-6 o-e O-4- -0-8 - UPPER WING H a 59 8-1 LOWER WING 2-15 8-3 1H Full Scale Model FIG. 49. Comparison of Pressure Distribution on Model and Full-scale Biplane. in terms of this unit. It will be seen that the L/D ratio increases continuously with aspect ratio. The actual figures THE PROPERTIES OF AEROFOILS 65 are graphed in Fig. 50, from which it will be seen that the value is about 10 for an aspect ratio of 3, and increases to about 15*5 for an aspect ratio of 8. The maximum lift coefficient remains practically constant, the increased efficiency at high values of aspect ratio being due to reduced drag coefficients. It will also be seen from this figure that as the aspect ratio becomes less, the angle of no lift occurs earlier. Since models of aerofoils and complete wing-spans are almost invariably tested with an aspect ratio of 6, it is only necessary to multiply the values given for the lift, and L/D Angle of Inc'derx* FIG. 50. -Effect of Aspect Ratio upon Lift/Drag Ratio. coefficients by the appropriate factor in Table XVI., in order to obtain the correct value for any aspect ratio between 2 and 8. The Relative Importance of the Upper and Lower Surfaces of an Aerofoil. The pressure distribution curves given in Fig. 42 show that at ordinary angles of flight the negative pressure or suction over the upper surface is much greater numerically than the positive pressure on the lower surface. In the case of the flat plate, the upper surface con- tributes about 75% of the total force normal to the chord over the greater part of the range of angles under consideration, 66 AEROPLANE DESIGN while for aerofoils the upper surface contributes practically all the normal force at from o to 4, and quite 75% of this force at 12. Since at these angles the force normal to the chord is scarcely distinguishable from the lift, it can be stated as a general rule that the lower surface of any aerofoil never pro- vides more than 25% of the lift. This is an important consideration from trie constructional point of view, in that it shows the necessity of securing the canvas forming the upper surface of the wing very firmly to the ribs in order to prevent it being torn away in an upward direction. There are no forces parallel to the chord in the case of the flat plate and in that of the aerofoil with flat undersurface excepting skin friction. For the cambered undersurface the lower surface contributes only I2j% of the total force at 12, while for angles below 7 the force on it is in the direction of positive drag and is therefore disadvantageous. An examination of the pressure distribution curves for the aerofoils and plate makes it possible to compare the variation of pressure distribution upon (a) a flat lower surface coupled both with a flat and a convex upper surface, and (b) a convex upper surface coupled both with a flat and a concave lower surface. As a result, it is found that the forces on the upper surfaces of aerofoils are only slightly affected by change of shape in the lower surface. For the lower surface, however, it is found that the percentage change due to variation of the form of the upper surface is considerable ; but as these forces are small in magnitude, this change has very little influence upon the total forces. These results demonstrate that the upper surface of an aerofoil contributes by far the greater part of the total force acting upon the aerofoil, and that the pressure distribution is practically independent of the shape of the lower surface, provided that it is not convex. As a corollary, the best form of upper surface can be deter- mined in conjunction with some standard lower surface, say a flat one, and when this has been completed, the lower surface can be varied without appreciably upsetting the results obtained for the upper surface. A detailed investigation upon these lines was carried out by the N.P.L. in order to determine the best form of aerofoil. The lift and drag of a series of aerofoils were measured, variations in the shape of these aerofoils being made according to the following plan : I. Aerofoils with a plane under surface, but with variable camber of upper surface. THE PROPERTIES OF AEROFOILS 67 2. Aerofoils possessing the same form of upper surface, but with variable camber of lower surface. 3. Aerofoils in which the position of the maximum ordinate was altered. As the results of these experiments are of considerable practical value in the design of aerofoils for specific purposes, they will be given fully. Determination of the Lift and Drag of a series of Aerofoils with plane lower surface and variable camber of upper surface. The variation of camber of these aerofoils was obtained by varying the height of the maximum ordinate .29 of chord - FIG. 51. Dimensions of Aerofoils of Variable Camber. kept in the same position at '29 of the chord from the leading edge from "063" to -437' '. This ordinate is then divided into ten equal parts and abscissae drawn in the positive and negative direction from each point of division. The lengths of these abscissae remained constant for the series of aerofoils. The scheme is shown in Fig. 51, and the resulting aerofoils are shown in Figs. 52 and 53. The numbers attached to the aerofoils are in order of the depth of the maximum ordinate. The length of each aerofoil was 15" and the width was 2-5". The velocity of the air stream in the wind tunnel during the tests was 20 miles per hour. The result of the observations is shown by Figs. 52 and 53. The aerofoil with the maximum ordinate begins to lift at an angle of 7, the maximum lift being obtained at an angle of 6. With diminishing camber 68 AEROPLANE DESIGN the angles of no lift and of maximum lift become greater, and the decrease of the lift coefficient after passing the critical angle becomes much less marked. For all the aerofoils the L/D curves show maxima between 3 and 4, but the actual values of these maxima vary greatly. As the camber changes, the L/D ratio approaches and passes a maximum in the neighbourhood of 15, the corresponding camber being about one in twenty. FIG NO I NO 2 HO 3 ANQUE OF INCIDENCE FIG. 52. Aerofoils with Variable Camber of Upper Surface. From an aerodynamical point of view, the most important characteristics of an aerofoil are : (i) The maximum L/D ratio obtainable. (ii) The value of the lift coefficient at the angle of maximum L/D. (iii) The ratio of the value of the lift coefficient at the angle of maximum L/D to the value of the lift coefficient at the critical angle. It will be seen from the curves that the aerofoils having a high maximum value for L/D ratio have a low value for the corresponding lift ; but since the ratio of this lift value to the THE PROPERTIES OF AEROFOILS 69 maximum lift coefficient is also low, such aerofoils are suitable for variable speed machines. Table XVII. was prepared to indicate the best camber. It was assumed that the aerofoils had been arranged at such angles of incidence as to give the same lift coefficient at the same speed. The lift coefficient corre- sponding to usual practice is about 0*25, and for this value the best L/D ratio is 15, and the camber required is '055. If for constructional purposes it is desired to use a larger camber, column 5 shows the extent to which this may be done without decreasing the L/D ratio by more than 10%. Afc-7 FIG. 53. Aerofoils with Variable Camber of Upper Surface. Lift coefficient absolute. "10 '20 25 3 '35 40 45 TABLE XVII. CAMBER. Camber for Corresponding Maximum maximum L/D. maximum L/D. lift. very small .. less than '02 . . . 043 15-0 - ... o'5 *55 15-0 ... o'57 06 14-5 '59 06 14*0 '59 06 I3'3 '59 07 I2'2 o'6o 08 in o'6o Camber for L/D - 10% decrease. o'o6 0-08 0-093 0*106 0*115 0-137 AEROPLANE DESIGN Determination of the Lift and Drag of a Series of Aerofoils with the same Upper Surface and Variable Camber of Lower Surface. The scheme of these aerofoils is shown in Fig. 54, together with the resulting aerofoils. Aero- foil 4 of the previous series (see Figs. 52 and 53) was taken as the basis, and camber given to the lower surface by gradually increasing the height of the maximum ordinate from the chord line according to the dimensions attached to Fig. 54. It was FIG. 54. Aerofoils with Variable Camber of Lower Surface. found that the L/D ratios are practically unaltered by camber of the lower surface. The value of the lift coefficient, as will be seen from Fig. 54, increases steadily with increase of camber ; but the variation is small, a maximum increase of 17% being obtained at the angle of maximum L/D. The critical angle is unaltered by increase of camber on the lower surface, and the fall in the lift coefficient after this angle is passed becomes less as the camber is increased. THE PROPERTIES OF AEROFOILS 71 Determination of the Lift and Drag of a Series of Aerofoils, the Position of the Maximum Ordinate being varied. The sections were all developed from one chosen section by altering the position of the maximum ordinate of the upper surface, the lower surface being kept plane, and are illustrated in Fig. 55. The column headed ' Ratio x/c' gives 72 AEROPLANE DESIGN the position of the maximum ordinate, and the column headed * Design index ' gives the value of the index ' a ' in the expression x = J c (o*76) a . The original series contained only members whose design indices were o, I, 2, 3, 4, and the other members were introduced as occasion required in order to preserve con- tinuity in the observations. The cutves obtained from the observations on the nine aerofoils are also shown in Fig. 55. The most important deduction from the experiments is that for the particular camber adopted (o'ioo), the greatest maximum T^- ~ occurs when the position of the maximum ordinate is at K x about one-third of the chord from the leading edge. The main variations in the lift curves occur at angles above 10. Below this angle the curves are of the same general character, although they differ widely at higher angles, and in certain cases are greatly changed by minute changes of the form of the section. It will be seen that for aerofoils o and i there is no defined critical angle, the lift following a continuous smooth curve having a flat maximum between 16 and 18. In the next aerofoil, design index i, a region corresponding probably to uncertain flow is observed between 17 and 18, the lift coefficient oscillating between 0*67 and 0*54. The next aerofoil, design index I J, shows this effect more strongly marked ; while suc- ceeding aerofoils show this peculiar dip in the lift curves becoming steadily wider and shallower. The wind velocity for these experiments was 28 feet per second. With an increased velocity this dip was practically eliminated. It is interesting to note that all the more complicated changes in the value of the lift coefficient occur with the aerofoils whose indices are between I and 2 ; that is, they correspond to a movement in the position of the maximum ordinate of '012 of the chord, and that the form of the curve is very sensitive to minute changes of the section. The sudden change in the lift coefficient at the critical angle is always accompanied by a change in the drag, an increase in lift being associated with decrease of drag and vice versa. This indicates that the change is due to a sudden alteration in the flow from an efficient to an inefficient type. T The ratio of -^/- increases as the maximum ordinate moves K x from the centre of the chord, until its position reaches a point about one-third of the chord from the leading edge. It would seem preferable, however, to avoid the uncertainty of flow above described and to use an aerofoil having its maximum ordinate THE PROPERTIES OE AEROFOILS 73 at about '375 of the chord from the leading edge. This results in a reduction of the maximum aerofoil, from 13*9 to 13*2. jr , for this particular type of Effect of thickening the Leading Edge of an Aerofoil. These experiments were devised in order to show the way in which the behaviour of an ordinary aerofoil is influenced by substituting a thickened for a sharp leading edge. The sections FIG. 56. Aerofoils of Variable Thickness of Leading Edge. of the aerofoils are shown in Fig. 56. All the aerofoils are identical behind the maximum ordinate, and the camber and chord remain unchanged throughout the series. The results of the observations are shown plotted in Fig. 56, from which it will be seen that the maximum L/D decreases steadily as the thick- ness of the nose increases, showing that the efficiency of an aerofoil section is impaired by thickening the leading edge. The lift is not greatly ^affected below angles of 8, but above this angle the form of the curve is sensitive to the increasing thick- ness of the nose. The final effect on the lift is to cause the critical angle to occur much earlier and to flatten out the lift curve after this angle is reached. 74 AEROPLANE DESIGN Effect of thickening the Trailing Edge of an Aerofoil. These experiments were undertaken in order to determine the extent to which an aerofoil can be thickened in the neighbour- hood of the rear spar without materially affecting its aero- dynamical properties, such extra thickness being very desirable in this region from a constructional point of view. The sections of the aerofoils used are shown in Fig. 57, No. 3 of the series being the same as the R.A.F. 6 aerofoil illustrated in Fig. 40. The observations are shown plotted in Fig. 57, from which it 10* OF INCIDENCE FIG. 57. Aerofoils with Variable Thickness of Rear Portion. appears that the lift coefficient is not much affected at angles greater than 7, while the L/D curves show a steady improve- ment as the thickness diminishes. Centre of Pressure. The position of the centre of pressure (C.P.) of an aerofoil is defined as the point at which the line of resultant force over the aerofoil section cuts the chord. Since the pressure distribution, and hence tlie total force over the aerofoil, varies with the angle of incidence in the manner already described and illustrated in Fig. 47, it follows that the C.P. will also vary in its position along the chord-line. It has been seen that with increasing angle of incidence up to the critical angle, THE PROPERTIES OF AEROFOILS 75 the pressure over the front portion of the aerofoil is greater than that over the rear portion, and as a result the C.P. moves forward. The importance of this fact from the practical point of view must be clearly realised, because the C.P. of a wing section may be regarded as the point at which the resultant lift of the supporting surfaces acts. The position of the centre of gravity (C.G.) of the machine, however, remains unaltered, hence, although for one particular angle of incidence the line of resultant lift can be arranged to pass through the C.G., for all FIG. 58. Travel of the Centre of Pressure. other angles there will be a Lift/Weight couple introduced. Increasing divergence from the position of coincidence of the C.P. with the C.G. will tend to make this couple greater, and consequently the system will become unstable. The function of the tail plane is to provide the necessary righting moment, in order that the machine may be capable of steady flight over the required range of angle of incidence. A knowledge of the variation of the position of the C.P. is therefore essential for a correct setting of the tail in order to obtain stability. It is interesting to recall in this connection that Lilienthal, in his glider experiments, obtained stability by moving his body over 7 6 AEROPLANE DESIGN the lower plane, thus countering the travel of the C.P. by a corresponding movement of the C.G. This travel of the C.P. has also an important bearing upon the design of the wing structure, for it gives rise to a variation in the stresses of the front and rear spar bracing systems as the angle of incidence increases. It is therefore necessary to stress the wing structure for the most extreme cases that occur over the range of flying angles, namely, (a) The most forward position of the C.P. (b) The most backward position of the C.P. Angle of Inc-dencc FIG. 59. Aerofoils with Variable Reflexure of Trailing Edge. The position of the C.P. is determined experimentally by measuring the lift, drag, and the moment about the leading edge of the aerofoil under consideration for various angles of inci- dence. A knowledge of the magnitude of the lift and drag enables the direction of the resultant force to be obtained for each position, and the moment of this resultant force being known, it is a simple matter to calculate the leverage of the moment. This fixes the position of the line of resultant force, and consequently the position of the centre of pressure. The moment and C.P. curves for the R.A.F. 6 aerofoil are shown in THE PROPERTIES OF AEROFOILS 77 Fig. 58. The curves of lift and drag for this aerofoil were given in Fig. 40. Reflexed Curvature towards the Trailing Edge. This research was undertaken principally with a view to determining the extent to which a reflex curvature towards the trailing edge of an aerofoil would tend to neutralise the rapid movement of the C.P. due to the change of the angle of incidence. The sections of the aerofoils used are shown in Fig. 59, No. I of the series being in the form of the R.A.F. 6. The point in the sections at which reflexing was commenced was at 0-4 of the chord from the trailing edge. The same brass aerofoil was used for all the sections, the form being altered behind the point of reflexure by means of moulded wax. The curves for L/D and travel of the centre of pressure are shown in Fig. 59, from which it will be seen that a practically stationary C.P. can be obtained with an aerofoil of this type by elevating the trailing edge by about 0*042 of the chord, while the point of reflexure may be at any point between 0*2 and 0*4 of the chord from the trailing edge. This effect, however, is only obtained at the sacrifice of about i2/ Q of the maximum L/D, and about 25% of the maximum lift. The elevation of the trailing edge, the rate of movement of the C.P., and the loss in the maximum value of the L/D ratio, are connected by approximate linear laws. Interference of Aerofoils. Mention has already been made of the superior efficiency of the monoplane from an aero- dynamical standpoint, due to the absence of interference effects as compared with the multiplane. There are three variables to investigate when dealing with this question, namely, gap, decalage, stagger. We have seen in Fig. 44 how the direction of flow of the air stream is affected when quite a considerable distance away from the leading edge of an aerofoil. It therefore follows that the placing of bodies or other aerofoils in close proximity to the first aerofoil will greatly affect the pressure distribution. When aerofoils are placed above one another/ as in the biplane and triplane, interference and modification of the air forces at once results. Gap. The distance between the superimposed surfaces is known as the gap, and the ratio of gap/chord is used as a measure thereof. The negative pressure or suction upon the upper surface of an aerofoil has been found to be very much greater than the positive pressure upon the under surface (see Fig. 47), and consequently we should expect to find that the 78 AEROPLANE DESIGN effect of placing one aerofoil over another is to reduce the lift and efficiency of the lower plane, and to leave the upper plane practically unaffected. This follows upon the consideration that the positive pressure on the under surface of the upper aerofoil, and the negative pressure on the top surface of the lower aero- foil, will tend to neutralise each other, whereas the negative pressure on the top surface of the upper aerofoil, and the positive pressure on the bottom surface of the lower aerofoil, will remain practically unaltered. The negative pressure or suction being so much more important, it follows that the upper aerofoil must be much less affected. This reasoning is borne out by the experimental investigations which have shown that practically the entire loss due to superposition is to be found in the reduc- tion of the lift and L/D ratio of the lower plane. Further, it may be deduced from this that wing-flaps are very much more effective when placed on the upper plane than they would be if on the lower ; also that in a combination of a high-camber upper plane, with a much flatter lower plane, the interference effects would be greatly reduced. Table XVIII. gives the biplane reduction factors for an average aerofoil, and is taken from an N.P.L. report. TABLE XVIII. REDUCTION COEFFICIENTS DUE TO BIPLANE EFFECT. Lift. L/D. Gap/Chord. 6 9 8 10 6 8 10 0*4 ... - 0*61 0*63 o'62 ... 075 o 81 0*84 0*8 ... 076 0*77 0*78 ... o'79 0*82 0*86 i'o ... o'8i 0*82 0*82 ... 0*81 0*84 0/87 12 ... 0-86 086 0-87 ... 0-85 0-85 0-88 i'6 ... 0*89 0*89 0-90 ... o'88 0*89 0-91 To obtain values for a biplane, multiply values for a single aerofoil by the factors given. Note that there is quite a considerable effect when the Gap is equal to the Chord. A more recent investigation carried out in the Massachusetts Institute of Technology* enables a comparison to be made between the lift and L/D coefficients and interference effects on the biplane and triplane. The biplane and triplane models had a constant gap between the planes equal to 1*2 times the chord length, and there was no stagger or overhang. A single aerofoil was first tested as a standard for reference, and then the addi- tional surfaces were introduced. The lift, and drag, and L/D curves for each case are show in Fig. 60. * Hunsaker and Huff. Reproduced by permission of Messrs. J. Selwyn & Co. THE PROPERTIES OF AEROFOILS 79 From comparison between the curves it will be seen that the triplane and biplane give nearly the same maximum lift at Lift Monoplane Lift Biplane Lift Tri plane. Angle o* Incidence FIG. 60. Aerodynamical Properties of Superimposed Aerofoils. about 1 6, but that for smaller angles of incidence the triplane lift is appreciably reduced. The lift coefficient for the 8o AEROPLANE DESIGN monoplane is seen to be superior to the other cases at all angles above zero. The drag coefficient for angles below 12 is very similar in each case, but at large angles of incidence the triplane has a materially lower resistance. The curves of L/D show the relative effectiveness of the wings. Thus, the best ratio is 17 for the monoplane, 13-8 for the biplane, and 12*8 for the triplane. These values refer to small angles of attack, and therefore correspond to a high flight speed. Table XIX. illustrates these points clearly, the biplane and triplane lift coefficients being expressed as percentages of the monoplane coefficients. TABLE XIX. COMPARISON OF LIFT COEFFICIENTS. Monoplane. Biplane. Triplane. Monoplane. Biplane. Triplane. Incidence. Lift Lift % Lift % L/D L/D % L/D % o ... -096 88'8 83-0 ... 8-6 73-2 70-8 2 ... "202 83-8 75'4 ... i6'8 74*7 69*8 4 ... -284 85-4 75-7 ... 16-8 82-0 76-1 8 ... '427 85*2 77'4 ... 13*8 81*9 80*4 12 ... *545 87^6 81 '2 ... io'o 95'o 89*0 16 ... -543 98-5 96-4 ... 4'5 124-0 145-0 Experiments were next undertaken to determine the distri- bution of load upon the three wings of the triplane made from aerofoils of R.A.F. 6 profile. The results are shown in Figs. 61 and 62. It appears that the upper wing is by far the most effective of the three, and that the middle wing is the least effective. This must be due to the interference with the free flow of air owing to the presence of the upper and lower wings. The results are conveniently tabulated as shown in Table XX. : TABLE XX. COMPARISON OF THE WINGS OF A TRIPLANE. Lift. L/D. Incidence. Upper. Middle. Lower. Upper. Middle. Lower. o ... 2'68 I'o 1-82 ... 3-63 i*o 2-30 2 ... 2-14 I'o 1-75 ... 3-18 i'o 2-13 4 ... 1*91 i'o 1^64 ... 2*59 i'o 1*69 8 ... 1-56 i-o 1-36 ... 1-49 i'o i'37 12 ... 1*56 1*0 1*31 ... 1*30 i'o i '34 16 ... 1*49 i*o 1*20 ... 1*22 i'o i'i7 It will be noticed that the middle wing has been taken as a standard of comparison, its lift and L/D being denoted by unity. A further important instance of interference is to be found in the case of the tail plane. The air stream is deflected from the main wing planes of a machine and takes a downward Upper Lower Rane Plame Tri plane Angle of Incidence Trip lane Biplane 20' 5- 10* IS* 20* Angle of Incidence FIG. 61. Lift and C.P. Coefficients for Superimposed Aerofoils. G 82 AEROPLANE DESIGN course. Consequently the angle of attack of the surfaces behind the main planes must be reckoned with regard to the actual direction of this deflected air stream. The tail plane Angle of lnctdet\oe FIG. 62. Lift/Drag Ratio for Superimposed Aerofoils. operates directly in the downwash of the wings, and this effect must be carefully considered when the setting of the tail plane is being determined. Investigations made by Eiffel and the N.P.L. upon this problem show that the downward direction of THE PROPERTIES OF AEROFOILS 83 the air stream persists for some distance behind the planes, and later experiments have shown that the angle of downwash is half the angle of incidence of the main planes measured, from the angle of no lift. Decalage. The term decalage is used to define the difference in the angle of incidence between two aerofoils of the same machine. For example, the upper plane of a biplane may be set at a different angle to the lower plane ; or the upper and lower planes of a triplane may be set at different angles to the middle plane ; and, again, the setting of the tail plane may be different from the inclination of the main planes. Decalage is illustrated in Fig. 63. It has been found experimentally that the effect of setting Ineidtnce of Ltircr Pi* CXelnq of Tail Ptgx FIG. 63. Decalage. the upper surface of a staggered biplane at about 2 less incidence than the lower surface results in a pronounced increase in the lift, and a small increase in the L/D ratio over any other arrangement. Such a result, however, is modified when different wing sections are used ; and there is room for considerable investigation into the problem of best wing combination, considering gap, stagger, decalage and interference effects. Decalage has the further advantage of reducing the instability of the C.P. curve, and even of stabilising the C.P. travel, if the angle between the surfaces is sufficiently great. Unfortunately this results in a loss in the aerodynamic efficiency of the system. Stagger. When the upper plane is set ahead of or behind the lower plane in the biplane or triplane arrangement the planes are said to be staggered, the amount of stagger being the horizontal distance between a vertical dropped from the leading edge of the upper plane and the leading edge of the lower plane or planes. Positive and negative stagger is illustrated in Fig. 64. In certain machines stagger has been adopted in order 8 4 AEROPLANE DESIGN to give increased visibility, but the constructional difficulties are naturally greater than in the no-stagger arrangement. Positive stagger leads to a slightly increased efficiency over the no-stagger position, but this increase only becomes apparent when the stagger is about half the chord. Under these con- No Sta09r >* FIG. 64. Stagger. ditions there is a gain in. the lift and the L/D of about 5%. Negative stagger, so far as present investigations go, would appear to be approximately of the same efficiency as the no- stagger arrangement. From the designer's standpoint, the question of stagger must be treated in conjunction with the amount of gap desirable, J*"" ""-** / ^"Vy. ^.S' / \ ,/ >^ 10* 1 V' / \ Mf* / / / j 6* f 1 ffl 2 /*' &4t so* I , ,. ' t y ( > Uf|--Ky FIG. 65. since stagger can be used to advantage when the gap is small in order to counteract the loss in aerodynamical efficiency due to interference. The Choice of an Aerofoil. Before concluding this chapter a short space can profitably be devoted to a brief THE PROPERTIES OF AEROFOILS % 85 outline of one or two simple methods of selecting an aerofoil which will be suitable for a wing section for various specific purposes. Curves of lift, drag, and L/D, and travel of the C.P., for some of the most successful aerofoils yet evolved, will be given at the end of this chapter ; and a careful examination of these curves, together with the following matter, will enable the choice of the most suitable aerofoil for certain definite conditions to be made. Having drawn the curves of lift and L/D ratio for an aerofoil as shown in Fig. 40, a further curve can be constructed by eliminating the angle of incidence. This is shown in Fig. 65. For the purposes of preliminary design work and for comparison this method of graphing wind-tunnel results is much more convenient than that shown in Fig. 40, as the angle of incidence is not of importance until the question of the actual position of the wing arises. The method of obtaining such curves is obvious from the figure, the corresponding value of the lift, and the L/D being taken at each angle of incidence. A further method of plotting results useful for preliminary design work is obtained by remembering that the landing speed of a machine depends upon the maximum lift coefficient of the section used. Thus, if V be the landing speed and K the maximum lift coefficient, W = K P A Also for any speed of horizontal flight V' W = K'pAV'2/T where K' is the lift coefficient at the corresponding angle of incidence. Hence, equating these two expressions we have KV 2 = K'V' 2 or V - V (|>) 4 '. .......... Formula 16 By means of Formula 16 the speed at various angles of inci- dence can be determined if the corresponding values of the lift coefficients are known. For example, if we take the R.A.F. 6 aerofoil, we see from Fig. 65 that the maximum lift coefficient is '6 approx., so that if a landing speed of 45 m.p.h. is desired, we have from Formula 16 *;-' so that by substitution of K' (the lift coefficient at any other angle of incidence), the speed at that angle can be obtained. 86 AEROPLANE DESIGN Formula 16 can also be put into the form y_ v that is, the ratio of the landing speed to any other speed may be expressed in terms of the lift coefficients of the aerofoil section. Combining this ratio with the L/D ratio for the aero- foil, a further graph can be obtained as shown in Fig. 66, the calculations for which are arranged in tabular form below. A ~X V ^ ^ S- A 3 / ' 16*: 7 A Values of & FIG. 66. S K' (V) V V o 074 16400 128 2 -173 7030 84 TABLE XXL CALCULATIONS OF V/V. 8 -423 2880 537 4 -275 4420 66-5 6 -354 3440 587 10 '49 6 2450 49-5 12 '5 6 4 2l6o 46-5 14 '593 2050 45-3 f77 "35 '535 ' 6 77 '7^7 '839 '9* '9^8 '994 16 18 600 -55 2O2O 2210 45 47 i '96 10*9 14*3 14*1 12*9 ii'4 io'4 9*3 6^9 4*1 From this curve the most efficient speed and the value of the L/D ratio for the wings at the maximum flight speed required can at once be determined. For example, since the maxi- mum L/D gives the value of V/V as 72, the most efficient flying speed so far as the wings are concerned = 45/72 = 62*5 m.p.h. Also if a maximum speed of 100 m.p.h. is required, the value of V/V' is then = 45/100 = "45, and for this value the curve shows THE PROPERTIES OF AEROFOILS 87 that the L/D ratio is only just over 8, so that this section is not suitable for a high-speed machine. The L/D ratio for the complete machine can only be deter- mined when the drag of the body has been added to that of the wings, but the curve shown in Fig. 66 will ^indicate at a very- early stage in the design whether the wing section chosen is suitable for the desired purpose. It is very convenient for design purposes to graph a large number of tests upon sections in this manner and to file them for future reference, indicating upon each graph the name of the section and the source from which the figures were obtained. All the curves should be drawn to the same scale upon good quality tracing linen, so that one curve can be readily compared with another for minute differences by superposition. The curve in Fig. 66 also shows that a machine can only fly horizontally at a high speed if the angle of incidence of the wings is much smaller than that for which the L/D ratio is a maximum. From what has already been said, it follows that for a machine to have a large range of flying speeds the wings must possess the following characteristics : 1. A large value for the maximum lift coefficient. 2. For small angles of incidence the value of the lift coefficient may be small, but the corresponding value of the L/D ratio must be large. 3. The section should have a large value of the maximum L/D, and the ratio of the maximum lift coefficient to the lift coefficient at the maximum L/D must be large. Practical considerations necessitate that the movement of the centre of pressure over the range of flying angles should be small in order to obtain longitudinal stability, and from a construc- tional point of view the depth of the aerofoil section must be such that an economical spar section can be adopted. Units. The units which are used in the published results of aerodynamic research work in Great Britain are known as absolute units, or absolute coefficients. From Formulae 13 and 14 we have Lift = K y A V 2 g whence K y = Absolute lift coefficient Lift . , , Formula 13 (a) g 88 AEROPLANE DESIGN and Drag = K X ^AV 2 & whence K x = Absolute drag coefficient Formula 14 (a) a A v 8 Similarly the moment of an aerofoil = M c A V 2 ^ Formula 17 where M c represents the absolute moment coefficient and b repre- sents the breadth of the wing chord. It is desirable that all measurements should be made in terms of the same units, whether the C.G.S. or the F.P.S. system is employed. For example, in the 'C.G.S. system, metres, metres per second, kilograms, square metres, etc., should be used ; and in the F.P.S. system, feet, feet per second, Ibs., square feet, etc., should be used. In order to obtain actual values from the absolute coefficients, the absolute values, which are of course independent of any system of units, must be multiplied by the remainder of the expression shown in Formulae 13, 14, 17 expressed in appropriate units. The value of in F.P.S. units for air at sea-level at a tem- g perature of 15 C., and at normal pressure, is '00237, while in the C.G.S. system under the same conditions it is '125. Con- sequently in the F.P.S. system, if we wish to convert absolute values of the lift coefficient to actual values, we have absolute value x -00237 x area in sq. ft. x square of velocity in feet per second while in the C.G.S. system we have absolute value x '125 x area in sq. ms. x square of velocity in ms. per second The Law of Similitude. Since the lift, drag, and L/D coefficients of an aerofoil vary with the speed, as shown by Fig. 39, it is not possible to pass directly from model tests to full-size machines. Lord Rayleigh called attention to this fact, and pointed out that the most general relationship between the quantities connected with aerodynamics could be expressed in the form F = ^V 2 L 2 / Formula 1 8 g THE PROPERTIES OF AEROFOILS 89 where v represents the kinematic viscosity of the air. For the V L condition of dynamic similarity to be satisfied - - must be the same for the model test and the full-scale machine. With a four-foot wind tunnel the scale of the models tested is generally about one-twelfth. Consequently, since the kinematic viscosity may be regarded as constant for the two cases, it would be necessary, in order to preserve dynamic similarity, to test the models at a speed of 1000 m.p.h. . This is obviously impossible, and it has therefore been suggested that a correction factor, known as the V L correction, should be applied to the results of model tests before they are applied to full-scale machines. The N.P.L. and others have investigated this question, but the results so far obtained are not conclusive. Although increase of L/D ratio was obtained with increase of speed, as shown in Fig. 39, this increase was not maintained, and a maximum value would appear to be reached with increase of speed. The latest work on the subject seems to suggest a motion in which the resistance decreases with an increase of viscosity, and Mr. Bair- stow suggests that an increase of viscosity may render this possible by making a different type of motion stable, and so reducing the turbulence of flow. Considering all the available data upon this point, it is apparent that it is at least on the safe side to test a model in the wind tunnel at a speed of from twenty to thirty miles per hour (30 to 44 feet per second), and then to apply the results so obtained without correction to full-scale design. This subject is essentially one upon which the designer must keep an open mind and modify his views as shown to be necessary by the results of the latest published researches into this subject, and by the results of his own applications of model figures to full-scale design. In this connection the findings of a special committee appointed to consider this matter are of interest. They are : 1. For the purpose of biplane design model aerofoils must be tested as biplanes, and for monoplane design as monoplanes. The more closely the model wing tested represents that used on the full-scale machine, the more reliable will the results be. 2. Due allowance must be made for scale effect on parts where it is known. In the case of struts, wires, etc., the scale effect is known to be large, but these parts can be tested under conditions corresponding with those which obtain on the full-scale machine. 90 AEROPLANE DESIGN 3. The resistances of the various parts taken separately may be added together to give the resistance of the com- plete aeroplane with good accuracy, provided the parts which consist of a number of separate small pieces (e.g.) the under-carriage) are tested as a complete unit. 4. Model tests form an important and valuable guide in aeroplane design. When employed for the determina- tion of absolute values of resistance, they must be used with discrimination and a full realisation of the modifications which may arise owing to interference and scale effect. Wing Sections. The dimensions and aerodynamic charac- teristics of some highly successful wing sections are shown in Figs. 67-76. All of these sections have been tested in actual aeroplanes and have proved themselves efficient in flight. They can therefore be confidently recommended for design purposes, the section for any particular machine being selected as explained in this chapter. THE PROPERTIES OF AEROFOILS o-i r o-i -*- o-i r OH -~o-i -" CM*- o-i -*- o-i - o-i 10 FIG. 67. Wing Section No. i. AEROPLANE DESIGN 8' 12" IG 20 FIG. 68. Wing Section No. 2. THE PROPERTIES OF AEROFOILS 93 [ osf os-K o-i f- 01 H*- o-i -+- 01 t- 01 -4- o i j~ Angle of Incidervce FIG. 69. Wing Section No. 3. 94 AEROPLANE DESIGN t i t i i 0-6 06 r\ o V B* 12' Angle ot Incidence FIG. 70. Wing Section No. 4. THE PROPERTIES OF AEROFOILS 95 16 20 FIG. 71. Wing Section No. 5. 9 6 AEROPLANE DESIGN Angle of Incidence. FIG. 72. Wing Section No. 6. THE PROPERTIES OF AEROFOILS 97 10 20* FIG. 73. Wing Section No. 7. H 9 8 AEROPLANE DESIGN -4- o-' j-oi -4-0-1 | FIG. 74. Wing Section No. 8. THE PROPERTIES OF AEROFOILS 99 16 14 12 10 t''1f"~1r"s 9 9 9 TITTTTTTT^ 20' FIG. 75. Wing Section No. 9. IOO AEROPLANE DESIGN FIG. 76. Wing Section No. 10. CHAPTER IV. STRESSES AND STRAINS IN AEROPLANE COMPONENTS. Moments of Inertia. The product of an area and its distance from a given axis is termed the moment of that area about the given axis. Thus in Fig. 77, if d& represent a small element of area of the surface s and y and x, the perpendicular distances of this area from the axes of x and y respectively, then d A .y = the moment of dA with reference to the axis of x dA . x = the moment of dA with reference to the axis of y FIG. 77. First Moment of Area. FIG. 78. Second Moment of Area. The total moment of the surface s about these axes is the sum of such elements as d A multiplied by the distance of each of these elements from the required axis, or Moment of s about the axis ox = ^dA.y Formula 19 Moment of s about the axis o Y = ZdA.x Formula 20 For many purposes the area of the surface S may be regarded as concentrated at a single point C, the position of the point C with reference to any axis being obtained from the relations and A x y = A x x = VdA.y or y A ^dA.x A Formula 21 Formula 22 102 AEROPLANE DESIGN where A represents the total area of the surface S, that is the sum of such elements as d A, that is S d A. The intersection of two such lines as A c and B c in Fig. 77, obtained by means of these two formulae, gives the position of the centroid c, which for a homogeneous lamina corresponds to the centre of gravity. The product of an area by the square of its distance from a given axis is termed the Moment of Inertia of the area about the given axis. Thus in Fig. 78, using the same notation as in Fig. 77, we have d A . jy 2 = moment of inertia of element d 'A about the axis of x d A . x* = moment of inertia of element dA about the axis of y and the total Moment of Inertia of the whole surface S is the sum of such elements multiplied by the squares of their respec- tive distances from the given axis, whence Moment of Inertia of s about o x = S d A . y 2 = I xx Formula 23 Moment of Inertia of s about o Y = 2 d A . x 1 = I YY Formula 24 The term ' moment of inertia ' is somewhat misleading, and, as will be apparent from Figs. 77 and 78, the term * second moment ' is much more applicable. The term moment of inertia is, how- ever, in general use. Now, in Fig. 78, if K be such a point that A x (yj= S^A./= I xx A x (X) 2 = S,' represents the distance from the line of reference x'x' of the mid-ordiriate of each of the sections into which the strut has been divided. Since the strut is 7" long, was divided by lines *i" apart, and since -05" has been adopted as the unit, the figures in the first column (y) will be the odd numbers commencing with i and running up to 139. The second column, headed '.ar/ shows the breadth of each of the mid-ordinates whose distance from the line of reference has been given in the first column. A diagonal scale can easily be constructed for reading off these lengths to any required degree of accuracy, Column three, headed ' a' represents the area of each of the sections, and is obtained from column two by multiplying each breadth by the depth of the section. Since, in the example, the depth of each section is constant, and equal to two units, column three is obtained from column two by multiplying by two. The total of column three gives % a, that is, the area of the section STRESSES AND STRAINS IN COMPONENTS 105 shown in terms of the unit employed. To obtain the area in square inches, we must therefore divide by the square of the unit, that is by 400, whence the area of the section is equal to lO'Oi square inches, as shown. The empirical formula for finding the area of the section illustrated is A = 2-5 /4 whence A = 10 sq. ins. so that the agreement is very close. The fourth column, headed ' ay] gives the first moment of each section about the axis of reference X'X'. Its total therefore represents ^ay, and by dividing this total by #, we obtain the position of the centroid of the section with regard to the line X'X 7 . As shown, this distance is S'8/'. Column five is obtained by multiplying column four by ' yl and gives the second moment, or moment of inertia, of the sections with reference to the axis X'X'. Dividing the sum % ay 1 of this column by the fourth power of the unit used, gives the moment of inertia of the whole section about X'X' in inch 4 units. The result, as shown in Fig. 79, is 178*18. Applying the principle of Parallel Axes to find the moment of inertia about the line through the centroid parallel to X'X', the figure 29*26 is obtained, as shown. The moment of inertia about an axis at right angles to X'X' can be found in exactly the same manner. Since, however, the section is symmetrical about Y V, it is only necessary to consider one-half of the section, and to multiply the results obtained by two, in order to obtain the correct results for the complete section. As will be seen from Fig. 80, the moment of inertia for the section about Y Y = 2-35 inch 4 units. The empirical formula for finding the moment of inertia of this section about Y Y is M.I. = -15 /* = 2*4 inch 4 units. The accuracy obtained in Figs. 79 and 80 is far greater than is generally required in practical work, since a wooden strut cannot be made so accurately as these figures show, and even if made so accurately would not retain its accuracy unless fully protected from atmospheric effects. Consequently the labour involved in preparing a table such as is shown in Fig. 79 can be considerably reduced by taking the distance apart of the sections 2" instead of *i", since the form of the section with reference to the axis X'X' does not change very rapidly. Since the form of the section changes fairly rapidly with reference to the axis Y Y, it is not advisable to increase the distances apart of the sections FIG. 79. Moment of Inertia of Streamline Section about Axis XX. Area of Sechor? DisVaTtce of Line tVough Cent-rcid frot? X'X of IrrerVia - gay* _ 28509652-1 20* 20*- 178-16 inch 4 unfa Moment' of IgerHa atouV XX --U -'Ay 3 * 178-18 - 149-92 = 29- 26 mcfc^ unite X' Y y x a ay ay* i ... 5*8 ir6 ... 11*6 11*6 3 .. 7'6 15-2 ... 45-6 136-8 5 ... 8-4 16*8 ... 84-0 420*0 7 .. 9'4 1 8-8 ... 131-6 921*2 9 ... 10*8 21*6 ... 194*4 1749-6 ii ... 11-8 23-6 ... 259-6 2855*6 13 ... 12-8 25-6 ... 332-8 4326-4 15 ... 13*8 27-6 ... 414*0 6210*0 17 ... 15*0 30*0 ... 510-0 86700 19 ... 16-0 32*0 ... 608*0 11552-0 21 ... 17-2 34-0 ... 722-4 15170-1 23 ... 18-2 36-4 ... 837-2 19255-6 25 ... 19-4 38*8 ... 970*0 24250-0 27 ... 20'0 40*0 ... 1080*0 29160-0 29 ... 21-2 42*4 ... 1229-6 35658*4 31 ... 22'2 44-4 ... 1376*4 42668-4 33 ... 23-2 46*4 ... 1531*2 50529-6 35 ... 24-4 48-8 ... 1708-0 59780-0 37 .- 25-0 50-0 ... 1850-0 ' 68450-0 39 ... 26-0 52-0 ... 2028*0 79092-0 FIG. 79. Moment of Inertia of Streamline Section (continued}. y X a ay ay* 41 27-0 54-0 2214*0 90774-0 43 28-0 56*0 2408*0 103644-0 45 29'0 58-0 2010*0 117450-0 47 29-4 58-8 2763-6 129889-2 49 30-4 60-8 2979-2 145980-8 51 31-2 62 '4 3182-4 162302*4 53 32-0 64*0 3392-0 179776*0 55 32-6 652 3586-0 197230*0 57 33-6 67-2 3830-4 218332-8 59 34'4 68-8 4059-2 239492-8 61 35-0 70 'o 4270*0 260470*0 63 35'6 71-2 4485-6 282592-8 65 36-2 72-4 4706-0 305890-0 67 36-8 73'6 4931*2 330390*4 69 37'4 74-8 5161-2 356122*8 71 38-0 76-0 5396-0 383116*0 73 38-4 76-8 5606-4 409267-2 75 38-8 77'6 5820*0 436500*0 77 39-2 78-4 6036-8 464833-6 79 39-4 78-8 6225-2. 491790-8 81 39'5 79-0 6399-0 518319-0 83 39"6 79-2 6573-6 545608-8 85 397 79"4 6749-0 573665-0 87 39'8 79-6 6925-2 002492-4 89 40*0 80-0 <.. 7120-0 . 633680-0 91 40*0 80-0 7280-0 662480*0 93 40*0 80-0 7440-0 691920*0 95 400 80-0 7600-0 722000-0 97 40*0 80-0 7760-0. 752720-0 99 39'8 79-6 7880-4 780159*6 101 39"6 79-2 7999'2 807919*2 103 39'o 78-0 8034-0 827502-0 105 38-4 76-8 8064*0 846720-0 107 38-0 76*0 8132-0 870124*0 109 37-6 75*2 8196-8 893451*2 in 37'o 74-0 8214*0 911754*0 113 36-6 72-2 8271*6 934690*8 115 35*4 70-8 8142*0 936330-0 117 34'6 69*2 8096-4 947278-8 119 33'4 66-8 7949' 2 945954-8 121 32-6 65-2 7889-2 954593'2 123 3i'o 62-0 7626-0 937998*0 125 29*6 59-2 7400*0 925000-0 127 27-6 55-2 7010-4 890320*8 129 25-6 51-2 6604-8 852019*2 131 23-2 46-4 6078*4 796270*4 133 20-4 40-8 5426-4 721711*2 135 I7'2 34'4 4644*0 626940-0 137 I2'2 24-4 3342-8 457963-6 139 4'6 9-2 1278-8 177753*2 4005*2 309724*8 28509652*1 io8 AEROPLANE DESIGN parallel to this axis, and as will be seen from Fig. 80, the labour involved in this case is not very considerable. y X a ay ay 7 ? f 39-9 279-8 279-8 279-8 3 138-0 27-0 828-0 2484-0 5 139-9 259-8 1299-0 6495-0 7 120-8 24 1-6 169! -2 1 1836.4 9 1 r 8 22 1-6 1994-4 I7949-6 1 1 103-7 2O7-4 2281 -4 35O95-4 1 3 a?'* 174-8 2272-4 2-9541-2 1 5 76-0 152-0 2280-0 34200-0 17 58-2 H6-4 1976-6 33779-6 1 9 360 72-0 1 366-0 25992-0 2001, -4 16273-0 f87655-0 . 187655 20* - 2-35 incV> 4 FIG 80. Moment of Inertia of Streamline Section about Axis Y y. Nomograms, sometimes called alignment charts in England, can be prepared for some of the formulae given in Table XXII. and many other formulae in use in aeronautics, from which the value of the moment of inertia, or other quantity for which the nomogram has been constructed, can be read off im- mediately within the limits of the graduations. Fig. 8 1 shows a nomogram constructed to give the moment of inertia of a rectangle, that is the quantity - To use nomograms it is very convenient to scribe a straight line on the under side of a large celluloid set square. Fig. Si is then used in this manner. Suppose that it is required to find the moment of inertia of a rectangle whose breadth is '6" and whose depth is 2" '. The line scribed on the set square is placed over the '6 graduation on the breadth scale and swung round until it is over the 2" mark on the depth scale. Where the line cuts the moment of inertia scale gives the answer, and as will be seen this gives the moment of inertia as '4 inch 4 units. The same nomogram can also be used to find the moment of inertia of a square placed either with its axis parallel to or diagonal to the line of reference, remembering that the reading on the STRESSES AND STRAINS IN COMPONENTS 109 breadth scale must be the same as the reading on the depth scale. It can also be used to find the moment of inertia of a hollow rectangle, I beam, channel, or hollow square, by FIG. 81. Nomogram for determining the Moment of Inertia of Rectangle, Square, Hollow Rectangle, Channel and ' I ' Sections. finding the difference between the moments for the whole and the missing portion. The following example will help to make this clear. no AEROPLANE DESIGN To find the M.I. of the box section illustrated in Fig. 82 : M.I. of whole section = -667 from nomogram. M.I. of missing portion = -137 M.I. of the box section = '53 inch* units. FIG. 82. Shear Force and Bending Moment. To obtain a clear idea of these quantities the following definitions must be carefully considered : The shearing force at any point along the span of a beam is the algebraic sum of all the perpendicular forces acting on the portion of the beam to the right OR to the left of that point. The bending moment at any point along the span of a beam is the algebraic sum of the moments about that point of all the forces acting on the portion of the beam to the right OR to the left of that point. Notice that since the beam is in equilibrium, the algebraic sum of the forces or the moments about any point considered on BOTH sides of the beam must be zero. Consequently the same value will be obtained for the shearing force or the bending moment, irrespective of whether we work from the right-hand end or the left-hand end. The cases illustrated in Table XXIIL, on pages 111-13, are of fundamental importance, and should be thoroughly well known before any attempt is made to apply the results to aeronautical design work. STRESSES AND STRAINS IN COMPONENTS in z o o ui _j u. UJ o h Z UJ o o Q Z u CO U p 1 ! 1 I JKI m tH ;- , l 1 ! 1 1 r Sit ir f S if i M it ill 4 * >J -. Jl Hi 1 1 1 1 <3 -^_ LO 112 AEROPLANE DESIGN X CO - DO CO VJllUUOJ IE 11 SI s * Is o c- u CD ll! 1 if il r- Jill *14* dUifr * iin Hi OD f^ij s4l 11 J^o I ^il [J u 01 \W 'li i!! MT r 1 1 ? jS ;_ O iU STRESSES AND STRAINS IN COMPONENTS 113 u II IKS Irli MIS llfl LI 0) ^ EH aii s s* *:* it i l*i! ?Ui | IV Q-5-5-* o CDS I? * I ft (0 < Hi CQ o O ti 4 AEROPLANE DESIGN Stresses in Beams. The assumptions made in the Theory of Bending should always be remembered, because any formula derived from the Theory of Bending rests upon these assump- tions. Consequently, when these assumptions do not hold good, the resulting formula cannot be applied with safety. Neglect of this almost obvious precaution is the root of practically all cases of discrepancy between theory and practice. In order to obtain a theory at all, certain assumptions have had to be made. Persons quite ignorant of theory find a formula in a pocket- book, apply it to a case (or cases) where the assumptions made in deriving the formula do not hold good, and when as a result failure occurs, the blame is laid at the door of theory. The chief assumption made in the Theory of Bending is what is known as Bernoulli's Assumption, namely : 1. Transverse plane sections of a beam which are* plane before bending remain plane after bending. The other assumptions made are 2. That Hooke's Law holds good. 3. That the modulus of Elasticity (E) is the same in tension as in compression. W 1 fa' FIG. 83. Distribution of Longitudinal Stress for I Section. 4. That the original radius of curvature of the beam is great compared with the cross-sectional dimension of the beam. In simple bending, the external forces producing bending form a couple which is balanced by the internal forces in the fibres of the beam. These internal forces form another couple, and are the resultants of the tensile and compressive stresses in the beam. Let Fig. 83 (a] represent a beam subjected to bending, whose cross-section is shown in Fig. 83 (b). Then, owing to the bend- ing moment M at a cross-section such as c D, the distribution of longitudinal stress will be as shown in Fig. 83 (c). The line N A, which passes through the point of no stress, is known as the neutral axis of the beam. Since /c J'c the neutral axis (N A) must pass through the C.G. of the section.. STRESSES AND STRAINS IN COMPONENTS 115 Considering two transverse sections of a beam which are very close together, it will be seen from Fig. 86 that, in order to fulfil Bernouilli's assumption, after bending, the bounding lines are no longer parallel, but that a layer such as A D has been stretched, while a layer such as B C has been compressed. It is obvious that there must be an intermediate layer, such as M N, which is neither stretched nor compressed. This layer is known as the neutral axis (N.A.). Produce A B, CD, Fig. 86 (b) to meet each other in o, and let the angle contained by these two lines contain a radians. Let the radius of curvature of the neutral surface M N = R, and let the height of any layer such as p Q from the neutral axis y. Then, from Fig. 86 (b) PJJ = (R + y)a = M N R a and the strain at a layer such as P Q, is equal to Extended length - Original length = (R+jy)a-Ra_j Original length Ra ~ R and the longitudinal tensile stress intensity at a distance y from the N.A. within the limits of elasticity = /' = E x strain These longitudinal internal forces form a couple which is equal to the bending moment at every cross section, and us known as the Moment of Resistance. Expressing this couple in terms of the dimensions of the cross section and equating to the bending moment, we have Combining these several results in one expression, we have " Formula 48 From this equation we see that - y R The ratio I/y is known as the modulus of the section. This modulus is generally denoted by the letter Z, the suffix V or 1 c* being added according as the beam is in tension or com- pression. n6 AEROPLANE DESIGN ,. SHEAR STRESS. The complementary maximum horizontal shear stress generally occurs at the neutral axis, the distribution for any cross-section being given by the expression F /" Y q / y , b . dy Formula 49 I bj v where q mean intensity of shear stress at a distance y from the N. A. F = shearing force on the cross section of the beam. I = moment of inertia of the cross section. b = breadth of the cross section, having a particular value outside the integral, but varying with the distance y inside the integral. FIG. 84. Distribution of -Shear Stress for I Section. A numerical example will make the use of Formula 49 clear. Consider the I section shown in Fig. 84 (a] : 1 At A, q = o 6 ^*,*-- .'. for the inner edge, of the flange and for the outer edge of web At c, on the neutral axis, +4-5) - .n? STRESSES AND STRAINS IN COMPONENTS 117 and since the evaluation of the integral gives rise in each case to^ 2 , the curve of shear stress is a parabola, whence the distri- bution of shear stress can be drawn from these figures as shown 7 86 -- T/O in Fig. 84 (). From this figure it is clear that the web carries most of the shear, and it is usual on this account to design the web on the assumption that it carries all the shear, which gives a result on the safe side. ii8 AEROPLANE DESIGN Relation between Load, Shear, Bending Moment, Slope, and Deflection. Let A B, Fig. 85, represent a beam carrying a continuous load w per unit of length, and x a length of this beam so small that whether w is constant or variable, for the distance x it can be regarded as constant. Then the forces acting upon this beam for the section con- sidered are as shown in Fig. 85. Equating upward and downward vertical forces, we have F + 2 F = F + w . S # whence & F = w . b x = w ............ Formula 50 b X that is, in words, the rate of change of the shearing force is numerically equal to the loading; and alternatively, the integra- tion or summation of the loading diagram between the correct limits gives the shear force curve. Again, equating moments about D for the external forces acting upon the section of length S x> we have and since S x represents a quantity of the first order of small- ness, products containing two of these small quantities can be neglected. Hence M + F.S# = M + 6M or & M ^ T- i - = r ............ Formula 5 1 o x or in words, the rate of change of the bending moment is equal to the shearing force ; and alternatively, integration of the shear force curve gives the bending moment curve. The curvature of a beam in accordance with the Theory of Bending is given by the relation A a-| 3 Formula 2 R ( ends fxn- jointed C*s>- 777" , 13 FIG. 88.- -Variation of Strut Formula, with Method of Fixing Ends. with a force P applied at each end of the rod, then if the deflection at a distance * x ' from A is ' ' yl or Let = M y = - - E I . ffiy dx* E I . P El then on substitution we have This is a differential equation satisfying the given conditions, and therefore a solution of this equation will also be a solution STRESSES AND STRAINS IN COMPONENTS 125 of the problem. Looking at this differential equation, we note that l y ' is a function such that its second derivative must be proportional to itself. This condition is satisfied by a sine or cosine function of the form y a sin (b x + c) ............ 2 where #, , and c are constants to be determined by the con- ditions of the case. Since this is a function of the sine, we see that the shape into which the column will be bent must be sinusoidal. Differentiating equation 2, we have dyjdx = abcos(bx + Ltd.'l FIG. 117. ' Vickers 16 D' Scout, fitted with 200 h.p. Hispano Engine. Facing page 144. DESIGN OF THE WINGS 145 1. The extra stresses produced by the failure of one of the members ; 2. The extra stresses produced in the building of the machine. The first requirement is of great importance in the case of military machines where an exposed part is liable to be shot away or seriously weakened by a bullet. The second is liable to occur owing to the uneven or excessive tightening up of the bracing system. The general overloading produced in what may be termed abnormal flight is to be allowed for in the design of the wings and attachments by assuming the load to be several times the flying weight of the machine, the multiplying factor varying for different aeroplanes according to the purpose for which they are designed. Attention must be given to the stresses induced by the breaking of a part, and a good design will provide against this wherever possible. Stresses due to counterbracing must be allowed for, but in building the machine it is essential to take the utmost care that these stresses are not unduly increased, and that in ' tuning up ' (a phrase sometimes used to describe the forcible straining of bad work into its correct shape), the mechanics are not given too free a hand. Experimental Investigation of the Stresses upon a Full-size Machine during Flight. One of the fundamental formulae of applied mathematics, which follows directly as a deduction from Newton's Second Law of Motion, states that Force = Mass x Acceleration. For an aeroplane in flight, the mass is practically constant, hence a determination of the forces set up in the wing structure will follow from a knowledge of the acceleration of the machine under various conditions. This principle has been adopted in the full-scale experiments carried out at the Royal Aircraft Establishment. An instrument called an Accelerometer is used to indicate photographically the acceleration of the machine in terms of the earth's attraction that is, in terms of the force of gravity. In addition to giving a measure of the resultant air force on the machine, the instrument also measures the time of rapid manoeuvres. Figs. 119 and 120 show records obtained from this instru- ment in actual use. Fig. 119 indicates the accelerations set up on an S. E. 5 Scout machine and a Bristol Fighter during a mock fight, while Fig. 120 indicates the accelerations and speeds of flight of a Bristol Fighter during various manoeuvres. It will L 146 AEROPLANE DESIGN be observed that in the case of the S. E. 5 the maximum stresses nowhere exceed three times the weight of the machine. In the case of the Bristol Fighter when manoeuvring, the maximum I b/D 2 ! '** - o } w- O stress occurs during a loop, and is less than four times the weight of the machine. The diagrams indicate that the stresses in the machines keep remarkably steady during flight. DESIGN OF THE WINGS 147 .b -c < PQ Accelerations in terms of g. 148 AEROPLANE DESIGN / Stresses. The stresses to which the members of an aero- plane are subjected, and which it is necessary to consider in design, are as follows : 1. Stresses set up in the wing structure during flight. These may be sub-divided into two classes : (a) Those due to Lift and Down forces. These forces are j>rincipally transmitted by the external bracing of the machine. (fr) Those due to the Drag forces. These forces are trans- mitted by the internal (or the drag) bracing. 2. Stresses set up in the wing structure and under-carriage during landing. The resultant landing shock may be resolved into a vertical component known as ' down shock/ and a hori- zontal component known as ' end shock.' The extent of these forces is dependent upon the slope at which the machine descends, and the nature of the landing-ground. 3. Stresses set up in the fuselage due to the operation of the control surfaces, and to the thrust of the airscrew. When rotary engines are fitted, there are also gyroscopic effects developed. The consideration of these stresses will be dealt with as follows : The stresses resulting from the various conditions arising in flight will be treated in this chapter ; those due to landing conditions will be considered in the chapter on the landing chassis ; while the fuselage stresses will be investigated in Chapter VII. Stresses in the Wing Structure. The process of de- termining the stresses in a wing structure consists of 1. Finding what proportion of the load is carried by the various parts of the structure. 2. Determining what stresses these loads induce in the members of the structure. In dealing with the first part of the problem it is necessary to know (a) the distribution of loading along the span of the wing ; (b} the distribution of the load over a section of the aerofoil. The first of these two considerations is largely dependent upon what is termed 'end effect.' Unless specially determined data for the wing section to be used is known, it is customary to assume the load grading near the wing tips as being para- bolic, the variation extending for a distance equal to the chord from the wing tips. We shall return to this point again later. The second consideration, the distribution of pressure over the DESIGN OF THE WINGS 149 wing section, has been fully considered in Chapter III. There remains to be determined what proportion of the load is carried by the spars at the minimum and maximum flying angles. This clearly depends upon the position of the centre of pressure of the section at these angles. The actual travel of the C.P. for the wing section adopted must be found by reference to its aero- dynamical characteristics. In general, it is found that at large angles of incidence, corresponding to slow speeds, the C.P. is at about 0-3 x chord from leading edge, while at small angles and high speed it is at about 0*5 X chord. As an example, consider the wing section shown in Figure i?i. For a travel of the C.P. from 0*3 to 0-45 of the chord, we have Maximum proportion of total load on front spar = (44 i2*8)/44 = 71 ; maximum proportion of total load on rear spar = (44 - 12) {44 = 73. FIG. 121. Care must of course be taken when determining these stresses that the most unfavourable condition is considered. Thus, in the above example, the front spars are stressed for the most forward position of the C.P. ; while the rear spars are stressed for the most backward position of the C.P. In this manner the maximum possible stresses in either frame are determined. Wing Loading. Wing loading may be defined as the ratio weight of machine : area of supporting surface. Hence from the fundamental equation (Formula i) we have - loading = K y V 2 A g The value of wing loading depends upon the maximum speed of the machine. For very fast machines the wing loading is high, while for a slow machine it is generally low. The advantage of a low value of wing loading is that it gives a large margin of safety against excessive loads. Such a machine will 150 AEROPLANE DESIGN require more engine-power to fly at a given speed than will a machine with high wing loading. The general tendency to-day is to adopt a higher wing loading than was formerly used, and the majority of machines in present use have a wing loading factor of from 7 to 10 Ibs. per square foot. Table XXV., giving particulars of some of the leading machines of the day, is very instructive in this respect. TABLE XXV. WING LOADING OF MODERN MACHINES. Machine. Wing Area. Weight. Loa ding. Sq. feet. Lbs. Lbs. / sq. feet. Airco 4 434 3340 ... 7-4 Airco 9 438 ... 3351 ... 7-6 Airco 10 ... ... 840 ... 8500 ... io - i2 A. R. (French) 484 ... 2750 ... 5 72 Avro ... ... ... 346 . . 2680 ... 8*23 A. W. Quadruplane ... 400 ... 1800 ... 4-5 Blackburn Kangaroo ... 868 ... 8017 ... 9*2 Breguet 528 ... 3380 ... 6-38 Bristol Fighter ... ... 405 ... 2630 ... 6-5 Bristol Monoplane ... 145 ... 1300 ... 8*97 Bristol Triplane ... ... 1905 ... 16200 ... 8*50 Bristol All Metal 458 ... 2810 ... 6*13 Caproni 990 ... 8730 ... 884 Caproni Triplane ... 2690 ... 14630 ... 5-45 Caudron 427 ... 3170 ... 7-42 Fokker Triplane 215 ... 1260 ... 5-9 F. F. Bomber (German)... 750 ... 6950 ... 9-3 Handley Page 6-400 ... 1645 ... 14000 ... 85 Handley Page V- 1 5 oo ... 2950 ... 28000 ... 9*5 Morane Parasol ... ... 145 ... 1440 ... 9*92 Nieuport ... ... ... 160 ... 1200 ... 7^47 S. E. 5 ... ... ... 249 ... 1980 ... 8*0 Sopwith 344 ... 2040 ... 5-93 Sopwith Camel ... ... 231 ... 1440 ... 6*2 Sopwith Dolphin... ... 258 ... 1910 ... 7*4 Sopwith Snipe ... ... 274 ... 1950 7'i Sopwith Triplane . 251 ... 1500 ... 6'o Spad 195 ... 1550 ... 8-08 Vickers' Commercial ... 1330 ... 11120 ... 8-4 Wing Weights. The wing weight varies as the wing loading per square foot of surface, and ranges from 0*5 Ibs. per square foot on small machines up to 1*4 Ibs. per square foot on very large machines. The following formula for the DESIGN OF THE WINGS 151 weight of aeroplane wings are taken from the 1911-1912 N.P.L. Report : 1. Weight of Monoplane wing = o 017 W(A)* + 0*16 A 2. Weight of Biplane wing = 0-012 W(A)i + 0-16 A where W = weight of machine less the weight of the wings. A = area of the wings in square feet. The second term represents the weight of fabric in above formulae. In Chapter I. we saw that the weight of the wing structure of modern machines is about 13 per cent, of their total weight. This figure represents the best that has been attained in practice so far, and is the result of much careful attention to detail, so that any improvement upon it will not be easy, but it should form a standard of reference to which to work. Stresses due to Downloading. In normal flight the re- sultant air force on the wings acts in an upward direction, thereby supporting the machine, in opposition to the force of gravity. Under certain circumstances, however, the pressure on the wings may be reversed in direction, as for instance when the elevator is depressed and the machine commences to dive. The force necessary to change the line of flight of the machine depends upon the radius of the turn with which the machine commences to dive. The centrifugal force upon the machine The reaction to this force must be provided by a down pressure on the wings. For example, let P = force on wings. W = weight of machine. V = its velocity. R = radius of flight path. V 3 / V 2 \ Then P - W - - = W i - ) j For the machine referred to in the consideration of factors of safety (p. 143), if flying at 100 m.p.h. and then suddenly directed downwards on a path of radius 170 feet, the downloading will be = 1200 I - i47 2 = 1200 (l -3'94) = - 2'94 X 1200 or the downloading indicated by the negative sign is about three times the weight of the machine. AEROPLANE DESIGN The effect of such downloading is to throw the down- bracing wires, which are shown dotted in Fig. 122, into operation. A further instance of the occurrence of downloading stresses is during the time that the machine is resting on the ground, the downbracing wires then being loaded by the weight of the wing structure itself. It is customary to design the wing structure for down- loading on the assumption that the down forces are one-half as great as the maximum lift forces. This is conveniently ac- counted for by adopting a factor of safety equal to one- half that used for the lift forces. Investigation of the stresses set up in the drag bracing of the wing structure will be dealt with later in this chapter, after the questions of duplication and stagger have been considered. FIG. 122. Stressing of the Wing Structure. The method of drawing the stress diagrams for the external bracing system of an aeroplane is as follows. The method adopted is similar for all types of machines, but as the biplane is the type in most general use, our attention will be confined to this type of structure for the present. The general procedure to be adopted will first be outlined, and then an example of its application to practice will be given. (i.) NORMAL FLIGHT. Let the total weight of the machine whose wing structure is to be designed be W, and let w be the weight of the wing structure alone. Then the load actually carried by the structure is W w, since the wings themselves are directly supported by the air pressure, and thus relieve the struts and wires of having to transmit any stresses due to their weight. The air pressure is transmitted by the wing fabric to the ribs, which in turn transmit the load to the spars which are braced to the body. An aeroplane being symmetrical about its longitudinal axis, it is only necessary to determine the stresses due to half the weight on one side of the machine. Consider a biplane as shown in Fig. 122.' Let T be the area of top plane and B the area of lower plane. DESIGN OF THE WINGS 153 We must first determine the proportion of the load carried by each plane. Owing to its greater area and also to biplane effect, the top plane will carry most of the load. Let e be the biplane coefficient, taken from the table given on page 78, for the particular ratio of gap/chord used. (VV - w) Then mean pressure over top plane = , - ^ and mean pressure over lower plane = P" = e P' Therefore load on top plane = W = T P' load on lower plane = W" = B P" The variation in load grading at the wing tips known as 'end effect ' must now be taken into account. We shall assume the load grading over the outer section of the wing to be parabolic. The general result of this is to reduce the effective area of the FIG. 123. planes. For this reason various shapes have been given to the wing tips in the endeavour to minimise this loss as much as possible ; but it is very doubtful if these special shapes are worth the extra trouble and labour involved from the practical point of view. Let T' and B' be the effective areas of the top and bottom planes respectively, obtained from T and B by sub- tracting the area lost through end effect. Then we have W Maximum pressure on top plane = W" Maximum pressure on bottom plane = B The curve of loading along the span can now be set out, since it is the product of the pressure per square foot and the chord length at that point. Curves for both the top and bottom planes must of course be drawn. They will be similar in nature to Fig. 123. The next step is to. determine the reactions at the supports. The spars approximate very closely to a continuous beam, and consequently the determination of the reactions involves the use of the Theorem of Three Moments as shown at D, Table XXII. 154 AEROPLANE DESIGN Graphical methods have been developed for determining these reactions, but for the simple loadings usually met with in aeronautical practice the Theorem of Three Moments is much quicker and simpler, and gives results which are quite satis- factory. Let w v w 2 , w^ etc., be the loading per foot run over each bay, determined from the load curve previously drawn. Considering first the top plane, let M A> M B) M c> &c., be the bending moments at A, B, c, etc. See Fig. 123. The bending moment at A is due to a varying upward load over a cantilever of length L p and must be determined by means of graphic integration if very accurate results are required. If the loading diagram be assumed parabolic over the outer section, the' bending moment may be easily calculated. With this assump- tion, M = W-, Lj x o . 4 Lj where W x is the average loading over this span. Then bending moments at B, c, D, etc., are deter- mined by applying the theorem of three moments. For the spans A B, B C, we have, referring to Fig. 41, M A L 2 + 2 M B (L 2 + L 3 ) + M c L 3 - \ (w 2 L 2 3 + 0/3 L 3 3 ) = o For the spans B C, CD, we have, by a further application of the theorem, M B L 3 + 2 M c (L 3 + L 4 ) + M D L 4 - \ (w, L 8 + w, L/) - o For the spans c D, D E, by a further application, M c L 4 + 2 M D (L 4 + L 5 ) + M E L 5 - I K L/ + w 5 V) - o Since the wing span is symmetrical, the support moment at D equals the support moment at E, and the support moment at A has previously been determined. Hence we have three equations to determine the three unknown moments at B, C, D. These can be easily obtained by successive substitution in the above equations. Knowing the bending moments at the supports, it is now easy to determine the various reactions by taking moments. Let R A , RB, Re, &c., be the reactions. Then taking moments about B for the reaction at A, we have W T 2 w l Lj 1 (o. 4 L! + L 2 ) + 2 2 - - R A L 2 - M B W. L 2 j (o . 4 Lj + L.J) + - - M, or R A DESIGN OF THE WINGS 155 Again taking moments about C for R B , we have (T \ W T 2 _? + L ) + 3 3 - R A (L 2 + L 8 ) - R 8 L 8 = M c Proceeding in this manner the reaction at each support can be obtained. The formulae look somewhat formidable, but their application is quite simple, and, with practice, both bending moments and reactions can be determined very quickly. The application of the theorem of three moments as used above assumes that the points of support are in the same straight line. In practice this is frequently not the case, the most notable difference being obtained after the process of tuning-up. Considerable errors are likely to be introduced in this manner, and if it is impossible to avoid this occurring a fresh set of bending moments must be obtained, assuming each point of support to be out of the straight line by say J" this will produce large differences. In this case the more general form of the theorem of three moments must be used, namely : 6A2*_ 2 + 6^ + MA ^ + 2 MB ( ^ + ^ + McLs ^2 Ij 3 + 6 E I U + f-M = o Formula 47 \J-/2 -L'3 / where A 2 , A 3 denote areas of free bending moment diagrams over second and third spans x. 2 denotes the position of the C.G. of A 2 from the support A x s denotes the position of the C.G. of A 3 from the support C 2 denotes the distance of B below A 3 denotes the distance of B below C A further proviso in the application of this theorem is that the bracing wires are attached in such a manner that the re- actions pass through the neutral axis of the spar. In practice this is not always easy to obtain, and in such cases the Bending Moment diagrams will be somewhat modified. Having determined the reactions at each support, the stress diagrams for the structure considered as a single vertical frame with pin joints can now be drawn as shown in Chapter II. An example of such a diagram is shown in Figs. 22 and 125. (ii.) DOWNLOADING. The procedure adopted in determining the stresses due to downloading is exactly similar to that out- 156 AEROPLANE DESIGN lined above for normal flight. In this case the reactions at the points of support will be downward. Reference has already been made to the fact that it is customary to design the wing structure for downloading forces of one-half those obtained in normal flight. As the application of the centre of pressure and factor of safety is being left over until the question of detail design of the members- is being considered, the reactions due to downloading may with advan- tage be set out equal in magnitude but opposite in direction to the lift reactions. The stress diagrams for downloading can now be drawn. It must be remembered in this case that it is the downbracing wires which are in operation. Fig. 126 illustrates a downloading stress diagram. ILLUSTRATIVE EXAMPLE. Before proceeding to show how to determine the detailed stresses in each member of the wing structure, we will illustrate the methods just described by means of a practical example, and draw the stress diagrams for the external bracing of the biplane shown in Fig. 124 (a) and (). The weight of the machine is 2000 Ibs. and the weight of the wing structure is 300 Ibs., the chord of the wings is 6 ft., span of top plane 40 ft., span of lower plane 31 ft. AREA OF TOP PLANE. AREA OF BOTTOM PLANE. Overhang = 2\ x 6 = 13*5 Overhang 2 x 6 = 12 AB = 4 x 6 = 24 B'C' = 39 BC = 6-5 x 6 = 39 C'D' - 36 CD = 6 x 6 = 36 IDE = J(2- S x 6) = 7'5 120 87 Distribution of load over upper and lower planes : Upward force to be distributed = 2000 - 300 = 1700 Ibs. = 850 Ibs. per side. Biplane effect : The ratio of gap/chord is unity, hence from Table on p. 78 the factor is 0*82 850 .*. Average pressure top plane = - = 4-44 Ibs./sq. ft. I2O + o2 X 07 and average pressure on bottom plane = 0*82 x 4-5 = 3^67 Ibs./sq. ft and load on top plane = 4*44 x 120 = 532 Ibs. load on bottom plane = 3*67 x 87 = 318 Ibs. Total = 850 Ibs. DESIGN OF THE WINGS '57 FIG. 1 c ;u D *kp r Line D'agnum of t-Whinft Sole -370 -C40 fSTo -KXO -WHO 4- lnjicar*sis Tens FIGS. 124 to 126. Method of setting-out Stress Diagrams. 158 AEROPLANE DESIGN Reduction of effective area due to end effect. Assuming parabolic loading over the outer 6 ft. ( = chord) of each plane, The equivalent loss in area = 6x6 x '33 = 12 sq. ft , and hence the effective area of the top plane = 120-12 = 108 sq. ft. arid of lower plane = 87 - 12 = 75 sq. ft. .*. Maximum pressure on top plane = - = 4/94, say 5 Ibs./sq. ft. I Oo Maximum pressure on bottom plane = =4*25 Ibs./sq. ft. The loading diagram for the planes can now be drawn as in Fig. 124 (c) (d), since load per foot run equals pressure at that point multiplied by the width of the chord, which in this example is constant and equal to 6 feet. Having determined the load distribution, we can proceed to find the Fixing or Support Moments. From Fig. 124 (c) the load on the overhang = (2*25 x 30 x '6) = 27*0 Ibs. Bending Moment at A due to this load = 27*0 x -f x 2^25 = 24-3 ft. Ibs. Applying Theorem of Three Moments to the spans A B, B c, we have 4 M A + 2 MB (4 + 6-5) + M c (6-5) - 1 (24-4 x 4 3 + 30 x 6'5 3 ) = o or 21 MB + 6-5 M c = 2350 (i) For the spans B c, CD, we have 6-5 MB + 2 M c (6-5 + 6) + 6 M D - i [30 (6- 5 3 + 6 3 )] = o or 6-5 M B + 25 M c + 6 M D = 3675 (2) For the spans CD, D E, we have 6 M c + 2 M D (6 + 2-5) + 2-5 M E - i [30 (6 3 + 2- 5 3 )] = o or 6 M c + 17 M D + 2-5 M E = 1730 (3) From symmetry M D = M E = c 19-5 Substituting for M D in (2) we have 6-5 M B + 25 M c + 532 - 1-85 M c = 3675 DESIGN OF THE WINGS 159 Substituting for M c in (i) we have r 2I M B + 884 - 1-83 M B = 2350 or i9'i7 MB = 1466 .-. M B = 77 ft. Ibs. Substituting this value in (5) = 3143 -jog = II4ft . lbs . 23'15 Substituting this value in (4) : M D = !^f^- 54 ft. Ibs. = M E We can now determine the Reactions. Taking moments about B for R A 27 {4 + 4 (2-25)} + 100 X 2 - 4 R A - M B or 132 + 200 - 4 R A = 77 whence R A = 64 Ibs. Taking moments about c for R B 27 {4'9 + 6-5} + ioo x 8-5 + 3^-_5)_ _ IO - 5 R A - 6-5 R B =M C or 308 + 850 + 634 - 672 - 6-5 R B = 114 whence R B = 155 Ibs. Taking moments about D for R c 30 x 6 2 27 (11*4 + 6) + ioo x 14-5 + 30 x 6'5 x 9^25 + - - 16-5 R A - [2-5 R B - 6 R c = M D or 470 + 1450 + 1800 + 540 - 1050 - 1940 - 6 R c = 54 whence R c = 202 Ibs. Taking moments about E for R D 27 (i7'4 + 2-5) + ioo x 17 + 30 x 6*5 x 11*75 + I ^ x 5'5 + - --^-^ - 19 R A - 15 RB - 8-5 R c - 2-5 R D =M E or 53^ + 1700 + 2290 + 990 + 94 - 1215 - 2330 - 1720 - 2-5 R D = 54 whence R D = 115 Ibs. Sum total of Reactions = 64 + 155 + 202 + 115 = 536 Ibs. The load on the top plane was found to be 532 Ibs. The slight difference is due to the fact that the loading was taken at 5 Ibs. per square foot instead of the more accurate figure of 160 AEROPLANE DESIGN 4-94 Ibs. per square foot. The total sum of the reactions should always be checked in this manner. In a similar manner the fixing moments and reactions at the lower plane supports can be determined. It should be noted that there are no lift forces over the portion D'E', .which repre- sents the base of the fuselage. The reactions are : R B ' 78 Ibs. ; R C ' 176 Ibs. ; R D < - 62 Ibs. The vertical reactions due to the lift forces being known, the next step is to draw the stress diagrams on the assumption that the frame is pin-jointed. The front elevation of the machine is set out to scale, and the stress diagram for the lift forces drawn in the usual manner, as shown in Fig. 125. In some cases it may be found more convenient to draw a diagram for both front and rear frames separately, by applying the requisite C.P. factor to give the maximum loading on each ; but in this example a single diagram will suffice, and the stresses in each frame can be determined by applying the correct coefficients afterwards. Having completed the diagram, the stresses in each member due to the lift forces in horizontal flight should be tabulated. The stress diagram for the downloading forces is shown in Fig. 126. General Procedure for Design of the Members of the Wing Structure. From the stress diagrams of the machine, considered as a single vertical frame and with a loading on the wings equal to its weight, the actual loads in the various members upon which . their design is, of course, based are determined by applying the necessary centre of pressure coeffi- cients and the requisite factor of safety. For all front-frame members of the wing structure the maximum stresses will be incurred with the most forward position of the centre of pressure during flight, and the maximum stresses in the members of the rear frame of the wing structure will be incurred with the most backward position of the centre of pressure during flight. As will be seen later, however, this condition of affairs may be modified by the method of duplication which is employed. By reference to the characteristics of the aerofoil which has been selected the travel of the C.P. is known, and hence the maximum proportion of the load which can fall on either front or rear frame can at once be determined in the manner shown on page 149, As the factor of safety to be adopted is complicated by the method of duplication employed, this question will now be considered. DESIGN OF THE WINGS 161 DUPLICATION OF THE EXTERNAL WING BRACING. The possibility of having one or other of the wing-bracing members shot away, or otherwise rendered inoperative in flight, makes it necessary to consider this eventuality when designing a machine. In certain cases it is desirable to provide an alterna- tive path whereby the lift reactions on the wing may be trans- ferred to the body. Such duplication provides an additional safeguard against the failure of a bracing wire or fitting due to faulty material or bad workmanship. Two methods of dupli- cation are in general use : I. Direct duplication, in which method two wires are inserted one behind the other ; and each one capable of taking two-thirds of the maximum load likely to fall on this member. In the event of one wire failing, the other wire will transmit the load ; but the system would now have a factor of safety of only two-thirds that FIG. 127. Transmission of Forces in Biplane Truss with Broken Wire. of its previous value. In large machines these two wires should be faired off to an approximate streamline shape, in order to keep down the resistance, which would otherwise be consider- ably increased. 2. Duplication through the Incidence Bracing. In this method the. load originally carried by the broken wire is transmitted through the incidence wire to the corresponding unbroken frame, and from thence to the body. Reference to Fig. 127 will make this method clear. Suppose, for example, that the lift wire C D' is broken. Originally this wire was transmitting the lift reactions at A, B, B', C, c', down to the body at D'. After it is broken these reactions are carried by the incidence wire C c' down to the point C/, whence they are transmitted up the rear strut C x c/, and thence by way of the rear lift wire C x D/ down to the body. In a similar manner if the rear wire C x D/ be broken, then the load previously carried by this wire is trans- mitted by the incidence wire q c' to the point c', and thence by AEROPLANE DESIGN way of the front strut and lift wire down to the body at the point D'. Since it is possible for any one of the external bracing wires, either ' lift ' or ' down,' to be put out of action, it is necessary to consider the case of each wire separately and to examine what effect will be produced upon the remaining members of the structure in each contingency. The factor of safety with one wire broken is generally taken as two-thirds that when all the wires are intact. From the diagram it is clear that the main bracing, incidence bracing, and drag bracing are all affected, so each will be considered in turn. (a) Effect on the Main Bracing. It has already been shown that in the event of the lift wire c D' being broken the lift reactions in the previous bays of the front frame will be trans- mitted to the rear frame by the incidence wire. This will result in a much greater load on the rear strut c x C/ the increased load being the sum of these reactions. The value of these reactions, however, depends upon. the position of the centre of pressure of the air forces, and thus there are three cases to be considered in the design of wing structures when this method of duplication is adopted. For the biplane truss illustrated in Fig. 127 these cases are : i. With the C.P. forward and reduced factor of safety. ii. With the C.P. rearward and reduced factor of safety. iii. With the C.P. rearward, the structure intact, and the maximum factor of safety. In case (ii.) the load transferred from the front frame will be less than in case (i.), but the loads due to the outer lift reactions on the back frame will be greater than in case (i.). Moreover, the reduced factor of safety used in the first two cases may lead to the conditions of case (iii.) being the criteria to adopt for design, and only a numerical determination will establish which condition produces the maximum stress in the strut and wire. The problem is not a difficult one, though somewhat more involved than that obtained when direct duplication is employed. The example given later in this chapter will help to clear up any difficulties. (b) Effect on the Incidence Bracing. When the wing struc- ture is intact there are no stresses in the incidence bracing due to air forces. Their function under such conditions is principally to make the structure rigid. With one of the main bracing wires broken, however, the corresponding incidence wire is called upon to carry its load and it must therefore be designed for this purpose. The load in the incidence wire will be the sum of the reactions in the frame to the left of the broken wire resolved in its direction. For example, if ab (Fig. 128) represent the sum DESIGN OF THE WINGS 163 of such reactions, then b c will represent the corresponding stress in the incidence wire. The horizontal component a c of tension in the incidence wire is taken by the drag bracing which must now be considered. (c) Drag Bracing. The general form of the wing -drag bracing is shown in Figs. 108, 139. It is general practice to assume the maximum drag force to- be uniformly distributed on the wings of a machine, and to be equal to one-seventh of its total weight, and to design on this basis. Here again the method of duplicating the lift and down bracing wires will largely influence the design. With the in- cidence wires in operation, there will be a component of their tension acting in the plane of the drag bracing, and hence this must be sufficiently strong to transmit the resulting shear along to the centre section. The simplest method of determining the stresses in the drag bracing will be to draw the stress diagram, FIG. 128. applying the forces due to the incidence bracing at the correct points of application. Referring to Fig. 127, if we suppose the front lift wire B c' to be broken, and the incidence wire B B/ to be transmitting the lift forces from the front to the rear frame : the resulting tension in B B/ will cause an unbalanced force in the direction of the arrow, and the frame would therefore become distorted unless some means were provided to counteract this force. The drag bracing wire B 1 ^ offers the best means of providing the necessary reaction and the shear will be trans- mitted along the path indicated by the arrows. The stresses set up in the drag bracing in this manner will be very much greater than those due to the drag forces alone, and the drag bracing will therefore need to be made correspondingly stronger. Change of Direction of Drag Forces in the Wing Structure. At first sight it would appear that the component of the drag forces in a wing structure always acted from the front to the rear of the wing. This, however, is not the case, as 164 AEROPLANE DESIGN was pointed out on p. 60. The following example further illus- trates this effect as influenced by the resistance of the bracing wires. From the R.A.F. 6 aerofoil characteristics we find that the ratio L/D at 12 angle of incidence is 11. Making an allow- ance for the resistance of the bracing wires, this ratio will be taken as 10. Next set out the values of the lift and drag perpendicular and parallel to the air stream respectively. From Fig. 45 it will be seen that the resultant force on the section is forward of the normal to the chord line. For such an attitude of the wings, therefore, the component of the drag forces is acting in a forward direction. For the purposes of design work it is best to assume that the drag forces act in such a direction that they cause the greatest stress in the spars. FIG. 129. A moment's consideration will show that for the back spar design the drag forces should be acting in the direction shown in Fig. 127, and that for the front spar they should act in the opposite direction. This is because the spars are almost in- variably designed to resist compression, and consequently the drag forces should act in such a manner as to increase this compression. Stagger. The effect of stagger upon the stresses set up in a wing structure is to introduce another factor to be applied to the loads obtained from the stress diagram of the vertical frame already discussed. Reference to Fig. 129 will make this clear. It will be seen from this figure that the vertical reaction of the lift forces is resisted by the lift bracing, which is inclined at an angle 6 to the vertical. As a result of this the stresses obtained from the diagram for the vertical frame bracing must be multiplied by the factor i/cos 6 for the case of the staggered machine. DESIGN OF THE WINGS 165 Moreover, it will be apparent that, on account of the line of pull of the lift wires not being in the same plane as that of. the lift forces, there will result a horizontal component of the lift in the direction of the drag bracing = L tan 0, where L is the lift reaction at the joint considered ; and the drag bracing must therefore be sufficiently strong to transmit the shear resulting from these horizontal components in addition to that due to the ordinary drag forces. This is illustrated in Fig. 129 (a). Similarly at the lower plane joints the lift reaction is trans- mitted by means of the inclined strut to the top joint where the lift wire is attached. Hence the stress in the struts obtained from the vertical frame diagram must also be multiplied by the factor i /cos 6. Also there will be a backward component = L' tan 9 to be taken by the lower plane drag bracing where Reaction* FIG. 129 (a). \L FIG. 129 (b). L' is the lift reaction at any lower plane point of support. (See Fig. 129 ().) The method of determining the stresses in the wing structure of a staggered machine may therefore be summarised as follows : (i) Lift Bracing and Interplane Struts. Multiply the loads obtained from the vertical stress diagram by the factor where 6 is given by the relationship tan = Stagger gap (ii) Drag Bracing. To the ordinary drag loads due to resultant air forces add a component equal to L tan 6 at each point of attachment of wing bracing, where L is the reaction at the joint considered, and draw stress diagrams for combined reactions. An example of this is shown in Fig. 146., It should be noted that in the case of the downloading 166 AEROPLANE DESIGN forces on a staggered machine the downbracing loads must be multiplied by the factor I/cos 0, while the horizontal component in the direction of the drag bracing will act in the opposite direc- tion to that when the lift wires are in operation. A separate stress diagram for drag bracing of a staggered machine must therefore be drawn for downloading conditions. Detail Design of the Wing Structure. The various cases likely to be met with in stressing a wing structure having been considered, the detail design of the various members can be investigated. From the previous paragraph we see that the actual maximum loads for which it is necessary to design each member can be obtained by applying : (#) The requisite C.P. coefficient ; (b) The necessary factor of safety ; and (c) Stagger coefficient in the case of a staggered machine, to the load obtained from the stress diagram drawn for unit load. These detail members comprise : 1. The lift and down bracing, j 2. The incidence bracing. J Bracing. 3. The interplane struts. 4. The drag struts and drag bracing. 5. The spars. The External Bracing. The most common method of bracing the wing structure is by means of high tensile stream- line wires or rafwires, of which particulars are given in Table XXVI. These wires are rolled out of circular section, down to the shape shown in Fig. 130 (a) and (), by means of specially shaped rollers, the process being termed ' swaging/ This process has the effect of making the steel very brittle, and also sets up initial strains in the material. It is therefore necessary to subject the wires to heat treatment, which consists of placing them in a bath of molten lead or in boiling salt solution, where they are allowed to remain until they have acquired the uniform temperature of the bath, after which they are removed and allowed to cool slowly. Previous to this heat treatment the yield point and the ultimate stress point are practically coin- cident, and there is no appreciable extension before fracture. The breaking stress of the wire in this condition is very often in the neighbourhood of 100 tons per square inch. After heat treatment the yield and ultimate stress points are much reduced, DESIGN OF THE WINGS 167 the latter being about 70 tons per square inch ; but there is now an extension of from 15% to 2Q/ Q on a gauge length of 8". TABLE XXVI. SIZE RAFW1RES TIE RODS TENSILE STRENGTH DiAof~Thn=ad L.H. or R.H. LentfhofTfesd Section of ^^ Sl-neamline Dia of Rod (circular) Ibs Major Axis Mirvar AKIS 4BA 1 0" 0-17" 0-06* o-io' 1050 2 B.A \ 1" 0-18" 0-07" O-I35 190O W B.5.F 1-3" 0-4" 0-09' o-ise' 3450 %2 " 1-4" 0-47" o-io" 0211" 4650 ?I6 " 1 5* 0-50" o-ir 0-234" 5700 %Z ' 1-6" 0-57" 0-13" 0262" 7150 %" " 1 7- 0-60* 014" 0-285 " 8500 %2 *' 1-8* 073" 0-15" 0-310" 10250 '5/* . /32 * 1 9" 075" 078" 0-16" 0-17" 0-340" 0360" 11800 13800 Xa " 2-0" O-80" 0-18" 0-380' 15500 (a) RaPwir wil'h Universal Fork Joinh (b) Rafwirc wil-h Plain Fork Joinf. (c) Tic-Rod wifh Plain Fork Joinf. FIG. 130. Bracing Wires and Tie-rods. The ends of the wires are left circular, and are finally screwed with right- and left-hand threads respectively, which i68 AEROPLANE DESIGN screw into plain or universal fork joints as shown in Fig. 130 (a) and (&). These fork joints are in turn pin-jointed to the wiring plates at the points of attachment to the wing structure. Another method of external bracing is by means of stranded wires. The strands manufactured by Messrs. Bruntons, of Musselburgh, are illustrated in Fig. 131 : a is a section which should be used only in those places where little wear takes place. These strands combine strength and flexibility, and can be obtained in any required size. They do not deteriorate so rapidly as a rafwire, because in the latter type of bracing there is a certain amount of crystallisation due to the vibration. Further, if a single tie-rod has a. slight nick upon its surface (c) 7 Srpands, each 7 Wires. 7Sl-pands, each 19 Wires FIG. 131. Bracing Cables (Brunton's). it is liable to snap under a sudden accidental, but not necessarily severe, strain. In the case of a strand such a nick would mean the severance of one or two wires only, and would not greatly impair the strength of the complete strand. Table XXVII. gives some details relating to the sizes, weights, and strength of the strands illustrated in Fig. 131. TABLE XXVII. PARTICULARS OF STRANDS FOR AIRCRAFT PURPOSES. Size. Circumference in inches. I'O Weight per 100 feet. Ibs. 3*33 5*5 8-0 12-5 16-66 Breaking strength. Ibs. 784 1332 1904 2576 5I5 2 7056 10080 DESIGN OF THE WINGS 169 These strands are tightened by the insertion of strainers or turnbuckles, a standard type of which is shown in Fig. 132. Details of these turnbuckles are incorporated in Table XXVIII. An illustration of their use is given in Fig. 138 at q, r, s.*jjfln general the barrel portion is made of gun-metal, and the eye and fork portions of steel. TABLE XXVIIL STRAINERS. WASHER FORK A FORK A. Diam. of Thread. Dimensions. A. B. C. D. E. F. G. H. J. K L. M. Tolerances. + '015 - 0. + '04 - 0. + 04 o. + -OC2 'OO2. + '015 o. + '015 - o. + '015 - 0. + 001 - o. + 015 - o. Min. B.S.F. Strength. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. i 95 cwt. i '5 1-6 i'o i 418 i '08 '7 "3 07 '54 '5 '2 A 7 2 ). 1 4 i'5 '875 7e 365 '95 65 '25 06 '475 438 '2 i 52 I '2 i '3 "75 i '30Q 8 '55 2 06 *4 '375 '2 A 34 I'O I '2 '65 A "252 '7 '475 *2 05 '35 312 '2 A 29 '9 I 'I '60 A 23 '65 '45 '2 *4 325 281 "2 i 2I'5 ,, 8 I'l 5 i 20 '56 42 *2 04 28 250 '2 zBA 'S ii '7 I 'I '4 2 BA i47 '43 '3 1 'IS 04 215 187 '2 4 BA 6'7 6 1.0 '35 4 BA *n 38 26 '12 04 '19 156 '2 Diam. of Dimensions. N. O. P. Q. R. s. T. U. V. W. X. Y. IZ. Thread - i ; '! Tolerances. + -oi + 01 + + '015 + -OI!+'OI - o. - 0. -'001. - o. - 0. - 0. ; Min. Strength. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. i Ins. Ins. Ins. i 95 cwt. 25 '17 96 86 498 64 '05 , 1-08 i A i A A 72 ,, *2 16 '91 8 436 56 '05 'I j "07 7* A H *V f 52 i, '2 16 '9 '7 '373 52 '025 'I -0 7 i & i A A 34 ,, 'IS '15 '71 '59 '310 '4 1 025 "07 ! '05 A A * l A A 2 9 *I **4 68 56 279 '38 025 07 -05 A A f A i '5 ii 075 -I 4 65 '53 248 *35 025 o 7 ; -05 i A f . A 2 BA "'5 075 .14 '53 '42 185 '3 025 7 '05 A iV J A 3BA 6' ,, 7 "M 48 '37 '154 245 025 07 -05 A A i A 170 AEROPLANE DESIGN 30 Riht- or- Left Hand All holes must not' be moce than -01 off 7 '55 '275 40 I 7 o '9 '5 875 fe '365 '96 '48 2 '06 48 24 '35 1 8 '3 '75 i '39 82 '4i 2 '06 '4i 205 '30 A 65 2 '65 A '252 68 '34 'i5 '05 '37 '17 '25 A '55 i '6 A '23 '58 29 i '04 29 'MS 225 1 '5 'i *5 i 20 '50 '23 '075 '04 <2 5 'US "20 2 BA '45 'i '4 2 BA i47 40 X9 *75 '04 *2 '095 Omitting expression for lateral load, this becomes = P( - r 2 V )/ +-|=/ (2) _ T / T , That is y t + e t = or v = whence The bending moment equation now becomes that is _ 2 Let f (/) = z and substitute, then f = - 0-05 (7) A representing the area of the section where the thickness is / . FIG. 136. FIG. 137. Compression Rib Sections. Investigations upon a number of struts have shown that this correction is practically negligible except for the wing struts of high-speed machines. For this reason the term containing w has been neglected in the illustrative example showing the practical application of the above theory to an actual strut, which is set out in full later in this chapter. The Drag Struts and Bracing. For small machines the general method of taking the shear due to the drag loads and components of the lift reactions is by means of compression ribs. These ribs have the correct contour of the wing section, and their web is made solid in order to resist the compression. The section of such a rib is as shown in Fig. 136. A stronger and better method than this is illustrated in Fig. 137, in which two ordinary ribs with lattice webs are placed i 7 8 AEROPLANE DESIGN adjoining one another. Such an arrangement is considerably lighter than that shown in Fig. 136. For larger machines it becomes necessary to build hollow wooden struts of circular or box section, or tubular steel struts may be used. It will generally be found that the wooden con- struction will prove the lightest for a given strength. Particulars of steel tubes which may be used for such purposes are given in Appendix. Reproduced by courtesy of ' Flight.* FIG. 138. General Sketch of Internal Bracing and of Interplane Strut Attachment. Their use and mode of attachment is shown in Fig. 138. The Drag Bracing wires usually take the form of small circular tie-rods screwed at each end, as shown in Fig. 1 30 (c\, particulars of which are given in Table XXVI. The general arrangement of a drag strut and fitting is shown, in Figs. 138 and 139. DESIGN OF THE WINGS 179 Design of the Spars. The spars are the most important members of the wing structure, and much care must therefore be exercised in their design in order that the necessary strength may be obtained for the minimum possible weight. The larger the machine and the deeper the wing section employed the more economically can the spars be designed, but even in this case it is not easy to reduce the weight of the spars alone to less than one-third of the total weight of the wing structure. In FIG. 139. Internal Wing Structure. small machines the weight of the spars may amount to as much as one-half of the total weight of the wing structure. A small percentage of saving in the weight of the spars will therefore be relatively of much more importance than a similar percentage saving in any other members of the wing structure. Spars are subject to (a) Bending stresses, (fr) Direct end loads. The bending stresses result from the uniform distribution of i8o AEROPLANE DESIGN the air forces along the wing section, and the spars therefore correspond, as shown in Chapter IV., to a continuous beam uniformly loaded and supported at various points by means of the wing bracing, and the complete bending moment diagrams are drawn upon this assumption by the methods outlined in that chapter. From the diagram so obtained the bending moment is shown at all points along the span, and the bending stresses can therefore be calculated from Formula 51. Complete bending moment diagrams for the top and bottom plane spars of a small machine are shown in Fig. 147. Bending stresses are set up by the lift forces., down forces, and drag forces. The drag forces are in the plane perpendicular to the other two forces, but they are usually so small that they can be neglected in ordinary practice. Direct end loads are the horizontal components of the tension in the wing bracing both external and internal. A reference to the stress diagrams for vertical frames shows that in normal flight the top plane spars are in direct compression owing to these end loads while the lower plane spars are in tension. In cases of downloading these end loads are reversed in direc- tion. For the end loads due to drag bracing it will be seen that with drag forces acting from front to rear, the rear spars are in compression and the front spars are in tension ; and that these directions are reversed when the drag loads act from back to front. The actual loads are obtained from the stress diagrams for the respective cases from which the resultant stress at any section of the spar is obtained by dividing the load by the cross- sectional area at that point The total stress in the spar at the point considered is therefore = /= - H Formula 66 I A If the spars are made of silver spruce, the total stress obtained from Formula 66 must not exceed 4500 Ibs. per square inch for compression, or 1 1,000 Ibs. per square inch for tension. CORRECTION FACTOR TO BE APPLIED TO THE BENDING MOMENT AT THE CENTRE OF SPANS. The effect of the dis- tributed load along the spans will be to produce a deflection at the centre of each bay relative to the points of support where the external bracing is attached. As a result the bending moment at the centre of the span will be increased if the end load is compressive, and diminished if the load is tensile, by an amount which is equal to the end load multiplied by the deflection. DESIGN OF THE WINGS 1 81 In order to allow for this a correction factor is applied to the bending moments at the centre of the spans, namely : - P Formula 67 2 7T 2 E I _ crippling load of spars considered as where P E * ji a strut, using Euler's formula and P end load. This factor will be greater than unity with compressive end loads and less than unity with tensile end loads. Some typical wooden spar sections are shown in Fig. 140 a, b, c. From the equation for bending stress (Formula 66) it is obvious that in order to keep the stress low the moment of inertia must be large, therefore the material must be concen- (a) (10 te) FIG. 140. Typical Spar Sections. (50 trated as far as possible from the neutral axis. This is why an 'I 1 (see '&' in Fig. 140), or a 'box' (see 'c' in Fig. 140) section is preferable to the rectangular section shown at ' a* in Fig. 140. It is on this account that the thin wing possesses unduly heavy spans, and of course where the depth is small the rear span is very difficult to design. The spar shown at * ' d* in Fig. 140 is made of No. 22 S.W.G. sheet steel, being built up of two corrugated channel sections to form the web, and riveted to two corrugated flange plates at either end. Such a spar can be made very light, is easy to manufacture, and it is very pro- bable that the near future will see a large development of steel spars built up in this manner. A very interesting type of spar section and internal wing arrangement is that adopted on the Fokker wireless triplane, which is illustrated in Fig. 141. As will be seen from this figure, the two spars (each of the box type) are placed very close together, and then the two box sections are united by a sheet of three-ply covering. As a result of this uncommon arrangement all internal wing bracing has been avoided. 182 AEROPLANE DESIGN O O t) O O, DESIGN OF THE WINGS 183 Practical Example of Wing Structure Design. The preceding work upon the detailed design of the wing structure members will now be applied to the machine for which the preliminary stress diagrams have already been drawn and shown in Figs. 125 and 126. The wing section chosen for this machine is the R.A.F. 6, shown in Fig. 40. NORMAL FLIGHT. The travel of the C.P. is from -3 to -55 of the chord that is, for a six-foot chord from 21*6" to 39'6" from the leading edge. FIG. 142. Spacing the spars as shown in Fig. 142, The maximum proportion of the load on front spar = 39' - * 2 ' 6 = 68% 39 and the maximum proportion of load on rear spar A factor of safety of 7 will be used throughout, and the method of duplication employed will be direct. The factor due to stagger = -- - = - = i '053 Hence the maximum load in the front frame members = load in vertical frame from stress diagram x 7 x -68 x ro53 = 5 x stress diagram load and the maximum load in the rear frame members = stress diagram load x 7 x '79 x 1*053 = 5*82 x stress diagram load. DOWNLOADING. The position of the centre of pressure for downloading may be taken at '25 chord, or at 18" from the leading edge. The maximum proportion of downloading on front frame = 39-9 = 77 =/ 39 whence proportion for the rear frame = 23% 184 AEROPLANE DESIGN The maximum load on the front frame downbracing = downloading stress diagram load x L x -77 x 1*053 = 2*84 x downloading stress diagram load and maximum load on rear frame downbracing = downloading stress diagram load x 3-5 x -23 x 1*053 = 0-85 x downloading stress diagram load. Proceeding to detail design and inserting two wires each capable of taking two-thirds of the maximum load, we have EXTERNAL BRACING. Stress FRONT FRAME. REAR FRAME. Load. Maximum Load. Size of Wire. Maximum Load. Size of Wire. Factor 5*0 Factor 5-82 Lift wire B c 1 440 2200 2 - 2 B.A. 2560 2 - 2 B.A. Lift wire c D 1 950 4750 2 - i"B.S.F. 5520 2 Factor 2*84 Factor '85 Down wire c B 1 435 1235 2 - 4 B.A. 370 2 - 4 B.A. Down wire D c 1 940 2670 2 - 2 B.A. 800 2 - 4 B.A. I DESIGN OF THE INTERPLANE STRUTS. The load on the interplane struts for various flight conditions follows directly from the stress diagrams. INTERPLANE STRUTS. Strut. Load from Stress Diagram. Maximum Loads. Front Frame. Rear Frame. AB 1 Lift forces Down forces + 80 Ibs. 80 + 400 Ibs. 227 + 465 Ibs. 68 BB 1 Lift forces Down forces - 142 ., - i55 7TQ 440 827 - 132 **, CC 1 Lift forces Down forces - 47 2 - 49 - 2360 - 1390 ,, - 2750 417 , D D 1 Lift forces Down forces + "5 > - 790 + 575 >i - 2240 670 670 DESIGN OF THE WINGS 185 The loads for which it is necessary to design each strut are underlined. With regard to the strut A B 1 , it is necessary to design this for the downloading forces because this strut is certain to fail under compression and not under tension. In order that the theory of the tapered strut already out- lined in this chapter may be fully appreciated, and the com- plicated results obtained made available for general design of struts, its use will be illustrated in the design of the strut CC 1 . Design of the Interplane Strut C C 1 , according to the tapered strut formulae given on pages 174-176 : Length of strut ...... 6ft. 4 ins. Maximum compression ... 2800 Ibs Fineness ratio ......... 3*5 : i Form of cross section as shown in Fig. 91. Area of cross section ... ... 2 '5 f 2 Least moment of inertia ... = 0*15 / 4 Taper to be from a maximum at the centre cross section down to the points of support. In order to find / and the correct taper of the strut, it is necessary to apply the theory shown on pages 174-176. At first sight this would appear to be a difficult process, but by taking each equation separately, and dealing with these portions in the manner indicated, the solution of the whole equation can be obtained without any knowledge of advanced mathematics, although, as will be seen, the process is somewhat lengthy. Consider first the expression ., - Jt\*\ l /A This expression represents the rate of change of the slope of the function with respect to /. In order to obtain this rate of change it is. therefore necessary to draw the curve of this function, and measure its slope at various values of /. Alternatively, but by taking much smaller intervals, the value of this slope could be obtained by tabular integration in the manner shown for bending moments and deflections in Chapter IV. The work 186 AEROPLANE DESIGN preparatory to graphing the function should be set out in the following manner : 1-25 r 5 1*75 2*0 1-56 2-25 3*06 4*0 2-44 5-06 9-40 16*0 3-92 5-63 7-65 I0'0 0-366 0*76 1-41 2-4 0-293 0-506 0*805 1*2 A = 2*5 t* 2*5 I = 0-15 / 4 0-15 I 7 ' r 5 P 2800 /A = CCQO A '204 0*13 0*0905 0-0665 0*051 p i - j^ 0-796 0-87 0-9095 0-9335 '949 I / P , 0-1193 ' 2 55 0*46 0-751 The curve for this function can now be plotted from these figures. The next step is to draw the tangents to the curve for the values of / taken. These tangents give the slope of the curve at these points, that is, they give the value of the expression dt{t\ /A as below : i 1-25 1-5 1*75 2-0 ~ 7T ) h '44 0-72 0-94 1-35 1*82 and from these values the graph of the expression can be drawn. (See Fig. 143.) The denominator of the equation A / A> if d vy lU P must next be considered, and examination of this expression shows that it is required to find the square root of the area of the curve represented by the expression i d fl DESIGN OF THE WINGS 187 between the values of / and / . The value of / is being sought, so that two or three other values of t must now be chosen and used to arrive at the value / required. The method of pro- cedure will be clear as the example progresses. Referring to the above expression it is seen that it represents the rate of change of slope of the square of the expression whose value has just been found multiplied by I/I. FIG. 143. It is therefore necessary to square the various values ob- tained for K- -A) and then to draw the curve for the results obtained, draw the tangents for the values of / under consideration, and then to measure the slopes of the tangents obtained as in the work dealing with the numerator. The values so obtained are then multiplied by i/I, and from the results the graph of the complete expression can be drawn. The tabular arrangement 1 88 AEROPLANE DESIGN of the work is shown below, and the various curves are shown in 143- / i 1-25 1-5 175 2-0 021 r 39 1-63 For the remainder of this chapter we shall replace these complicated expressions by the following symbols : _ Expression. Symbol. P K /A 7(1- yjr] L t \ /A/ d f i / p \| M = 3-5 ' 6-3" 6*125 " 5*95" 5'6" 5*25" 4*375" 3*5" *J N.B. x - 1*2 K( K 1 -. /A)/] \dt it r 'i[ M\( '-A )F> DESIGN OF THE WINGS 191 The shape of the strut can then be drawn out from the values obtained above for x and /, as illustrated in Fig. 145. DESIGN OF THE DRAG BRACING. In accordance with standard general practice the total drag force to be distributed FIG. 144. Graphical Evaluation of Tapered Strut Formula. over the planes will be taken as one-seventh of the weight of the machine. Hence drag force 2000 = 286 Ibs. = 072 Ibs. per square foot. This load is assumed equally carried by the front and rear spar. The plan of the wings must next be drawn out and the spacing of the drag struts decided upon. A suitable arrange- 192 AEROPLANE DESIGN ment is shown in Fig. 146. The forces at the joints due to the drag forces is then determined in the following manner : Area of outer section of top plane from joint A to tip = 14 sq. feet approx. Drag load on this area = 14 x 0*72 = 10 Ibs. This is equally distributed between the front and rear joints of A and A 1 Area of wing between A and B = 4 x 6 = 24 sq. feet. . * Drag load over this area = 24 x 0*72 = 17 Ibs. approx. Half of this acts at A and A 1 , and the other half at B and B 1 . Proceeding along the span in this manner the drag forces at each of the joints are obtained, and the results should be tabulated as under : TOP PLANE. a b - c ' d e f g Area of drag bay ... 14 24 21 18 18 18 15 Drag load Ibs. ... 10 17 15 13 13 13 n f Front o'o 8'o 7 % *o 6x 6's 6'o Reactions I Rear 9-0 8'o 7*0 6-5 6-5 6'o BOTTOM PLANE. j k I m n Area of drag bay ... 14 20 18 18 18 Drag load Ibs. ... 10 15 13 13 13 n f Front 8-5 7-0 6x 6-15 3-0 Reactions ...{* * I Rear 8-5 7-0 6-5 6-5 3-0 In addition to these loads there will be horizontal com- ponents due to the inclination of the lift bracing owing to the stagger arrangement. These are obtained by multiplying the lift reactions at each point of support by the factor stagger 6 3 The components thus obtained are added to the reactions due to the drag forces and the drag bracing stressed in the DESIGN OF THE WINGS 193 usual manner. In adding the lift components it is first necessary to apply both the C. P. and safety factors to Curve 2. Jnfeyal Curve of ( Meridian Curve Of Strut- ti Sfe V,CM Of Si-rut FIG. 145. the original reactions. The coefficients to be applied are as ^follows : (a) Centre of pressure forward : Front frame = 7 x '68 x "33 = 1-59 Rear frame =7 x '32 x "33 = 0*75 (b) Centre of pressure backward : Front frame = 7 x '21 x -33 = 0*49 Rear frame = 7 x -79 x -33 = 1*84 194 AEROPLANE DESIGN These coefficients applied to the lift reactions give the following horizontal loads which must be taken by the drag bracing : TOP PLANE. Joint A B c D Lift reaction on vertical frame 64 155 202 115 (a) C.P. forward, hori-) Front 102 246 321 183 zontal components/ Rear 48 116 151 86 (b) C.P. back, hori- \Front 31 76 99 56 zontal components/ Rear 118 286 372 212 BOTTOM PLANE. Joint B' c' D' Lift reactions on vertical frame 78 176 62 (a) C.P. forward, hori-^ Front 124 280 98 zontal componentsj Rear 58 132 46 () C.P. back, hori-| Front 38 86 30 zontal components/ Rear 144 324 114 STRESS DIAGRAMS FOR DRAG BRACING. The reactions thus obtained are added to the local drag reactions, and the stress diagrams for the drag bracing can then be drawn as shown in Fig. 146. It will be observed that in the case of normal flight the drag bracing wires shown in full lines will alone be in operation, whereas in the case of downloading the dotted bracing will be in action. It is customary to make the two bracing wires in each bay similar, so that it is not necessary to make a stress diagram for the drag bracing under down- loading forces. Further examination shows that it is only necessary to stress the drag bracing for the lift conditions with the centre of pressure forward. The stress diagrams therefore reduce to one for each plane. The factor having been applied to the reactions before drawing the diagrams, the individual stresses in the members can be read directly from the diagram and tabulated for reference as shown. DESIGN OF THE DRAG BRACING. Circular tie-rods, of which particulars have already been given in Table XXVI. and illustrations in Fig. 1 30 (c\ may be used. From the table the sizes necessary are : TOP PLANE. Bracing wire ... 14-15 16-17 18-19 20-21 22-23 Load Ibs. ... + 280 + 800 +770 + 1450 + 1470 Tie-rod size 4 B.A. 4 B.A. 4 B. A. 2 B. A. 2 B. A. DESIGN OF THE WINGS 195 LOWER PLANE. No. 4 B.A. tie-rods throughout. DESIGN OF THE DRAG STRUTS. For such a small machine as that under consideration it would be preferable in practice to insert compression ribs, but for purposes of further illustration we will assume that steel tubes of thin gauge are to be used as compression members. The size of tubes necessary will there- fore be determined by use of Formula 62 (Rankine's Formula), where the constants have the following values : f c = 2 1 tons per square inch. a = 1/7500 L = length = 39" TOP PLANE STRUTS. Drag Strut ... 1-14 15-16 17-18 19-20 21-22 23-24 Load Ibs. ... -no -430 -560 -890 -1050 -1250 Diameter of tube, 22S.W.G. ... J" f" r F F BOTTOM PLANE STRUTS. Drag Strut ... 6-n 12-13 14-15 16-17 Load Ibs. ... -130 -220 -490 -660 Diameter of tube, 22S.W.G. ... i" 4" I" f" DESIGN OF THE SPARS. The stresses in the spars are due to bending and direct end loads. In order to determine the bending stresses it is necessary to draw the complete bending moment diagrams for the top and bottom planes. The fixing moments at the supports have already been obtained. In order to complete the diagrams, the free bending moment diagrams are drawn upon each span for uniform loading. As shown in Chapter IV., these free B.M. diagrams will be parabolas with their maximum ordinates equal to w / 2 /8 Hence in the case under consideration the maximum B.M. ordinates are : TOP PLANE. Span ... ... ... ... ab be cd de Maximum B.M. Ibs. ft ... 50 158 135 23-4 LOWER PLANE. Span ... ... ... ... b' c c d' Maximum B.M. Ibs. ft. ... 135 115 i 9 6 AEROPLANE DESIGN The parabolas and the fixing moments are set out to the same scale, and the net bending moment diagrams are shown shaded in Fig. 147, from which figures the value of the bending moment at any point along the span can be read off directly. f 57lbs \ZA I A.J ai Lmear Scale Ffeacrions due to Stagger Joint- A B c D Front* FcAme wa 46 321 185 Rw>r . >8 116 151 at Re-vchons due to Stagge Joiaf B' C' D' Front' Fc4mc 124^ 280 as Rea-r . 58 isa 46 FIG. 146. Stress Diagrams for Drag Bracing. To the values thus obtained the requisite factor of safety and the centre of pressure coefficients for the front and rear spars must be applied. LIFT FORCE FACTORS. Factor for front spar = 7 x -68 = 4-76 Factor for rear spar = 7 x 79 = 5*53 DESIGN OF THE WINGS 197 Tabulating the bending moments for various positions along the spans, we have the following tables : TOP PLANE. B.M. Maximum B.M. Position. on diagram. Front spar. Rear spar. Ibs. ft. Ibs. ft. Ibs. ft. Joint A 25 ... 119 ... 138 Middle of span A B ... negligible ... Joint B 77 ... 366 ... 426 Middle of span B c ... 65 ... 310 ... 359 Joint c ... 114 543 - 6 3 Middle of span c D ... 50 ... 238 ... 276 Joint D 54 257 298 Middle of span D E ... 30 ... 143 ... 166 LOWER PLANE. Joint B' 16 ... 76 ... 88 Middle of span B' c' ... 75 ... 357 ... 415 Joint c' 105 ... 500 ... 580 Middle of span C'D' ... 45 ... 214 ... 249 Joint D' 38 ... 181 ... 210 Middle of span D' E' ... 38 ... 181 ... 210 DOWNLOADING. The bending moments on the spars due to downloading forces follow in the same manner by applying the correct factors for the centre of pressure and safety. In general it is necessary to consider the lower spars only for downloading forces, as they will then be in compression, whereas in normal flight they are in tension. The much greater strength of spruce in tension as compared with compression, may result in the necessity of designing the lower spars for downloading forces in spite of the reduced factor of safety employed. DOWNLOADING FORCE FACTORS. LOWER PLANE SPARS. Factor for front spar = 3-5x77 = 2-69 Factor for rear spar = 3-5 x '23 = '805 B.M. Maximum B.M. Position. on diagram. Front spar. Rear spar. Ibs. ft. Ibs. ft. Ibs. fif. Joint B' 16 ... 43 ... 13 Middle of span B' c' ... 75 ... 202 ... 60 Joint c' 105 ... 283 ... 85 j- Middle of span c' D' ... 45 ... 121 ... 36 Joint D' * ... 38 ... 102 ... 31 Middle of span D' E' ... 38 ... 102 ... 31 i 9 8 AEROPLANE DESIGN The correction factor for end loads P, Formula 67. P E - P has not been applied to the bending moments at the centre of the spans in this example. This correction should not be omitted in actual practice. Bending- Moment Diagrams - Load Factor <= I 4 D Scales :- Linear Bending Monwnt- "TopPUne Spars IB' |c' Bottom Ranc Spars Sher Force Diagrams load factor = I Force Scale _ ?OQ zoo Ibs. lop Plane Spans FIG. 147. Design of the Wing Spars B.M. and S.F. Diagrams. The bending moments along the spars having been de- termined, the next step is to deduce the direct end loads upon the spars resulting from the tension in the lift and drag bracing. These are read off from the stress diagrams. In the case of the loads obtained from the lift and downloading stress diagrams, factors must be applied as in the preceding work. The loads DESIGN OF THE WINGS 199 due to drag forces are read off directly. It will be observed that from the drag bracing stress diagram for the top plane, shown in Fig. 146, the front spar is in tension and the rear spar in com- pression. The result of this is that the drag forces on the front spar tend to reduce the direct load in the front spar, since the top plane spars are in compression due to the lift bracing, while they increase the direct load on the rear spar. DIRECT END LOADS ON THE SPARS. (Top Plane Compression due to lift bracing.) Span AB BC CD DE Front spar 214 1760 4950 495 Ibs. Rear spar 249 2040 5750 5750 Ibs. LOADS DUE TO DRAG BRACING. Span b c d e f g Front spar o +220 +820 +1360 +2360 +3360 Ibs. Rear spar... -220 -820 -1360 -2360 -3360 - 3360 Ibs. RESULTANT END LOADS ON TOP PLANE SPARS. Front spar Lift ... -214 -1760 -1760 -4950 -4950 -4950 Ibs. Drag ... + o + 220 + 820 +1360 +2360 +3360 Ibs, Total ... -214 -1540 - 940 -3590 -2590 - 1590 Ibs. Rear spar Lift ... -249 -2040 -2040 -5750 -5750 -5750 Ibs. Drag ... -220 - 820 -1360 -2360 -3360 - 3360 Ibs. Total ... -469 -2860 -3400 -8110 -9110 -91 10 Ibs. In a similar manner the direct loads upon the lower plane spars under normal flight conditions can be calculated. With downloading forces the same procedure is adopted, the end loads being read off the respective stress diagrams for the external and drag bracing under these conditions. The work should be set out in exactly the same manner as shown above for the top plane spars. DESIGN OF THE SPARS. The tables of direct loads in the spars indicate that the top rear spar will be the most heavily loaded, and therefore it is selected in order to illustrate the 200 AEROPLANE DESIGN method to be adopted in the general design of a spar. The following figures relating to this spar have been obtained : Span ...... b c d e f g Maximum B.M. 426 426 630 630 298 298 Ibs. ft. Direct end load -469 -2860 -3400 -8110 -9110 -9110 Ibs. It is necessary to select spar sections capable of carrying these loads safely. The depth of the spar is already fixed by the aerofoil section chosen. This in the present case is the R.A.F. 6 with a 6 ft. chord, and with the spars in the position shown in Fig. 142 the depth for the front and rear spars is limited to about y$". This depth will therefore be taken, the spars will be made of I section, and will be lightened out towards the wing tips by diminishing the depth of the flange. The dimensions of the spar at various points along the span can now be determined, and a suitable series of sections are indicated in Fig. 148. Considering the section for bay e, 9 M.I. = (2 x 3'5 3 - 1-5 x 2 3 ) = 6" 14 inch 4 units A. - 7 - 3 = 4 inch' 2 units Bending stress = J3 x I2 x r ?5 ' = 2155 Ibs. per square inch. Direct stress = -5ll 4 = 2027 ibs. per square inch. Total stress 4182 Ibs. per square inch. The maximum compressive stress being 4500 Ibs. per square inch for a spruce spar, this section is evidently satisfactory. Considering the section for bays/*, g^ d, M.I. = 5*2 inch 4 units A. =3*25 inch 2 units Maximum bending stress in bays/and = 1205 Ibs. per sq. in. Maximum = direct stress in bays f and g = 2805 Ibs. per sq. in. Total stress = 4010 Ibs. per sq. in. Maximum bending stress in bay d = 2540 Ibs. per sq. in. Maximum direct stress in bay d = 1045 Ibs. per sq. in. Total stress = 3585 Ibs. per sq. in. This section is therefore suitable for these bays. DESIGN OF THE WINGS 201 Considering the section for bays a, b y c, M.I. = 4*55 inch 4 units A. = 2*87 inch 2 units Maximum bending stress = 1965 Ibs. per sq. in. Maximum direct stress = 997 Ibs. per sq. in. Total stress = 2962 Ibs. per sq. in. As will be seen, this section is very much stronger than re- quired ; but as it is not practicable to make the thickness of the flanges less than that indicated in the figure, this section should be used over the outer portion of the wing. Finally, the strength Seofton far Baya die Sccfton for Bays Bay e Tob Rear Spar FIG. 148. Design of Wing Spar. of the spar in shear should be investigated. On referring to the shear force diagram shown in Fig. 147 it will be seen that the maximum shear occurs at the joint C, at which point the shear is equal to 202 x 79 x 7 = 1 1 20 Ibs. whence the shear stress approximately 1120 3*5 x 2 = 1 60 Ibs. per sq. inch since the section is rectangular at the joint in order to accom- modate the fitting. A shear stress of 800 Ibs. per square inch is permissible across the grain of spruce, so that the sections are quite safe as regards shear. 202 AEROPLANE DESIGN This completes the design of this spar, and by following a similar procedure in the case of the remaining spars the requisite sections can be determined and the spar detail drawings pre- pared. Having settled the spar sizes, the design of the members of the wing structure is finished and a close estimate of the total probable weight when manufactured can be obtained. This should be compared with the weight assumed for the initial stressing of the wings, with which it should agree fairly closely. Before leaving the question of wing-stressing, an example will be given of the alteration in the stresses when the method of duplication through the incidence bracing is adopted. It is supposed that the front lift wire c D' (Fig. 127) is broken, and that the load originally carried by this wire is now transmitted by the incidence wire c c' to the rear frame. What will be the effect of this upon the stresses in the members of the bay CD? There are three cases to consider : 1. All wires intact, the C.P. back, and a factor of safety of 7. 2. Front wire C D' broken, C.P. back, a factor of safety of 66 x 7 3. Front wire C D' broken, C.P. forward, factor of safety 66 x 7 To determine the size of the incidence wire to transmit the load : the maximum load to be transmitted occurs with the C.P. forward and has a value = sum of the front reactions as far as the bay c D = (64 + 155 + 78 + 202 + 176) x '68 x -66 x 7 = 675 x '68 x '66 x 7 = 2140 Resolving this load in the direction of the incidence wire we have Maximum load = - - =2250 Ibs. cos and the size of the incidence wire must be sufficient to carry this load. Maximum load in the rear lift wire q D/. Considering the cases enumerated above separately we have 1. Vertical load to be taken by wire = 6 75 x -79 x 7 = 3740 Ibs. 2. (i.) Load on the rear frame = 3740 x '66 = 2490 Ibs. DESIGN OF THE WINGS 203 (ii.) Load transmitted from front frame by incidence wire = 675 x '21 x 66 x 7 = 66ilbs. .-. Total load = 2490 + 66 1 = 3151 Ibs. 3. Load on rear frame = 675 x '32 x '66 x 7 = zoiolbs. Load transmitted from front frame = 675 x '68 x *66 x 7 = 2140 Ibs. /. Total load =3150 Ibs. which is nearly the same as in the second case. The maximum vertical load is therefore seen to be 3740 Ibs., occurring when the C.P. is back and all the wires are in. This is consequently the condition for which the design must be carried through, and thus the design which has just been shown for direct duplication holds good because the broken wire does not alter the maximum load. There will also be an additional load to be carried by the drag bracing due to the horizontal component of the tension in the incidence wire. To determine which condition will produce the maximum load in the drag bracing the horizontal reactions for each of the three cases must be calculated as was done for the lift bracing, and separate stress diagrams must be drawn from which the maximum load occurring in the drag bracing members is deter- mined. The direct end load in the spars will also be affected by the altered stresses in the drag bracing ; and these must like- wise be determined from the stress diagrams and the maximum load thus obtained combined with the correct bending moment in the manner previously described. Design of the Wing Ribs. In communicating the air forces from the wing surface to the spars, the ribs act as small girders, and it is necessary for design purposes to examine the loads acting upon them in flight. The load on each rib is obtained by determining the maximum load over the wing surface and dividing by the number of ribs (;/). This load will be the sum of the reactions previously obtained multiplied by the factor of safety adopted. The distribution of the load over the wing section will be similar to that obtained upon an aero- foil tested in a wind tunnel, and the design of a rib is based upon the results of pressure distribution experiments such as have been described in Chapter III. When applying these results to a wing rib the load curves must be drawn for the most unfavourable conditions of incidence 204 AEROPLANE DESIGN which are likely to occur. For example, the maximum load on the leading edge of a wing will occur at large angles of incidence, whereas the maximum load on the rear portion of the wing will occur when the angle of incidence is small. In Fig. 149, obtained from pressure distribution experiments, the load curves over a wing section for angles of incidence of 2j- and T2| are shown. During flight the wing loading W/n will be the same in each case. It is therefore necessary that the mean height of both diagrams should be the same, hence the pressure ordinates have to be altered in a constant ratio until this result is obtained. The primary shear diagrams are next obtained by tabular- graphic integration from the load curves, commencing at the front of the section for the 12^ curve and at the rear of the section for the 2| curve (Fig. 149 b}. By further integration of the shear diagram the first bending moment curves are obtained as shown in Fig. 149 (V). The position of the spars having been decided upon, the final bending moment and shear force diagrams are obtained in the following manner : i2j incidence. The centre of pressure corresponding to this loading is at about '28 chord. Taking moments about B, see figure 149 (<:), RjO+j') = Py where P is the total load on the rib . R P}> ; RI = JT7 P x and R = x + y The bending moment at the rear spar due to R-^ = R! (x + y} = Py From B set off a distance B c to represent P^ on the same scale as the bending moment diagram ordinates. Join A c. The bending moments on the rib between the spars are given by the difference in ordinates of the straight line A C and the bending moment curve. These have been drawn to an enlarged scale in Fig. 149 (d\ Similarly the final shear force curve can be drawn now that the spar reactions are known. This is shown beneath the bending moment curve. The same procedure must be carried through for the load distribution at 2j incidence, and the final bending moment and shear force curves drawn preferably on the same base as those DESIGN OF THE WINGS 205 for the \2\ incidence, as shown in Figs. 149 (d and e). The curve which indicates the maximum bending moment or shear . I2i" Incidence Rib for 2 Angles of Incidence Sl">ar Force Moment Rnl Bendfng Moment' " Incidence 12? Incidence Final Sheaf Fcx-ce FIG. 149. Determination of B.M. and S.F, Diagrams for Wing Ribs. force at a particular section must be used when designing the rib at that section. In developing the bending moment and shear force diagrams along these lines, care is necessary with 206 AEROPLANE DESIGN the various scales employed at each stage in order that the final diagrams may be correctly graduated. Having determined the bending moment and shear force over the rib, the detail design follows in the usual way. If M be the bending moment at the section considered, / the maximum allowable stress in the material, y the distance of the outer fibre of the rib from the neutral axis, and I the required moment of inertia, then A suitable section with this moment of inertia is then set out. It is customary to assume that the rib flanges alone take all the bending moment, while the web takes all the shear, as explained in Chapter IV., when dealing with the stresses in beams. The web may be designed as follows : Let d be the depth of the web / be the width or thickness / s the safe shear stress Then the shear load that the web will carry = d x / x / s whence . _ Shear force at the section ~Tx~f~ This relation fixes the thickness of the web, while the wing section itself fixes d. Generally it will be found that in all machines, except very large ones, other practical considerations will fix the sizes of the rib, which, if then tested for shear strength, will be found to be amply strong. Wing Assembly. Several illustrations of typical aeroplane wings and their fittings have been shown in the illustrations given in this chapter, in order to show the various details and the method of assembly. The principles of construction are similar in every case. The wing is built up on the main spars. These main spars are fixed at the correct distance apart, and then the ribs, which have been constructed on special formers in order to give them the correct profile of the aerofoil selected, are slid along the spars until they reach their allotted place. A distance of from 15" to 20" is generally allowed between each main rib, but between the leading edge and the front spar a DESIGN OF THE WINGS 207 number of small intermediate ribs are fixed alternately with the main ribs in order to withstand the more intense pressure which occurs over this portion of the wing. Each rib is then glued and screwed to the spars. If compression ribs are used to take the drag loads, they must be strung on in their correct order with the other ribs ; while if tubular or box struts are used, these are now inserted. A number of small stringers are threaded through the ribs, and serve to make them more rigid laterally. The drag bracing wires can next be attached to their respective fittings, and the skeleton wing is then com- plete and ready, as shown in Fig. 141, for its covering of fabric. This is bound round the wing and sewn to the ribs, and then covered with three or four coats of dope, in order to render the wing taut and weather-proof. CHAPTER VI. RESISTANCE AND STREAMLINING. Resistance. The total resistance of an aeroplane that is the force which is balanced by the airscrew thrust at the flying speed of the machine is made up of two parts : 1. The drag of the wings. 2. The resistance of the remaining parts of the machine. Under the second heading is included the effect of the inter- plane struts, the external wires, the body, the chassis, the tail, and all other fitments exposed to the wind. In a normal machine flying at the most efficient speed, these component resistances are approximately equal, so that in de- signing an aeroplane it is as important to assess the resistance of the exposed parts correctly as it is to know the drag of the wings. From one point of view it is more important, because, with increase of speed, the wings adjust their angle of incidence so as to keep Ky-AV 2 constant, and if the normal flying o angle is not far from that giving maximum L/D, then the increase of wing drag may be small ; but since the resistance of the exposed parts, other than the wings, varies approximately as the square of the velocity, a relatively small increase of speed produces a considerable increase in the magnitude of the resistance. Unfortunately, it is much more difficult to estimate the second component than it is to calculate the first component, as very little is known of the manner in which the various parts of an aeroplane interfere with one another. For example, one part may effectually screen another from wind pressure, or the rela- tive velocity with which one part engages the air may be more or less than the velocity of the machine. Also the slipstream from the airscrew increases the resistance of those parts of the machine placed in its stream. Further, it is necessary to correct an estimate of the value of the resistance of the remaining parts of a machine in the light of the actual performance of the machine in flight. Of course, if the complete aeroplane were reproduced as a scale model, it would be possible to predict this body resistance accurately from wind tunnel tests, but the construction of the .model is obviously a difficult and at the same time an expen- Reproduced by courtesy of Messrs, Handley Page, Ltd. FIG. 151. Front and Rear Views of -1500 with Wings folded back. Facing page 208. RESISTANCE AND STREAMLINING 209, sive matter. The designer should, therefore, try to arrange for the reduction of the second component to an absolute minimum. In so doing he will be guided by knowledge of the resistance of bodies of various shapes placed in the air-stream of the wind tunnel under more simple conditions. A brief consideration of the following example will empha- sise the importance of concentrating on minimising the second component, and will also show that if considerable weight has to be added in order to reduce the resistance of a given part, it may frequently be an advantage to tolerate this additional weight. Say a complete machine has a gliding angle of one in eight. This means that the overall efficiency or ljft/drag ratio of the machine is 8. Now, if the second component can be reduced by I Ib. by some means, then the machine will lift another 8 Ibs. for the same speed. The overall efficiency of a FIG. 152. Flow past a FIG 153. Flow past a Streamline Flat Plate. Shape. machine must be borne in mind, therefore, when comparing struts, bodies, etc., of different resistances and different weights. It follows from this that the more efficient an aeroplane, the more it pays to improve its efficiency. The resistance of any body placed in a current of air (this is, of course, equivalent to its resistance to motion through still air) is composed of two parts, namely : 1. The excess of air pressure in front of the body over that behind it, and 2. The skin friction. It is shown subsequently that the amount of skin friction is small where the surface is small, and the considerable resistance therefore of some small bodies is almost wholly due to the first part. This excess of pressure is caused by a discontinuity of flow due to the abruptness of the body giving rise to a * dead- air' region of diminished pressure in. the rear of the body, and to increased pressure on the face of the body owing to the forward velocity of the air being reduced. This point will be made clear by reference to Fig. 152, which shows the flow of air past a normal flat plate. At B one streamline is evidently 210 AEROPLANE DESIGN brought to rest, its velocity head being entirely converted into pressure, but the pressure thus set up will evidently diminish towards the edges of the plate as the stream divides to flow round the plate. A is the dead-air region of diminished pressure. In seeking to diminish resistance, the principle is to elimi- nate as far as possible this region of 'dead air/ and to make the air flow round the body, thus preventing any discontinuity in the flow. The body is then said to be ' streamlined,' that is, it possesses a contour that the streamlines of flow can easily follow. Its limiting resistance is then the frictional resistance of the air flowing over the surface ; and as this is a small quantity, the saving of resistance which can be obtained by efficient stream- lining is large. Taking again the example of the normal flat plate of circular section, it is found necessary, in order to stream- line it, to fit on both a nose and a tail, so that we arrive at a form somewhat similar to that of a fish or a bird. The more or less pointed nose eases the streamlines away from the disc without materially checking the flow, and a longer more or less pointed tail eases them back again after passing the plate or disc. Fig- J 53 depicts the flow past the plate streamlined by the addition of a nose and a tail. As will be seen, the region B has been eliminated, and the region A considerably reduced. Variation from the (V 2 ) Law. The resistance due to excess of air pressure varies with the square of the wind velocity (V), while that due to skin friction varies as V 1 ' 85 . So long as the skin friction is small in amount in comparison with the resistance due to disturbance of the streamline flow, then the total resistance varies very approximately as the square of the velocity (V). In the case of a good streamline form, how- ever, where the skin friction is comparable with the resistance due to the excess of air pressure, it may confidently be antici- pated that the total resistance will vary according to an index of V much less than 2. Hence if it is assumed that the resist- ance of a streamline section is given by the general formula R = K A V 2 , then the coefficient K will diminish as the velocity is increased. Experiments carried out on strut sections by Eiffel and the National Physical Laboratory have shown that this is actually the case. Fig. 154 gives the results of tests carried out by Eiffel on the three strut sections shown. As will be seen, the drag coefficient diminishes in each case with increase of speed, but with the more perfect streamline shapes the diminution is very much RESISTANCE AND STREAMLINING 21 greater than with those of a more imperfect form. This reduction of resistance at high velocities should be taken account of when estimating the resistance of good streamline shapes. Also in making a comparison between the efficiency of such shapes tested at different speeds, an allowance should be made for this effect. The advantage to be derived in making a strut section of good streamline shape is well illustrated in Fig. 154, for it will be seen that the resistance of strut No. 3 is four or five times greater than that of struts Nos. I and 2 at the speeds of flight corresponding to those used in normal flying conditions. FIG. 154. Variation of Drag with Change of Velocity. Streamlining. The designing of a good streamline form is an exceedingly delicate matter. The greatest scope is offered with airship bodies. The fuselage of a tractor aeroplane and the floats of a seaplane would offer an equally good field, but unfortunately the former has % to accommodate an engine, and the latter have to be capable of easily leaving the water, both of which considerations result in a modified form. A fair field is, however, offered by the various struts of an aeroplane, and it is convenient to examine the matter from this point of view, although, of course, most of the following remarks apply almost equally well to any other solid body. Increase of resistance for small increases in size varies approximately as the increase of projected area upon a plane normal to the wind. It must be 212 AEROPLANE DESIGN borne in mind that a very large change in size may involve a scale effect only to be investigated by experiment. The strength of a strut depends on the moment of inertia of its section. The form of section giving a maximum value of this with minimum weight is the circular section of hollow form. The circle is a partly streamlined form of section, the resistance being, according to Eiffel, some 60% of the resistance of a flat rectangular plate of the same dimensions, for sizes usually obtaining with aeroplane struts. A further large reduction in resistance may be obtained by simply elongating the circle in the direction of motion into an ellipse, that is, by giving the section a ' fineness ratio.' Fineness ratio is defined as the ratio o oo DC HavilUod FIG. 155. Strut Section. of the length of the section to its maximum breadth. The saving to be effected in this way is about S% f the resistance of the cylinder when the fineness ratio is 2, and a further IS/ can be obtained by increasing the fineness ratio to 5. For a really good shape it is best to use a fineness ratio of about 3, or just a little over, and to keep the maximum thickness well towards the nose, say one-third of the length of the section back, and to keep the run of the ' contour ' fairly flat at about this point. It is easy to produce a strut on these lines having a resistance of only 15% of the equivalent cylinder, or only 9% of the equivalent rectangular plane. It is of practically no importance whether the ends of the section are pointed or not, and it is usually most convenient to have well-rounded ends. An instructive series of tests was carried out upon a number of struts by the N.P.L. in order to determine the best form of RESISTANCE AND STREAMLINING 213 strut when taking into consideration both weight and strength as well as resistance. Some of the sections are shown in Fig. 155, and of these the Bleriot, Farman, and De Havilland were taken from struts in use on machines. The results are set out in Table XXIX. It will be noted that there is a considerable range of shape of section for which the equivalent weights vary but little, while some of those sections which have been used on actual machines have an equivalent weight of from 150 to 180 Ibs. The substi- tution of struts of 'Beta' section for those on the Farman biplane would have enabled it to carry 79 Ibs. more useful load, without any addition to the horse power required. TABLE XXIX. RESISTANCES OF STRUTS. Resistance of Weight of 100 ft. Maximum Equivalent weight 100 ft. of strut of strut, thickness for of struts of Type of strut. @ 60 ft /sec. maximum equal strength. equal strength. Lbs. thickness i". Inches. Lbs. Circular, i" diam. 43 ... 23-4 ... I '005 295 De Havilland ... 2 5'5 29-2 0'99 1 80 Farman .. 22-9 36'0 0-905 ... '54 Bleriot 237 ... 37'2 ... 0-92 162 Baby 7'9 59'4 0-822 ... 79 Beta 6-9 88-1 0718 ... 75 B.F. 34 7-2 ... 133-0 ... '65 84 B.F. 35 6'3 ... 128-0 0-677 84 The effect of yawing is to increase the resistance of a strut considerably on account of the additional side force exerted by the air. Inclination of Struts. An investigation into the effect of inclining struts to the air stream, such as will occur for example in the struts of staggered machines, has also been made. It was found that for streamline shapes there is very little alteration in the resistance, but that for blunt-nosed sections the resistance was greatly reduced owing to the increased length of section in the air stream. Resistance of the Body or Fuselage. Since this forms the largest item in the consideration of resistance it will be con- sidered at some length, so that when a fuselage of a new design has been drawn out, an estimate can be made of its probable resistance. The resistance of the body will vary approximately as the square of the velocity, and as has already been observed since the drag of the wings remains practically constant for 2I 4 AEROPLANE DESIGN various flight speeds, the question of the resistance of the body relative to that of the wings becomes of increasing importance as the speed of flight is increased. This item must of course be reduced as much as possible, and more especially is this necessary in the case of very high- speed machines. The necessity for adopting good streamline shapes is at once evident, and it is to the realisation of this fact in practice that the modern development of high-speed machines MM. N?3. /1 "~T t 4-7" 4-1 ^f-T I ! -O 1 * 01 -* t NTS. FIG. 156. Shapes used by Eiffel in determining best form of fuselage. ' is to a large extent due. A convenient method of comparing the resistance of various types of fuselages is to express these resistances in terms of a flat plane normal to the wind. For an efficient type of body the equivalent normal plane should be of a very much smaller cross section than the maximum cross section of the body. In many cases in practice, however, the resistance of the body is more than half the resistance of the equivalent flat plane, but with efficient design the maximum RESISTANCE AND STREAMLINING 215 resistance should not be more than one quarter (25 per cent). In order to obtain such a desirable result it is necessary to avoid, as far as is practically possible, all projections and corners likely to cause disturbances in the air flow. All members exposed to the air stream must be * faired ' to a streamline shape, a process which calls for the exercise of a considerable amount of care and patience, but which is amply repaid in the reduced resistance obtained. In the absence of definite figures relating to the particular machine under design, the calculation of body resistance requires the computation of the resistance of each element, for which purpose it is convenient to have the tabulated results of the resistance of different kinds of fuselages, wires, chassis, wheels, and other components. Most of the data available for this purpose is the outcome of experiments carried out by M. Eiffel and by the National Physical Laboratory. Some experiments carried out by Eiffel upon the shapes shown in Fig. 1 56 will form a very useful introduction to this subject. These shapes consist of a nose, a cylindrical centre portion, and a conical tail. The results of the tests may be summarised (i) The blunter the nose the greater is the resistance. N.B. Nose of section I is of streamline form. (ii) For the same nose and tail the resistance diminishes as the length of the central portion is reduced, (iii) Diminution in the length of the tail leads to a slightly increased resistance. (iv) With streamline shapes the resistance varies with the velocity according to an index less than 2, the skin friction forming a considerable portion of the total resistance. TABLE XXX. RESISTANCE COEFFICIENTS FOR FUSELAGE SHAPES. Equivalent normal plane coefficients. Body. 32-8 f.p.s. 65*6 f.p.s. 98-4 f.p.s. 131-2 f.p.s. I 0*015 0*0139 0*013 0"OI2 II 0*0147 0*0141 0-0133 0*0121 Ill o 170 0*0161 0*015 00135 IV 0*0132 0*0123 0*0115 o'oio3 V. Round end foremost... 0-0152 0*0143 0-0137 o 0103 V. Round end behind . . . 0*0182 o 0167 0-0164 0*0103 N.B. Observe that the resistance coefficient diminishes consider ably as the air speed increases. 2l6 AEROPLANE DESIGN Turning from the general question of the resistance of bodies made up of geometrical solids, the question of the resistance of Square and Circular Square bod^ wif>> fore and afr Mind shields (d ^ Circular bod^ wirti and afr wind FIG. 157. Aeroplane Bodies. various types of fuselages met with in aeronautical practice will next be considered. RESISTANCE AND STREAMLINING 217 Aeroplane Bodies. An investigation was made by the N.P.L. into the effect of various modifications of the form of aeroplane bodies upon the resulting forces and moments. 0.25 35- Angle of Yaw FIG. 158. Comparison of Bodies of related Cross Section. A comparison was first made of bodies of square and circular cross section, after which these bodies were modified by the 2l8 AEROPLANE DESIGN addition of wind shields of various types. For the first com- parison the square section body was taken as the basis. Then the relation between the three bodies was such that the circular sections at all points along bodies were respectively the in- scribed and circumscribed circles of the square section. The SQUARE: CROSS SECTION FIG. 159. scheme is shown at the top of Fig. .158. Throughout this investigation it was assumed that the position of the centre of gravity was 5*8 inches behind the nose of the body, this figure being taken as a fair mean position after a consideration of a large number of types of existing machines. RESISTANCE AND STREAMLINING 219 The experimental results are shown plotted in Fig. 158, and it is somewhat surprising, taking into consideration the available amount of stowage space for engines, etc., the greater con- CIRCULAR CROSS SECTION FIG. 1 60. venience of attachment of such details as the wings and chassis, and the much greater ease with which it can be constructed, that the square section should prove to be the best type of body for 220 AEROPLANE DESIGN general use. Ip order to obtain the same amount of stowage space it would be necessary to go to the size of the circum- SQUARE: CROSS SECTION Wind Spfced Y(0 N(b) 5* 10 15* E0 25 T>0~ Angle of Yawfyr) FIG. 161. scribing circular section, and, as will be seen from Fig. 158, while this body is nearly as good as the square section generally, in RESISTANCE AND STREAMLINING 221 the case of the Yawing Moment curve near the origin there is a much greater slope for the circular section, so that a much larger rudder would be necessary in order to counteract the negative righting moment due to the body. Moreover, the CIRCULAR CROSS SECTION Wind Spc d 40 f p.S Y(d)' Y(aV FIG. 162. square section body possesses a much greater value for the lateral force than the circumscribing circular section, which in practice is equivalent to an addition to the area of the fin, and acts as a corrective to sideslip. 222 AEROPLANE DESIGN The models for the general series of tests are shown in Fig. 157, namely, (a) Perfectly plain body square and circular sections ; (b) Cockpit and pilot added square and circular sections ; (c) Cockpit, pilot, fore and aft wind shields added square section ; (d) Cockpit, pilot, fore and aft wind shields added circular section. Tests were also carried out upon models possessing a rear wind shield only, but, as was to be expected, the results showed that wind shields, both fore and aft, are preferable in all respects. The curves plotted in Figs. 159-162 show that small modi- fications in the shape of the bodies do not affect either the forces or moments to any great extent, with the one exception of the longitudinal force. As will be seen from these figures, this force is particularly sensitive to small changes of shape, more especially so at large angles of yaw. The designer should therefore aim at keeping this longitudinal force as low as possible, while giving the pilot as much protection as possible from the wind, consistent with a good forward view. The development of a satisfactory transparent screen totally en- closing the pilot would be of considerable utility in ensuring his comfort upon long-distance journeys, and of distinct advan- tage from an aerodynamical standpoint, but mechanical or other means would have to be devised to keep it clear in all weathers. RESISTANCE AND STREAMLINING 223 Deperdussin Monocoque Fuselages. Eiffel has tested two types of fuselage similar to those shown in Figs. 163 and 164. It will be seen that the bodies differ in the arrangement of the nose portion, the one being fitted with a rotary engine and top cowl only ; while in the other the engine was totally enclosed save for a small aperture between the propeller boss and cowl to admit the cooling air. The models were to one- fifth scale and of the following dimensions : Length, 2*94 ft. ; diameter of No. I (Fig. 163), 0*525 ft.; diameter of No. 2 (Fig. 164), 0-588 ft. The tests were made at speeds varying from 80 to 90 f.p.s. In the first series of tests no airscrews were fitted to the models, and the results were as follows : TABLE XXXI. MONOCOQUE FUSELAGES WITHOUT AIRSCREWS. Fuselage. Resistance at 60 m.p.h. No. I ... ... ... ... 22*6 Ibs. No. 2 ... ... ... ... 19*0 Ibs. In the second series of experiments made upon these models, airscrews were fitted and allowed to rotate with the engine under the influence of the moving air, the conditions thus approximating to those occurring during a glide with the engine switched off but rotating. The results were ^as follows : TABLE XXXII. MONOCOQUE FUSELAGES WITH AIRSCREWS. Type of fuselage. Resistance at 60 m.p.h. No. i 65 Ibs. No. 2 43-8 Ibs. It will be seen that the introduction of the airscrew increases the resistance very considerably. This is due to the increased pressure on the fuselage resulting from the airscrew wake and to the disturbance in the air flow over the entire surface. Fig. 165 represents another type of body in which the model was fitted both with a tail plane and an under-carriage, and was to one-twelfth scale. The resistance of the model was found to be 0*1365 Ibs. at 30 m.p.h. The corresponding resistance of the full-size body 24*5 ft. long complete is 218 Ibs. at 100 m.p.h. B.E. 2 and B.E. 3 Fuselages. A very complete investiga- tion was made by the National Physical Laboratory into the forces and moments acting upon the models shown in Figs. 166 and 167. Since the results are also of great utility in considering questions of stability in addition to their value in estimating body resistance, they will be given in entirety. 224 AEROPLANE DESIGN Scale of Mode! w- Figs. 1 66 and 167. Forces and Moments on Model Fuselages. Measurements were made of (i.) Lift and drag for various pitching angles with zero angle of yaw ; (ii.) Pitching moment about a horizontal axis perpendicular to the wind, with zero angle of yaw ; (Hi.) Drag, lateral force, and yawing moment, about a vertical axis for different angles of yaw with the pitching angle zero. RESISTANCE AND STREAMLINING 225 The results are exhibited graphically in Fig. 168. It will be noted, in the case of the longitudinal force curves, how the longitudinal force rises rapidly with increase of angle of yaw in the case of the B.E. 2 body. This is probably due to the pro- jecting head and shoulders of the aviators when there is a small angle of yaw. In the B.E. 3 there is no such effect, and the longitudinal force varies very little for small angles of yaw. The curves show that the B.E. 3 body is of a much better form than the B.E. 2, its drag at zero angle being only half that of the Pitching Angle Pitching Angle FIG. 1 68. Forces and Moments on B.E. Fuselages. latter in the same position. The moment curves show that in all cases the bodies are unstable in their symmetrical position if supported at the C.G., that is, for small angular displacements there is a moment tending to increase the angle of displacement. The wind speed throughout all these tests was 30 f.p.s. A further series of experiments was carried out on the two models shown in Figs. 169 and 170. Both these models were tested with and without the rudder, and readings were taken of the lateral force, drag, and yawing moment for various angles of yaw. After these tests had been completed on the model shown in Fig. 169, the recesses round the crank case were faired with 226 AEROPLANE DESIGN plasticine and the drag at zero yawing angle determined. The drag was found to be reduced from 0*016 Ib. to 0*0148 lb., a reduction of 7*5%. The general results indicate that model, Fig. 169, is slightly better than the B.E. 2, but not so good as the B.E. 3 body. The curves of model, Fig. 170, are similar to the others in general form, the chief difference being in the curve of yawing moment without rudder. For this body a restoring moment is obtained Rg. 170 for displacements from the symmetrical position as regards yawing motion about the C.G. This is probably due to the two small fins just in front of the rudder itself, which were not removed in the test without the rudder. In Table XXXI II. the resistance in pounds of the four full- sized bodies at 60 m.p.h. without rudder or elevator planes is given. The four bodies are not very different in over-all length, hut in order to allow for this difference the value of the resist- ance, divided by the square of the over-all length, has been given. The figure so obtained is a fair criterion of the relative FIG. 171. LONGITUDINAL &c L/KTEIRAL FORCES ON MODELS il 02. -01 \ of Yav/ I r 15 2 Q f. \ Rudder in pos 1 ? I -0-4 -03 ~ -10 Force .._.b. A " B ^ WfH> Rluddcr in poe^ Rudder -0-1 -02 228 AEROPLANE DESIGN efficiency of the bodies as regards resistance in the normal flight position. The actual resistances were calculated by assuming the drag to vary as the square of the velocity and as the square of the linear dimensions. TABLE XXXIII. COMPARISON OF FOUR FUSELAGE BODIES. Body. Drag at 60 m.p.h. Drag/Length 2 . B.E. 2 ... ... 54*0 Ibs. ... ... o'io2 B.E. 3 ... ... 25*8 ... ... 0*041 Model 4 ...... 35-3 ...... 0-080 Model 5 ...... 18-4 ...... 0-054 Resistance of Wires. The results of a large number of tests show that the resistance of a wire may be expressed in the form, R = K^ V 2 ............... Formula 68 where d = the diameter of the wire, V = the velocity of the air relative to the wire in feet per second, R = the resistance per foot run, K = a multiplying constant, which depends upon the product dV, in accordance with the principle of dynamic similarity. For values of dV less than 0*15, K decreases with increase of */V, and for values of d^J greater than 0*15, K increases with dV. It is with the latter portion that we are chiefly con- cerned in aeronautics. Table XXXIV. gives the values of K for increasing values of d V, and is taken from results of tests at the N.P.L. TABLE XXXIV. VALUES OF K WITH INCREASE OF dV. dV 0-5 1*0 1*5 2'o 2-5 3*0 K *ooi2 '0013 '00137 '00141 '00144 '00145 These experiments covered a range of speed of from 9 to 25 feet per second, and the diameters of the wire varied from 04" to 0-2 5 ". More recent experiments at the N.P.L. have been made at speeds up to 50 feet per second and upon wires up to f " in diameter. The results are shown graphically in Fig. 172, where the value of the constant K is plotted against the product d V in F.P.S. units. RESISTANCE AND STREAMLINING 229 For example, to find the resistance of a J" wire at 100 m.p.h., d = "25" = '02083' V = 100 m.p.h. = 1467 f.p.s. f/V = 146*7 x '02083 = 3'6 From Fig. 172, the value of K corresponding to 3*06 = '00145. .'. Resistance of " wire = '00145 x '02083 x J 46'7 x J 46'7 = 0*65 Ib. per foot run 5 (-0 d V it 15 20 Fbof Secorjd Uijite FIG. 172. Values of k with increase of //V The values given in Table XXXIV. are for smooth wires, For stranded cables or ropes these values must be increased 20%. If the struts or wires are inclined to the direction of motion of the air, the resistance may be very much diminished owing to the change in shape of section, as the following table shows. TABLE XXXV. RESISTANCE OF INCLINED STRUTS AND WIRES. Inclination of strut to wind 80 6o c 90 60 70 00 50 40 30 Percentage of Normal Resistance for Constant Projected Length. Circular section ... 100 96 88 76 61 45 31 Streamline section 100 97^5 91 83 70^5 55*5 45*5 20 21 44 2 3 AEROPLANE DESIGN The percentage resistance for the struts at all angles is given in terms of their resistance when normal. The last line shows, as was to be expected, that for a streamline strut or wire there is. not such a large gain due to inclination as for a strut or wire of circular section. Resistance of Flat Plates. The resistance of a flat plate normal to the wind, apart from scale effect, depends upon the compactness of its outline. For example, the best form of FIG. 173. Wind Forces on Wheel of Landing Chassis. outline is a circle, while the worst form is one having many re-entrant angles. The following formula by the N.P.L. gives the resistance of square plates for values of V L between I and 350, where V is the velocity in feet per second, and L is the length of the side in feet. R = -00126 (VL) 2 + -0000007 ( VL ) 3 Formula 69 For rectangular plates the results obtained by this formula must be corrected by the use of the factors in Table XIV. Resistance of Landing Gear. When designing a landing gear, care should be taken that all struts and tubes are enclosed RESISTANCE AND STREAMLINING 231 in a streamline fairing in order to cut down the resistance as far as possible. , M. Eiffel has measured the resistance of several full-sized landing chassis wheels, the results of which are embodied in Table XXXVI. below. TABLE XXXVI. RESISTANCES OF LANDING WHEELS. Dimensions of T> f Equivalent Type of wheel. tyre Kesistanc. normal ^ (mms.) at82f.p.s. const nt . Deperdussin ... 725 x 65 ... 3*88 ... 0*92 Farman (uncovered) 610x77 4' J 9 I ' Farman (covered)... 610 x 77 ... 2^07 ... 0*49 Dorand ... ... 530 x 80 ... 2*57 ... 0*68 Astra Wright ... 450 x 53 ... 1*27 ... o'8o It will be observed from this table that the effect of covering the Farman wheel is to reduce its resistance by $0%. Tests at the National Physical Laboratory to find the resistance and the lateral force upon the wheel shown in Fig. 173 gave the results shown in that figure. Effect of Airscrew Slip Stream. The effect of the air- screw slip stream upon those members situated within it is to increase the velocity of the air impinging upon them. This means a corresponding increase in their resistance, which must be allowed for when making an estimate of their resistance and the total resistance of the machine. The slip stream effect is not easy to estimate, the relative increase in resistance being much greater at low speeds than at high. The slip stream is regarded by some designers as a tubular body of air of external diameter of approximately 0*95 times the airscrew diameter, and internal diameter of 0*2 times the airscrew diameter. All members included within this annular cylinder are exposed to the increased velocity. In the absence of more definite results bearing upon any particular design under consideration, the curve shown in Fig. 174 can be used to give a rough estimate of the increase in resistance. The value of the Tractive power of an airscrew at a given speed is obtained from the equation, F / V \ 2 ~~| Tractive power = k i - (^7-7] N 3 D 5 Formula 70 where N = number of revolutions per second, D = diameter of the airscrew, V = forward speed of the machine, p = experimental mean pitch of the airscrew, k = constant whose value can be determined experimentally. 232 AEROPLANE DESIGN If D be the diameter of the airscrew used, then the value of the fraction r .^. lve ^^ er can be calculated for the speed under (Diameter) 2 consideration. The value of the corresponding slip stream co- efficient is at once obtained from the curve in Fig. 174, and the resistance of each component falling within the slip stream must be multiplied by this coefficient. Resistance of Complete Machine. The following table shows the estimation of the resistance of the various parts of an aeroplane. The machine under consideration is the B.E. 2, total weight 1650 Ibs., with a speed range of 40-73 m.p.h. TABLE XXXVII. ESTIMATE OF BODY RESISTANCE OF B.E. 2 AT 60 M.P.H. STRUTS. 8 6' o" x ij" @ -85 Ibs./sq. ft. ... 4/2 Ibs. 44' o" x ij" @ ,, ... 14 63' o" x ij" { ... i -6 - 7-2 WIRING. 220 feet cable @ 10 Ibs./sq. ft. ... 29-5 70 feet 12 G.H.T. wire@ 10 Ibs./sq. ft. 5 '6 52 strainers, estimated ... ,,. 3*0 38T Rudder and elevators ............ 2'o Body with passenger and pilot ......... 40*0 Axle @ '85 Ibs./sq. ft ............. 2*0 Main skids and axle mounting, estimated ... ro Rear skid, estimated ... ... ... ... '5 Wheels ............ 3-5 Wing skids, etc. ............ io'o - 59' I0 4'3 Exposed to a slip stream of 25 feet per second. Body .................. 40 Ibs. 4 4' o" struts ... ... ... ... ... i '4 fof 3 'o" ............... -8 50' o" cable ......... *-. t ... 67 30' o" H.T. wire ... ... ... ... ... 2*4 Rudder and elevator ..... . ...... 2*0 Rear skid ... ... ... ... ... ... 0*5 Fittings ... ... ... ... ... ... 2*0 55-8 Increase in resistance due to slip stream = 357 357 Whence total resistance of machine = 140-0 Ibs. RESISTANCE AND STREAMLINING 233 The following data will be found useful in estimating the resistance of the different members of a machine. TABLE XXXVIII. RESISTANCE OF AEROPLANE COMPONENTS. Resistance at 100 f.p.s. Component. Normal area. Streamline struts ... 1*3 Ibs. per sq. ft. Streamline axle .. ... ... ... 1*5 Round smooth cable ... ... ... 10 Stranded cable 12 Landing wheels ... ... ... ... 4*0 Fuselage 3-4 Tail skid ... ... ... 5-9 Tail plane and rudder ro R.A.F. wires 3*25 Wing skids 5 - o Tail plane and aileron levers 2-5 Experimental Measurement of the Resistance of Full- size Machines. Two methods have been adopted for the measurement of the resistance of actual machines (i) By measuring the gliding angle of the machine with the engine switched off and the propeller stopped. Let = the gliding angle then we have tan = Formula 71 Drag In gliding flight the lift is given by the relationship Lift = W cos ; . . Formula 7 2 so that the drag is very easily calculated. The value of the drag thus obtained represents the total resistance of the machine, that is the wings and the body, at the speed of flight considered. The resistance of the wings can be calculated directly from the area of the supporting surface and the characteristics of the aerofoil used by applying Formula 14. Corrections must be applied for speed and scale effect and also for interference effects. By subtracting the resulting re- sistance of the wings from the total resistance obtained previously, the body resistance is determined, and may be compared with that used in the original estimate for the purposes of preliminary design. The principal difficulty en- countered in this method results from the very rapid change which occurs in the density of the atmosphere as the machine descends, and which will give rise to serious errors unless its effect is eliminated. 234 AEROPLANE DESIGN (ii) By determining the thrust of the airscrew during a series of climbs. The direct measurement of this thrust by means of a thrust meter offers the most convenient and accurate method of deter- mining the resistance of a machine, but the difficulty of obtaining a reliable instrument has so far prevented the results secured from being of an entirely satisfactory nature. It has therefore been necessary to deduce the thrust from particulars of the horse- power and the airscrew efficiency of the power unit employed, under conditions similar to those encountered during the test. f FIG. 174. Slip Stream Coefficient. Having obtained this information and carried out the climbing tests, the thrust necessary to overcome the drag of the machine can be determined and the body resistance deduced, as in case (i). Skin Friction. When a fluid flows smoothly over a stream- line body such as an efficient airship envelope, or a thin flat plate placed edgewise and assumed to have no head resistance, a certain resistance is still felt against the relative motion of die body and the fluid, which is termed the skin or surface friction. A thin film of air covers the actual surface of the body, being entangled in the 'roughness' of its outer layers, and imprisoned there by the outer pressure of the air. The frictional force felt RESISTANCE AND STREAMLINING 235 is partly due to the continuous shearing which takes place between this film and the stratum of fluid adjacent to it. It is therefore a function of the viscosity of the fluid. The coefficient of viscosity is defined as the force required to maintain a plate of unit area at unit velocity when it is separated from another plate by a layer of fluid of unit depth. Stokes showed that so long as the motion was sufficiently slow to avoid eddies the frictional resistance varied as the first power of the velocity. Allen showed that this stage was followed by one in which the index of the velocity was 1*5. In the range of velocity common in aeronautical practice the index appears to lie between 1-5 and 2. Lanchester and Zahm further de- veloped the fundamental equation, and from the experiments carried out by the latter, in which the skin friction of a large number of smooth surfaces in a current of air was measured, it was found that the resistance increased according to the power 1*85 of the velocity. Zahm therefore developed the following equation, connecting skin friction with the length of the plane and the velocity. p oc L 7 V 1 - 85 Formula 73. V = the velocity in feet per second. L = the length of the planes in feet. p = the tangential force per square foot. Lanchester has shown that to express the resistance of a plane in terms of the linear size and kinematic viscosity, the relation Roc viL r V r Formula 74 Where R = resistance per unit density , . ... u coefficient of viscosity v = kinematic viscosity = " = _ __ p density L = linear size V = velocity holds for an incompressible fluid when q + r = 2. Expressing Zahm's equation (Formula 73) in terms of R, it becomes R oc L 1 ' 93 V 1 ' 85 Formula 73 (a) whereas, in order to satisfy Lanchester' s equation (Formula 74), the indices of L and V should be equal. Lanchester has therefore adopted the following expression for a smooth surface : R a v l L 1 9 V 1 ' 9 Formula 74 (a) 2 3 6 AEROPLANE DESIGN Assuming that the exponent varies with the nature of the surface, as has been found to be the case by actual experiment, Formula 74 (a) may be written in the form R oc v 2 - n L n V n Whence F = Rp = K,ov 2 - n L n V n Formula 75 For any one surface it is convenient to neglect the length and embody its value and the value of p and v in one constant, when the equation becomes F = K.V n Formula 75 (a) The value of K depends, of course, on the units employed, and both n and K may vary with the surface for even so-called * smooth ' surfaces. An exhaustive series of experiments have been carried out at Washington to determine the values of F, n, K in the simplified Formula 75 (a) above. Plate glass was used as a standard surface, since it is very smooth, and can be readily duplicated. The various fabrics were attached to this surface by a special varnish, to obtain as smooth a surface as possible ; and experiments were made at velocities of 30 to 70 m.p.h., and the forces measured with great accuracy. The results obtained are shown in Table XXXIX., where F is in Ibs. per sq. ft., and the resistance factor (R.F.) is the ratio Observed resistance : Resistance of Glass Plate. TABLE XXXIX. SKIN FRICTIONAL RESISTANCES. Nature of surface exposed. PLATE GLASS. FINE LINEN, uncoated. One coat of aero varnish. i ! Three coats of 1 aero varnish. Three coats of ' aero varnish ' one coat of . spar varnish. AEROPLANK FABRIC, rubber surface. | n ... r8i i -94 1-84 1-89 r*4 I-8 3 K x IO T ... 1 66 128 I6 3 129 J 53 165 m.p.h. 1 3 F 0079 0095 '0085 0082 0081 0084 R.F. ... i*b 1*205 I '08 1 1*042 1-031 1*070 40 F oi33 '0161 0141 0138 oi35 0142 R.F. ... I'O 1*234 i'o8o i"o6o 1*034 I-082 50 F ... 0199 0249 0218 0208 0204 0215 R.F. ... I'O 1*254 1-098 1-048 i 1-028 1-083 60 F ... 0276 0361 0309 0295 0287 0299 R.F. ... I'O I*305 rn8 1*067 1-038 I -08 1 70 F .. 0364 '0496 0424 0403 0387 0393 R.F. ... I'O 1*362 1-162 1-108 1 i "06 1 1-079 RESISTANCE AND STREAMLINING 237 With the aid of this table the actual skin friction of an aeroplane wing surface can be easily calculated. It will be found that it is a very small quantity. It is only in the case of airships where relatively low head resistance is combined with a large surface area that the effects of skin friction are found to be considerable. Zahm's experiments upon a series of surfaces of width w resulted in the following equation : F = -000007 7 8 wL 093 V 1 ' 85 Formula 76 where F is the friction in pounds at a speed of V feet per second. For double-sided planes this value must be doubled, but when estimating the surface friction of streamline shapes the single value alone may be employed. Zahm's formula for the skin friction of a fuselage is F = -00000825 A 0925 V 1 ' 85 Formula 77 where A is the superficial area in square feet, and V the velocity in feet per second. CHAPTER VII. DESIGN OF THE FUSELAGE. Weights. Before proceeding to consider the general and detail design of the fuselage, it is necessary to examine more closely the question of the weights of the various components of an aeroplane. With this object in view Table XL. has been prepared from an analysis of a large number of machines of various types, and gives the percentage weights of the different portions of the machine arranged in groups. The weights of the individual members of a group will, of course, vary considerably in different designs. TABLE XL. PERCENTAGE WEIGHTS OF AEROPLANE COMPONENTS. 1. The Power Plant. (a) Engine ... ... ... ... 2o - o (b} Radiator 2-5 (c) Cooling water ... ... ... 2 - o (d) Tanks and pipes ... 3-0 (e) Airscrew ... ... ... ... 2*5 30*0 2. The Glider Portion. (a) Wings ... 13-0 (b) Wing bracing ... ... ... 3*0 (c) Tail unit 2-0 (d) Body... .... ... ... ... 13*0 (e) Chassis or undercarriage 4-0 35' 3. Useful Load. (a) Fuel ... ... ... ... ... 20 m o (b) Passengers and cargo 14*0 34-0 4. Instruments, etc. ... i'o Table XLI. gives particulars concerning the chief Modern Aero Engines of various types. I W N - - , O VO ONOO I O O I I N OO cooo co i 11 s M LT-l t-T) ri cio t^ . 00 . oo co rn 1 rs, 1 1 fO CO CO M M LO M OO O w a 3 S M in . rn 1 M_> u>"i i i O i i i O O pNOO I^M *co I vb i ON '- \ I ON I IV CO "^ ^f ^O i i iy~> ur> ON vO ON 1-1 NO vo NO NO H -^- i-r> \O OO NO "" OO \O OO i-o O I M OO J^ I O NO i-O O ON ON ON O .-< ON Tf "~> t^MO OO ' OO O MD ' MD ioOO OO LO\O NO K t^MD O - O n co oo M oo oo rj - -1-i-iON --' OOO oo .ON Weight ngine d sqq COCO IT "->oo vo O <-< >-o O rO"^ ONOO Fuel umpti s. per .P. h no NONO ONOOOMOOLO ?So o o - o i -ooooooo'booooo C-l |OO-'~>i-rii_oi-r>coc^cOMr>r)NOcoco " "5 O O NVO t^ L/-> LT% LO l-TV LO l-O sjapuqXojo ' t^ ON ON ON ON ON ONOO oooooo M NO NO M M M NO c-J \O 'Tj "T2 UO U R-C vO SuijooD jo poq:j3j\[ 'Wd'H OOOOOOOOOOOOOOOQOOOO urii_oijriO l -'~>OO'- / ~>OOOO l -oOOOOOOO \rira OOONOOOOOO AME OF ENGINE. CO 00 " bo ex o C w < PQPQUUOS ^ J tapering away at both ends. If a single length of good quality is not obtainable, two pieces may be spliced together or con- nected by a clip. Wherever exposed to wearing conditions such as foot-rubbing, engine heat, etc., they should be sheathed or otherwise protected. As far as possible, holes in the longerons should be avoided, in fact it is good practice not to pierce them at all. If pierced to carry any strain-bearing bolt, the bolt should be supported externally by means of flitch plates. A form of clip can be easily devised which satisfies the requisite conditions to take the wires which radiate from the junction of the longeron with strut, and further keeps the strut in position. From the general arrangement made in the drawing office the fuselage will usually be set out full size on a large board in the shops. The longerons may be bent and the size of the struts cut off to the very accurate shapes and dimensions given by the latter drawing, which also supplies the local angles of the longeron and struts. The longerons are bound in pairs at their rear ends to the sternpost, which provides a strong fixing for the rudder and the hinge spar for the elevators (the rear spar of the tail plane). The sternpost may be either of wood or a steel tube, and may also conveniently serve for the fixing of the tail skid and of the vertical fin. At their forward end the longerons are bound to the corners of a square or rectangular plate, which, if the engine is overhung, serves as an engine bearer plate (see Fig. 183), or provides a fixing for the extension shaft for the airscrew. The square section given to the fuselage by the longerons and struts is far from ideal from the standpoint of streamlining, especially on the top. The flat top may be rounded off by means of fabric stretched over formers. These formers may be built up very lightly out of reinforced plywood, lightened out until there is very little material left. They may be mounted over the top struts, and connected together by means of a few longitudinal strips. Fabric should be used wherever possible as a fuselage covering, and three-ply should be avoided owing to its much greater relative weight. In this connection it is important to remember the results of the experimental investigation into the relative merits of the square and circular cross-section aeroplane bodies given in the DESIGN OF THE FUSELAGE 243 preceding chapter. As will be seen from Fig. 184, it is possible to build up a fuselage of streamline shape without using a square section as the basis. Two or three of the top bracing-panels will require to be omitted on account of: 1. The pilot. 2. The passenger, if any. 3. The fuel tanks. 4. Perhaps, the engine. This means collectively a serious weakening, and the panels thus mutilated should be strengthened by all convenient means. Points of attachment of heavy weights, such as pilot, pas- senger, tanks, etc., should be made to the longerons at the cross panels. This principle must be the chief guide in setting out the fuselage, and it should be remembered that bending moments are to be avoided as far as possible in any member of the body. Further, where a compressive force, as from the wings, comes on the fuselage a specially strong strut should be arranged to take the strain directly, and, where a tension may be applied, a special tension member should be introduced. The sizes of these members are calculated in the usual manner. The general arrangement of the fuselage may be conveniently set out by drawing a section longitudinally, and then making drawings of each cross panel. The formers, wherever used, should be included in the transverse sections. It is useful to have cardboard models of a pilot to the scales most generally used in the office, as there is frequently a question as to clearance between some portion of the pilot's body and the various fittings in the cockpit. The engine is the limiting factor in considering the design of the fuselage aerodynamically. If a radial engine is used it may be hung on the nose of the fuselage and partially protected by a cowl, or it may be totally enclosed some way back. The second method gives much greater scope with regard to the streamlining of the fuselage, although it is impossible owing to considerations of balance to get the Engine back far enough. It has the great drawback, however, that it greatly increases the weight owing to the extension of the airscrew shafting, bearings for same, and extra engine-bearing plates of large size. It is further liable to accidental fires. In the case of the heavier type of the fixed cylinder engine its^ greater weight will necessitate a less forward position, leading to greater ease in aerodynamic design. On account, however, of the shaft, and consequently the airscrew, being at the bottom of the engine instead of in the middle, it will be difficult to totally enclose the 244 AEROPLANE DESIGN engine without unduly increasing the height of the chassis, still keeping to the minimum section of body. The cylinder heads are often left exposed in this type of engine and may spoil an otherwise good design, more especially if the heads happen to be placed in the airscrew slip-stream. In important cases, and in fact whenever possible, different designs should be tested in the wind tunnel and compared from the point of view of weight and cheapness to manufacture. As we have pre- viously seen, a small saving in head resistance is likely to be of great importance in high-class work. When considering the question of skin friction of a fuselage, Zahm's formula may be used, namely : Skin-friction = '00000825 A ' 925 y1 ' 85 where A is the superficial area in square feet V is the velocity in feet per second. Assuming ordinary dimensions, it will be seen on applying this formula that air friction accounts for about half of the total resistance of a good fuselage shape. The total head resistance of the fuselage will then vary as a power of the velocity between 1*85 and 2. Wind-tunnel tests would be useful to a designer in assessing the true value of this index for a particular case. The importance of keeping the maximum cross-sectional area low should not be lost sight of. It is frequently argued, and it is often true, that it is better to waste space by increasing the maximum cross dimensions of the fuselage, if by so doing the ' lines ' of the fuselage may be improved. The principle is a sound one within certain prescribed limits, but it is easy for an enthusiast in streamlining to increase the maximum section of the fuselage by 2" to 3" all round, thus increasing the maximum cross-sectional area by some 30%. This means that the coefficient of head resistance of the thicker shape would have to be improved by the same amount (30%) in order that the en- larged body should have as small a resistance as the original body, and more than this amount if improvement is to be attained. The size, therefore, should be kept as small as possible consistent with housing the engine and pilot and without unduly exposing parts of either to the wind. This last consideration will limit the size of the cockpit opening and lead to a small shield, a few inches only in height, being placed on the forward half of the cockpit opening in order to spill the air over the opening. It is interesting to note in this connection that totally en- closing the fuselage cockpits results in a greatly reduced resistance. For example, the Vickers' Commercial machine, DESIGN OF THE FUSELAGE 245 the fuselage of which will be shown later in'this chapter, is 10% faster than its prototype, the Vickers' Vimy Bomber. This and other types of fuselage used in modern practice are shown in Figs. 175-184. In the fuselage shown in Fig. 175 it will be observed that the longerons are supported in their correct position by means of three-ply formers in the front portion and by means of struts and wires in the rear portion. An excellent type of fuselage is that of the Bristol Fighter, 'which is shown in Fig. 176. The front portion, comprising the engine-bearers, is composed of tubular steel, while th~ remain- ing portion of the structure is built up of wood braced together with small tie-rods. The depth of the beam increases towards the centre of the machine and thereby helps to keep the bending stresses low throughout. In order to afford a comparison between the fuselages of two FIG. 177. Fuselage of Handley-Page (0-400) Machine. machines which are being largely adopted for commercial work, and to illustrate the different methods employed in practice in building large fuselages, the fuselages of the Handley-Page and the Vickers' machine are shown in Figs. 177 and 178. As will be seen, the Handley-Page follows the construction of the types already illustrated, while the Vickers' machine ' exhibits a totally different type of construction. As adapted to commercial uses the passenger cabin of the latter machine is of considerable interest. It may be mentioned in passing that the sole modification of the well-known war machine of the Vickers Company, the Vimy Bomber, for commercial purposes, lies in the use of a different fuselage. As will be seen from Fig. 179, the shell of the cabin is built up of oval wooden rings of three-ply box section, the formers being shown in the back- ground of Fig. 179. The cover of the cabin is made according to the ' Consuta ' patent of Saunders, of Cowes, and is con- structed of thin layers of selected wood, the grain being placed diagonally, and then glued and sewn together, the rows of stitch- ing running in parallel lines about ij" apart. The strength of this material is very great, giving a high factor of safety to the 246 AEROPLANE DESIGN cabin, and enabling all cross-bracing wires to be dispensed with in the interior of the cabin, as shown in Fig. 180. An exterior view of the completed cabin is shown in Fig. 18 1. Fig. 182 illustrates the fuselage of the Fokker Biplane. It will be seen that it is built up of thin steel tubes which are welded together. The longerons are fixed relatively to one another by means of cross struts butt-welded to the longerons and by means of bracing wires. This method of construction has not proved very successful up to the present, mainly on account of the difficulties of welding and brazing, and it is found FIG. 182. Example of Steel Fuselage. that for a given weight a wooden fuselage is stronger. It is more probable that the steel fuselage of the future will be constructed by the methods usually adopted in other engineer- ing structures, namely, by means of steel channels and angle irons. By. this means the expensive steel sockets and fittings necessary at the joints of a wooden structure can be avoided, while the work involved in pressing out a steel channel or angle iron to the desired shape is very much less than the process of producing such a member in wood. Moreover, it can readily be lightened when necessary and desirable by punching holes in it. The Sturtevant Company of America have already pro- duced fuselages upon these lines, and apparently with some success. One type of their large battle-planes is fitted with a steel fuselage which, complete with steel engine-bearers and DESIGN OF THE FUSELAGE 247 bracing, weighs only 165 Ibs. It is estimated that a wooden structure of equal strength would weigh over 200 Ibs. The Sturtevant fuselage has been found quite satisfactory in a series of prolonged tests, and there is little doubt that in process of time the use of steel or light alloys in this direction will be very greatly extended. Fig. 183 illustrates the fuselage of a small scout machine, namely, the Sopwith Camel. It follows the usual girder type of construction. The longerons at the forward end fit into a pressed steel engine bearer which carries the rotating engine with which this machine is fitted. The engine cowl is attached to the circular tube seen at the fore end. Fig. 184 illustrates the fuselage of the German Pfalz, and gives an excellent idea of the body formers and the position of the longerons for obtaining a good streamline shape. FIG. 183. Stressing the Fuselage. The method of stressing the fuselage will now be briefly considered. For the general type of fuselage structure such as is shown in Figs. 176, 177, 178, 182, 183, the determination of the stresses in the various members is not a difficult matter once the external loads upon the structure have been estimated. It is customary to stress the rear portion of the fuselage for the tail load alone, this being considered as an isolated load acting at the point of attachment of the tail plane to the fuselage. The tail plane may be subject either to lift or to down load, thus causing the fuselage members to be subject to reversed stresses. The usual method of designing the tail plane is to assume it to be subject to a uniform load per square foot of area. This having been decided upon, the total load applied by the wind forces through the tail plane upon the fuselage is easily determined. It is common practice to assume a loading of from 15 to 25 Ibs. per square foot of tail surface, either up or down forces, the larger figure being adopted where a high factor of safety is desired. The principal load upon the front portion of the 248 AEROPLANE DESIGN FIG. 179. FIG. 180. FIG. 181, Reproduced by courtesy of Messrs. Vickers, L,td. Construction of Passenger Cabin for Vickers' Commercial Machine. Facing page 248. DESIGN OF THE FUSELAGE 249 fuselage is that of the power plant, which includes engine, radiator, fuel, tanks, etc. Having decided upon the position in which these items are to be fixed, the structure can be stressed in the usual manner. An example of stressing the fuselage is shown in Fig. 185. It has been assumed that the machine is fitted with a tail plane of 35 square feet in area, and of weight 40 Ibs. The maximum down load on the tail plane is to be 25 Ibs. per square foot. The total load acting at the rear end of the fuselage is therefore 35 x 25 + 40 = 915 Ibs. This is dis- tributed equally on each side of the structure, making 458 Ibs. to be applied on each girder. A side elevation of the fuselage 458 FIG. 185. Stress Diagram for Fuselage. is next drawn out as in Fig. 185 (a), and the stress diagram for the rear section then follows, as shown in Fig. 185 (). From the table of stresses prepared from this diagram the necessary sizes of the members in the rear portion of the fuselage can be determined. Referring first to the longitudinal members, it will be found that they must be designed for compressive loads. For this purpose the Rankine Strut formula may be used, the constants being taken from Fig. 89. In using Rankine's formula, it generally happens that the area chosen gives a crippling load much above or much below the value required. To obtain a close approximation, several values have to be tried, though other considerations frequently intervene to fix the sizes. 250 AEROPLANE DESIGN The vertical struts are generally of the same thickness as the longerons at their junction to the latter, but may with advantage be spindled out intermediately. A section such as that shown in Fig. 185 (d) will be suitable. The sizes should be checked by means of Rankine's formula modified as in the case of the longerons ; in fact, it is a good plan if one is engaged on much strut work to graph this formula for various standard sections, so that much tedious computation is avoided. The front portion of the fuselage may next be considered. The principal load occurring on it is the engine and radiator. For this design the weight is about 800 Ibs. Half of this load is carried by each side and in turn distributed over the struts which carry the engine bearers. The stress diagram can then be drawn as in Fig. 185 (c). It will be seen that the loads are so light that the longerons will take them comfortably, and in practice the nose portion is very rarely stressed. It must be remembered, however, that they have also to take the vibration due to the engine, so that it is inadvisable to reduce them in size, as the structure might shake to pieces. The fuselage shown has three-ply wood J" thick over the entire front of the structure. There remains to be considered the stresses in the centre portion of the body when the machine lands. Half the maximum landing shock will be taken by each landing wheel. This force may be resolved into two components along the under-carriage struts, and these components will set up a direct compression in the vertical struts of the fuselage, and place the portion of the longeron between them in tension. When designing this por- tion of the fuselage, the effect of this tension must be considered, and care taken to see that the longeron is sufficiently strong for this purpose. Where possible it should always be arranged that the principal loads act upon the vertical struts so that the longerons are not called upon to act as beams, but are only subject to direct tensile or compressive stresses. Design of the Engine Mountings. Before commencing to examine the various types of engine mountings, a few notes as to the problems involved will be useful. We shall first consider the stationary vertical type of engine, this being perhaps in most general use. The engine itself consists of 4, 6, or sometimes 8 cylinders placed one behind the other in a straight line on top of a common crankcase. This arrangement of cylinders makes for a somewhat long engine bed, which must be very rigid if misalignment is to be avoided. Some types of engine have been supported by transverse members running DESIGN OF THE FUSELAGE 251 through the crankcase from side to side, but in the majority of cases the two sides of the crankcase are provided with horizontal flanges running the whole length of the engine, or else with brackets projecting out from the sides at intervals, designed to be bolted on to longitudinal engine bearers resting on the body structure of the aeroplane. The problem confronting the designer is to provide a structure which, while rigid enough to ensure that the engine itself is not subjected to any bending stresses, is yet sufficiently flexible to transmit the vibration of the engine to the mounting, and yet damp out these vibrations before they reach the structure of the aeroplane ' fuselage ' proper. In this connection it should be remembered that apart from such minor considerations as vibration, which should be reduced to a minimum in a modern engine, there are two main loads to be considered. One is the weight of the engine, which is always acting, while the other is the thrust or pull of the airscrew acting only when the engine is running, and varying from a maximum when the engine is going 'all-out' to a minimum when it is throttled right down. There is also the reverse thrust when the machine is diving and the air pressure on the back of the screw is driving the engine. It will thus be seen that these two main loads give one ver- tical component and one horizontal component. Neither is constant, for during a vertical nose-dive with engine running, the weight of the engine is acting along the same line as the thrust, both tending to pull the engine out of the fuselage in a forward direction. Moreover, as we have already seen, the horizontal component varies both in magnitude and direction. In general, however, we may consider the two components as being vertical and horizontal respectively. In normal flight the resultant of these two components will have a forward inclination of approximately 45. This may be illustrated as follows : Consider an engine of the average ver- tical type weighing, say, 5 Ibs. per h.p. this is somewhat high, but will illustrate the point and the thrust obtained with an airscrew of average efficiency as 5 Ibs. per h.p., it will be seen that the vertical and horizontal components are approximately equal in magnitude, and their resultant will therefore have an inclination of approximately 45. For a 100 h.p. engine weighing 5 Ibs. per h.p. and giving a thrust of 5 Ibs. per h.p., the resultant will therefore be about .700 Ibs. acting at an angle of 45. During a vertical dive the weight component will be parallel to the thrust component, and hence for the same engine the pull tending to tear it out of the fuselage 2 5 2 AEROPLANE DESIGN will be about 1000 Ibs., that is, twice the weight of the engine. When diving with engine off', the thrust will operate against the weight and thus reduce the forward pull on the engine bearers. There are several different ways in use for transmitting the load from the longitudinal bearers to the body structure proper. In some machines the engine is supported at each end only, while others have three or four points of support. The question Reproduced by courtesy of ' Flight. ' FIG. 186. FIG. 187. of the number of supports to employ depends largely upon the size of the engine. Wood is the most common material used for the direct support of the engine, this being largely on account of its greater resiliency, which acts to a certain extent as a 'shock absorber,' and thereby lessens the vibration. We shall now consider several practical examples of engine FIG. 188. mountings. The arrangement of engine bearers on an Albatross biplane is shown in Fig. 186. The two longitudinal members are of ash, and are supported by transverse members connecting them to the upper and lower longitudinals of the fuselage. The front transverse member takes the form of a pressed steel frame lightened in places and joining a capping plate over the ends of the four longitudinals which converge somewhat at this point. The next support is joined by a ply-wood member 20 mms. thick, cut out in places for lightening purposes. From the DESIGN OF THE FUSELAGE 253 point on the lower longitudinals where the front landing chassis struts are attached, two supporting transverse members radiate. One of these, which is of the same thickness and general con- struction as the preceding one, slopes forward, while the other, supporting the rear end of the engine, has a backward slope. The thickness of the latter member is 25 mms. In the Curtiss biplane the engine is supported by two trans- verse members only, and for the comparatively light engine employed this is quite adequate (see Fig. 187). The front support takes the form of a steel plate lightened out, and with the edges turned in to stiffen the plate against buckling. At the rear the engine bearers rest on a transverse member, which is in turn secured to the upright body struts. Each bearer is clamped to the transverse beam by two bolts as shown in sketch. As the engine overhangs the front chassis struts, the bracing Reproduced by courtesy of 'Flight.' FIG. 189. of the sides of the fuselage has to be sufficiently strong to with- . stand landing shocks, and for this reason the wiring of the front bays is in duplicate. An excellent type of engine bearer for either vertical or V-type stationary engines is shown in Fig. 188. Another interesting type of mounting is that fitted to the all- steel Sturtevant biplane (Fig. 189). Here it will be observed that the engine bearers are of ash, supported by members of channel steel. Four supports carry each bearer, three running to the point where the under-carriage struts are attached to the fuselage longerons. In addition to their forward slope the channel steel supports are inclined inwards, thus effecting a very rigid bracing of the engine in every direction. An additional consideration in the mounting of air-cooled stationary engines is that of providing the necessary cooling effect. It seems probable that ultimately the air-cooled engine will, on account of the large reduction in weight resulting from the absence of radiators and water tanks, supersede the water- cooled type, so that, although the water-cooled engine is now 254 AEROPLANE DESIGN used almost universally, it is well to keep in sight the ad- vantages to be derived from an efficient air-cooled engine. It will be realised that since the cylinders are usually placed in two rows of 4, 6, or 8 each, according to size of engine, the front cylinders will have a shielding effect upon the rear ones, which, as a consequence, will be insufficiently cooled, and this will lead to trouble. The method usually adopted in the tractor type of machine is to direct the air by means of deflector plates so that it enters the space between the two rows of cylinders from the front, is prevented by a vertical partition from escaping at the rear, and is thus forced by the pressure of the incoming air out through the spaces between the adjacent cylinders. In the pusher type of machine the difficulty was overcome by mounting a large enclosed centrifugal fan on the front end of the crankshaft. The space between the cylinders was covered by an arched roof of aluminium running from the tops of the cylinders on one side to the tops of the cylinders on the other. The Vee between the last two cylinders was covered by a vertical aluminium plate. When the fan sucked the air into the space between the rows of cylinders the only escape for the air was the small spaces between adjoining cylinders, which were thus cooled on three sides the inner side, the front, and the back while the outer sides of the cylinders were cooled by the air current due to the forward speed of the machine. This method proved very satisfactory for the ' pusher ' machine. Considerable diversity of practice occurs with the mounting of rotary engines, but the different methods may be divided into two categories: (I), those in which the motor is supported be- tween two plates ; (2), those in which the motor itself is over- hung. This latter method allows of ready accessibility of the engine when repairs are necessary, but is probably slightly heavier than the double bearer mounting, owing to the necessity of using a thicker gauge material. Fig. 190 illustrates an example of the first method. The plates are pressed from sheet steel and all the edges are flanged in order to prevent buckling. The front plate takes the ball race through which the airscrew shaft runs, while to the rear bearer is bolted the back plate of the engine. Great care is necessary in cutting out the lightening holes, and these should be such as not to materially diminish the strength of the plates. The general arrangement of the front part of the fuselage will be clear from the drawing. An example of the overhung method is shown in Fig. 191. In this case the back plate of the engine is attached to the DESIGN OF THE FUSELAGE 2 55 front of the front engine bearer, while the rear bearer acts as a support to an extension shaft which passes through both bearers. The plates are pressed out of sheet steel either by machine or hand, and care is necessary to ensure that they are attached to the longerons in a suitable manner. A problem of considerable importance in connection with the housing of rotary engines is that of obtaining sufficient cooling effect with a minimum of head resistance. The method generally adopted is to fit a cowl or sheet metal shield over the engine. In the majority of machines only the upper part of the engine is covered. Reproduced by courtesy of ' ''Flight FIG. 190. FIG. 191, Radiators. With the increasing demand for larger engine power and the difficulties encountered in air-cooling methods the question of water-cooling has become of great importance,, and it will be useful to consider this subject briefly. The type of radiator in most general use for aero engines has de- veloped principally from the motor-car radiator. According to the present practice there can be but little doubt that the honeycomb type of radiator holds the lead, whether it is mounted in the fuselage or elsewhere. In the absence of wind- tunnel tests it is difficult to say whether the square tube, round tube, or other formations are best as regards the ratio of cooling capacity to wind resistance. COOLING AREA OF RADIATORS. The 'following method was adopted by Lanchester in order to determine the area of cooling surface required. The heat units disposed of per square foot of single service may be expressed by the equation : o-2 4 ECPVT H = - Formula 78 2 where E = double surface coefficient of skin friction = -008 C = normal plane resistance coefficient = *6 V = velocity of air stream, T = temperature difference, say 120 Fahr. P = -078 AEROPLANE DESIGN Taking velocity of the air stream equal to 50 ft/sec, then the heat units disposed of per sq. ft. of surface 24 x -008 x -6 x '078 x 50 x 120 ,, , TT TT - Fahr. H.U. per sec. = -27 and the horse-power equivalent = ~- - = '4 nearly (Note that 780 is the work equivalent of i Fahr. Heat Unit. Hence under the above conditions, and for a velocity of 50 ft. /sec., about 2^ square feet of radiator surface are required per h.p.) The above values for the coefficients were obtained from experiments carried out on a motor-car radiator, and the results were in good agreement with general practice. FIG. 192. The increased speed with which the radiator on an aeroplane meets the air stream will bring down the cooling surface area inversely as the speed, and the above formula may be used to determine the necessary radiator surface required. It is found practically that if the radiator is placed imme- diately behind the airscrews, about 1*6 to r8 square feet of area per 100 h.p. is required ; whereas, if it is placed so that it gets the full effect of the slip stream, about I square foot of area per 100 h.p. is necessary. In the Airco 9, the radiator is arranged in the floor of the front portion of the fuselage, so that a greater or less portion can be exposed at will, according to the pre- vailing conditions. A form of honeycomb radiator which is being largely used on modern aeroplanes is shown in Fig. 192. The water spaces consist of a series of semi-circles and quarter-circles. Each air DESIGN OF THE FUSELAGE 257 channel is formed of a strip of brass, the ends of which are folded over and joined by a machine. The strips are then placed in a press and given their correct shape, after which the honeycomb is completed by soldering the ends of the adjoining strips together. A point in favour of this type of radiator is that it can be built up in units or sections of almost any size. Further, repairs can be quickly and cheaply effected, a damaged section being simply removed and a new one substituted. The Denny Jointless Honeycomb Radiator is constructed by the electro-deposition of pure copper, so that the re- sultant radiator is all in one piece, thus eliminating soldered joints. A useful system which has recently been taken up is one in which a radiator is built up of a number of standardised units in such a manner that for a given engine a certain number of units are employed. If the same engine is used on a faster machine fewer units are employed ; if on a slower one, more units. The advantage of such a system is that instead of a different size and shape of radiator varying for each type of machine, and even in the same machine according to the engine fitted, the standard unit can be turned out in quantities irrespective of the machine to which it is to be fitted. This naturally leads to rapidity and economy of production. Some German machines have been fitted with radiators in the top centre section of the main planes, but it is very doubtful whether the increased complications of such a system are worth while, either from a practical or an aerodynamical standpoint. Gyroscopic Action of a Rotary Engine and the Air- screw. Before concluding this chapter it is desirable to say a few words upon this subject, since the effect is to pro- duce a sideways twist upon the engine-bearer end of the fuselage. From the practical point of view this twist can be easily provided for, and from the pilot's point of view it is found that the usual controls are ample so far as handling is concerned. This gyroscopic action will arise in the case of an aeroplane when for example a tractor machine whose airscrew and engine, viewed from the C.G. of the machine is rotating in a clockwise direction, attempts a right-handed turn. An external couple about a vertical axis is set up owing to the applied air forces, and the axis of the engine and the airscrew tries to set itself in a line with this axis, so that there will be a tendency for the machine to dive. 258 AEROPLANE DESIGN The magnitude of this gyroscopic couple is given by the expression Couple = - ft. Ibs Formula 79 where I is the moment of inertia of engine and airscrew about its axis of revolution in absolute units. O is the angular velocity of the engine and airscrew about their axis of revolution in radians per second. w is the angular velocity of the machine in radians per second at which precession is forced. Experiments were carried out at the Royal Aircraft Estab- lishment in order to determine the magnitude of the gyroscopic couple upon an aeroplane fitted with a 100 h.p. Gnome Engine. The moment of inertia of the engine was found by weighing the parts and measuring their distance from the centre, the result being checked experimentally by measuring the period of oscillation of the engine when suspended by three wires. Similarly the moment of inertia of the airscrew was determined by suspending it bifilarly and measuring its period of oscillation. The results obtained for a 100 h.p. Gnome Engine were: Weight, 270 Ibs.; M.I., 114 Ibs. feet 2 ; Speed, 1200 r.p.m. And for an airscrew: Weight, 30 Ibs. ; M.I., 150 Ibs. feet 2 ; whence total moment of inertia for engine and airscrew is 264 Ibs. feet 2 . The gyroscopic couples due to the precessional movements involved both in turning and pitching were determined as under. (a) Gyroscopic Moment due to Turning. In the first of these cases the aeroplane was turned completely round in 20 seconds, involving severe banking and a very sharp turn. Angular velocity of machine 2 7T ,. = w = = "314 radian per sec. Angular velocity of engine and airscrew = il = 2 TT x ~ - = 125*8 radians per sec. oo whence Moment due to gyroscopic couple 264 x 125*8 x "I\A. f = - LJ? = 324 ft. Ibs. 32-2 (b) Gyroscopic Moment due to Pitching. The problem in this case necessitates the determination of the maximum angular DESIGN OF THE FUSELAGE 259 velocity about the axis of pitch when the elevator is suddenly deflected to its full extent. The limit of this will be determined by that velocity at which the pilot is just about to be lifted from his seat. Let M' be the mass of the pilot. 6 angle of the path. V the velocity of the machine along the path. r the radius of curvature of the path. M' V 2 Then M' g cos d = - whence angular velocity _ V _ g cos ~ r ~ y This is a maximum when 6 is zero. For the machine considered in case (a) V is 100 f.p.s. o> = 32-2/100 = -322 radian per sec. and the moment due to gyroscopic couple _ 264 x 125-8 x '322 32'2 = 330 foot Ibs. which is of approximately the same magnitude as that due to turning. These figures indicate that there is a twist set up in the engine structure which will be communicated to the wings and must be opposed by a movement of the control surfaces. As previously pointed out, this must be borne in mind when designing the fuselage. CHAPTER VIII. DESIGN OF THE CHASSIS. Function of the Chassis. The most important duty of the chassis is to provide for the attainment along the ground of a sufficient speed to lift the aeroplane into the air. This is very easy to design for, but the other duties of a chassis are in some respects equally important, and must not be lost sight of. They are : 1. To make easy a good landing. 2. To protect the airscrew at all times from contact with the ground. 3. To provide and permit of easy 'taxying' along the ground. 4. To form a firm base upon which the machine may safely stand at rest in a wind. 5. To save the machine from damage, as far as possible, in the case of a bad landing. Forces on Chassis when landing. The design of the Chassis is one of the most difficult problems confronting the aeronautical designer, for, while it is desirable to obtain sufficient strength in the landing gear for the machine to be able to land itself when gliding down at its normal gliding angle, it must at the same time be comparatively light and must offer as little resistance to the air as possible. It will be useful first to enumerate the forces acting upon a machine when landing. Referring to Fig. 193 (a) they are : 1. The weight of the machine W acting vertically down- wards through the C.G. 2. The lift w remaining on the wings by virtue of the forward velocity. 3. The head resistance R. 4. The resistance of the ground acting at the point of contact of the wheels and equal to ^ P where fj. is the coefficient of ground friction, and may be taken as 0*16, and P is the reaction at wheel. 5. The force due to the momentum M depending upon the velocity and the weight, and acting through the C.G. of the machine. DESIGN OF THE CHASSIS 261 The fifth force causes a couple M . d (d being the perpen- dicular distance from the line of action of M to the point of contact with the ground), about the point of contact tending to overturn the machine. It is clear, therefore, that the position of the wheels is an important factor in this connection. The higher the C.G. of the machine, the farther forward must the wneels be placed. Method of locating Fore and Aft Position of Chassis. It will be sufficiently accurate for our present purpose to assume that both the lift of the wings and the resistance of the machine act through the C.G. Then with the notation shown in Fig. 193 (a), in order to prevent nose-diving of the machine on landing, it is necessary that the moment P x must be at least equal to the moment JUL P y. Consequently we have the condition P.x > p-Py that is - > u Formula 80 y or tan > p Taking the value of p given above and substituting tan 6 = o'i6 whence 6 = 9 In order to allow for landing on soft ground or upon a slope this angle is generally made equal to about 14, with the axis of the body in the horizontal position. A second method of determining the fore and aft position of the wheels of the chassis is to utilise the gliding angle of the machine in the following manner : Let C (Fig. 193 b) represent the position of the C.G. of the machine under consideration. From C drop a perpendicular C A upon the ground line X X. The distance of the ground from the horizontal line through the C.G. is governed by the amount of clearance given to the airscrew, twelve inches being a usual allowance for the purposes of preliminary design. Then set out the angle A C B equal to the gliding angle of the machine. As in the first method, an extra allowance must be made for unforeseen contingencies, 5 being a common figure in this respect. Therefore set out the angle BCD equal to 5, the point of intersection D with the ground line X X giving the required position. 262 AEROPLANE DESIGN A third, but still more empirical method, is illustrated in Fig. 193 (c). In this method an angle of 75 is set off from the chord of the lower main plane in order to arrive at the required position. (a* ^>.p FIG. 193. Methods of locating the Fore and Aft Position of the Chassis. General Principles of Chassis Design. The first point to decide is the necessary factor of safety. A chassis cannot be made strong enough for aerodynamic reasons to stand up to everything that may possibly occur. Provided that the pilot DESIGN OF THE CHASSIS 263 lands squarely on the two wheels, shock occurs in one of two ways : 1. He does not flatten out quick enough ; or 2. He flattens out too early, and so * pancakes ' down from a height. Considering the first case and assuming that the machine comes into contact with the ground as in a natural glide without having flattened out at all, Let v be the vertical component of V, the forward velocity of the machine, / be the vertical retardation on meeting the ground, d be the give of the landing gear in feet, a the angle of descent to the horizontal, P be the mean reaction at the wheels during landing, w be the lift remaining on the wings, equal say to two-thirds of W, the weight of the machine, we have p = w + ^/ - S 3 Now the vertical retardation is given by v 2 v i that is /= - = 2 d 2 d whence P = W ( -33 + V l sin2 \ V 2df / .. Formula 81 By the use of this formula the reaction at the wheels during landing can be ascertained. Suppose a machine of weight W to be descending at a slope of i in 6 at a speed of 50 m.p.h. To find the reaction at the wheels for a give of 10" in the landing gear : Substituting in Formula 81 we have 64-4 x -834 - 3-03 w Assuming the shock absorber to be of rubber and the force to decrease uniformly from a maximum to zero, the maximum force on landing will be twice the average force. Hence maximum force on landing = 6-06 W 264 AEROPLANE DESIGN that is, the landing gear must be sufficiently strong to with- stand a ground reaction of about six times the weight of the machine. From Formula 81 it is obvious that the mean reaction will decrease according as the give of the landing gear increases, and hence it is necessary to provide a material capable of absorbing the maximum amount of energy. This may take the form of rubber cord, steel springs, or pneumatic cylinders. Rubber cord is to be preferred for small machines, because it is light, cheap, easily workable, and easily replaced ; and it has the further great advantage of not causing an elastic rebound as in the case of steel springs, owing to the energy of the shock being suffi- ciently absorbed by the viscosity of the rubber. Formula 81 also shows the advisability of adopting as low a value as possible for the landing speed of any machine under design. It will also be observed from the above example that while it is a comparatively simple matter to design a chassis for a machine possessing a low landing speed, it becomes increasingly difficult to do so as the landing speed increases. An efficient chassis becomes heavy as the loading of the wings is increased, ample wing area leading to a saving in weight and resistance. A compromise must evidently be made in most designs, and the latitude to be allowed the pilot must be assessed according to the particular experience and practice of the designer. An important consideration to bear in mind is -the necessity to preserve the machine, even if the chassis is smashed. In this connection the skid type of chassis would be very advantageous were it not for its great resistance at high speed. Types of Chassis. The various designs may be roughly divided into two groups : (a) Those with a longitudinal skid placed in front of, and forming part of, the chassis ; and (b) Those without such skids. Of these two groups by far the greater number of modern machines belong to the second group, while most of the earlier types ^f machines possessed chassis belonging to the first group. In the earlier types the skid stretched from behind the C.G. of the machine to either just behind of, and in some cases even in front of, the airscrew, thus forming a backbone to the chassis. This skid was fixed just a few inches above the ground, and provided a rigid stop to the elasticity of the chassis, thereby DESIGN OF THE CHASSIS 265 preventing the airscrew from touching the ground. This type is illustrated in Fig. 194. The objection to its use is the heaviness and large air re- sistance of the type of chassis involved, and the danger, either in a bad landing due to the pilot not flattening out soon enough, or the presence of rather rough ground, of the nose of the skid striking the ground directly, or even impaling itself. In the event of this happening, it is easy for the aeroplane to turn completely over, owing to the high position of the C.G. above the ground. The best way of guarding against this contingency is to protect the forward part of the skid or skids with small extra wheels. These wheels may be very much smaller than the landing wheels proper, but a considerable addition to the weight, and, more important still, a great increase in the head FIG. 194. Central Skid Chassis. resistance, cannot be avoided, and therefore these wheels are seldom used. The longitudinal skid may be either a single central skid fixed between the two landing wheels, or may be made up of two lighter skids placed one in front of each wheel. The single or central skid type of chassis has the long central skid fastened to the longerons of the fuselage by two pairs of struts which form a V one J ust behind the airscrew, and the other just in front of the C.G. (See Fig. 194.) The forward struts should be given a rake, so that their lower ends are in front of their upper ends, and the panels formed by skid, fuse- lage, and struts should be cross-braced by cables. This will provide for the longitudinal backwards component of the force of a bad landing. The axle can also be made divided, each half being hinged at the skid, and having a cable or swivelling- rod connection from just inside the wheel to about half-way along the skid. With this arrangement, a telescopic shock absorber 266 AEROPLANE DESIGN is needed from the axle to the longerons. The skid should be well curved up in front so as to minimise the chances of impaling. This type of chassis is suitable for heavy work, and can be made strong enough to withstand rough handling and very bad landings. The V struts and shock absorbers at the axle together form a triple V or M shaped girder, which can be designed to withstand a large sideways force, such as occurs in a bad landing on one wheel. High resistance is unavoidable owing to the multiplicity of its parts, so that it is unsuitable for high-speed work unless great strength is required. If the skid is continued under the airscrew, protection is afforded to that member. There is one point of weakness which requires attention : that is, the inability of the forward chassis struts to withstand any considerable transverse force. As wide an angle FIG. 195. Double Skid Chassis. V as possible should be sought for here, and consequently, if the longerons converge considerably towards the front of the fuselage, the forward struts may require to be attached to them some distance back from the engine plate or fuselage nose, at a position where the fuselage is still wide. Although the forward rake of these struts helps in this respect, their bottom ends may nevertheless be a long way behind the airscrew in the case of an overhung rotary-engined machine. Consequently it is not always possible to prolong the skid under the airscrew, owing to the large bending moment which may occur in the skid due to this leverage. Apart from this, it will be found advisable to use a strong skid in this central type. The double skid- type of chassis (Fig. 195) is comprised of two lighter longitudinal skids, connected to each other and the fuselage with struts, the whole structure being braced together in the usual manner. Telescopic shock absorbers are out of place in this design, DESIGN OF THE CHASSIS 267 elasticity being provided by wrapping rubber cord round the outer ends of the axle, just inside the wheel and the fixed trans- verse strut. This type is lighter, and has less resistance than the central skid type, owing to the decreased number of parts, and to the abolition of the telescopic shock absorber ; but it is not so flexible, and is weaker in many ways, and therefore not suitable for heavy rough work. On the other hand it is stronger than the central design as regards side force well forward, as its braced square panel forward is much superior to the simple V 5 but it may be argued on the contrary that it is more liable to such a force. It is not much use to prolong the double skids (a) 196 Landing Chassis Details. forward past the airscrew. The prolonged central skid is a protection chiefly against a localised hillock on the ground ; the double skids, if prolonged, would only be useful against a bank, as a small hillock might easily pass between them and cause an accident. The chassis just considered is a half-way house between the heavy, single skid type, and the light, skidless design which con- stitutes the second group. (See Figs. 197 and 198.) In this group there is no limit stop or positive protection for the airscrew. The axle is connected to the lower longerons by a pair of struts on each side, forming two V' s longitudinally placed. The two struts forming each V are united at their lower ends by a steel 268 AEROPLANE DESIGN fitting, which also provides a vertical slot for up-and-down movement of the axle. See Fig. 196 (d). If this slot be made with a beaded edge, and the axle fitted with four collars, the axle can be arranged to act the part of a transverse strut. The resistance and weight are, in this case, reduced to a minimum for a fixed chassis ; there are, indeed, only five members, and two of these are in protected positions. Skill, however, is required in manoeuvring over the ground so as to keep the airscrew safe, and, with this end in view, the wheels may with advantage be set somewhat further forward than with the other types, in order to give a larger tail moment. When landing on one wheel there is generally a resulting side blow on the chassis, and in order to resist this it is cus- Fig. 197. Landing Chassis of Bristol Fighter. tomary to introduce cross bracing between the axle and the fuselage. This type is eminently suitable for light, high-speed work, and owing to its great advantages as regards light weight and low resistance, its use should always be carefully considered before another type is adopted. Some excellent examples of this type of chassis are shown in Figs. 196, 197, 198. Fig. 196 shows the details of the chassis of the Airco 4 machine. Each pair of side struts is made of solid wood, and at their lower ends vertical strut shoes, which carry the wheel, axle, and fittings for the rubber shock absorbers, are fixed. The axle itself Fig. 196 (c and e) rests between two cross struts of wood, which are shaped to a streamline form as shown. The total weight of this chassis is 119 Ibs. DESIGN OF THE CHASSIS 269 The landing chassis for the Bristol Fighter is shown in Fig. 197, and Fig. 198 shows this type as used in a standard American training machine. The method of construction is Reproduced by courtesy of ' Flight. ' FIG. 198. Wright-Martin Landing Chassis. clearly shown in these figures. An illustration of a chassis suitable for heavy machines is shown in Fig. 199, which depicts the complete chassis of the Handley-Page machine O-4OO. FIG. 199. Chassis of Handley-Page Machine. Stresses in Chassis Members. The determination of the stresses in the V type of chassis offers little difficulty. The stresses in the struts are obtained by resolving the reaction 270 AEROPLANE DESIGN J P at the wheel along the required direction, or by drawing a stress diagram as shown in Fig. 200, care being taken to use the component of the reaction in the plane of the struts, as determined by means of Formula Si. The maximum side shock likely to occur on the machine may be taken as one-fifth of the maximum vertical reaction at each wheel. The cross bracing should be designed to withstand this load. FIG. 200. Stress Diagram and Bending Moment Diagram for Landing Chassis with Divided Axle. The stress on the axle may be simply determined in the following manner : i. Continuous Axle (Fig. 201). Let the distance between the centre H of the hub and the centre S of the shock absorber elastic be a, and that between the latter point and the central longitudinal plane of the machine be b. The bending moment on the axle is due to the couple \ P a, and increases from zero at the hub centre to J P a at the point s. From S it is constant until the other shock absorber centre is reached, whence it decreases to zero at the centre of the other hub. The axle in this case may well be made of uniform strength, except for the portions H to S, which may be strengthened against shear. In DESIGN OF THE CHASSIS 27 F the event of landing on one wheel, if Q be the load on the wheel,, the bending moment increases from zero at H to Q x a x 2 b 2 b + a at S, and from there decreases again to zero at the point of attachment of the other shock absorber. There is then a tension in the other shock absorber equal to Q a I (2 b + a) y which must be provided for. B.M DiACRAH FIG. 201. B.M. Diagram for Landing Chassis with Continuous Axle. 2. Divided Axle (Fig. 200). In this case the bending moment gradually diminishes from ^ P a at S to zero at M. The axle may therefore be made lighter towards the centre so long as it is able to take the constant shearing force between the central plane and S, equal in magnitude to Qa/ (a + b). Alternatively the portion on each side of S subjected to large bending moment may be reinforced. It is important to keep S as near to H as possible. With a divided axle it is necessary to haye a wire leading from the junction of the cross bracing wires to the central hinge fittings, in order to counteract the downward force at the hinge due to the load on the wheels. Shock Absorbers. The two main types of shock-absorbing: devices are (a) Rubber shock absorbers. b Oleo shock absorbers. 2*72 AEROPLANE DESIGN The first type is used much more extensively than the second, principally on account of its lightness, ease of construction, and its cheapness. It has this disadvantage, however, that the mechanical properties of rubber vary very considerably, and are likely to deteriorate with time. Some interesting experiments upon a rubber shock-absorbing device were carried out by Hunsaker at the Massachusetts Institute of Technology. A shock absorber of the type shown in Fig. 202 was fitted with twelve rubber rings 2" x 2" x 5/16" each passing over a J" steel pin. Table XLII. shows the elongation corresponding to a series of loads. FIG. 202. TABLE XLII. ELONGATION OF SHOCK A] Load. First loading. Second loading. 1000 Ibs. 22" 48" 2000 7 8" I'25" 3000 1*40" 1-98" 4000 2'o8" 2'45" 3000 r 9 8" 2-26" 2000 1-57" 174" 1000 68" 70" Third loading. 52" i'37" 2*12" 2 "60" 2-45" Finally the absorber was tested to destruction and failed at 9750 Ibs., with a corresponding extension of 5*06". The pins failed by shearing with the rubber rings still unbroken. The effect of hysteresis is brought out clearly in the above figures, the rubber not contracting as rapidly as it extends. The area of the hysteresis loop represents the work done on the rubber which is not restored and is a measure of the shock-absorbing quality of the rubber. Moreover, it will be seen that the hysteresis loss diminished with each loading. The stress in the rubber rings when the bridge failed was 650 Ibs. per square inch. In a subsequent test they failed at a stress of 900 Ibs. per square inch. Reference to Formula 81 shows that an increase of 'give' of the shock-absorbing device reduces the reaction at the wheels, DESIGN OF THE CHASSIS 273 hence greater elasticity of the rubber is needed to reduce landing shock on the chassis. Such increase of elasticity would have the further advantage of relieving the jolting of the machine when running along the ground. DESIGN OF A RUBBER SHOCK ABSORBER. In order to determine the number of rubber rings required to absorb the landing shock of a machine it is necessary to equate the work done in stretching the rubber to the energy of the machine at the instant of landing. For example, to take the case of the machine referred to on page 263. Kinetic Energy of the machine on landing = (W - w) (V sin a) 2 = -33 W x i2 2 2g 6 4 = 74 W Potential Energy of the machine on landing = (W - w) x ' give ' of gear = -33 W x '834 = -275W whence Total Energy = 1-015 W To determine the stretch of the shock-absorbing device we apply Formula 3, so that PL Extension AE and if n be the number of rings of rubber cord of f " diameter, and the diameter of each ring assumed to be 6", we have = ' I4 % For good quality shock absorber elastic E is taken as 300 Ibs. per square inch. In the above expression P = average load on the shock absorber = half total reaction at wheels = 51-51 W , -142 x 1-51 W whence x ^ - - - n 3 FIG. 203 (a) Reproduced by courtesy of l Flight: FIG. 203 (b}. FIG. 203 (c) Types of Shock Absorbers. FIG. 203 (ff). Shock Absorber Device, and Streamlining of Axle of Landing Chassis. Reproduced by courtesy of ' Flight.' 1 FIG. 203 ( e ) 276 AEROPLANE DESIGN Further equating the total energy to the work done 1-015 W = P- 12 Substituting for x and P 1-51 W(-i42 x 1-51 W) 12 n whence n = '0267 W Formula 82 3 B -A 1-015 FIG. 204. Therefore, for a machine weighing 2000 Ibs., 54 rings of rubber of the size mentioned would be required. In this calculation no allowance has been made for the energy absorbed by the resilience of the pneumatic tyres. In practice, the most general method of ap- plying the rubber is to coil a long length around the axle and the main chassis struts respectively. The number of turns required will correspond to the number of rings as calculated from the above formula, plus any turns necessary for starting and ending. Illustrations of rubber shock-absorber devices are shown in Fig. 203. OLEO SHOCK ABSORBERS. These in general consist of two telescopic steel tubes, the outer one serving as a cylinder in which oil is maintained at a fixed level, whilst the inner tube is attached to the body and acts as a piston. The inner tube or piston see Fig. 204 (B) carries a spring- loaded valve which covers ports round the lower part of the tube. The cylinder below the valve is filled with oil. When landing occurs and the tubes are compressed, the oil passes through a series of small holes into the upper cylinder. If the shock of landing exceeds a certain figure, the spring- loaded valve opens and provides an additional passage. Particulars of an Oleo Leg, designed by the R.A.F., are given in the Report of the Advisory Committee for Aeronautics, 1912-13, as follows : During flight the lower tube or cylinder con- taining the oil drops through 1 1 inches. When the machine (a B.E. 2) strikes the ground the oil first passes into a small central air chamber and after- wards through three 4 millimetre exit holes. If DESIGN OF THE CHASSIS 277 the velocity of impact with the ground is sufficiently great so that the resistance of the oil passing through these holes is enough to raise the oil pressure to 640 Ibs. per square inch, the spring-loaded valve opens and allows the oil an additional passage. The arrangement was designed so that after the first two inches of travel the resistance remains constant, and equal to two and a half times the weight of the machine. The vertical travel, excluding the first 2", was 13", so that if the pressure remained constant the total energy absorbed was 4300 ft. Ibs., and consequently the machine could land without damage with a vertical velocity of 13 feet per second. In addition to the oil and air gear, strong spiral springs carry the weight of the machine when running along the ground. The Tail Skid. The tail skid, which is situated at the rear end of the fuselage, provides the necessary point of support for the rear portion of an aeroplane when the machine is resting on FIG. 205. the ground. There are several different types of skids in actual use, examples of which are shown in Figs. 205-208. The simplest form is that shown in Fig. 205. In this type the movement of the tail skid relative to the body of the machine is confined to the vertical plane. The skid itself is usually made of ash, and is provided with a steel shoe. The shock absorber device is composed of rubber cord. Another example of this type is shown in Fig. 206. Another type of tail skid is that in which the tail skid is free to rotate about a vertical axis. The skid can then set itself at any angle laterally, and can therefore follow the curve traced out by the aeroplane in its motion over the ground. Such movement makes it much more convenient to manoeuvre the machine when ' taxying ' before flying or after alighting. Examples of this type of skid are shown in Figs. 207, 208. FIG. 206. Example of Tail Skid for Large Machine. FIG. 207. FIG. 208 DESIGN OF THE CHASSIS 279 In Fig. 207 the shock absorber device is of rubber, which is placed in tension when the machine is upon the ground. A variation of this* type is shown in Fig. 208. In this skid the shock-absorbing device is composed of steel springs which are subject to compression when in use. Such an arrangement is more efficient mechanically than the previous types shown, for it leads to a much smaller reaction at the fulcrum. Streamlining the Chassis. In general, the question of streamlining the chassis must be carefully considered, and all parts should be given a streamline form wherever possible. The resistance of the under-carriage is generally an important item, particularly in the case of high-speed scouts. Attempts have also been made to design a chassis which can be drawn within the fuselage during flight. CHAPTER IX. DESIGN OF THE AIRSCREW. Methods of Design. The problem of airscrew design has been approached by analytical arid by empirical methods The two principal methods of analytical attack are : (i.) Examination of the air flow through the airscrew and the determination of expressions for its change in momentum and energy. (ii.) Consideration of the actual forces set up upon the blades. The first method may be called the classical method, and was the method developed by Rankine, Froude, and others for application in the first instance to the design of marine propellers. The second method is known as the blade element method, and has been specially developed for application to airscrews, the pioneers in this case being Drzewiecki and Lanchester. In both cases the premises are somewhat obscure, and the conclusions therefore correspondingly uncertain. It should be noted, however, that the second method is concentrated on the airscrew itself from first to last, and so leaves the designer with something definite to work upon, even if the data are not suffi- ciently accurate for his purpose. The design of airscrews, there- fore, is best attempted along the more modern line of thought, which is generally known as the Blade Element Theory. This method is based upon laboratory experiments upon aero- foils, about which a vast amount of information is available from the force shape point of view ; whereas very little definite information concerning the flow of air past an aerofoil is avail- able in a quantitative form that can be applied to the design of an airscrew. The first principles of the method are familiar to most engineers : an element of the blade is taken at any radius from the centre of rotation, say a section of width dr at a radius r, as shown in Fig. 209 ; and this section is then studied separately while still considered as joined to the whole. It is assumed that this element operates like a small aerofoil, the aerodynamic properties of which can be easily determined, as explained in DESIGN OF THE AIRSCREW 281 Chapter III. The whole airscrew is then treated as a summa- tion of such elements, and the forces on the whole airscrew as a summation of the forces such as those upon the small element studied, applying to successive subdivisions of the blade such corrections for velocity, leverage, &c., as may be necessary. It is first essential to realise the path of the blade. This, being a motion of uniform rotation and of uniform translation, is a helix, similar to a very deep-cut coarse pitch screw thread. Fig. 210 is a sketch of such a path. It should be noticed that as the radius is increased, so the resultant or helical velocity is increased. The forward velocity of the airscrew, however, is uniform for every point on it, and therefore the pitch is constant, Fi<5 909 'l00f CP (- TTD unless expressly made otherwise. The geometric pitch is the length along the axis which the airscrew would move for one revolution, if the fluid through which it moved suffered no translation, as for example if the screw worked in a fixed nut. If the fluid behaved in this manner we could represent the motion of our element as taking place along the hypotenuse of the triangle shown in Fig. 21 1 ; having for its sides the pitch (D), and the length of the particular circumference chosen (2 wr). The motion of the wing tip would of course be given by sub- stituting TT D for the latter side, where D is the diameter of the airscrew. The above conditions of working are most nearly realised in practice when the airscrew is so working that there is no thrust on the shaft. The axial advance for one revolution 282 AEROPLANE DESIGN under these circumstances is termed the experimental mean pitch (/). Tn practice, however, the airscrew is usually exerting a pull, and the axial advance per revolution is considerably less owing to the velocity impressed on the air. If V feet per second be the translational velocity of the airscrew, revolving at N revolutions per second, the actual effective pitch is V/N feet. The amount by which this quantity is less than P is termed the * slip,' and averages between 20% and 30%. It is convenient to think of this in terms of the dimensions of the airscrew, and so the term ' slip ratio ' is introduced, and we have P - Slip ratio = N The true path of the blade element is as represented in FIG. 212. FIG. 213. Fig. 212, where the dotted lines show the variation from Fig. 211. Now the blade element theory is built upon aerofoil data, and remembering this, we may define the position of the element as shown in Fig. 213. A is the angle of the helix, and is equal to the angle whose tangent V is the angle of incidence of the aerofoil element to its path, R is the resultant air force upon it, is its inclination to the perpendicular to its path, L is the aerofoil lift along this per- pendicular to the path, D is the drag or component of R along the path. The thrust t is the component of R, along the direction of the translational velocity, that is, along X Y. Hence we have / = R cos (A + 0) DESIGN OF THE AIRSCREW 283 The resistance to rotation, that is, the torque q on the shaft, is proportional to the component of R along the line xz (the direction of rotative velocity), so that q = R sin (A 4- ) r and the efficiency of the element R cos (A + 0) V ~ R sin (A + 0) 27rrN = cot (A + 0) tan A . ........... Formula 86 (a) Aerofoil data are given in the form of the absolute co- efficients of the Lift and the Drag. From these we can obtain the values of the absolute coefficients of resultant force, K r and of 0, by making use of the following relationships : K r = V K x 2 + K y 2 K x and = tan~ r TT~ Ky If b is the breadth of the blade where the element is chosen, the thrust on each element may be written d"p= Kr - b dr v 2 cos (A + 0) 6 where v = velocity along helical path V V 1 - _lJL sin A whence Jflp= K.^- cos (A + sin 2 A and total thrust = cos (A " P T^ A/ ^ K x * K y V i + (K) Formula 83 where n = number of blades. 284 AEROPLANE DESIGN Similarly the torque q on any element is given by q = K r . . dr . z/ 2 sin (A + 0) . r o - K / /K x \ 2 p V, 2 /y i + IjF" ) ' ' ^ >* ^^^ i T ' S" and total torque = Q - Formula 84 An examination of these expressions for the total thrust and the total torque of an airscrew indicates that in order to evaluate FIG. 214. Aerodynamic Characteristics for Airscrew Sections. the integrals, it is necessary to know the aerodynamic charac- teristics of a number of suitable aerofoil sections. The data given on pages 67 - 70 relating to the sections illustrated will be found very useful in this connection. From the data there given the curves shown in Fig. 214 have been drawn. It is seen from Fig. 53 that the most efficient angle of incidence, that is, the angle giving the maximum ratio K y /K x , for these sections, is in the neighbourhood of 3. Consequently the angle of attack (0) for the airscrew elements, if these sections be used, should be 3. Using this angle of incidence, the curves shown DESIGN OF THE AIRSCREW 285 in Fig. 214 have been drawn, giving the values of K y and K y /K x plotted against the ratio Thickness/Chord. The value of this proceeding will be evident shortly. It should be observed here that on account of the loss in efficiency of an airscrew when climbing, it is sometimes advisable to adopt a smaller angle of attack than the angle of maximum efficiency, in order to mini- mise this loss when climbing. Such an arrangement leads to reduced efficiency at top speed, which is counterbalanced by an increased efficiency when climbing. The advisability of such procedure naturally depends upon the desired performance, and the designer must compromise in order to get the results desired v fi-ar, Ccrtr*) Fig. 215. Design Data for 2-bladed Airscrews. in any particular case. It is on this account that an airscrew in which the angles of the blade are adjustable in accordance with the conditions of flight prevailing would offer considerable advantages, and would lead to a much-increased all-round efficiency. This problem of the variable pitch airscrew is re- ceiving much attention, and will undoubtedly be solved in the very near future. A second factor involved in the evaluation of the integral is the ratio / max . This ratio represents the width of the blade at any section in terms of the maximum width of the blade, and its value is dependent upon the plan form adopted. Figs. 215 and 216 represent an analysis of several types of 286 AEROPLANE DESIGN airscrews which have proved successful in practice, the curves showing (a) The ratio bjb m3LK (b] The ratio maximum thickness/chord, for all points along the blade. These curves, together with those shown in Fig. 214 give all the necessary data for the design of a successful airscrew, and their application to such design will be fully explained in this chapter. Referring to the expression for total torque (Formula 84), it is seen that by inserting the values of K y , K x , - ON N c^ O r 3 . to o o M co O M t>* t^* CO to o CO CO b O M IO !>. oo CO M to to M O CO b b b to r^ M M oo 1-1 TJ- N CO co ON co vO O t> M to tfi OJ 8 I t 81 + + oj U 01 + "> >> X >> DESIGN OF THE AIRSCREW 293 These values are next plotted against the radius r as shown in Figs. 217 and 218, and the areas enclosed by the curves very carefully measured. From these curves it was found that / x sin 2 A cos (A + sin 2 A dr = 6-91 dr = 3-17 hence the torque required to drive the airscrew = 4 x "00237 x '72 x 190^8 x 190*8 x 6 - 9i x ma 5 but the torque available from the engine _ 70 x 375 x 55 2 7T X 1000 60 = 1375 Ibs. ft. Equating these values Anax = 1375/1720 = 0*8 feet = 9 - 6 inches FIG. 218. Thrust Curve for Airscrew. The maximum blade width having thus been determined, the dimensions of each section follow at once from the third and fourth lines of table on page 292, namely : 294 AEROPLANE DESIGN Section A B c D E F G H Blade width ins. 6'8i 7-69 8-64 9-36 9*6 9-31 8'o6 5*76 Thickness ins.... 2-18 177 1*43 ri2 0*88 0-76 0*65 0*49 The thrust of the airscrew = 4 x '00237 x 72 x i9o'S x 190*8 x o'8 x 3*17 = 630 Ibs. Therefore the efficiency 630 x 190*8 ""^"375 = 83-57. In practice it is found that the efficiency of an airscrew is generally slightly higher than its calculated value ; hence it is probable that this figure would be more than realised if tested under the conditions assumed. The general lay-out of the airscrew can now be developed as shown in Fig. 219. Each section must be drawn in its correct position along the blade and with the required angle, A + 3, to the axis of the blade. The following points should be observed when sketching the aerofoil sections : (a) The centre of area of the sections should lie on the blade axis. (b) The respective positions of the centre of pressure of the sections should be arranged about the blade axis so as to eliminate as far as possible all twist upon the blade. For this reason a symmetrical blade is unsuit- able, because the centres of pressure in this case would all lie on one side of the centre line of the blade. By adopting some such shape as that shown, this unbalanced effect is avoided. (c) Sections near the boss of the airscrew are designed chiefly from considerations of strength, and the adoption of a convex instead of a flat face is a help in this direction, while the aerodynamical loss resulting from such altera- tions at these sections is practically negligible. The contour lines of the blade can now be constructed. This is an operation which demands great skill, and depends for its success principally upon the experience of the designer. Indeed, it may be said that after the design of a few successful airscrews DESIGN OF THE AIRSCREW 2 95 an airscrew designer will no longer need aerodynamical data to assist him, but will be able to produce an efficient airscrew merely by ' eye.' In normal flight the 'slip' of the airscrew may be somewhat greater than that corresponding to maximum efficiency, in which case small variations in the rotational speed will be accompanied by appreciable variations in the value of the thrust. By adjusting the torque of the engine, allowance can be made for any small discrepancy between the calculated and the actual behaviour of the airscrew. PLAN OF BLADE FIG. 219. Lay-out of an Airscrew. HI Stresses in Airscrew Blades. An airscrew blade is generally subjected to the following forces : 1. A tension due to centrifugal force. 2. A bending moment. 3. A twisting moment. It is therefore obvious that an accurate calculation of the combined stresses at any point along the blade is a matter of 296 AEROPLANE DESIGN considerable difficulty. We can, however, obtain satisfactory results by considering each stress separately. In a well-designed airscrew the stresses due to twist will be quite small, since particular care will have been taken in the design to eliminate all twist as explained previously. The stresses due to centri- fugal force and bending moment may be determined as follows : Let Fig. 220 represent the airscrew blade, and let 'a' be the cross-sectional area of a section of the blade distant ' x' from FIG. 220. FIG. 221. the centre. Then the centrifugal force set up by a small element of the blade at this distance weight of element x (velocity) Let iv be weight in Ibs. of a cubic foot of the material of which airscrew is made Then volume of element = a . dx weight of element = w a .dx Ibs. velocity of element = 2 ?r n x feet per second, hence centrifugal force due to element _ w a . d x 4 7T 2 2 # 2 g x The stress at any section L L 1 of cross-sectional ' a ' f , 4 TT* n- w . f = z . i ax.d x ga ' The value of the integral can best be determined graphically by taking values of the product a x along the blade and plotting them on an ( x' base. The area of the curve thus obtained will enable the stresses due to centrifugal force to be determined with considerable accuracy. A simpler but not so accurate a method is to assume the blade of constant section over a certain distance, and to treat each such length separately by determining its weight and mean distance from the centre of rotation. Adopting this method, the stresses due to centrifugal force in the airscrew just designed are obtained by tabulation as shown FIG. 225. Appearance of Airscrew Laminations before shaping. Reproduced by courtesy of Messrs. Oddy, Ltd. FIG. 228. Finished Airscrew. Facing page 296. DESIGN OF THE AIRSCREW 297 below. The area of each section should be determined by graphical summation, Simpson's rule, or by the use of a plani- meter. The weight of a cubic foot of mahogany is taken as 35 Ibs. TABLE XLIII. STRESSES DUE TO CENTRIFUGAL FORCE. Section ... ... A B c D 8-24 6-98 49-4 41-9 767 794 Area sq. ins. Volume cu. ins. Centrifugal force S C ... Stress = ~r~ A B 9'94 9'o< 59'6 54'3 5*4 655 5202 4688 4033 3266 524 519 490 469 E 5^4 33'8 758 2472 438 F G H 4*72 3*49 i'88 28*35 20 '95 11*28 732 613 369 1714 982 369 364 282 196 Rad.us (if) 1 1.00 i. r f! FIG. 222. S. F. and B. M. Diagrams for Airscrew. Stresses due to Bending. The stresses in the blade due to bending are set up by the air pressure exerted upon each element of the blade. Consider a section of the blade such as is shown in Fig. 221. The maximum tensile and compressive 298 AEROPLANE DESIGN stresses will occur in the outer layers of the material, and to find their values we require to know the bending moment at the section considered. The value of this bending moment is determined by drawing the load grading curve. The ordinates of this curve are obtained from the thrust grading curve by dividing the thrust at each section by cos (A + 0). Since this is a mere ratio the thrust grading curve can evidently be used with a different vertical scale. The curve of shear force over the blade follows directly from the load grading curve by graphic or tabular integration, whichever is preferred. Similarly integration of the shear force curve gives the bending moment curve. This work is quite straightforward, but if graphical integration is used care must be taken to see that the correct scales are obtained. Fig. 222 (a be) shows these curves for the airscrew under consideration. The bending moment at each section is then read off from the curve, and the stresses obtained from Formula 51. The moment of inertia of the sections and their centre of gravity can be readily determined by the use of the graphical method outlined for a streamline strut in Chapter IV. The stresses due to bending are tabulated in Table XLIV. below. TABLE XLIV. STRESSES DUE TO BENDING. Section ...A B c D E F G H I/y c in. 8 ... 2*46 1*83 i'34 0*89 0^565 o'4i 0^26 0*105 I/Xt in. 3 ... 3*70 2*75 2*02 i '34 0*85 o'6i4 0*39 o'i6 B.M. Ibs. ft. ... 365 286 214 146 87 40 15 4 B.M. Ibs. in.... 4390 3440 2570 1750 1040 480 180 48 Compressive Stress. My c /I Ibs. per sq. in. 1790 1880 1920 1970 1840 1170 690 460 Tensile Stress. Mj t /I Ibs. per sq. in. 1190 1250 1270 1310 1225 780 460 300 The maximum stresses at each section can now be obtained by adding together the centrifugal and the bending moment stresses of Tables XLIII. and XLIV. It will be observed that the tension due to centrifugal force will diminish the compressive stress in the blade, but increase the tensile stress. For mahogany or walnut the maximum per- missible working stress is 2000 Ibs. per square inch. The stresses obtained are seen to be well within this figure. DESIGN OF THE AIRSCREW 299 TABLE XLV. MAXIMUM STRESSES IN THE AIRSCREW BLADE. Section A B c D E F G H Compressive stress 1266 1361 1430 1501 1402 806 408 264 Tensile stress ... 1714 1769 1760 1779 1663 1144 742 496 The best materials at present available for the construction of airscrews are walnut (black) and mahogany. Walnut is heavier than mahogany, but is not so liable to warp ; on the other hand, mahogany takes the glue better. The ultimate tensile strength of either material is about 4 tons per square inch, so that a working stress of 2000 Ibs. per square inch may be considered as quite satisfactory. The method of building up an airscrew will be apparent from a study of Fig. 223. The thickness of the laminae varies from |" to ii". Much controversy has raged round the question of the relative merits of the two- and the four-bladed airscrew. The four-bladed airscrew is probably better balanced and slightly more efficient than the two-bladed variety, but the latter type is easier to build, and can also be made stronger at the boss. The Construction of an Airscrew. As already stated, the timber most frequently used for the construction of an airscrew is either mahogany or walnut. The wood should be thoroughly seasoned, straight-grained, and quite free from knots. The use of curly-grained timber will cause the blades to cast, and should therefore be avoided. The planks from which the laminae are to be cut should be stored for several weeks in a room where the temperature and atmospheric conditions are the same as those prevailing in the workshop. The first operation is the sawing out of the laminae. The dimensions at the various sections are obtained from the drawing. A template giving the shape of each lamina should be prepared in either three-ply wood or in aluminium sheet. A margin of about J" in the case of a four-bladed airscrew and of about " in the case of a two- bladed airscrew should be allowed all round to compensate for any errors in gluing in position or change of position due to warping. The planks should then be marked off from the templates, the grain running longitudinally and parallel to the flat surface. The laminae are best cut out with a band saw and then planed to the correct thickness. In the case of the four-bladed screw the lamina are half 3 oo AEROPLANE DESIGN lapped at the centre, as shown in the sketch, Fig. 224. The surfaces which receive the glue should be ' toothed ' lengthwise along the blade. The laminae are then supported at their centre on a balance and the heavier ends marked. The gluing together of the laminae can now be commenced. The first two laminae are thoroughly warmed by placing them in contact with a hot plate, and then their adjacent surfaces are quickly covered with best Scotch, French, or Lincoln glue, after which they are clamped together in their correct relative positions by means of a number of hand-screw clamps. Clamping should be commenced at the centre of the block, working outwards to the tip. In this manner a failure of the glued joints at the boss is avoided. It is also very essential that the temperature of the glue room should be maintained at a uniform temperature of about 70 F. throughout the entire process of gluing, otherwise an opening of the joints is FIG. 223. FIG. 224. probable after the airscrew has been completed. The remaining laminae are then glued in position one at a time, a period of at least eight hours being allowed between the addition of each lamina. The process of warming the lamina and clamping it down on to the block is exactly similar to that described above for the first two laminae. In order that the position of each lamina relatively to the block may be correct, it is customary to locate it by measure- ment from the preceding one, and then a small piece of wood is glued on the projecting portion of the lower lamina, and the side of the freshly added lamina is pressed up tight against this small block. Before the addition of each lamina the block is checked for balance and then the light end is balanced up by placing the heavier end of the next lamina upon it. In this manner the block is comparatively well balanced at the completion of the gluing-together stage. After gluing, a number of small pegs are driven into the blades at positions which have been DESIGN OF THE AIRSCREW 301 indicated upon the drawings. They should be a moderate driving fit, and glued into position. The boring of the hole in the boss is the next operation. This is best performed upon a boring machine, the cutter being run at a high speed in order to obtain accuracy. The block is then set aside for several days in a room the temperature cf which is exactly the same as that prevailing in the glue room. During this period the tendency of the blades to cast or warp will be taken up. Fig. 225 shows a 4-bladed airscrew at this stage of construction. The ' roughing-out' stage is now commenced, and the blades are shaped down to within about J" of their final dimensions. A further period of several days is then allowed for the timber to take up any change of state. In this manner a more accurate and more permanent contour is finally obtained. Lastly, the final shaping of the airscrew is proceeded with. The correct , shape and angle of the blade sections are obtained by the use of FIG. 226. Template for checking Airscrew Section. steel or aluminium templates such as are shown in Fig. 226. The balance of the airscrew should be frequently tested during this latter process and more material removed from the heavier blades. The blade surfaces are finally smoothed up by means of glass-paper and the airscrew should now be almost perfectly balanced. One type of balance for testing airscrews is shown in Fig. 227. A thin steel tube is passed through the airscrew hub, and the airscrew is then lifted on to the balance, the outer ends of the tube resting on two knife-edged plates. Another method is to support the tube inside roller bearings carried on a wall bracket. The remaining operations are (a) The drilling of the bolt holes through which pass the bolts which attach the airscrew to the steel boss, which in turn is serrated and fits on to a correspondingly serrated shaft attached to the engine shaft. (b) The final varnishing of the airscrew in order that it may withstand climatic conditions. 102 AEROPLANE DESIGN The first of these operations necessitates the use of a drilling jig for which a standard airscrew boss can be used ; while for the second, two or three coats of good boat varnish with an oil base should be used. The final balance of the airscrew is effected by adding extra varnish to the lighter blades. A completed airscrew is shown in Fig. 228. The practice of adding brass tips to the airscrew blades was FIG. 227. Testing the Balance of an Airscrew. largely adopted during the war. These tips are bent to the correct shape on a former, and then riveted to the blades by means of copper rivets. They serve as a protection to the outer leading edges of the airscrew, but greatly increase the stresses at the roots of the blades due to centrifugal force. The sheathing of the blades with fabric has also been largely adopted. CHAPTER X. STABILITY. Definition. The stability of an aeroplane considered from the most general point of view would involve a discussion of all those qualities which enable a machine to be flown in safety under all the varying conditions likely to be met with in flight in all weathers. The stability of an aeroplane as studied in this chapter will be considered from the more limited standpoint of the following definition : ' If a body be moving in a uniform manner relative to the surrounding medium, then the motion is said to be stable, if when any small disturbance takes place in the medium, the forces and reactions set up in the body tend to restore the body to its original state of motion ; while if the forces due to a small initial disturbance tend to produce a* further departure from the original state of motion, then the motion is said to be unstable.' Applying this definition to an aeroplane, it is seen that a machine will be inherently stable if after a sudden dis- turbance in its flight path it is able to regain correct flying attitude without any assistance on the part of the pilot. For an aeroplane to be completely stable it must possess both statical and dynamical stability. An aeroplane is stati- cally stable if righting moments are called into play which tend to bring the machine back to its normal flying attitude if deviated therefrom temporarily. These righting moments will, however, set up oscillations, and the machine will be dynamically stable only if these oscillations diminish with time and ulti- mately die out, leaving the machine in its normal flight attitude. It is therefore essential to establish statical stability before making an investigation of dynamical stability. The question of stability is closely inter-connected with the question of controllability. A machine possessing a large amount of inherent stability is sometimes difficult to control, or in the words of the pilot, is said to be ' heavy on the control.' It is generally necessary to make a compromise between the two factors. For fighting purposes manoeuvrability is of the utmost importance, and it is essential that a war machine should answer very rapidly to its controls, and consequently the question of 304 AEROPLANE DESIGN inherent stability is not of such vital importance as in the case of the commercial machine. Reference to the particulars given in Chapter XIV. with regard to the Bristol Fighter and the S.E. 5 illustrates that fighting machines have been designed possessing a large degree of both inherent stability and ma- noeuvrability. It will be readily appreciated that in the case of long distance flights an aeroplane which continually tends to depart from the normal flight path, owing to minor disturbances, requires constant attention on the part of the pilot, and imposes upon him a very severe strain, which it is both possible and desirable to avoid. The mathematical theory of stability, with respect to an aeroplane in the restricted sense of the above definition, has been developed principally by Lanchester* and Bryan.f The application of Bryan's theoretical results to a particular machine was very ably carried out by Bairstow,* and much of the sub- sequent matter is based upon his work. The theory is very complex and those desiring a fuller treatment of the subject should consult the references given below. Our aim in this chapter is to outline the theory and to show its application to the results of tests upon models, and then to indicate the method whereby the stability of a completed machine may be predicted from these wind channel tests. The investigation of stability can be summarised as under : Summary of Procedure : A. Theoretical determination of the equations of motion by mathematical reasoning in terms (i.) Of the velocities of the C.G. of the machine along the axes of reference ; (ii.) Of the angular velocities of the machine about the same axes. B. These expressions contain a number of constants termed Derivatives, which can be divided into two classes (i.) Resistance Derivatives which depend merely on the shape and size of the machine, and not on its motion ; (ii.) Rotary Derivatives which depend upon the motion of the machine. Both classes of derivatives can be determined analyti- cally, and also by means of model tests. * Aerodonetics (Constable & Co.). t Stability in Aviation (Macmillan). % N. P. L. Report ', 1912-1913. STABILITY 35 C. Substitution of the values obtained for the derivatives in the general equations of motion developed under A leads to a solution in many important cases, and consequently the nature of the motion can be investigated. D. The investigation of the small oscillations occurring about the steady motion of an aeroplane leads to a classifica- tion into two groups, each determined by three equations of motion. These groups are : (i.) The group representing motion in a vertical plane, and determining the nature of the longitudinal oscillations upon which the longitudinal stability of the machine depends. (ii.) The group representing motion about the plane of symmetry, and determining the nature of the rolling and yawing oscillations upon which the lateral stability of the machine depends. Stability Nomenclature. The N.P.L. system is illustrated in Fig. 229 and tabulated in Table XLVI. The axis O x corresponds to the axis of drag of the machine in normal flight. The axis O z corresponds to the axis of lift. The axis Oy is perpendicular to the plane x o z. O is the centre of gravity of the machin *. Rotation about the axis O x is termed ROLLING. Rotation about the axis Oy is termed PITCHING. Rotation about the axis O z is termed YAWING. The linear velocities in the directions of the axes are denoted by u, v> w respectively, and the angular velocities about these axes are denoted by/, ^, r respectively. TABLE XLVI. STABILITY NOMENCLATURE. Axis, i N ^ rf UX1S. Name of S >' f ' nbo1 <'~- \ fo f c r e. Name of angle. Symbol for angle. Name of moment. Symbol for moment. T O x Longitudinal Longitudinal X Ro f Rolling Or Lateral Lateral Y Pitch 8 Pitching M O z Normal Normal Z i ' Yaw i/, Yawing N The signs of the forces are positive when acting along the positive directions of the axes indicated by arrows in Fig. 229; the angles and moments are positive when turning occurs or tends to occur from Qy to O^ ; Q z to O.r ; Ox to Oj. In order to define the angular position of an aeroplane, 3 o6 AEROPLANE DESIGN Euler's * System of Moving Axes' is adopted, the motion of the machine being referred to a system of axes fixed in the machine itself. If the motion of these axes be known with reference to FIG. 229. Axes of Reference. any set of axes fixed in space, then the motion of the aeroplane is completely known. In Euler's method this fixed set of axes is chosen so as to coincide with the moving body axes at the FIG. 230. instant under consideration. Hence the fixed axes are con- tinually being selected and discarded during motion. This method has the advantage of enabling the difficulties of referring the motion to a set of axes fixed in space to be avoided, but STABILITY 307 possesses the disadvantage that it cannot be used for allowing the flight path of the machine to be continuously traced out. The Equations of Motion. ( a ) LINEAR ACCELERATIONS. Let OX, O Y, o Z be axes fixed in the machine occupying positions O X, O Y, O Z, and O X 1 O V l O z l at successive instants, as in Fig. 230. Let u, v, w be the velocities of the machine along the axes OX, O Y, O Z ; and u + S , v + v, w -}- w the velocities of the machine along the axes O x lf O Y lt O z r The position of the axes relative to each other is obtained by first rotating the machine through an angle S i/< about o Z, secondly rotating the machine through an angle S about the new axis of Y, and lastly by rotating the machine through an angle of 8 < about the new axis of X. Increment of velocity along fixed direction o x = ( + 3 u} cos b 6 cos b 4 + (w + % ^ sin b ft - (v + & v} sin b ^ - u = u + ?>u + wb6 - v $ i// - u whence neglecting second and higher orders of small quantities. Acceleration in the direction o x bt - dt at dt Similarly the increment of velocity along o Y = (v + 8 v) cos b cos b \L + (u + 8 u) sin b - (w + b w) sin b

, i//, with the fixed axes Q x v Oy v O z, as shown in Fig. 233. Let the velocity of the C.G. of the machine along the body FTG. 233. axes be , v, and w respectively, and the angular velocity of the machine about these axes be p, q, r. The general equations of motions will then be m (u + w q v r) = m X m (v + u r - wp) = m Y m(w+vp - Formula 87 y/ 2 - p h z + r h l = ; M where u = -= ; and similarly for v, etc. m = mass of the aeroplane /& 2 =^B-rD-/F *k-rC-/B-fD-' STABILITY 311 A B C D E F are the moments and products of inertia. In the general problem the air forces X, Y, Z, and the air moments L, M, N, are functions of the velocity components, and of 0, ^, and ;//, and a disturbance from the normal flying speed and attitude causes a change in each of these quantities. If U be the normal flying speed and the disturbance be small, then u, v, w, /, q, r, 9, 0, ;// are small compared with U, so that we can write X = /(U, K, v, w t p t q, r, 6, 0, i/,), which can be expanded into the approximate form X = u X u + v X v + w X w +/ X p + q X q + r X r + X + g sin which is a linear function of the small quantities u t v t w y p,q, r> 0. The coefficients of these small quantities are the derivatives, which represent physically the slope of the curve of X upon a base of u, v, w,p\ q, r respectively. In a similar manner we have Y = u Y u + v Y v + w Y w + / Y p + q V q + r Y r + Y - g cos sin Z = u Z u + v Z v + w Z w + / Z p + q Z q + r Z r + Z - g cos cos L = u L u + v L v + iv L w + p L p + q L q + r \ , r + L M = M u + v M v + ze/ M w -f / M p + ^ M q + r M r + M N = N u + v N v + w N w + / N p + q N q + rN r + N Before proceeding to form the equations for small oscilla- tions, it should be observed that from considerations of the symmetry of the aeroplane, eighteen of the derivatives will be zero. For this reason the derivatives X, Z, M disappear when the suffix is v,p, or r, and the derivatives Y, L, N disappear when the suffix is u, ^v, or q. Separating the equations of steady motion from those for small oscillations by writing (U + u) for u, (V + v} for v, (0 + 1 ) for 0, etc., and omitting those derivatives whose value is zero, and combining formulae 87 and 88, the equations become m[u + (W+ w)(Q+g) - (V + p) (R + r)} X +^sin(0' + 8)] m(v + (U + u)(R + r) - (W + a/)(P +/)] = m{v Y v +/ Y p + rY r + Y - ^cos(0'+ 0)sin('+ i[w + (V + )(P +/) - (U + Z - g cos (0' + 0) cos (' + p A - ^ F - r E - r h. 2 + ^ // 3 = m [v L v + / L p + r L r + L ] // B - r D - / F - / A s + r ^ = m [u M u + w M w + ? M q + M ] / C - p E - q D - ^ ^ + / h. 2 = m [v N v + / N p + r N r + N ] 312 AEROPLANE DESIGN By limiting the conditions to those occurring in steady flight In a straight line in the plane of symmetry .r O s, this plane being vertical, these equations can be still further simplified. For such conditions the dash attached to an angle being used to denote the angles for flight under such conditions. The .terms such as X , Y , Z , &c., are included in the con- ditions of steady motion. For equilibrium in steady flight X,, and Z are balanced by the thrust of the airscrew and force of gravity respectively, and since there is no side force on the machine the various moments are zero. For steady motion, therefore, when axis of machine is at an angle 9' to the direction of flight, X + g sin 6' = o ; Z - g cos 0' = o ; Y = o ; L = M = N = o; 9 = = o Hence, neglecting small quantities of the second order, the equations of small oscillations reduce to + W<7 = uX u + wX w + N v + / N p + rN r ] The oscillations being small, it can be assumed that the displacements are proportional to Xt , so that the rate of change of each of the quantities u, v, w,p, q, r, is proportional to X, or - = u = X u and so on. at Also by a suitable choice of axes W can always be made zero, and the generality of the equations is not affected thereby. Further, by writing the moments and products of inertia, represented by A, B, etc., in the form m k^, m B 2 , etc., where X- A , , etc., represent the radii of gyration about the respective axes, it is possible to eliminate the mass of the machine from the equations, Formulae 89. The resulting equations can be divided into two groups representing the Longitudinal and the Lateral Oscillati -ns- respectively, and are best expressed in the form of t\vo de- terminants, namely Formulae 90. STABILITY i. LONGITUDINAL OSCILLATIONS. X - X u , - X w , ' - XX q - - Z u , X - Z w , - X (U + - sin - M u , - M, X ( - M q + X = o Formula 2. LATERAL OSCILLATIONS. X - Y v , cos - X Y - L v - N v X ( - L p + X / X ( - N p - X X (U - Y r ) + g sin 0' - X(L r + X* E 2 ) ......... .... Formula 92.. Bryan has shown that the solution of these equations can be written in the form AX 4 + B X 3 + C X 2 + D X + E =o A l X 4 + B! X 3 + Gj X 2 + D t X .+ E T - o Formula 93, For stability the quantity A must be negative if real, or have its real part negative if it is complex, in which cases the amplitude of the oscillations diminish with time. The condition that the real roots and the real parts of imaginary roots of Formula 93 may be negative, is that the coefficients A, B, C, D, E, shall each be positive, and also that the quantity B C D - A D' 2 - B 2 E generally known as Routh's Discrimi- nant shall be positive. In this manner Bairstow has derived. the following values for the coefficients from Formulae 91, 92. LONGITUDINAL OSCILLATIONS. A = K 1; - B - - M c - z w , U + Z q + X u j Q ' *n~ ^>u j -**-w M w , M q M u , M q Z u , Z w I) m - X u , X w , Xq - g M u , - sin B' z u , Z w , U + Z q M w , cos 6 M u , M w , Mq E = - g X u , X w , cos B' Z u , Z w , s : n 0' M M 9 . Formulae 3'4 AEROPLANE DESIGN A- LATERAL OSCILLATIONS. c,= T 1? 2 _ A -'--'y A A > A NA 2 A V 5 ^E J ** Y v , Yp , o T I T? 2 My j i^p j : H i N N p Y Y , , Y r -U D 1 = Y V ,Y v 3 > x r u Ly,L p > *- N v ,N p ,N r ' E! = - g cos LV , Lr N v , N r T 1? 2 T l^y , "-A ) *^P N b 2 "NT **V I ^E > *';! L v , - K E 2 N v , - K c 2 N N N sin 0' L r N r N v , + g sin 6 Formulae 95. The application of these formulae to the investigation of the stability of an aeroplane appears a formidable task, but it will be shown subsequently that several of the derivatives included in the above expressions are of minor importance and may be neglected. This results in much simpler expressions. The Resistance and Rotary Derivatives Before pro- ceeding to the solution of the biquadratic equations it is necessary to consider the manner in which the resistance derivatives depend upon the dimensions of a machine. Simple mathematical expressions can be deduced for most of these derivatives, enabling a much clearer conception to be formed as to their dependence upon the form of the machine. The experi- mental value of the derivatives for the model of a Bleriot monoplane constructed to a scale of one-twentieth full size, its shape and leading dimensions being shown in Fig. 234, was carried out by the N.P.L. The model experiments were carried out at a speed of 30 feet per second, while the normal speed of the prototype was 65 m.p.h. (95*4 f.p.s.). The weight of the actual machine was 1 800 Ibs. To convert the model results to the full-size machine the following conversion factors were therefore used : X 20" STABILITY Force on machine = force on model x ( 2-2 \ V 30 / = 4040 x force on model Moment on aeroplane = moment on model x ( -2Lf 1 x 2o 3 V 3 / = 80800 x moment on model It should be noted that the methods to be adopted in the investigation of the stability of any machine will be upon similar lines to those outlined here for the Bleriot monoplane. The various derivatives will be considered in turn and the method of their determination fully explained. A. Derivatives affecting Longitudinal Stability. X u This is the rate of change of horizontal force with forward speed. Let the forward speed of the machine increase from 'U to U + u It may here be pointed out that as the motion of the aero- plane is in the negative direction along the axis of X, the sign of U will always be negative in actual flight. It is also convenient to have an expression for wind velocity relative to the aeroplane, although it only varies from the velocity of the machine in its sign. For this purpose we shall use the symbol 'U, which, of course, is connected with U by the relation 'U = - U 3 i6 AEROPLANE DESIGN The equilibrium forces other than those due to the airscrew vary as the square of the forward velocity, hence the horizontal force or drag ;;/ X becomes m x / ru ~ frr \ = m ^ ( i - 2U \ Differentiating this expression with respect to u, we get whence ~ 2 where X is the drag per unit mass. X *i CVu, \ Chas -6* -<** -2 2 4* Angle of Pihch 6 8 FIG. 235. Forces and Moments on Model Bleriot-type Monoplane. The experimental determination of X is carried out as follows : The model is supported in the wind channel, and measurements are made of the longitudinal force X that is, the force along the airscrew axis for varying angles of incidence. STABILITY 317 The observations made upon the Bleriot model covered a range of pitch from 8 to -f 14, and are shown graphically in Fig. 235. The value of X when the angle of pitch is zero is the required value of X From Fig. 235 this is seen to be o - o62 Ibs. for the Bleriot monoplane, and therefore it will be 4040 x "062 = 250 Ibs. for the full-size machine. Consequently 5 * 32*2 * o 95*4 IGOO = - 0-0935 Generally X u may be expected to lie between 0^05 and X w Variation of longitudinal force due to a normal velocity of the machine relative to the wind. The effect of a small upward velocity of the machine is to reduce the angle of incidence of the wings. If w be this small normal velocity, then the reduced angle of incidence This variation is equivalent to a small angle of pitch d9 away from the equilibrium position, and in the limit we may write = ~ /..dy.t g .'v. d -%.y Total Moment = \ m . / . L p = -pP.'M.bAff d y g dt J J o That is, for wings of rectangular plan form Ley O , TT d K. y , , q D = ^ ! . U . * . U , U * gm di The value of L p will be considerably affected by 4 End Effect,' and in order to obtain the most accurate results it is necessary to take this loss into account. The greater the aspect ratio of the wings the less important does this correction become. For small machines L p 'varies from 200 to 400, while for machines of 20,000 Ibs. in weight its value approaches 2000. An example of this is shown by applying formula to the case of the Bleriot machine. Then " 3 x ~?6 x 95 ' 4 x ' 45 x 5rs x 12 x x = - 195 326 AEROPLANE DESIGN ' The experimental value of the derivative was 167 and the discrepancy is largely due to the fact that no correction was made for * end effect.' The aspect ratio of the Bleriot machine is low, and consequently there would be a considerable reduction in the average dY^^di over the wing surface. Assuming the value of the lift coefficient to vary in accordance with a para- bolic law over the outer section for a distance equal to the wing chord, the average value of K y for the Bleriot wing surface will be approximately 0*83 x max. K y . The value of the deriva- tive L p would now become 195 x 0*83 = 162, which is in very close agreement with the experimental value. A much more accurate method of taking into account ' end effect ' is to solve the integral /*-&#? graphically, and to substitute the value thus obtained in the general formula. Such a method would of course necessitate an accurate knowledge of the variation of the lift coefficient over the whole span. The experimental method of determining L p is to mount the model of the machine upon the balance in such a manner that it is free to rotate about a horizontal axis. The model is oscillated by means of a spring against the damping present due to the wind forces and frictional losses in the apparatus for a period of from 20 to 40 seconds. The oscillations are photo- graphically recorded for several wind speeds and provide a means of estimating the damping coefficient due to the relative wind, and this is the derivative required. N p The variation of Yawing Moment due to Rolling. It has been seen that the value of L p depends upon the slope of the lift curve for the wing section employed. It follows that the yawing moment due to rolling must depend chiefly upon the slope of the drag curve. The value of N p may tneretore be written 7 T7- bd* 3 g m di The effect of the body and fins will be very small in most machines. The ratio of the slopes of the Lift and Drag Curves at angles slightly greater than those giving maximum Lift/Drag is usually in the neighbourhood of 10, hence the value of N p will be about one-tenth that of L p at these angles. Also since the slope of the drag curve may become zero, the value of STABILITY 327 N p when the machine is flying at the angle of minimum drag of the wings must be zero. N p is found to vary between o and 40 for small machines, and increases up to 300 in large machines. The experimental determination of N p is a somewhat difficult matter, the method adopted being similar to that used for the determination of L r which will be described later. For the Bleriot machine the experimental value was 24. Using the formula we have di 005 x 57'3 whence 00237 3 56 = - x -^ x 95-4 x -005 x 57-3 x f- x (-- x 20* 12 = 22 Y r Yariation of Lateral Force due to Yawing. This derivative has practically no effect upon the stability of an aeroplane, and its value can therefore be neglected. r FIG. 239. L r Yariation of Rolling Moment due to Yawing. The value of this derivative is largely dependent upon the wing surfaces of a machine. Its experimental determination is not easy, as it is necessary to produce a forced oscillation of known magnitude about one axis of rotation, and to measure the corresponding oscillation about a second axis of rotation perpendicular to the first. The model is arranged to be free to rotate about the axes of roll and yaw, the rolling motion being controlled by a stiff spring so that the model can oscillate in sympathy with an impressed force of suitable period. An oscillation is then set up about the axis of yaw, and the period of oscillation about the axis of roll is adjusted until resonance is obtained ; the required data can then be deduced from a know- ledge of the amplitude of the oscillations. The experimental value for L r for the Bleriot model is found to be 54. 328 AEROPLANE DESIGN MATHEMATICAL DERIVATION OF L r In yawing, the outer wing of the machine is moving faster than the normal speed of the machine, while the inner wing is moving slower. This will cause an increased lift on the outer wing and a diminished lift on the inner wing. If r be the angular velocity of yaw, then the increased speed of any element distant y from the axis of z = d'U = ry (see Fig. 239), and the increased lift on this element = K y t / (U' 0> V. - U 2 ] b . dy ,i = K y 2 . 'U ry b . dy V and Moment of Element - K y 2. 'Ur/ .' o whence J m r L r = K y ^- 'U r A .f* , dy o +J O That is, for wings of rectangular plan form L r = 4__ Ky'UJ*/ 8 $gm For the Bleriot machine the calculated value of L r = 1 x 002 37 x 0-^8 x 0=5-4 x x (---] x 20 4 3 56 12 VX2/ = 58 as compared with the experimental value 54. L r varies from 50 in small machines to 600 in large machines. N P Variation of Yawing Moment due to Yawing. This derivative depends upon wings, body, and fins, and its value must therefore be determined experimentally. The method adopted is similar to that used in the determination of L p . Its value may be expected to vary between 20 and 100. For the Bleriot machine its value was found to be 31. Application of Derivatives to Stability Equations. From this enumeration and consideration of the derivatives it is now necessary to turn to the question of the method of their application to the stability equations. A. LONGITUDINAL STABILITY (Period of Oscillation). Bairstow has shown, from an examination of the relative numerical values of the coefficients in the biquadratic equation, STABILITY 329 that it can be factorised to a first approximation and expressed in the form This approximation is sufficiently accurate if - and A are less than C C 2 20 R C* and A D is less than 20 These conditions are generally satisfied by modern machines, but should be checked before proceeding further with an analysis of stability. In Formula 96 the first factor represents a short oscillation which in most aeroplanes rapidly dies out and is not of much importance. The second factor represents a relatively long oscillation, involving an undulating path with changes in pitch, forward speed, and attitude. It is termed by Lanchester the " phugoid oscillation. ' These long oscillations should diminish in amplitude with time, in which case the motion is stable and the aeroplane will return to its original flight attitude if tempo- rarily deviated therefrom by accidental causes. The motion is unstable if the amplitude increases with time. Eliminating the various resistance derivatives of negligible value, the formula for the coefficients in longitudinal stability (Formula 94) can be written. A = kj B = - (Mq + X u b 2 + Z w / b 2 ) C - Z w M q - M w U + X u M q + k (X U Z W - X W Z U ) D = - X u M w U + Z u M q X w - X u Z w M q E = - -M w Z u ............ Formula 94 (a) By substitution of the values of the various derivatives in the above formula, the periodic time of each oscillation is easily determined. A very short oscillation indicates great statical stability, and the machine will very rapidly resume its normal flying attitude. Such a machine would be very uncomfortable for flying purposes on account of the violent changes in motion. It is preferable that an aeroplane should have a heavily damped oscillation of long period, such that the resumption of the normal flying attitude takes place very gradually. The aim in design should therefore be to ensure that the righting moments on the machine are just sufficient to give static stability, and to 330 AEROPLANE DESIGN depend upon large damping surfaces for dynamic stability. It is probable that longitudinal stability may be secured at all speeds by the use of a sufficiently large tail plane. Longitudinal Stability of the Bleriot Machine. Collect- ing together the various quantities and the values of the derivatives affecting the longitudinal motion of this machine, we have m = 56 X w = 0-152 b =5 feet Z w = - 2-43 X u = - 0-935 M W - 2-21 Z u = - 0-672 M q = -- 175 The values of the coefficients are therefore A = S' 2 = 25 B = - (- 175 + " '935 x 25 + - 2-43 x 25) = 236 C = (- 2-43 x - 175) - (2-21 x - 95-4) + (- 0-0935 x " 175) + 25 (- 0-0935 x - 2'43 - o -I 5 2 x - -672) = 636 D = (0-0935 X 2'2I X - 95-4) + (- 0*672 X - 175 X 0*152) - (- '935 x - 2-43 x - 175) = 77 E = 32-2 X 2'2I X - 0-672 = 48 Substituting these values in Routes Discriminant 236 x 636 x 77 - 25 x yy 2 - 236' 2 x 48 = 8-7 x io 6 Since all the coefficients and Routh's Discriminant are positive, the aeroplane is longitudinally stable. The periodic time of the short oscillation is determined from the first factor of Formula 96. Substituting the values obtained above X* + ^6 x + 636 = Q 2 5 25 that is X 2 + 9*44 A + 25-4 = o whence X = 4-72 i"j6i The imaginary roots indicate an oscillation of periodic time -- =3*6 seconds approximately and the time to damp 50% - seconds = o-ic; seconds. 4-72 STABILITY 331 The periodic time of the long oscillation is determined from the second factor of Formula 96. Substituting the values obtained above / 76 __ 236 x 4 8 x ' ~J 636 or X 2 + 0*092 X + 0*0754 = o whence X = 0*046 0*271 / The period of the longitudinal oscillation is therefore = 23 seconds 0*2 /I and the disturbance is reduced to half its value in , seconds, that is in about 1 s seconds. 0-046 The mathematical treatment given in the foregoing para- graphs has been extended by the N.P.L. to show the motion of this aeroplane during recovery from gusts and movements of the controls. Fig. 240 shows the disturbed longitudinal motion due to a single horizontal gust. As a result, the velocity of the aeroplane relative to the air is increased by a small amount, u . This increase rapidly dies away, and after 5 seconds becomes zero ; the velocity goes on decreasing for a further 5 seconds, reaching its minimum value at the end of 10 seconds. This velocity then increases again for a period of about 10 seconds before commencing to diminish again. The changes appear to follow a periodic curve of rapidly decreasing amplitude, such as would be obtained, for example, from the projection of a logarithmic spiral, and after about 50 seconds are completely damped out. The change of velocity of the machine normal to the air is w, and, as will be seen from Fig. 240, this commences from zero, reaches a maximum value of about *2 , and then dies away rapidly in the same manner as u. Curves for q the angular velocity of the machine (shown dotted), and for the angle of pitch, are also shown to a greatly enlarged scale. It will be seen that the pitch angle increases for about 5 seconds and then diminishes again, being finally brought to zero through a series of periodic changes of decreasing amplitude and of 22 seconds period. The corresponding case in practice arises when the machine is struck by a horizontal gust. The lift on the wings will be momentarily increased and the machine will begin to climb ; that is, there will be a component of velocity w normal to the direction of flight. The nose of the machine will be inclined 33 2 AEROPLANE DESIGN upwards ; that is, an angular velocity q is set up, and the angle of incidence of the wings is increased by an amount 9. The result of the gust, however, will be to reduce the velocity of the machine, and after it has passed the lift on the wings will be insufficient to support the machine. It therefore commences to 20 24 28 32 36 40 44 48 52 56 So TJME IN SECONDS FIG. 240. Disturbed Longitudinal Motion of an Aeroplane (Single horizontal gust). TTME IN SECONDS. FIG. 241. -Disturbed Longitudinal Motion of an Aeroplane (Single downward gust). fall, and in so doing picks up speed again. On account of its momentum, however, its velocity increases to a greater extent than is required for equilibrium, and the machine will then flatten out and commence to climb again, the cycle of opera- tions being repeated until the oscillation dies away through the damping out, owing to the action of the control surfaces. The STABILITY 333 motion is therefore seen to be stable, and the machine settles down to its original speed relative to the wind in less than a minute. A second curve, Fig. 241, was prepared to show the effect of a downward gust upon the machine. By combining the results of these two diagrams, it is possible to find the effect of a steady gust of wind striking the machine in any direction in the plane of symmetry. Recent investigations upon the stability of full-size machines by the use of cinematography, show that the mathematical theory is borne out with considerable accuracy in practice. B. LATERAL STABILITY. The factorisation of the biquad- ratic equation for lateral stability as deduced by Bairstow is Ef \ / /~D' \ 2 A ' f" 1 ' \ P""" / C* 1 T?'\ "D' T\' "H 1 I X -4- \ ' t I A 2 4- I 1 X 4- - *~ D 7 / \ A' B' / L VB 7 D/ (B') 2 - A'C'J Formula 97 which approximation is sufficiently accurate if T?' T7' T : and , are less than B D 20 and B' D' - (C') 2 is less than ( C ' 20 The value of the coefficients for lateral stability in horizontal flight given in Formulae 95 be can reduced to the simpler expressions A' - kj k must be negative. None of the quantities involved in A', B', C' is liable to vary in such a way as to render this expression negative for ordinary conditions of flight, and the equation represents a rolling of the aeroplane, which is heavily damped by the wing surfaces. In the case of a stalled machine, how- ever, this motion would lead to trouble, since the damping effect produced by the increased lift on the downward-moving wing will no longer operate. Under such circumstances a movement of the wing flaps will no longer produce any righting moment. The third factor may be written approximately and the motion represents a damped oscillation of period A A B/ 2 7T Y and damping 3C' C 2 B' The third oscillation consists of a combined yawing and rolling motion, and for stability the amount of fin surface above the C.G. should not be excessive, while there should be sufficient fin surface on the tail. It will be seen that these requirements clash with those for spiral stability, but it is possible by a careful adjustment of the surfaces to satisfy both conditions. Lateral Stability of the Bleriot Model. Applying the equations of motion to the case of the Bleriot Model, its lateral stability may be investigated. v The derivatives concerned are Y v = 0*108 N p = 0*44 L v = 07 Lp = 167 N p =24 L r = 54 N r = - 31 The radii of gyration of a machine can be calculated from 336 AEROPLANE DESIGN the scale drawings in the manner indicated for a streamline strut in Chapter IV., and in the case of the Bleriot were found to be / A (radius of gyration about axis of roll) = 5' k c (radius of gyration about axis of yaw) = 6' Substituting these values in the stability equation, the values of the coefficients are found to be A' is 900; B' is 6780; C is 5580; D' is 6640; E' is - 68 whence Routrfs discriminant = B'CD' - A'D' 2 - B' 2 E' = 2 I '5 X 1C 10 The coefficient of E' being negative, the machine is laterally unstable. Considering the first factor of the equation, we have / = - E'/D' = - (- 68/6640) = -0102 As / is positive, the motion is not oscillatory, and the amplitude will increase and double itself in time, = o'6q/'oio2 = 68 seconds. Considering the second factor, we have /= - 671 This represents a steadily damped motion, which will be reduced to half its value in 0-69/6-7 1 = o'i seconds. The third factor of the equation becomes and the roots are p = - 0*416 0^963 i The period of oscillation will be 2 ^ =6-5 seconds 0-963 and the amplitude will be reduced to one-half in 0-69/0-416 = 1-65 seconds. We thus see that the machine under consideration is spirally unstable, which is shown by the fact that the coefficient E' is negative, that is, L V /N V is less numerically than L r /N r Refer- ence to the mathematical expressions for these derivatives will STABILITY 337 show that in order to eliminate the spiral instability it is necessary to have L v large, that is a good dihedral. N v small, that is a smaller rudder. The other two derivatives are difficult to control, but N r may be increased by adding equal fin areas in front of and behind the centre of gravity of the machine, and this will not affect the value of N v A graphical representation of this lateral or asymmetric motion, prepared upon the same lines as for the longitudinal motion, is shown in Figs. 242, 243. Since this lateral motion is unstable, they differ essentially from those shown for the longitu- o s to H 25 046 II 16 30 34 29 3? 36 4O 4< 48 TIME IN SECONDS FIG. 242. Disturbed Lateral Motion of an Aeroplane. dinal motion. Fig. 242 shows the effect of suddenly banking the machine through an angle . It will be seen that after a slight subsidence the angle of bank .increases continuously, and after 40 seconds exceeds its original value by more than 60 per cent At the same time the velocity of side-slip (v) also increases rapidly in a negative direction. The machine there- fore turns to the right, the angle of bank together with the velocity of side-slip increasing, and the machine falls with increasing speed. Fig. 243 shows the effect of a side wind v" striking the machine on the left-hand side. The sideways motion is very rapidly damped down, but after about seven seconds commences 338 AEROPLANE DESIGN to increase again very gradually, and unless the controls are altered this velocity of side-slip will continue to increase. The velocity of roll (p) grows very rapidly at first, but after two or three oscillations is reduced almost to zero before commencing a gradual increase, which will necessitate an alteration of the controls if it is to be checked. Longitudinal Stability of a Biplane. A more recent investigation of the longitudinal stability of a machine was carried out at the Massachusetts Institute of Technology by Hunsaker, and is described in the U.S.A. Advisory Committee Report for 1914. The machine was a Curtiss Biplane, and the Time in Seconds FIG. 243. Disturbed Lateral Motion of an Aeroplane. model which is shown in Fig. 244 (a b c] was made one- twenty-fourth full size, and geometrically similar to its prototype. The leading dimensions of this machine are as follows : Weight, iSoolbs. Total wing area, 384 sq. ft. Area of tail, 23 sq ft. Area of elevator, ipsq. ft. Area of rudder, 7*8 sq. ft. Span, 36 ft. Chord, 5 ft. 3 ins. Gap, 5 ft. 3 ins. Length of body, 26 ft. The model was mounted on the balance, with its wings in the vertical plane, and the Lift, Drag, and Pitching Moment were measured for various angles of wing chord to the wind. These results are exhibited graphically in Fig. 245, the STABILITY 339 forces being given direct in Ibs., and the moments in Ibs. inches. The wind velocity was 30 m.p.h. The axes of reference are assumed fixed in the aeroplane EL FIG. 244. Model Curtis Biplane. and moving with it in space, with the origin at the centre of gravity. For steady horizontal flight at a given attitude the axis of ' z' is vertical, and the axis of ' x* is horizontal. Angles of pitch departing from the normal flying attitude will be 340 AEROPLANE DESIGN denoted according to the table by 9. For equilibrium 6 is, of course, zero. At high speed (79 m.p.h.) the axis of ' x' was horizontal, FORCES &, MOMENTS ON MODEL Wind Spe d 50 m.p h 4- 8* Angle of Incidence FIG. 245. Forces and Moments on Model of Curtis Biplane. and made an angle of i with the wing chord ; while at low speed (45 m.p.h.), with the axis of ' x* still horizontal, this axis made an angle of 12 with the chord. The axes are fixed by the equilibrium conditions for flight, and differ for each normal flying attitude STABILITY 341 It was found convenient for wind-tunnel purposes to measure the lift and drag about axes always vertical and horizontal in space. To transform these axes to those re- Angle of Rt-cVi (9) + 3" *7* \ Angle of Incidence (i.) Caoel V/-79n>p.-h. I f i FIG. 246. Forces and Moments on Model of Cuitis Biplane. quired for stability investigation the following relationships are used : m Z = L cos 6 + D sin m X = D cos S - L sin By the use of these formulae and reference to Fig. 245, Tables XLVIL and XLVIII. were calculated. 342 AEROPLANE DESIGN TABLE XLVIL (CASE I.) Speed, 79 m.p.h. Angle of attack (i) i' i 6 L D Z X - 4 - 5 - 0-08 '"5 - 6-4 77 i '35 'IO2 24-9 7-76 4 3 765 118 54'9 5'6 8 7 *'I3 165 81-0 1-9 12 ii J'39 270 lOO'O '7 16 i5 1-48 428 109-0 - 2-05 TABLE XLVIII. (CASE II.) Speed, 45 m.p.h. Angle of attack (/) 12' i 61 L D Z X 8 - 4 13 165 26-1 5-68 10 - 2 28 *2I 29-6 5-33 12 O '39 27 3 2> 4 6*29 14 2 45 348 34' 6*92 16 4 48 428 35' 2 7-56 These results are shown graphed in Figs. 246 and 247 From these curves the values of JX dZ dM 7-6 -TV 70 are read off, and the values are then inserted in the formulae for derivatives X W ^Z W and M w Case L =, _ 57.3 dX ~\r Je _ 57'3 x - "65 - 115*5 X 2 = '162 Note that U is negative, as explained previously. 7 _ 57'3 "IT 57'3 d~B - "5'5 x 4 - 3'95 STABILITY 343 _ 57-3 dM U dQ 57'3 x - ii5'5 x 4 = 174 2 x Drag ~ - TT - m U _ 2 x -104 x 2 4 2 x (V/3o) 2 32 x - "5*5 : -128 2 x 2 X 32*2 - "5*5 = - -557 By experiment the value of M q was found to be 1 50. The radius of gyration of the machine about the axis of pitch was experimentally found to be 5*8 feet, which gives us at once the value of b . By substituting in the various values of the derivatives the values of the coefficients are found to be A = '5'8 2 - 34 B = - (- 15 - "128 x 34 - 3-95 x 34) - 289 c = 3'95 x 15 + i'74 x 115-5 + 34('i2 x 3-95 + '162 x -557) = 834 D= -128 x 1-74 x 113*5 + '557 x 150 x -162 + '128 x 3-95 x 150 = IJ 5 E = 32-2 x 174 x -557 3i Substituting these values in Routh's Discriminant we get that the discriminant = 289 x 834 x 115 - 34 x ii5 2 - 2892 x 31 = 18 x io 6 Since Routh's Discriminant and all the coefficients are positive, the machine will be longitudinally stable at the speed considered, namely 79 m.p.h. 344 AEROPLANE DESIGN The short oscillation = X 2 + 8-5 X + 24-5 = o whence \ = 4-25 2*54 i The period = P = 2 7r/2'54 = 2*5 seconds The time to damp out 5o/ = 0-69/4-25 = -16 seconds The long oscillation = X 2 + '125 X + "0374 = o whence / = - '063 'i83/ The period of this long oscillation = P' = 34'3 seconds and time to damp 50 / o = / = io'8 seconds The small oscillations are thus seen to be unimportant while the long oscillations are strongly damped. The aeroplane should therefore be very steady at this speed. Case E \ V 45 mph I L . Angle of Pil-ch (6) FIG. 247. Forces and Moments on Model Curtis Biplane. Case II. Speed, 45 m.p.h (66 f.p.s.) ; incidence, 12. Pro- ceeding in a similar manner, the values of the derivatives at this speed are found to be X u = - -189 Z u = -972 M w = 2-15 X w = - -236 972 Z- = 736 M q = - 106 STABILITY 345 and the values of the coefficients are A = 34 C = 243 E = 67-2 B = 137-5 D = *7'4 whence Routh's Discriminant = 137-5 x 243 x 17-4 - 34 x i7*4 2 - i37*5 2 x 67-2 = - 7 x io 5 which being negative indicates that the machine will be unstable at this speed. The short oscillation = X 2 + 4*04 X 4- 7 '14 = o whence X = - 2*02 1*75* and the period = 3'59 seconds and the time to damp out 5o/ o = 0*342 seconds The long oscillation = X 2 - 0*085 X -f '276 = o whence X = 0*043 "5 2 4* and the period =12*0 seconds and the time to double amplitude = 16 seconds. The machine is thus seen to be unstable at a speed of 45 m.p h., and it is essential that the pilot should keep a firm hold on his elevator control. CHAPTER XL DESIGN OF THE CONTROL SURFACES. Controllability and Stability. The question of the rela- tion between the control and stabilising surfaces was briefly considered in the preceding chapter on stability, and it was stated that the degree of controllability of a machine was determined generally by the duties for which it was to be used. For fighting purposes it is necessary that the machine should answer very quickly to the controls, and hence its static stability must be small ; whereas in the case of a large commercial machine with which long journeys must be undertaken, the static stability can with great advantage be considerably in- creased. In general it is preferable to keep the static stability as low as possible, and to obtain dynamic stability by using large wings and stabilising surfaces. The problem of static stability can be considered as under. In nearly all the modern machines the stabilising surfaces are : (a) The Tail Plane and Elevator. (b) The Rudder and Fin. Of these the elevator is also the control surface for longitudinal flight and the rudder for directional flight, while ailerons or wing-flaps control the rolling motion of a machine. The Tail Plane and Elevator. The tail plane and elevator in an aeroplane of normal design are essentially those members which are intended to give the machine that longi- tudinal stability which the wing surface alone lacks. It will be remembered from Chapter III. that over the range of normal flying angles the C.P. of an aerofoil moves forward as its angle to the wind direction is increased. The resulting effect upon the machine is illustrated by the diagrams in Fig. 248. If it be assumed, as shown in case (), that at that particular instant the line of lift passes through the C.G. of the machine, then there will be no moment upon the machine, and conse- quently no load upon the tail. If, however, a upward gust of wind strikes the machine, the angle of incidence of the wings to the resultant wind direction will for a very short time be reduced, the C.P. will move backward to the position shown in case (a), and a pitching moment will be set up upon the machine DESIGN OF THE CONTROL SURFACES 347 which must be counterbalanced by the tail plane surface if equilibrium is to be restored. Similarly in case (c\ if the angle of incidence relative to the wind direction has been temporarily increased, then a stalling moment will be set up and the tail plane will be called upon to produce a righting moment. Co) r FIG. 248. Direction of Load on the Tail Plane. Moreover, by reference to the fundamental equation (For- mula i), it will be seen that Normal force g = W cos / or / W cos i = v - v= A for case (b) 348 AEROPLANE DESIGN from which it follows that the attitude of the machine for equilibrium varies with the speed. The righting moment to be exerted by the tail plane and elevator will therefore depend upon the speed at which the machine is flying. Conversely, an adjustment of the tail plane and elevator will alter the pitching moment upon the machine, and so lead to an alteration in its attitude and speed. A further deduction from the preceding paragraph is that the successful design of a tail plane for a particular machine will depend largely upon its speed range ; for it is easy to see that, in a fast machine with a wide speed range, if the tail plane is sufficiently large for the upper limits of speed, then it will be inadequate for slow flight unless 'more weight and head resist- ance are allotted to it than would in most cases be advisable. In the absence of an easy method of making a variable-area tail the real solution to the above difficulty some compromise must be made in practice. In certain cases it is sometimes found necessary to displace the line of thrust of the airscrew so that it no longer passes through the C.G. of the machine. The unbalanced moment resulting will need to be corrected, and this duty also falls to the lot of the tail plane. It will thus be seen that a large number of factors enter into the design of the tail plane and elevator. The duty of the tail plane does not require it to be cambered, a flat plane being all that is necessary, though some stream- lining at the leading and trailing edges may help towards lessening resistance. A streamlined tail plane can generally be designed that will offer no extra head resistance, but will afford greater thickness and greater strength, combined with better accommodation for a strong hinge-spar for the elevator lift The tail, moreover, need normally exert no lifting force ; but this non-lifting or 'floating' tail will only be so at certain angles that is, at certain speeds of the machine. When the disposition of the tail plane is such that the tail exerts a downward force, it is found that the stability of a machine is increased, and this arrangement is frequently adopted in modern design. Should, however, the line of thrust of the airscrew fall below the horizontal line through the C.G. of the machine, an upward load on the tail is necessary. Either arrangement causes an increase of gliding angle, and may, if carried to excess, decrease the useful angular range of the machine, owing to the proximity in one direction of the critical angle. In considering the tail plane as a stabilising surface the area of the elevators should be added to the area of DESIGN OF THE CONTROL SURFACES 349 the fixed part. If the fixed part is symmetrical in section, the elevators, in the case of a floating tail, will exert zero lift when in the same straight line. A floating tail is not at o angle of incidence to the flight path, but positively inclined at some less angle than the wings owing to the downwash, the rate of change of momentum vertically of which is the lift for horizontal flight. Again, the drag of the whole machine is the rate of change of momentum horizontally of the disturbed air. Thus the tail operates in a region where the air is in a state of motion downwards and forwards relative to the surrounding L 743'- 4 1_ . tll ... x Scale of Model Va FIG. 249. Model of B.E. 2 Biplane. atmosphere in gliding flight. It has been found experimentally that the angle of downwash from the main planes is approxi- mately one-half the angle of incidence of the main planes measured from the angle of no lift. This will give the position of the tail plane when ' floating.' Reduction of Effectiveness of the Tail Plane due to Wash from the Main Planes. A method of investigating the effect of the downwash of the main wing surface upon the moment exerted by the tail plane was to determine experi- mentally the pitching moment upon the model of a complete machine for a large number of angles of pitch. The tail plane was then removed from the model, and a similar series of experiments conducted in order to determine the pitching 350 AEROPLANE DESIGN moment on the model without its tail plane. Measurements were then made of the longitudinal and normal forces upon the tail plane and elevator alone at various angles of incidence. A comparison between the results obtained in these three cases enables the effect of the downwash of the main planes to be determined. From an investigation of this nature the N.P.L. found that both the normal force and pitching moment for the tail plane in its normal position are reduced approximately to one-half the values they show when the tail plane is tested separately that is, interference due to the downwash from the main planes reduces the slope of the pitching-moment curve in this ratio, and consequently the necessary area of the tail FIG. 250. Contoured Plan and Sections for Tail Plane 3. plane is double that to be otherwise expected. These results are of such practical importance that they are reproduced here for the purposes of reference. A scale drawing of the model is shown in Fig 249, from which it will be seen that it is of the B.E. type. Experiments were carried out with a series of different tail planes, the results obtained being of a very similar character, those given here referring to the tail-plane section designated T.P. 3, the contours of which are shown in Fig. 250. As will be seen, this section is very similar to that adopted in general practice upon modern machines. The effect upon the value of the pitching moment on the model of a change in the position of its CG. was also observed, and the indications showed that in order to obtain a machine of reasonable longitudinal stability without unduly increasing DESIGN OF THE CONTROL SURFACES 351 the length of the fuselage and the area of the tail plane it is necessary to have a down load on the tail. The position of the C.G. of the model relative to the chord in the experiments here- with recorded was at '41 of the chord from the leading edge. In the first series of tests the longitudinal force, normal force, and pitching moment were measured for the complete machine for angles of pitch ranging from 23 to + 17 at a wind speed of 40 feet per second. These tests were repeated with the elevator x -OB -09 -20 FIG. 251. Longitudinal Force Curves for Complete Machine. set at inclinations of -45. - 30, - 15, - 10, - 5, o, + 5, + 10, +15, +30, +45, respectively. The curves corre- sponding to inclination 5, 10 follow the general lines of o and 15, and are omitted for the sake of clearness. The inclination of the elevator is taken to be positive when it is. turned downwards. The tail plane and elevator were then removed from the model and tested separately over the same angular range at the same wind velocity. The results are shown graphically in Figs. 251-259. 352 AEROPLANE DESIGN no 15 OS N o 2 o -OS Normal Force Curves for- 45" 30 I 5* 30 ^ Tad Plane -20 -15 ~IO -5 O -5 to" Angle of Pitch . * FIG. 252. Normal Force Curves for Complete Machine. FIG. 253. Pitching Moment Curves for Complete Machine. DESIGN OF THE CONTROL SURFACES 353 '100 -02 ~2O tS Angle of Pitch 9 . FIG. 254. Longitudinal Force Curves for Tail Plane Alone. FIG. 255. Normal Force Curves for Tail Plane Alone. A A 354 AEROPLANE DESIGN Angle of Pitch 6. FIG. 256. Pitching Moment Curves for Tail Plane Alone. -20' -/J -to' -5 0' 5 IO Ang\e of Pitch 6 . FIG. 257. Longitudinal Force Curves for Effective Tail Plane. DESIGN OF THE CONTROL SURFACES 355 -3 FIG. 258. Normal Force Curves for Effective Tail Plane. FIG. 259. Pitching Moment Curves for Effective Tail Plane. 356 AEROPLANE DESIGN The Elevator. From the preceding remarks it will have been observed that the function of the elevator is of a twofold .nature : (i.) To regulate the speed of flight ; (ii.) To correct any variation in the attitude of the machine which may arise from the action of gusts or other causes. It will be apparent that the position of the elevator required to maintain equilibrium when the machine is flying at its top speed will generally be quite different from that required when stalling. In addition it is necessary to have a range of positions in order to correct for disturbances at these speeds, hence the maintenance of the elevator in such attitudes involves a con- siderable strain upon tne pilot. Two methods have been adopted in practice to reduce this strain, namely : (a] The elevator is balanced by means of an extension pro- jecting in front of the hinge spar, or by placing the Binges close to the C.P. of the elevator load. {b) The function of the elevator in regulating the speed of flight may be transferred to the tail unit as a whole by making the latter adjustable. This method is now common practice on most machines, and reference will be made to it subsequently. The elevators are fitted to the rear of the tail plane, and elevate or depress the tail of the machine as actuated by the pilot. They do not provide the whole of the lifting force them- selves by virtue of their inclination to the wind, but are very much more efficient because they induce a lift of like sign in the surface to which they are fixed, owing to the fact that when rotated from their mean position they form, with the fixed surface, a kind of rudimentary aerofoil. The centre of pressure of this lift which forms the controlling force is not, therefore, necessarily upon the elevators at all : it may be somewhat in front of the hinge spur. This does not mean that the force required to be exerted by the pilot for turning the elevators is in any way diminished ; the distance between the C.P. of that part of the total force which is distributed over the elevator itself from the hinge spar must be considered separately in this connection. The form of elevator which will be easiest to turn will be that variety which is not hinged to a fixed surface; the total area being sufficient to provide for the stabilising moments required. In this case the pivot should be arranged at the mean position of C.P. travel during rotation. Owing to the intervention of the critical angle, it is of no use arrange for a greater angle of incidence being given to the DESIGN OF THE CONTROL SURFACES 357 elevators than 25, and even this amount may well be reduced. The elevators are likely to be called upon most when the tail plane itself is already set, by reason of the general inclination of the machine, at a large angle in the direction in which rotation of the elevators is carried out by the pilot. A con- siderably smaller rotation than 25 will then bring about the critical angle. In some cases it may be necessary to guard against this, as a considerable fall in lift may occur from over- rotation, a calamity the cause of which the pilot in time of emergency cannot be called upon to appreciate. For this reason the rotation may well be limited to between 15 and 20 degrees. It is of course better to provide ample surface with small rotation than a meagre surface with a large rotation. There is one great danger which must be guarded against, namely, that the pilot should be able to exert too great a control longitudinally. This is of fundamental importance in the case of nose-diving ; in which case, as we have already seen in Chapter V., the wings may be very greatly overstressed if the pilot should intentionally or accidentally flatten out the nose-dive too quickly. tr-T ' J K ' ^/^yy! }^2AL^^^^ l^-> fc I FIG. 260. Equilibrium of a Machine in Flight. Tail Plane Design. From these experimental results we can with advantage consider their application to general design, and for this purpose it is necessary to draw a diagram of the various forces acting upon a machine in normal flight. (See Fig. 260.) Let A B represent the mean chord of 'the wings ; X w the resistance of the wings ; X b the resistance of the body, chassis, &c. ; X t the resistance of the tail ; Z vv the normal load on the main planes ; Z t the normal load on the tail plane ; T the thrust of the airscrew. 358 AEROPLANE DESIGN Of these quantities the mean chord of a biplane is determined in the following manner. First find the length and position of the mean chord of the wing surfaces, taking into account the variation of chord over the surface and the amount of the dihedral angle. Let C D (Fig. 261) represent the mean chord of the top wing surface, and EF the mean chord of the bottom wing surface. Join A E, D F. Then draw the line A B such that CA : AE^DB : BF _ effective area of bottom plane effective area of top plane the effective areas of the top and bottom planes being deter- mined as shown in Chapter III. This mean chord represents the chord of an imaginary monoplane surface equivalent aero- dynamically to the several planes of a multiplane. FIG. 261. Equivalent Chord. FIG. 262. The values of the resistances of the wings and tail are easily determined from a knowledge of their aerodynamic characteristics. The resistance of the body and the point at which its resultant may be taken to act is determined by summing up the resistance of the various components included for a given speed, as shown in Chapter XIII. By taking moments of the various resistances about some fixed point, the position of the resultant is found. The airscrew thrust at any speed is determined from particulars of the airscrew which is to be used. The normal force on the wings can be calculated from the wing characteristics and the t> area. Now taking moments about the C.G. of the machine (see Fig. 260) Z t (I - t) = Z w (a - b) + T/- (X*d + X bf + X t g) Formula 98 DESIGN OF THE CONTROL SURFACES 359 This formula gives the requisite moment to be exerted by the tail plane to secure equilibrium. By substituting the values of the quantities for the range of speeds over which the machine is required to operate, a series of tail moments are obtained, from which it is possible to choose the tail-plane area and setting which will best satisfy the given conditions. The moments set up by the 'tail plane when the machine ib disturbed from its position of equilibrium must be such that they always tend to restore the position of equilibrium, but for ease of control it is essential that the righting moments should be comparatively small with small displacements from the position of equilibrium, while they should increase with increase of displacement. It is necessary in deciding on the size of tail plane required for a given machine to consider it in conjunction with the length of the fuselage. As will be seen subsequently in relation to the rudder, there is an advantage in a fairly long distance between the C.G. of the machine and the tail ; but for stabilising quality of the tail plane, moment only is of importance. A curve may be drawn representing the moments of lift of wings at various angles of incidence owing to the travel of the centre of pressure about the C.G. of the machine, assuming the machine to swing while in a straight path under its inertia. A similar curve will show the correcting couples due to the change cf angle of the tail. The first curve may be subtracted geomet- rically from the second, and thus may be obtained a righting couple curve which is an index to the statical stability of the machine. Determination of Dimensions of the Tail Plane and Settings of the Elevator. In the following paragraphs the design of a tail plane'is fully carried out, since it is only by such a method that the nature of the problem involved can be fully understood and grasped. The machine for which this tail unit is designed will be the one for which the wing-bracing stresses were worked out in Chapter V., the weight being 2000 Ibs., the effective area of the supporting surfaces 366 square feet, and the wing characteristics those given in Table XLIX. The C.G. of the machine is assumed to be at "32 of the chord from the leading edge of the wing, the distance between the C.G. of the machine and the centre of pressure of the tail being 16 feet. The tail-plane section to be used will be the T.P. No. 3, for which the contours arc given in Fig. 250. The chord of the wing is 6 feet, and the angle of incidence relative to the body axis is 4. 360 AEROPLANE DESIGN TABLE XLIX. WING CHARACTERISTICS. Inclination of wing to wind ... o 2 4 6 3 Absolute lift coefficient (K y ) ... 0*09 0*205 0-298 0-37 Pos 11 of C. P. (fraction of chord) '575 '4 2 5 '35^ '3 2 9 Pos 11 of C.P. relative to C.G. chord ( behind, + in front) -0*255 - 0*105 -0-038 -0*009 Ditto (feet) ...... -i'53 -0^63 -0-228 -0-054 Pitching moments (Ibs. ft.) ... -3060 -1260 -456 - 108 Inclination of wing to wind ... 8 10 12 14 Absolute lift coefficient (K y ) ... 0-441 0-514 0-573 0-598 Pos 11 of C.P. (fraction of chord) 0-312 0-302 0-292 0*28 Pos 11 of C.P. relative to C.G. chord (- behind, + in front) +0-008 + 0-018 4-0-028 +0-04 Ditto (feet) ...... +0-048 +0-108 +0-168 +0-24 Pitching moments (Ibs. ft.) ... +96 +216 +336 +480 The normal force on the wing may, with sufficient accuracy, be assumed to be equal to the weight of the machine. The small variation in the load due to the tail-plane pressure is also ignored. The pitching moments on the machine due to the travel of the C.P. are shown in Table XLIX. The function of the tail plane is to introduce opposing moments to these, so that the total pitching moment upon the machine in normal flight is zero. Other forces besides the wings and the tail plane modify the pitching moment on a machine, as is seen by reference to Fig. 260 ; but, for the sake of simplicity and clearness, these will be neglected in the present case. In an actual design, however, they must not be ignored, and each of the forces shown must be considered. It is therefore assumed that the C.G. of the machine lies along the mean chord of the wings, and that the line of thrust and body and wing resistance passes through the C.G. Hence c, d,f,g in Formula 98 are each equal to zero, so that the only forces producing a pitching moment upon the machine are the normal forces upon the wings and the tail plane respectively. Moment due to the tail = Normal force on tail x distance of tail C.P. from C.G. of machine y P A' V 2 / g DESIGN OF THE CONTROL SURFACES 361 Where 6' = angle of incidence of tail plane to relative wind A' = area of tail plane y = rate of change of normal force on tail plane with angle ^ of incidence. / = distance from C.P. of tail plane to C.G. of machine. From the experiments on the tail plane No. 3 it was observed that the angle of downwash from the main planes is approximately one-half the angle of incidence of the main planes measured from the no lift position. This for the section employed is 2, so that the following table can be prepared : TABLE L. Inclination of wings to wind ... o 2 4 6 8 10 12 14 Inclination of wings measured from angle of no lift ... 2 4 6 8 10 12 14 16 Angle of downwash ... ... i 2 3 4 5 6 7 8 At this stage it is necessary to determine A' and 9' by trial. A value of 'A' must be assumed and the necessary tail setting B r calculated as in the following manner. If the result obtained by this assumed value is unsatisfactory, a fresh value for A' must be taken, and the calculations repeated until a satisfactory setting results. In this connection it is very useful to refer to some such table as that given on page 439, in which the dimensions of the tail plane for several successful machines are shown. From this table an estimate can be formed in most cases of a probable suitable size. A tail plane of span 16 feet and a chord 4 feet will be assumed for the tail plane under con- sideration. Two cases will be considered : (a) A tail plane of variable angle of incidence relative to the body axis. (b) A tail plane of fixed angle of incidence relative to the body axis. (a) In this method, by adjusting the angle of incidence of the complete unit, equilibrium at varying speeds of flight is obtained without the use of the elevator, the latter being used solely to perform corrective manoeuvres about the position of equilibrium. Moment due to tail = 6' -/ x -00237 x 64 V 2 x 16 d By reference to the curves shown in Fig. 255, it will be seen that the normal force on the tail plane increases with the angle 362 AEROPLANE DESIGN of incidence according to a straight-line law and the value of the slope dZ This refers to the model of area '1275 square feet at a wind .speed of 40 feet per second '*' 158 = ~df X <00237 x ' I275 x 4 ~ whence -- = -033 dff that is, the rate of change of the absolute lift coefficient for the Tail Plane Section No 3 when interference effects are absent is 033. Interference effects reduce this figure by approximately one-half. Hence we may write the moment due to the tail = 0' x -Jz5 x -00237 x 64 x 16 V 2 - -o 4 0' V 2 Also V' 2 = W 2000 Ky X -00237 X 366 From the values given by the relationships A and B a table of the following nature can be prepared : TABLE LI. Angle of incidence of wing. V2 Moment due to tail. Moment ^ required. V = oment required Moment o ... 25600 1024 ft' - 3060 3 2 II2OO 4480' ... - 1260 - 2-82 4 .-. 7740 ... 310^' ... 456 ... - 1-47 6 ... 6220 249 6' ... 1 08 - 0-43 8 ... 5200 208 6' ... + 96 + 0-46 10 ... 4480 ... 179 V ... + 216 -f I'2 12 4020 ... 161 0' ... + 336 ... + 2-08 14 ... 3850 . 1560' ... + 480 + 3'oS DESIGN OF THE CONTROL SURFACES 363 The angle of the tail plane relative to the downwash of the machine must therefore vary between 3 and + 3. The angle required for determining the travel of the variable gear is that relative to the body axis. The angle of the body axis is 4 less than that of the wings, whence Table LI I. can be prepared from Tables L. and LI. TABLE LI I. Angle of wing to wind ... ... o 2 4 6 Angle of body axis ... ... ... -4 -2 o 2 Angle of downwash relative to body axis -5 -4 -3 -2 Angle of tail plane relative to bo iy... 2-0 i'i8 i'53 r '57* Angle of wing to wind ... ... 8 10 12 14 Angle of body axis :. 4 6 8 10 Angle of downwash relative to body axis ... ... ... ... - i o i 2 Angle of tail plane relative to body... 1-46 1-2 1-08 1-08 It is thus seen that a variation in the angle of incidence of the tail plane and elevator relative to the body axis of from + 2 to + i is sufficient to ensure equilibrium at all angles of flight with no deflection of elevator relative to the tail plane. Examples of Tail Plane Incidence Gears are shown in Figs. 274 and 275. (b) Tail-plane setting fixed relative to the body. With this method equilibrium at various speeds is obtained by the use of the elevator. It is therefore necessary to choose some inter- mediate position for the tail-plane setting in order that the requisite elevator deflection may be small, thereby ensuring that sufficient additional moment may be secured for manoeuvring. From an examination of the tail-plane settings in the case just considered it is probable that a fixed angle of incidence of i J relative to the body will be suitable. Using this figure the angle of downwash relative to the tail plane is as shown in Table LIII. TABLE LIII. Angle of incidence of wings ... o 2 4 6 Angle of downwash relative to tail planed' ~ 3i ~ 2 i ~ T i ~ i Moment due to tail plane ... -3590 -1120 -464 -124 Moment required ... ... -3060 -1260 -456 - 108 Moment to be exerted by elevator +530 140 +8 + 16 364 AEROPLANE DESIGN Angle of incidence of wings ... 8 C 10 12 14 Angle of downwash relative to tail plane 6' ... ... ... \" ij 2^ 3^" Moment due to tail plane ... + 108 + 358 +402 +540- Moment required ... ... + 96 + 216 +336 +480 Moment to be exerted by elevator 12 142 66 - 60 Additional Moment due to Deflection of Elevator- Referring to Fig. 255 it will be seen that the increase in normal force due to the deflection of the elevator at a fixed angle of pitch is approximately proportional to the angle of deflection over the range of angles of pitch from 11 to -f 5. On plotting this increase of normal force the variation of lift coefficient for the section per degree movement of elevator is found to be *oi8. Allowing a decrease of 50% due to its opera- tion in the downwash of the main planes, a value of is deduced. Hence the increase of normal force on the tail due- to the deflection Q" of the elevator = 6" x '009 x -00237 x 64V 2 whence additional moment = -0219 V 2 0" and the required deflection of elevator at each speed in order to produce equilibrium fl" = Moment to be exerted by elevator 0219 V 2 .*. 6" = + o'95 - o'57 + o + o'i2 - o - io - 1*45 075 - 0*70 It will be seen that the deflection of elevator required for equilibrium is almost negligible when a correct tail-plane setting has been secured. Consequently, in a case such as the present no advantage is to be derived by installing a tail-incidence gear, such a device being generally of more value in the case of large machines. So far the method of design has been limited to the deter- mination of the righting moments necessary to produce equili- brium at a particular angle of incidence. Before finally deciding upon the tail plane it is essential to examine whether equilibrium will be restored should the machine be temporarily deflected from its normal flight attitude by a gust or other cause. For stable equilibrium it is necessary that the moments set up by DESIGN OF THE CONTROL SURFACES 365 the tail plane in the event of such disturbance are sufficient to overcome the unbalanced moment set up by the wings and to .restore the machine to its original flight attitude. For such to be the case, the moment due to the tail plane must increase at a faster rate than that due to the wings. Now, from Fig. 260 Moment due to wings - K y A V 2 (a - b) o and Moment due to tail = K' y A' V 2 / Since p/g (V 2 ) is common to both expressions, it follows that the moment due to the wings is proportional to K y A (a b) and the moment due to the tail proportional to K' y A' /. By plotting these exnressions on a pitch-angle base, curves such as are shown in Figs. 263 and 264 result, and an examination of such curves enables conclusions to be drawn as to the probable dynamic stability of the machine. For the machine previously considered, the values for K y A (a b} for different wing angles are shown in Table LIV., and for the tail plane the values of K' y A' / are shown in the same table, where v - dtf for fixed angles of ij to body. Incidence of wings TABLE LIV, 2 4 6 c 9 205 298 '37 -i'53 -'63 -228 054 .. -5'4 -47 '4 -24-9 7'3 K y A (a-b) x 366 ... Angle of tail pla r e relative to downwash 0' K'y = 0' x -0165 K'y x 1 6 x 64 Incidence of wings K y A(a-t>) x 366 ... Angle of tail plane relative to downwash 0' ... ... J ij 2^ 3^ K'y = 0' x '0165 ... ... '0082 '0248 '0413 '0578 K'y x 16 x 64 -8-4 -25-4 -42*3 -59*2 -35 - 2 2 - i J -r -0578 - -0413 - '0248 - "0082 59' 2 42-3 25'4 8-4 8 10 12 14 442 5i4 '573 598 048 108 168 24 7-76 20-3 35*2 5 2 *5 3 66 AEROPLANE DESIGN These values are shown plotted in Fig. 263. The moment due to the tail plane is a straight line, whose position may be shifted (corresponding to a movement of the elevator) such that equilibrium may be obtained at each angle of incidence of the wings. The slope of the resultant curve obtained by combining the two curves will indicate the nature of the equilibrium at the various attitudes of flight. These curves are shown in Fig. 263, and it will be observed Case I . A' = 64 sQ.fr. I - 16 fh_ I WingCoeff. FIG. 263. Case I. that the slope of the resultant curve for which equilibrium is obtained at angles of from 5 to I2 C is of opposite sign to that for angles 2 to 5. Between angles of 5 to 12 the equilibrium is such that an alteration in the angle of incidence of the wings sets up a pitching moment which causes the machine to revert to its original position that is, the equilibrium is stable. Between angles 2 to 5 an alteration in attitude of the machine sets up a pitching moment which tends to increase the deviation from the DESIGN OF THE CONTROL SURFACES 367 equilibrium position, hence the equilibrium is unstable, and unless a correcting moment is introduced by a movement of the elevator, the attitude, and consequently the speed, will be per- manently altered. It is therefore apparent that, to secure stable equilibrium at all angles of incidence, the slope of the curve due to the tail plane must at all points be greater than that of the curve due to the wings. The slope of the tail-plane curve is directly proportional to the area of the tail A', and the length -60 ng Coeff FIG. 264. Case II. from the C.G. at which it is acting (/) ; therefore by increasing either of these factors stable equilibrium can be secured in the above case. For the present purpose an increase in / from 1 6 to 20 feet will be adopted. The values of K' y A' / will then be as shown in Fig. 264. In this case the equilibrium is stable throughout the complete range, of flying angles. In actual design work this latter operation must be carried out, before determining the tail settings, otherwise in the event 3 68 AEROPLANE DESIGN of any alteration in the main dimensions being necessary, as was the case in this example, the whole of the preceding work would have to be repeated. The order in the present chapter is due to reasons of clarity, it being easier to understand the principles involved after the treatment of the tail settings. Having thus secured a tail plane to give stable equilibrium at all speeds, an investigation into the longitudinal stability of the machine should be carried out according to the method shown in Chapter X. A satisfactory result will in all pro- bability be secured, and the dimensions of the tail plane can then be embodied in the general design of the machine. Fin and Rudder. The fin and rudder form the stabilising and control surfaces for directional flight. In certain cases the Scale of Model io. FIG. 265 Model of S. E. 4 Body. rudder may be used without a fin, and under these circum- stances it performs the dual function of a stabilising and con- trolling surface. Such an arrangement, however, should be limited to small machines. On rotation of the machine about the axis of yaw, the rudder and fin, together with all members of the aeroplane which present a side area to the line of flight upon rotation, produce a horizontal force transverse to the line of flight which is known as the lateral force. Just as curves can be drawn for longitudinal ' static ' stability, so also can curves be drawn in the case of members contributing lateral force during or owing to directional change ; the moments of lateral forces about the C.G. of the machine being plotted. The rudder, and fin if any, must then be sufficiently large to ensure that there is always a small positive residual moment DESIGN OF THE CONTROL SURFACES 369 that is, a moment which tends to restore the machine to its proper direction. The balance should be right in the case of a tractor machine when the engine is working, as then the pro- peller exercises more fin effect than in gliding flight. Lateral force is to be avoided as far as possible, owing to the fact that it produces side slip. For this reason, therefore, a long fuselage carrying a small rudder is an advantage. The same remarks as were made in reference to the elevators are applicable, but to a less extent, owing to the smaller area, with regard to the inter- action of the rudder and fin. The rudder may with advantage be given a moderate amount of aspect ratio, particularly in the case where there is no fin. -20 -5 O" Angle of Yaw FIG. 266. Yawing Moment on Model of S. E. 4 body. Wherever possible, the lateral force on a machine should be deduced from experiments on a model of the machine being designed. An example of the experimental determination of the lateral force and yawing moment upon a model is shown in Figs. 266 and 267. The model was of the S. E. 4 body, and was one-tenth full size, the overall length with rudder in position being 25". This model is shown in Fig. 265. Measurements of the lateral force and yawing moment about the C.G. of the machine were made for four modifications of the model, namely : (a) With fins and rudder straight. (b) With rudder set over at 10. (c} With fins on and rudder off. (d) Without fins and rudder. B B AEROPLANE DESIGN The curves (a), (), (d) in all figures show the progressive effect of the decrease of rudder and fin area. The lateral force is reduced to about 70% of its value by removing the rudder, and to about 40% of its value by removing fins as well The yawing moment is reduced to about 35% by removing the rudder, and changes sign when the fins are also taken off, showing that the body is unstable as regards yawing about the C.G. when there is no fin area at the after~end. In the event of such experimental information being unob- -25 -so -13 -10 -5 5 FIG. 267. Lateral Force on Model of S. E. 4 body. tainable for a machine under design, the following figures may be used : TABLE LV. ITEM. Streamline wires Streamline struts Fuselage Rudder and fin Lateral Force in Ibs. per Degree Yaw at 100 f.p.s. o'8 Ibs. per square foot side area. 07 Ibs. per square foot side area, o'n Ibs per square foot side area. 0*6 Ibs. per square foot side area. Landing chassis and Wheels ... 0*8 Ibs. per square foot side area. From a side elevation of the machine the amount of side area presented by the various members may be assessed and the lateral force then calculated. DESIGN OF THE CONTROL SURFACES 371 Lateral Force due to the Airscrew. When investigating the lateral stability of the B.E. 2 machine it was found experi- mentally that the total side force upon the machine when the airscrew was rotating was considerably greater than a calcula- tion of the known lateral forces had indicated. This led to an investigation of the possible action of the airscrew as a fin, and a mathematical theory was developed by Mr. T. W. K. Clarke, B.A., A.M.I.C.E., from the following considerations. Referring to Fig. 268, it will be seen that the effect of a side wind will be to cause a difference in the velocity of the airscrew blades relative to the air according as to whether they are moving in the direction of the side wind or moving towards it that is, if u be the velocity of the side wind, and v the velocity of the blade, then the velocity of the blade relative to the wind will be v u and v + u respectively. The result of this will be that the angle of FIG. 268. Lateral Force on Airscrew. attack for the lower blade will be increased (see Chap. IX.), while that for the upper blade will be diminished. This increase in the angle of attack and the increased velocity of the lower blade relative to the air produce a greater pressure on the lower blade, while upon the upper blade there is a decrease in pres- sure : hence the side components of the forces on the blades no longer balance each other, and it is this unbalanced force which causes the variation of total side force according as the airscrew is rotating or motionless relative to the machine. An experimental verification of the theory was carried out by the N.P.L., and it was found that the results were in good agreement with those calculated. For an airscrew of 9 feet diameter and speed of 900 r.p.m., such as was used on the B.E. 2, the lateral force was found to be io - 3 Ibs., with an angle of yaw of 5 and a translational speed of 100 f.p.s. For angles up to 25 the force is approximately proportional to the angle of yaw. After an estimate of the lateral force per degree of yaw 372 AEROPLANE DESIGN upon the various members of a machine has been made, the yawing moment about the C.G. can be determined, it being the sum of the various lateral forces multiplied by their respective distances from the C.G. This calculation will en- able the f minimum ' size of fin and rudder area required to give directional stability to be estimated. The maximum size of fin is fixed by considerations relating to spiral instability, as was indicated in Chapter X. The condition for spiral instability there stated was that the numerical value of the ratio of the derivatives L V /N V should be greater than the ratio of the derivatives L r /N r . The limiting condition for stability will therefore be reached when = L r N r and this will give a value for the maximum fin permissible. The value of L v is dependent upon the dihedral angle of the machine, and hence this forms the readiest means of securing spiral stability, as in most cases it is simpler to assume a size of fin by reference to successful machines of a similar size. The dihedral angle can then be varied, if necessary, to give the required degree of stability. The determination of the above derivatives necessitates model experiments for the best results, though in the absence of such data they may be calculated to a fair degree of approximation in the manner shown in Chapter X. Ailerons or Wing Flaps. Control about the axis of roll is provided generally by means of hinged ailerons or flaps. In the early stages of aviation the method of warping the wings was adopted, but this practice has now been almost abandoned. When dealing with the elevator, it was stated that the effect of rotating the elevator with reference to the tail plane was to increase or decrease the lift of the combined surface. In the case of the aileron a continuous high lift/drag ratio is of prime importance. The angle of rotation must therefore be small and the flaps consequently large. In actual practice we find a wide range of aileron areas according to the width and the amount of rotation employed. The moment required depends upon the span, the area of the wings, and the transverse moment of inertia of the whole machine. The area itself also depends upon the speed and the speed range. In designing the ailerons, attention must be paid to the yawing moments induced by using the flaps. We have previously seen that a reflexed trailing edge diminishes the resistance of a wing. If the flaps are interconnected, therefore DESIGN OF THE CONTROL SURFACES 373 the resistance of one wing will be diminished while that of the other is increased, thus necessitating simultaneous use of the rudder. When the rudder is used to produce a turning move- ment, banking is necessary to prevent side-slip. If the machine be banked by use of ailerons or warping of the wings, the upper wing-tip will have the flap pulled down and so will have an increased resistance ; while the reverse is the case for the lower wing-tip. But the upper wing-tip is required to travel faster 3 -4 -5 -6 Lift Coefficient (Absolute) FIG. 269. Comparison of Lift Drag Rated for Aerofoil with and without Flaps. than the lower one, and thus the rolling control opposes the rudder action. In an extreme case the size of the rudder might have to be increased to take account of this. Experiments relating to an aerofoil with a hinged flap are described in the 1913-1914 Report of the Advisory Committee for Aeronautics. Although these experiments were not made in connection with the question of aileron surface, they are of considerable interest in this respect, and therefore the most important results are here given. The section of the aerofoil used in these experiments was similar in form and aerodynamic 374 AEROPLANE DESIGN characteristics to that shown in Wing Section No. 6. The dimensions were 18" by 3", and the flap extended over the whole length of the rear part of the section, being 1*155" wide (385 chord). The gap in the aerofoil due to the hinges was rilled with plasticine, as it was found .that the drag was greater if this was not done. Angles of pitch were in all cases measured from the chord of the original aerofoil, so that when the flap angle was zero the section corresponded exactly with that shown in Fig. 72. Fig. 269 represents the value of the L/D ratio plotted against lift coefficient for the most efficient equivalent aerofoil that is, the combination of angle of incidence of chord with FIG. 270. Movement of C.P. for Aerofoil with Hinged Flap compared with Original Aerofoil. angle of flap to give maximum efficiency as compared with the original section. The angle of pitch of the wing and the relative pitch for the flap are marked on the figure, and the corresponding curve with zero angle of pitch of flap are shown on the same figure. In the neighbourhood of points of greatest efficiency it will be noticed that an alteration of flap angle gives a better lift without alteration of the pitch angle of the wing itself, while maximum lift is obtained with the flap at a large angle of pitch and the incidence of the wing itself smaller than at the corresponding point for the original section. The centres of pressure for these most efficient combinations are shown in Fig. 270. For values of lift coefficient between '13 and '3 the C.P. moves backward with increase of lift coefficient. The equivalent aerofoil is therefore stable over the most important DESIGN OF THE CONTROL SURFACES 375 portion of the range of lift coefficient used at ordinary flying speed . From these experiments the following important conclusions as to the advantages of using a wing with a hinged rear portion can be deduced : 1. That the increased maximum lift coefficient obtained permits of a considerable reduction in the landing speed. 2. That by adjusting the flap to correspond with the minimum drag for the particular angle of incidence at which the machine travels over the ground, the distance necessary to take off may be reduced by 13%. 3. The distance required to pull up after having landed can be diminished by setting the flaps at a large angle of pitch. 4. The maximum flying speed can be increased by giving to the flaps a small negative angle at the corresponding angle of flight for maximum speed. 5. A slight gain in climbing speed can be obtained by the use of flaps. The mechanical dfficulties involved in securing a hinged wing section are considerable, but it is apparent from these experiments that a considerable gain would accrue in aero- dynamical efficiency. Balance of the Control Surfaces. When referring to the elevator it was stated that, in order to reduce the strain upon the pilot when working the controls, the method of balancing the control surface was desirable. Similar considerations make it advisable to balance both the rudder and the ailerons upon large machines i.e., machines of greater weight than 5000 Ibs. This may be effected in two ways (see Fig. 272) : (i.) By adding an extension which projects in front of the main portion of the surface, (ii.) By placing the hinge about which the control rotates some distance behind the leading edge. In both these methods the air forces upon the surface in front of the hinge produce about the hinge a moment of opposite sign to that of the main control surface, and therefore reduce the effort required to move it by twice this amount. The first method has been most frequently employed in practice in this country, but the second method possesses many more advantages, and has been used on the latest Handley-Page AEROPLANE DESIGN machine (V-I5OO). In order to decide upon the area of the extension required, it is necessary to determine the moment of the main surface about the hinge for various angles of deflection and then to select an extension which will produce the necessary degree of balance over this range of angles. For most accurate results a model of the wing and flap with extension should be made and tested in the wind tunnel, the size of the extension being altered until satisfactory. If such a procedure is not possible, the size of the extension must be deduced from whatever results are available. The Angle FIG. 271. Moments about Hinge of Wing Flap. following example indicates a rough method of arriving at the necessary dimensions of an extension to a wing flap for balancing purposes. Fig. 271 gives the moments about the hinge for the wing flap described on page 374. The increase of moment per degree of deflection of flap at o pitch of wing = '0004 ft. Ibs. The section of the extension used will be that of the Tail Plane No. 3, whose dK y /d8 "033, and the average position of whose C.P. will be taken at 0*23 chord from the leading edge. Then for balance the moment of the extension about the hinge must equal '0004 ft. Ibs. per degree that is, the load on the extension x 2 x c = '0004 ft. Ibs. DESIGN OF THE CONTROL SURFACES 377 The C.P. coefficient is -23 ; therefore the distance from hinge at which the resultant force upon the extension may be assumed to act is = (i - -23) a - 1-155" = 77 a - 0*0963 feet whence 2 x -033 x '00237 x 40* x a x (7 7 a - 0*0963) = '0004 Taking an approximate value for b of 1-5, we have 0-3130(770 - '9 6 3) = * 00 4 or 77 a 2 - 0*0963 a = 0*00128 Solving this quadratic a = '206 feet = 2*47" so that the dimensions of extension necessary to balance the flap upon the assumptions made = 1.5" x 2*47". In practice 111 m i A 1 1 i ^ FIG. 272. Balanced Ailerons. the dimensions 1-5" x 2'$" would be adopted. Such an exten- sion increased to the same scale as the wing and flap would give the required balance upon a full-size machine. The above calculation is given to indicate the method to be used for calculating the size of a balancing extension. It is obvious that the assumptions made cannot be strictly justified, since the position of the C.P. of the extension varies with the angle of incidence. Moreover, end-effect will be important at the wing-tips. Nevertheless, such a calculation is extremely useful in the initial stages of a new design as affording an indication of the probable size required. This can be altered subsequently, if experience shows it to be necessary after a trial flight has been made. For securing balance by the second method, recourse must also be made to wind-channel experiments. An examination of the pressure-distribution diagrams over the rear portion of a wing section will be of much assistance in this direction. In 378 AEROPLANE DESIGN general the shape is found to be approximately triangular, as will be seen by reference to Fig. 48. For a triangle the C.P. will be at one-third the altitude from the base line, hence it can be inferred that the position of the hinge for full balance in this method should be at one-third the chord of flap from its leading edge. In order to avoid the possibility of the flap being over- balanced, and hence tending to increase its deflection relative to the remainder of the section, it would be advisable to place the hinge at about 0*3 times the flap chord from the leading edge. These methods of balancing are applicable to each of the control surfaces. Construction of Control Surfaces. The controlling sur- faces are built up much as portions of wings. A strong leading edge serves to take the hinges, and forms as it were a ' back- FIG. 273. Dual Control System. bone ' from which may radiate ribs of light construction to the trailing edge. The~ trailing edge, in the case of the wing-flaps, will be similar to the rigid part of the wings. In the case of the elevators, and more especially in the case of the rudder, thin tubing of about 10 mm. diameter will be found useful. Short levers of streamlined section are fixed to the backbone or spars, and connected by wires to the control lever in the pilot's cockpit. The wires, or rather thin cables, may be guided round corners by pulleys or bent copper tubing. They should be covered as much as possible, so as to diminish head resistance, and should therefore be placed inside the wings, inspection doors being provided where necessary in the wings ; and for the DESIGN OF THE CONTROL SURFACES 379 TAI LPLANE1 & ELEVATORS FIN FIG. 274. Tail Unit Components. (See also p. 412.) IS Cacfc* Cockpit- FIG. 275. Variable Incidence Gear for Tail Plane. 380 AEROPLANE DESIGN tail unit should enter the fuselage as soon as convenient. The success with which the designer balances his controlling surfaces is shown by the consequent lightening of the control lever, and more especially is this the case with large machines. The design of the control is a simple matter when the loads spread over the control surfaces have been assessed for the highest speeds and the greatest angles, and their moments about their hinges obtained by locating the c.p. The pulls in the connecting wires follow at once, and the problem of the Reproduced by courtesy of ' Flight.'' FIG. 276. Double Tail Unit for Large Machine. control lever reduces to the question of a short cantilever pro- jecting vertically from a horizontal pivot tube. If the elevators are attached to levers at the ends of this tube that is, near the sides of the fuselage the tube must be designed for combined torsion and bending. The fuselage should be strengthened where it takes the strain of the control lever and rudder bar. The rudder bar, arranged for the feet, may be conveniently made of wood with metal fittings. The vertical pivot should be made adjustable along the fuselage, with lengthened strips for the rudder cables, so as to be available, without any inconvenience, for pilots of different sizes. CHAPTER XII. PERFORMANCE. Definition. The factors upon which the performance of a machine depends that is, the maximum speed which it is able to attain at varying altitudes and the time it takes to reach such altitudes were considered at some length in Chapter I., and those pages should be again consulted at this stage. In this chapter the relation between these various factors will be investi- gated in their effect upon the increase or decrease of the efficiency of the aeroplane, and some practical methods of actually determining the performance of machines in flight will be given. The primary object of the aeroplane is to transport a certain useful load from one point to another, using the air as a medium of travel. It must therefore be provided with a surface capable of developing the necessary reaction to overcome the force of gravity, while offering at the same time a minimum resistance to forward motion through the air. The development of the modern wing section, as outlined in Chapter III., had for its aim the accomplishment of this purpose. The area of the sup- porting surfaces that is, the wings must obviously be such that the -reaction upon them is equal to the weight of the machine. This reaction, as we have already seen, is termed the lift of the wings. Such lift is always accompanied by a resistance or drag at right angles to it. The drag of the complete machine is made up of two parts namely, the wing resistance and the body resistance. It is the aim of the designer to obtain a wing section in which the ratio of L/D is a maximum for the speed at which he proposes the machine under consideration should fly. The efficiency of the wing surfaces is largely influenced by ' end effect,' and in order to obtain a highly efficient wing the aspect ratio must be high. Increase of aspect ratio leads to a corresponding increase in the weight and constructional diffi- culties, so that apparently a definite limit for greatest efficiency is soon reached. The maximum aspect ratio adopted to-day is in the neighbourhood of 10. All resistances other than that of the wings are grouped 382 AEROPLANE DESIGN together under the term ' body resistance.' This resistance varies approximately as the square of the speed, but a factor which leads to considerable uncertainty in this direction is the slip stream from the airscrew. In the case of the tractor machine the body moves directly in this slip stream, and its resistance is thereby increased relative to the remainder of the machine. Moreover, interference between the various com- ponents leads to a further modification, which it is very difficult to estimate. The body resistance becomes of increasing relative importance as the speed of flight becomes greater and greater. Examination of the resistance curves for the wings shown in Fig. 278 indicates that the wing resistance remains fairly constant over a considerable range of speeds, whereas the body resistance increases as the square of the speed. The minimum total resistance will occur when the wing and body resistance are approximately equal in amount, and the speed corresponding to this will be the most economical for flight. Turning next to a consideration of the Engine power, the first point to be observed is that the thrust exerted by the engine must be sufficient to propel the wing surfaces through the air at such a speed that their reaction overcomes the weight of the machine. The forces tending to prevent the attainment of such speed are the wing and body resistances referred to above, hence for horizontal flight at a particular speed it is necessary that the engine thrust must be at least equal to the drag of the machine. An estimate of the horse-power required necessitates, therefore, a knowledge of the resistance to be overcome at the various speeds of flight. It is also essential that a reserve of power is available in order that the- machine may climb. The excess of power supplied over that required for horizontal flight is the horse-power available for climbing, and this being known a simple calculation will enable the rate of climb to be determined. The maximum rate of climb will correspond with that speed at which the excess power is greatest. The prediction of performance necessitates a knowledge of the resistance of the machine at various flight speeds, together with the excess horse-power available at that speed. The following paragraphs show one method of determining these quantities for a machine of new design. The Resistance of the Machine. (a) Wing Resistance. The wing resistance is deduced directly from the aerodynamic characteristics of the section employed. It is first necessary to determine the lift coefficient corresponding to various speeds of flight. For this purpose the fundamental equation (Formula PERFORMANCE 383 13) is used. This gives K y for varying values of V. From the wing-section characteristics the value of the drag coefficient corresponding to each K y is obtained, which gives the corre- sponding flight speed. The drag coefficient and the speed of flight being known, the wing resistance follows from the funda- mental equation (Formula 14). These values should be plotted on a speed base. (b) Body Resistance. The estimation of the body resistance is a matter of some difficulty, and recourse must be had to wind- tunnel data wherever available. It is advisable to make a point of collecting resistance data whenever an opportunity occurs, particularly the resistances of a complete machine, such as that, for example, given in Chapter VI. for the B.E. biplane. The figures given in Chapter I. will also be useful in this connection, and will serve as a base upon which to make comparisons. The accuracy of the entire performance calculation is directly dependent upon this estimate of body resistance, and hence it is essential that it should be carried out as carefully as possible, due allowance being made for those parts in the airscrew slip stream. If the machine has already been built, it is possible to deter- mine the body resistance by noting the gliding angle. If this be 9 we then have the relationship tan 9 = D/L where D represents the total drag of the machine, and L the lift of the wings. From the total drag the resistance of the wings is subtracted, and the remainder will be the body resistance. This is in most cases the more accurate method, it being usually found that the estimated ' body resistance ' is higher than the observed body resistance. The body resistance (R) being known for any speed v, then at any other speed (V) the resistance is obtained from the equation R V'' Resistance at speed V = -!- v* Add the body resistance at any speed to that of the wings for the same speed, and the total resistance of the machine for that speed is obtained. A curve of total resistance against speed can then be drawn. In determining the wing resistance of a machine, allowance must be made for the following factors : 1. Effect of slip stream of airscrew. 2. Interference effects. 3. The stagger and aspect ratio of the wings. 384 AEROPLANE DESIGN Horse-power required. (a) To overcome wing resistance : H.P required = Resistance x Velocity 55o - K x ( P /*-)AV* x V/550 550 Knowing K x corresponding to each flight speed V, the wing H.P. can be calculated. (If) To overcome body resistance : similarly, H.P. required =?^. v* 55 RV 3 From this equation the ' body ' H.P. can be determined for several values of V. It will be found most convenient in calcu- lating the total horse-power required to use the tabular method of setting out the work. Horse-power available. As stated in Chapter I., this depends both upon the engine and the airscrew. The efficiency of each of these units depends largely upon the conditions under which it is working, and data respecting the variation of engine-power, with altitude and revolutions per minute, should be obtained from tests carried out independently or by the manufacturer. It is found that the brake horse-power of an engine varies almost directly as the density of the air, and is practically independent of its temperature. The efficiency of the airscrew varies with its forward speed, and may be calculated with a considerable degree of accuracy in the manner set forth in Chapter IX. The horse-power required to drive the airscrew at various speeds can be determined from its torque curve. If Q be this torque, then the horse-power required to drive the airscrew at n revolutions per second _ 2 TT n Q 55 This horse-power should then be plotted on the base of ' n ' and . superposed on the B.H.P. curve of the engine similarly plotted on a base of ' n.' The intersection of these curves gives the maximum horse-power available at that translational speed of the airscrew. The efficiency of the airscrew being known, it is PERFORMANCE 385 then an easy matter to determine the maximum available horse- power to drive the machine. On plotting (i) total H.P. required to overcome resistance of machine at ground level ; (2) power available at ground level, on a speed base, it can be seen from the curve what power is available for climbing, and also the fastest climbing speed. Similarly, by calculating the above quantity for the density corresponding to the maximum height it is desired to fly at, it will be seen what is the highest speed attainable at this height. Rate of Climb. The difference between the ordinates of the H.P. required and the H.P. available curves shows the H.P. available for climbing at each speed. The maximum difference will give the maximum rate of climb to be expected, which can be calculated thus : Let P represent the horse-power available for climbing. Then rate of climb in feet per minute = r = P x 33ooo/weight of machine The points at which the H.P. curves intersect determine the maximum and minimum flying speeds respectively, and thus a calculated estimate of the performance of the machine is obtained. The nearness with which these results approach those experimentally obtained will be a measure of the accuracy of the estimated resistances and other assumptions. Performance Calculations. An example of the method of predicting the performance of an aeroplane, such as should be carried out when designing a new type of machine, will now be given. For this purpose the particulars of the machine referred to in Chapter V. will be used. It is required to deter- mine the maximum speed and rate of climb of this machine at ground level when fitted with a 150 H.P. engine and an airscrew possessing an efficiency of 80%. The estimated resistance of the machine less wings that is, its body resistance is equal to I5olbs. at 100 feet per second. The characteristics of the wing .section are as follows : CHARACTERISTICS OF THE WING SECTION. o 2 4 6 8 10 12 14 K y ... '097 -192 -273 -347 -421 -492 '555 '59 K x ... '0167 '0165 '0199 '0261 '0355 '0452 '0551 '0742 c c 386 AEROPLANE DESIGN From these characteristics the wing resistance at varying speeds is calculated in the following manner. From formula w , . . Substituting f> A'K 2000 2-100 - - - = -2 00237 X 366 X Ky Ky Notice that the effective supporting wing area A' is used in this formula. From this relationship the velocity of flight corresponding to varying values of K y , and consequently of the angle of incidence, can be calculated, whence the determination of the wing resistance is obtained by using the formula R w = K x P A V 2 (VV 2 + v L = constant W 2g Let P be the so-called dynamic pressure transmitted by the inner tube. At the mouth of this tube the velocity is reduced to zero, hence W W 2 g Or P -/ = (V) 2 W 2g The quantity (P p)jw is the difference in pressure measured by the manometer. Denoting this by H, the above expression may be written (V)2 = 2 -H In order to allow for errors in construction, the law of the Pitot tube is generally written (V) 2 = K . 2 ^ H ............ Formula 99 where K is a correction factor to be determined by calibration of the actual tube. It is generally equal to unity, 392 AEROPLANE DESIGN The advantages of the Pitot tube over other instruments are : 1. Absence of any kind of friction, as there is practically no displacement of air along the the tube. 2. Comparative ease of reproducing exactly similar tubes, thus obviating the necessity of individual calibration. 3. Very small wind resistance. If the potential pressure head in Formula 99 is read on a gauge containing a liquid of density ' dl while the density of the air current is p, the above equation takes the form V 2 = K.2-, c 0) 1 Correction I 59'I 31-0 i6i'5 Sg-2 5100 31-0 0-879 80 83-6 3-6 o 12.3-4 28-6 5*5 937 5100 31-0 0-879 8< / 8 2-8 3 62*0 32-3 168-5 93- 5050 31-0 0-881 8S 88-1 3'i 4 1247 32-3 2I'0 95 6 5000 31-0 0-882 86 8* !-8 2-8 Mean = 3*1 TABLE LXI. AIRSPEED AT HEIGHTS. j. C i e-f .?! JO o ^ '53 r; iti, r-} 1> fit *-* c/J rt 1 11 is. 55 -S c -SB g V-i C ^ S2 O -^ o g ^ 'ex < o SL Q p-s O'S h & g 3000 39 '935 2900 95 98 ioi'5 1280 3000 103-0 1290 5000 35 875 4900 93 96 1. 02 '5 1280 6500 100-5 1250 7000 30 -821 6900 88 9 1 100-5 1240 9200 24 -767 9000 81 84 96*0 1220 10000 9 6 '5 I2I 5 10800 I 9 '73 1 10400 80 83 97-0 1220 12800 17 682 12600 72 75 92*0 1200 13000 94-0 1180 13800 12 '664 13400 68 72 88-5 1180 ! 1 15000 86-0 1 1 60 15200 8 -636 14800 64 69 86-5 1 1 60 i i D I) 4O2 AEROPLANE DESIGN TABLE LXII. RATE OF CLIMB TEST. Height in feet aneroid. Observed tem- perature (F.). % Standard density. Observed time (minutes). Ratt of climb feet per minute. 1000 37 lOI'O I '00 814 2000 38 97-2 2'10 7 l8 3000 36 94.2 370 622 4000 36 907 5 '40 544 5000 36 87-4 7-25 495 6000 33 847 9'4 435 7000 3 82-1 11-9 389 8000 26 79*9 14*25 347 9000 26 77-6 17*00 312 IOOOO 23 747 20-25 294 1 1 000 21 72-2 23-6 264 12000 20 69-8 27-4 216 13000 1 7 677 3 r 9 182 14000 12 65*9 37'9 139 15000 8 64-1 45' 2 5 IOI CHAPTER XIII. GENERAL LAY-OUT OF MACHINES. The process of laying out an aeroplane varies consider- ably in different drawing offices, some of the methods adopted being excellent examples of correct procedure, while in many other cases 'rule-of-thumb' ideas prevail. As a painful illus- tration of the latter method, or perhaps more correctly lack of method, we outline how a machine is designed in an aeronautical drawing office with which we are acquainted. The designer decides upon a new machine, and guesses at the complete weight. The area of the wings is then obtained by dividing this weight by seven, probably in order to give an approximate loading of seven pounds per square foot. A draughtsman is then summoned, and instructed to get out a side elevation, and sometimes a plan, being told the type of engine, the number of passengers, the quantity of petrol to be canied, the stagger and the chord. The draughtsman is then compelled to arrange his weights by a process of wangling, and usually puts in the size of the fin, rudder, and tail plane by ' eye/ This drawing is then passed on to the fuselage expert, who runs out a drawing of this unit, while the first draughtsman stresses the wings and settles the section, spars, and other details. The whole staff is then employed on preparing detail drawings for the shops, and on completion the building of the first machine is commenced. At this stage the designer has a walk round, and promptly proceeds to alter the majority of the details, fresh drawings are prepared, and the process is repeated. After several alterations the machine approaches completion, and then possibly a general arrangement is prepared. In the hands of a genius such a manner of lay-out may produce excellent results, but it is neither scientific nor up-to-date. In order to produce an economical, successful, and scientific design, requiring very few alterations while in course of manu- facture in the shops, careful attention mu>t be paid to the points outlined in the preceding chapters, and we will now show how to apply these chapters to the design of machines. For the purpose of illustration we will consider the des ; gn of a biplane having a speed of 120 miles per hour at io,oco feet, a landing speed of 50 miles per hour, a radius of action of 480 miles, and capable of carrying at least 500 Ibs. of useful load, excluding fuel. Referring to Table XL., we see that the useful load repre- AEROPLANE DESIGN sents approximately 14% of the total weight of the machine, so that we have Total weight = x 500 = 3570 Ibs. For the present this weight will be taken as 3500 Ibs., and at once from Table XL. we get the following estimate of the weights of the various components : I. THE POWER PLANT. (a) Engine (b) Kadiators (c) Cooling water ... (d) Tanks, c etc (e) Airscrew, etc. ... II. THE GLIDER. (a) The wings (b} The wing bracing (fj The tail unit ... (own wires Outer F and R 1 30 x 0*09 x 4 0*325 1-055 7*75 8-17 Inner F and R 112x0*11x4 0*342 1*110 7*25 8*05 /ing struts 36 X I X 2 0-50 2*50 3'5 875 iileron levers ... 8x|x 4 0-166 0-415 1 1*0 4*56 TOTAL 25*72 116 23 Centre of Resistance of Components out of Slip Stream = IL 2 ^ = 4*52' from datum. E E 4 i8 AEROPLANE DESIGN TABLE LXVI. BODY RESISTANCE. (!N SLIP STREAM.) i > , 8 |.a 1 ITEM. Size. Area. g~ RxH REMARKS. Sq. ft. .2 ctf "o as 5 *~ 06 O Ii Inches. Wheels Q Axle 54x2 075 1-125 1-25 1-41 Front struts (chassis) 30 X I'25 X2 0-52I 0-680 2-75 1-87 Rear struts 3O X I '25 X 2 0*521 0-680 2-75 l-8 7 Bracing lift Wires front ... 42XO'I4X2 6-082 0-266 7-25 i'93 Bracing lift Wires rear Body Tail plane Tail-plane levers 42XO'II X2 II4X3 0-064 2-38 0-I67 0-208 55 lbs.* 2-38 0-418 7-0 6-25 5'75 5'75 1-46 344-00 13-70 2-40 * Deduced from expts, relating to Aeroplane bodies', Chap. VI. Tail skid 18x2 0-25 1-480! 3'5 5-19 Rudder and fin ... 36X3 075 0-750 6-5 4-88 Central wing Struts front ... 36x n x 2 0-55 0-715 8-75 6-25 Central wing Struts rear 38 x i -o x 2 0-50 0-65 8-50 5'53 Central wing bracing 54 x -09 x 2 0-067 0-216 875 1-91 TOTAL ... 64-57 392-40 Centre of Resistance of Components in Slip Stream = = 6*08' from datum. 64-57 In order to arrive at a correct estimate of the resistance it is necessary to take into account the variation of the resistance due to the slip stream of the airscrew. For this purpose the curve given in Fig. 174 may be used, together with Formula 70. The following particulars relating to the airscrew it is proposed to use are also necessary for the evaluation of this formula : Diameter 10 feet, experimental mean pitch lO'i feet, number of blades 2, revolutions per minute 1650, k = 4'6 X io~ 7 Sub- stituting in Formula 70, Tractive Power 4-6 x 10 - ' ' V '["0! *,,)'] si X 10' 60 960 I - GENERAL LAY-OUT OF MACHINES 419 From this relationship the values given in Table LXVII. can be calculated, and the slip-stream coefficient determined from Fig. 174. TABLE LXVII. CALCULATION OF SLIP-STREAM COEFFICIENT. V ft. /sec. ... 60 80 100 120 140 160 180 V 2 ... ... 3600 6400 10000 14400 19600 25600 32400 V 2 /772oo ... -047 -083 -130 -187 -254 -332 -420 i - (V 2 /772oo)... -953 -917 -870 -813 746 -668 -580 P ... 915 880 835 780 715 640 556 Tractive power /IT ^ 9' X 5 8 - 8 8 '35 7' 8 7' r 5 6 '4 T56 (Airscrew diameter) 3 Slip-stream coeffct. 274 2*45 2*16 r88 1-65 1-5 1-37 The resistance of all components affected by the slip stream from the airscrew must be multiplied by these factors. From Table LXVI. it is seen that the estimated resistance of the components affected by the slip stream is 64*57 Ibs. at 100 feet per second. The resistance at other speeds = -4_57_ x x s iip- s tream coefficient io 4 and the resistance of the components out of the slip stream 2572 x V 2 io 4 From these two equations the body resistance of the machine at various speeds can be calculated, as shown in Table LVIII. TABLE LVIII. BODY RESISTANCE OF MACHINE AT DIFFERENT SPEEDS. V ft. /sec ... ... ... 60 80 ioo 120 140 160 180 Components in slip-stream 64 101 139 175 209 248 286 Components outside slip- stream ... ... ... 9*2 16*5 257 37 50 66 83 Total resistance R B ... 73 117*5 l ^5 212 259 314 369 Wing|Resistance. From the fundamental equation W = K y A'V 2 8 4 20 AEROPLANE DESIGN the necessary lift at any speed can be obtained by putting into the form W P A ' V 2 g 350 00237 x 500 x V 2 2960 - V 2" and when K y has been determined, the corresponding K x can be read directly from the curve of aerodynamic characteristics. Knowing K x , the drag of the wings R w can be determined at each speed, and by adding this result to the resistance of the body the total resistance of the machine at ground level can be determined, as shown 'in Table LXIX. [TABLE LXIX. CALCULATION OF TOTAL RESISTANCE. V ... ... 70 80 100 120 140 160 180 K y ... ... '605 '461 '296 '205 'J5 1 ' TI 5 ' '091 K x ... ... "051 '0294 '0137 '0127 '0118 '0118 "0127 (54o)K x V 2 = R w 320 241 175 234 296 387 527 g R B + R w 393 35 8 340 446 555 7i 896 Horse-power. The horse-power required at the various speeds is obtained by the use of the formula . , Resistance x Velocity Horse-power required = 55 and the variation in engine power will be assumed to follow the law of the curve shown in Fig. 277. The maximum efficiency of the airscrew will be taken as 80% at a forward speed of 1 20 miles per hour. The rate of climb in feet per minute Horse-power available x 33000 Weight ot machine Table LXX. can now be prepared. GENERAL LAY-OUT OF MACHINES 421 TABLE LXX. HORSE-POWER REQUIRED AND AVAILABLE, AND RATE OF CLIMB. V (ft. per second) ... 70 80 100 120 140 160 180 R V - = H.P. required ... 50 52 62 97 141 204 293 55 V/V max . ......... -398 -455 -569 -682 795 -908 ro. Power factor ...... -56 '64 76 '86 '93 '98 ro 320 x -8 x F p ...... 143 164 195 220 238 251 256 H.P. available ...... 93 112 133 123 97 47 o Rate of climb (ft. per min.) 876 1055 1250 1160 914 443 o The effect of the variation of the slip stream will be to alter the position of the centre of resistance in a vertical plane, and it is therefore necessary to determine its position both at top speed and at slow speed. These positions are obtained by combining the information given in Tables LXV., LXVI. Resistance of components out of the slip stream at a speed of 80 f.p.s. = 2572 x '64 = i6'5 Ibs. at a speed of 160 f.p s. = 2572 x 2*56 = 66*0 Ibs. acting at a distance of 4-52 feet from datum line. Resistance of components in slip stream at a speed of 80 f.p.s. = 64*57 x '64 x 2^45 = ioi 2 Ibs. at a speed of 160 f.p.s. = 64-57 x 2-56 x 1-5 = 248 Ibs. acting at a distance of 6 - o8 feet from datum line. Resistance of wings at a speed of 80 f.p.s. = 240 Ibs. at a speed of 160 f.p.s. = 400 Ibs. acting at 675 feet from datum line. Taking moments about datum line, For a speed of 80 feet per second H' - 74-5 + 615 + 1620 = 6 . 4 357 For a speed of 160 feet per second , _ 298_+^o8 + 2700 = 714 so that there is a variation in the vertical position of the centre of resistance of o % i6 feet or 1*92 inches over the speed range. Knowledge of the position of the centre of resistance enables the final balancing up of the machine to be obtained and the direction of the tail-loading determined. In this case the line 4 22 AEROPLANE DESIGN of pull of the airscrew acts at a distance of 6*25 feet from the datum, so that the resulting thrust-resistance couple will be very small, and a small up load on the tail will correct for this effect. The tail-setting for various flight speeds is next calculated in the manner shown in Chapter XI. The various performance curves for the machine are shown in Fig. 288, from which it will be seen that the estimated flight speed is 117 miles per hour, as against 120 miles per hour required by the design. It will be noted, however, that in the calculation of the resistance and the available horse-power, no allowance has been made for the variation due to the change in VetoaVy ( .pa) FIG. 288. Performance Curves. the density of the atmosphere. As pointed out in Chapter I., these two items will have a neutralising effect upon each other, and as the resistance has, if anything, been over-estimated, it is extremely probable that the desired performance of 120 miles per hour at 10,000 feet will be achieved upon the trial flights of this machine. In order to reduce the labour of design work to a minimum, it is very desirable that a careful record should be kept of all machines designed and built. For this purpose some such table as that shown in Fig. 289 should be prepared and rilled in as the various particulars become available. Data relating to some of the most successful machines yet built is given in Chapter XIV., which will form a nucleus upon which the embryo designer can build his own designs. ENGINE : Type B.H.P. No. fitted Airscrew r.p.m. Kngine r.p.m. Fuel p. B.H.P. Oil p. B.H.P. Range Speed Ground 10000ft. 15000ft. Landing Climb to 5000ft. to 10000 ft. to 15000 ft. to 20000 ft. Ceiling R.P.M. Total H.P. AEROPLANE Type No. of wings Span Chord Top wing Second middle Third Bottom Incid'ce Dihedral Gap Stagger mis. hrs. Time Rate Overall lenjrth Height Gap Chord Distance from L.E. of lower wing to elevator hinge Stability Longitudinal Lateral STRUCTURAL UNIT AREAS WINGS Top plane Second plane... Middle plane Third plane Bottom plane Ailerons Struts (No. = ) External bracing wires WEIGHT Ibs. Wt. Sq. ft. %wt. TOTAL WINGS ... CONTROLS Tail plane Elevator Fin Rudder TOTAL CONTROLS ... 1 Fusels Chassi Tail si Contr< ige dd )ls TOTAL BODY ... TOTAL WEIGHT OF STRUCTURAL UNIT r ' POWER UNIT ENGINE Engin Airscr Radia Engin 3 dry ew tor, piping, and water 3 accessories POWER UNIT ... W p Fuel tanks and piping Oil tanks and piping . Fuel Oil TOTAL WEIGHT OF POWER UNIT USEFUL LOAD Crew ... Passengers . Instruments W.T. do. Luggage Cargo ... Spares Sundries . ... TOTAL WEIGHT OF LOAD UNIT TOTAL WEIGHT OF MACHINE FIG. 289. CHAPTER XIV. THE GENERAL TREND OF AEROPLANE DESIGN. * Soon shall thy arm, unconquered steam, afar Drag the slow barge, or draw the rapid car ; Or on wide waving wings expanded bear The flying chariot through the field of air.' The Botanic Garden, by Erasmus Darwin, published 1791. IT seems hardly credible, when one surveys the present science of aeronautics, that it was only in 1903 that the Wright Brothers were making their first experiments in aerodynamics and their flights with gliders at Kitty Hawk. A brief resume of the leading facts of aeronautical history is of more than ordinary interest. In 1848 a small model was made by Stringfellow which flew for about forty yards under its own steam, but it was not really until the late nineties of the last century that serious attention was devoted to the problem of the 'heavier than air' flying machine. At that time the two most prominent investigators in this new field of science were Langley in America and Hiram Maxim in England. Langley's machine had a wing- surface area of 70 square feet, a steam engine of one horse- power weighing 7 Ibs., the whole arrangement weighing 30 Ibs, It was designed to carry no passenger, and flew under its own steam-power upon two occasions in 1896; the lengths of the flights being respectively one-half and three-quarters of a mile. Maxim's machine of the same date was much more ambitious in conception. The wing-surface area was 4000 square feet, the steam engine was of 360 horse-power, weighing 1200 pounds, and the whole machine weighed 8000 pounds. It was designed to carry three passengers, and on its trial was anchored down to rails to prevent actual flying. The check rail, however, was torn away and the machine wrecked on its trial. In 1897 Ader constructed an aeroplane weighing complete iioo pounds, the power unit being a steam engine of 40 horse-power and weighing nearly 300 pounds. This engine was capable of pulling the machine along the ground for short distances, but no flight was accomplished. Meanwhile Langley was still experimenting in America, and produced in 1903 a full-size aeroplane as the result of his researches. With his power-driven models the FIG. 297. The Bristol Monoplane. FIG. 298. Bristol Fighter fitted with Wireless. To foUo w page 424^ GENERAL TREND OF AEROPLANE DESIGN 425 method of launching from the top of a house-boat had been adopted with successful results, but when applied to the full- scale machine this plan proved a failure, and Langley abandoned his efforts in this direction. Then on September i/th, 1903, the Wright Brothers, after many years spent in experiments, succeeded in flying a power- driven machine as stated in Chapter I., the machine weighing- 750 pounds and being equipped with a 16 horse-power petrol motor. This first flight lasted but twelve seconds. Rapid progress was now made, and in 1908 Wilbur Wright made his sensational flights in France, and although he was at first treated as a * bluffer,' a flight lasting for over ninety minutes at Le Mans in the September of that year dispelled all doubts about actual flight. Since that date the main air marks to record are the crossing of the English Channel by Bleriot on a monoplane in July, 1909 ; the great development and expansion of aeronautics, owing to the War, from 1914-1918; the flight of a -1500 Handley-Page from Ipswich to Karachi (India) by stages from December I3th, 1918,10 January i6th, 1919; the unsuccessful attempt of Hawker and Grieve to cross the Atlantic on a Sopwith machine on May I9th, 1919; the crossing of the Atlantic on June I4th-i5th, 1919, by Alcock and Whitten- Brown in a Vickers-Vimy-Rolls from St. John's, Newfound- land, to Clifden, Ireland, a distance of about 1900 miles, in 16 hours; the flight of Captain Ross-Smith and three com- panions from Hounslow (England) to Port Darwin in Australia in a Vickers-Vimy-Rolls, a distance of 11,300 miles, between November I2th and December loth, 1919. Civil aviation opened officially in England on May 1st, 1919. Table LXXI. shows the results obtained by private enterprise during the six months ending October 3ist, 1919. TABLE LXXI. PROGRESS OF CIVIL AVIATION IN ENGLAND, MAY I5TH TO OCTOBER 3isT, 1919. Number of hours flown ... ... ... ... 4,000 Number of flights ... ... ... ... ... 21,000 Number of passengers ... ... ... ... 52,000 Approximate mileage ... ... ... ... 303,000 Total number of accidents ... ... ... ... 13 Number of fatal accidents 2 It will thus be seen that more than 25,000 passengers were carried for every one fatally injured, so that flying can be re- garded to be quite as safe as any other form of locomotion, while offering the advantage of much greater speed. 426 AEROPLANE DESIGN The Bleriot Machine. The machine used by Bleriot in his cross-Channel journey was known as a No. XI. type mono- plane. The fuselage was of open wooden framework braced by steel wires throughout. The two halves of the main plane were set at a slight dihedral. The span was 28*5 feet, the chord 6'5 feet, and the total surface area 151 square feet. The tail plane consisted of a fixed plane at the rear of the fuselage of area 17 square feet. The elevators were placed on each side of this fixed tail plane, their total area being 16 square feet. The rudder was rectangular in form, fixed beyond the end of the fuselage, and having an area of 4/5 square feet. Lateral stability was maintained by warping the main planes, as in the case of the Wright machines. The power plant consisted of a three-cylinder 25 h.p. Anzani air-cooled engine, driving a two-bladed airscrew nearly 7 feet in diameter at 1350 r.p.m. The total weight of the machine was about 700 Ibs., and its maximum speed 40 miles per hour. Reproduced by courtesy of 'Flight.' FIG. 290. Avro Triplane, 1908. Avro Machines. One of the pioneers in England was A. V. Roe, whose early experiments in aviation have led to the development of A. V. Roe & Co., Manchester and South- ampton. A study of the various machines produced by this firm illustrates the progress of aeroplane design in an interesting manner. The ' Bull's Eye, 1 as the triplane with which Mr. Roe carried out many experiments on the Lea Marshes in 1908-1909 was called, weighed only 200 Ibs. It had a surface area of about GENERAL TREND OF AEROPLANE DESIGN 427 300 square feet, while the engine was a 10 h.p. Jap. The fuselage was triangular in section, the pilot being situated some distance behind the main planes. The main planes could be swivelled round a horizontal axis in order to vary the angle of incidence. These main planes could also be warped in order to maintain lateral stability, while directional control was main- tained by the rudder at the rear of the tail planes. The triplane tail was of the lifting type, and was rigidly attached to the rear end of the fuselage. Fig. 290 gives a very good idea of the general appearance of this machine. The first Avro biplane appeared in 1911. It was fitted with a 35 h.p. Green engine, only the nose portion of the fuselage being covered with fabric, while the body was triangular in shape as in the triplane. The tail plane was of the non-lifting Reproduced by courtesy of ''Flight' Fig. 291. Avro Biplane, 1911. type fitted with elevators, and lateral stability was obtained by warping the main planes. (See Fig. 291.) In Fig. 2Q2A is shown the Avro 504 K, which is a modifica- tion of the Avro 1913 machine. This machine has been used as the standard training machine for pilots of the Royal Air Force, and is practically the only early machine still in general use. The Avro Spider is shown in Fig. 2926, and embodies an entirely different type of wing-construction. As will be seen, the struts are arranged similarly to the struts in the Wireless Biplane Truss shown in Fig. 101, and the side elevation is of the same type as that illustrated in Fig. 1 14 (the Nieuport 'V'). Fig. 2920 depicts the Avro Manchester Mark II., which repre- sents the probable commercial machine of this firm. It is a twin-engined biplane, and follows orthodox construction except that the ailerons are balanced by means of an auxiliary plane mounted on two short struts from the main aileron and placed slightly ahead of it. 428 AEROPLANE DESIGN ARMADILLO r~^r~^ W MANCHESTER MAR*]! ARA Avro Machines. FIG. 292. Armstrong-Whitworth Machines. FIG. 294. FIG. 300. 0-400 Handley Page. Reproduced by courtesy ot Messrs. Handley Page, Ltd. FIG. 301. Front and Side Views of -1500 Handley Page. GENERAL TREND OF AEROPLANE DESIGN 429 In order to illustrate further the general trend of aeroplane design, the most interesting machines of the leading aeronautical firms of Great Britain will be briefly reviewed so far as par- ticulars are available. The general dimensions of the machines dealt with are summarised in Table 72, and their performance and weights are given in Table 73. The line drawings of these machines are due to the courtesy of Flight and have all been prepared to the same scale, so that direct comparison is possible. Airco Machines. The design of the Airco machines has throughout been the work of Capt. G. de Havilland. They were on this account formerly termed 'de H. machines,' and under this appellation earned a well-deserved reputation during the War. The first of these machines made its appearance early in 1915, and was a two-seater machine of the pusher type fitted with a 70 h.p. Renault engine. It was followed by the de H. I A, practically identical in dimensions and construction, but fitted with a 120 h.p. Beardmore engine. The performance of this machine was quite good for the engine power available. (See Fig. 293 A). The de H. 4 machine was one of the most successful machines produced during the war, and was used for all pur- poses. It is a tractor biplane of ^ood clean design. Various types of engine have been fitted to this machine, the first being a B.H.P. 200 h.p. The engine power has been gradually in- creased until at the present time some of these machines are fitted with 450 h.p. Napier engines. The engines most fre- quently fitted are the Rolls-Royce 250 h.p. and 350 h.p. types. (See Fig. 293 B.) Since the war the passenger accommodation on this machine has been enclosed to form a cabin capable of seating two persons, and in this form the de H. 4 was used for many journeys between London and Paris in connection with the Peace negotiations. An Airco 4 R (de H. 4 fitted with the 450 h.p. Napier 'Lion') won the the Aerial Derby in 1919. The de H. 5 is a small tractor scout in which the chief aim in design appears to have been the provision of a clear field of vision for the pilot. The most notable constructional feature of this machine is the large amount of negative stagger, and perhaps it was due to this fact that the machine was not easy to handle. (See Fig. 293 c.) The de H. 9 in its main dimen- sions was largely a v copy of the dfc H. 4, the chief difference being in the fuselage. The pilot's cockpit is placed further back, and by fitting a vertical engine the front portion of fuselage has been given a much better shape. (See Fig. 293 D.) A modified 9, fitted with Napier ' Lion ' engine, piloted by Capt. Gather- 43 AEROPLANE DESIGN Scate of F- (ff 5 0' 10' SO' 40 50 60 AtRCO IQ* FIG. 293. Airco Machines. GENERAL TREND OF AEROPLANE DESIGN 431 good, broke eighteen British records in one flight on Nov. 15th, 1919. This machine has attained a speed of 155 m.p.h. The Airco IOA was designed for long-distance bombing combined with all-round performance. Table LXXIII. shows how well this aim was achieved. It is a twin - engined machine, the Liberty engines being placed out on the lower wing structure, one on each side. (See Fig. 293 E.) Armstrong- Whitworth Machines. Since the A.W. Quad- ruplane is the only example of this type of machine which has been constructed by British aeronautical engineers, its leading features are of considerable interest. On trial it was found that its performance was slightly inferior to that of contemporary triplanes of the same engine power, and much inferior to that of small biplanes. The load per brake horse-power is somewhat high, and it is possible that the fitting of a more highly powered engine would lead to a considerable improvement in its per- formance. (See Fig. 294 A.) The Armadillo is noteworthy from the fact that the fuselage entirely fills the centre section of the wing structure (see Fig. 294 B); while in the Ara machine there is a slight gap between the top of the fuselage and the top plane. (See Fig. 2940.) Both of these machines are single- seater tractors, and, as Table LXXIII. shows, their performance under test was good. Bristol Machines. Although the monoplane is the most efficient type of aeroplane aerodynamically, it fell into disrepute about 1912 on account of several fatal accidents which occurred in use, owing principally to structural defects. It is therefore very creditable that Captain Barnwell, the designer of the Bristol machines, has produced, in the face of much opposition and prejudice, such a pleasing and efficient monoplane as is shown in Figs. 296 A, 297 (p. 424). As will be seen, the wing section employed allows of deep spars, the wing being fitted with aileron surfaces instead of with warping arrangements as is usual in monoplane practice. Especial care has been devoted to streamlining, and openings are provided in the inner portion of the wings near the sides of the fuselage, resulting in a further increase in the natural range of visibility of the monoplane type. The Bristol Fighter (Figs. 296 B and 298, p. 424) illustrates a machine designed primarily for fighting purposes. The F 2 B, as it was also called, was very largely used for fighting, scouting, and other purposes during the war, and the illustrations show the modifications that have been made in the design of the fuselage and other components in order to render this machine efficient 43 2 AEROPLANE DESIGN -MONOPLANE- -FIGHTER - FIG. 296. Bristol Machines. GENERAL TREND OF AEROPLANE DESIGN 433 for its specific purpose. In particular it will be observed that the lower plane is situated well below the fuselage, resulting in a somewhat more complicated arrangement of the chassis. Pilots report that this machine is very responsive to its controls, while it also possesses a large amount of inherent stability. The Bristol Triplane (Fig. 2960) is a four-engined machine driving two tractor and two pusher airscrews. It was primarily designed for bombing purposes, but is being adapted for other uses. Handley Page Machines. From the very inception of his firm Mr. Handley Page has pinned his faith to the future of the large aeroplane. The first Handley Page bombing machines did not make their appearance until December, 1915, and it was not until August of the following year that the first squadron of the O-4OO type was formed at Dunkirk. From that date until the conclusion of hostilities, all heavy night bombing on the Western Front was performed with these machines. The V-15OO type was designed originally to bomb Berlin, but is now being adapted for commercial use. One of these machines has carried forty- one passengers to a height of 8000 feet. A line diagram of the O-4OO type is shown in Fig. 299, a front view of the O-4OO in Fig. 300, while front and side views of the V-I5OO type are shown in Fig. 301 (p. 424). Photographs illustrating the position of the wings in their folded- back position were shown in Figs. 3, 151. Sopwith Machines. The Sopwith Tabloid was originally built as a side-by-side two-seater for Mr. Hawker, who has since achieved fame as a first-class test pilot, and whose attempt to be the first airman to cross the Atlantic on a Sopwith machine was only frustrated through radiator failure. In the Tabloid machine lateral control was effected by means of wing-warping. This machine first demonstrated the possibilities of the small single- seater biplane as a rival to the monoplane for high-speed work, while retaining a large range of flying speeds. (See Fig. 302 A.) The ij Strutter is so designated because of the type of wing- bracing employed. The top plane was in two halves bolted to the top of a central cabane, and the spars, are provided with extra support in the shape of shorter struts running obliquely from the top longerons to the top plane spars. This machine is also interesting owing to the fact that it was fitted with an air brake taking the form of adjustable flaps inserted into the trailing edge of the lower plane close to the fuselage. Another feature incorporated in the ij Strutter was the tail-plane variable incidence s F F 434 AEROPLANE DESIGN gear. (See Fig. 302 B.) The Sopwith Pup follows the general lines of the ij Strutter and the original Tabloid. It handles remarkably well, and, as will be seen from Table LXXIIL, O- 4OO o FIG. 299. Handley Page, 0-400. possesses a very low landing speed. (See Fig. 3020.) The Sopwith Camel was so called from the hump which it possesses on the forward top side of its fuselage, due to the fitting of two GENERAL TREND OF AEROPLANE DESIGN 435 _J_5 53 J T^ HC Scale of )flf 5' O' 10' 20' XT FB -TRi PLANE. FIG. 302. Sopwith Machines. 436 AEROPLANE DESIGN fixed machine guns both firing through the airscrew. It achieved a great reputation during the War, but as a sporting machine the Pup is preferable in many respects. (See Fig. 302!).) The Dolphin (see Fig. 302 E) differs considerably from the Camel in structural arrangements. It will be seen in the illustration that a double bay arrangement of struts has been adopted, the gap has been diminished, and negative stagger introduced. The radiator was divided into two portions, placed one on either side of the fuselage opposite the pilot's cockpit, each radiator being fitted with deflectors. The Sopwith Triplane was designed solely to afford good visibility and manoeuvrability. As will be seen from the figures and tables, the wing chord has been considerably reduced, and single ' I ' struts have been fitted between the planes in place of the more usual pair. Vickers Machines. At the commencement of the war the Vickers Gun Bus (F.B. 5) (Fig. 303 A) was practically the only fighting aeroplane in existence. It was a pusher machine, the Vickers gun being mounted in the nose of the nacelle, from which position a very wide range of unobstructed fire could be obtained ; and its arrival on the Western Front established for the time being the aerial supremacy of the Allies. The F.B. 7 (Fig. 3036) was brought out in August, 1915, and was one of the first twin-engined machines to take the air. It is par- ticularly interesting as being the prototype of the now famous 'Vimy Bomber.' In the experimental model of the F.B. 16 trouble developed owing to the weakness of the leading edge of the main planes. Investigation showed that this weakness resulted from an inadequate factor of safety for the high speed attained by this machine. After remedying this defect, the machine was tested officially and showed a performance better than that obtained by contemporary machines of a similar type. The Vimy Bomber (Figs. 30313, 304, p. 432) was remarkable for its small size when compared with its large lifting capacity. It is claimed that this machine is stable both longitudinally and laterally. The engines are placed out on the wing structure directly over the landing chassis. The fore part of the fuselage is constructed of metal tube and the rear part of special wooden tube. This machine, as used for crossing the Atlantic, is shown in the Frontispiece and Fig. 304, and it is noteworthy that with the fitting of additional fuel tanks only, it succeeded in accom- plishing the first direct flight across the Atlantic. An exactly similar type machine accomplished the first flight to Australia. Fig. 305 (p. 432) shows the Vimy as adapted for commercial work. Details of this machine have already been given in Chapter VII. GENERAL TREND OF AEROPLANE DESIGN 437 Boulton and Paul Machines. Illustrations of two of the machines manufactured by this firm are shown in Figs. 306, 307 (p. 444). It will be noticed from Table LXXIII. that the load r.6-5 V F.B. 7 Scale of P*e^ '050 10 20 SO 40 50 fcO 70 F.B. f6 FIG. 303. Vickers' Machines. per horse-power for the passenger machine is only 7-8 Ibs., which, coupled with particular care in the remainder of the design, in a large measure accounts for the remarkable per- formance of this type. 4,3 AEROPLANE DESIGN TABLE LXXII. TYPE WING SPAN. WING CHORD. WING AREA. OF Over- all MACHINE. length Top. Middle. Bottom. Top. Middle Bottom Top. Middle Bottom Total. AVRO feet feet feet feet feet feet feet sq . feet sq . feet sq. feet sq. feet 504 K Spider ... 28-92 2O "5 36-0 28-5 36-0 4-83 4-83 171-5 - 21-5 i 6-00 1 2-5 162-0 - I58-5 -j 46*0 330-0 208-0 ManchesterMk.il 60 -o 60*0 7-5 ' 7-5 430-0 - 387-0 817*0 AIRCO IA 29-0 41*0 41-0 5'9 5-9 187-0 - 175-25 362-25 4 30-0 42*39 42*39 5'5 5-5 223-0 - - 211-0 434-0 5 22-0 25-67 25-67 4'5 4-5 lll'2 100-9 2I2-I 9 30-83 42-39 42-39 5'5 5'5 223-0 - 2II*O 434-0 IOA 39-62 65-5 65-5 7-0 7-0 429-2 - 408-2 837-4 A-WHITWO RT H (2)27-83 ( 2 )3'58 (2)92-6 Quadruplane ... Armadillo 22-25 18-83 27-83 2775 27-83 2775 3]58 (3)3-58 3*58 ~~ 4'5 102-6 (3)92-6, 102-6 I2t-,'0 "I25-0 398-4 250*0 Ara 20-25 27-42 27-42 5-25 4'5 147-0 - - IIIO'O 257-0 BRISTOL Monoplane 20-33 3075 5-92 i 145.0 - 145-0 Fighter... Triplane 2575 81-67 8l-67 39'25 78-25 8-5 *~s I-! 202-5 202-5 650*0 650-0 605-0 405-0 I905-0 HANDLEY PA GE 0-400 62-85 I OO'O 70 - o 10-0 IO'O 1020-0 i - - 625*0 1645-0 SOPWITH Tabloid 20-33 25*5 25-5 5-12 5*12 128*3 j - < "3"0 241-3 l Strutter 25*33 33-5 33' 5 5'5 5*5 183-0 - I7O-O 353*0 Pup 19-33 26-5 - 26-5 5-12 5-12 132-0 122-0 254-0 Camel ... 1875 28-0 28-0 4-5 4-5 125-0 II5.O 240-0 Dolphin 22-25 32-5 32-5 4'5 i 4'5 i3 2>0 - I3I-O 263-0 Triplane 18-83 26-5 26-5 ^26-5 3-25 3'25 3' 2 5 84-0 72-0 i 75-0 231*0 VICKERS F.B. 5 27-17 36-5 36-5 5*5 5*5 I97-0 - 185-0 382-0 F.B. 7 36-0 1 181-0 640*0 F.B. 16 25-0 22-33 5'5 4-i7 ! 126-0 81-0 2O7-O vi y i 43*54 67-17 67-17 10-5 10-5 686-0 644-0 I330-0 BOULTDN & PA UL : ; Scout 20-0 29-0 29-0 ! 5'37 4'i2 152-0 - 114-0 266-0 Passenger 40*0 59 'o 59-0 8-0 6-5 432-0 - - ; 338-0 770-0 S.E. 5 20*92 26-62 26-62 i 5-0 -- | 5-0 130-0 - - ; 113' 249-0 I i GENERAL TREND OF AEROPLANE DESIGN 439 TABLE LXXII. INCIDENCE. Gap. Stagger DIHEDRAL. AREA OF CONTROL SURFACES. Top Middle. Bottom. Top. Middle/Bottom Tail Aileron p l an e. Elevator Total. Fin. Rudder Total. feet feet 1 sq. ft. sq. ft. sq. ft. sq. ft. sq. ft. sq. ft. sq. ft. 4-5 - 4-5 ! 5-5 2-17 2'5 -- 2-5 45'5 26-0 18-0 44-0 ! 90 9-0 O'O O'O 4'22 2'OO 0-0 0-0 22-0 I5-2 10-4 25-6 - 7'8 7-8 4-0 4-0 7-25 o-o 2-5 2-5 * 124-0 50 -o 35-0 85-0 12-0 16-0 28-0 j 5-5 5-5 5-87 o-o 3-0 3-o 64-0 37 '5 23-0 60-5 i 37 '5'4 s 19-1 3-0 5 -50; i-o 3-0 3-0 82-0 38*0 24-0 62-0 5'4 13-7 19-1 2'O 2'O 475-2-25 4-5 4'5 46-4 13*4 12-2 25 6 2-2 6-3 8 '5 3 -0 3-0 5-50 i-o 3-0 3'o 82-0 38-0 24-0 62 ; 5'4 137 19-1 7-0 - 7-0 7-0 O'O 4'5 - 4'5 118-0 75"5 33'* 1086 IO'O 2575 3575 (2) 3' (2) i'5 3'o (3) 3 3' 2-67 1-42 i -5 (3) i-S i 5 6 7 -2 16 o 16-0 i '9 8-0 9'9 2-25 - 1-0 3-92 0-71 0-0 2'0 36-0 17 o 14-0 31-0 1-6 6-0 7-6 2-75 1-25 3-88 0-96^-5 ,1-5 20'4 25-0 24-0 490 2-5 iro I3-5 O'O _ i 2-0 18-0 20 -o 15-0 35-o 5-0 4'5 9 "5 i '5 1*5" 5-42 1-42 3 -5 3 -5 5' 22*2 23-2 454 107 7-2 17-9 2'5 2-5 2-5 7-21 2-0 2-0 2-0 192-0 96-5 85-0 181-5 28-2 25-0 51-2 I I'O 4-0 4'0 I23-5 65-3 1888 14-7 45'7 60-4 i-o c 1-0 1 4-5 0-92 i -5 *'5 28-0 1 1 -8 n-8 23-6 r8 41-3 6-r 2-45 2-45 i-5 I i'5 4-42 2-0 I'5 ?/ ~ I' 4 / 52-0 35-5 21-5 22-0 23-0, 1 1 -8 570 3'5 34-8 3'5 7-25 4'5 1075 8-0 2-0 2-0 5-4 1-5 0.0 5' 36-0 14-0; 10-5 24-5 ! 3' 49 7'9 2-o J 2-0 ; 4-25 - I'O- 2 '5 2-5 38-0 17-0 13-5 30-5 3'5 80 2'O 2'O 2'O 3-0 J '5 2-5 2'5 2-5 34-0 14-0 9-6 236 2-5 4'5 7-0 4'5 4'5 6-0 O'O I'O 1-0 57-0 560 24-6 80-6 8-5 13-25 2175 3-0 3-0 7-33 0-92 3-0 3-0 93 o 42-0 28-5 705 20 -o 20-0 2-0 2-0 3-92 2'5 j I'5 1 '5 23-5 18-5 15-3 33-8 6-5 6-0 I2'5 3'5 3-5 io-o o-o 3-0 30 242-0 ! 1 14-5 630 177-5 2X 2X 48-5 1 ! ... i 1 ! 5-o - 5-o 4-6 1-5 5-0 5-0 I 30-2 ! -5-1 15-1 30*2 6-5 6-0 12-5 1 j I i 440 AEROPLANE DESIGN TABLE LXXIII. TYPE OF ENGINE, WEIGHT OF MACHINE. Fuel capacity. Range. MACHINE. Name. No. H.P. Empty. Loaded. Hours. -Miles. L1JS. J.bs. AVRO 504 K Le Rhone I no 1230 1820 3 -0 225 Spider Manchester Mk. II. ... Clerget Siddeley I 2 I 3 600 963 4580 1517 7160 3-0 3'8 330 450 AIRCO I A Beardmore I 1 2O 2400 4 Rolls-Royce I 370 3340 4-0 500 5 Le Rhone I 1 10 1490 2'0 200 9 Lion (Napier) .. I 420 3725 IOA Liberty ... ... 2 800 ! 8500 5' 650 A-WHITWORTH Quadru plane Clerget ... I I3O 1140 I800 Armadillo B.R. 2 ... I 230 1250 1860 275 Ara Dragonfly I 320 1320 1930 3-25 450 BRISTOL Monoplane Le Rhone I 1 10 1300 Fighter Triplane... Rolls-Royce Siddeley I 4 250 j 1000 9300 2800 16200 ; HANDLEY PAG E 0-400 Rolls Royce 2 700 ; 8000 14000 7-0 650 SOPWITH Tabloid Gnome 80 730 I 120 3*5 320 i Strutter Le Rhone 1 10 1280 22OO Pup Gnome ... loo 856 1297 2'O 2OO Camel B.R. i J 5 1470 2-6 3o Dolphin ... Triplane Hispano Suiza ... Clerget I 200 130 1406 1 100 1880 1540 2'O 230 310 VICKERS F.B. 5 Gnome ... I 100 1220 2O5O 4'5 330 F.B. 7 Gnome ... 2 200 2130 3200 2'5 200 F.B. 16 Hispano Suiza ... I 200 1380 1880 2-25 300 Vimy : Rolls-Royce ... 2 700 6700 12500 1 I'OO IIOO BOULTON & PA UL Scout B.R. 2 I 230 1230 I92O 3-5 440 Passenger Napier Lion 2 900 4OOO 7OOO 3-0 S.E. 5 Hispano Suiza .. I 200 1980 2 250 GENERAL TREND OF AEROPLANE DESIGN TABLE LXXIII. 44 1 SPEED. CLIMB. Ceiling. Feet. Landing speed. m.p.h. LOAD. ( 1 round level. At 1 0000 ft. At 15000 ft. Minutes to Per sq.ft. PerH.P. 5000 ft. 10000 ft. 15000 ft. 90 75 65 6-2 5 l6'0 40-0 35 5'52 ! 16-5 1 20 no 4-0 9*5 22*0 19000 40 778 i 1 1 "6 125 119 112 1 1 '15 40*0 17000 45 876 | 11-9 * . 89 - IO'O 6-6 20'0 133 J26 9 - o 16-6 23500 52 7'4 9*4 1 02 8 9 12-4 27-4 17000 50 7-0 13-6 140 135 8.2 14-6 25300 8'5 8-8 124 117 " II'O 20-5 20000 55 10 '0 io'6 105 99 17-0 25000 4-5 '3*9 125 113 6'5 24000 55 7 '4 8-0 150 145 4'5 28000 5 7 '5 6-0 130 117 _ 3-5 9-0 19-0 _ 49 8-97 10-8 125 "3 irs 21-5 48 6-92 1 1 '2 1 06 93 35-0 55 8-5 16-2 85 < 8-5 20 g o 92 _ 36 4.66 14-0 103 8-0 18-9 41-5 l6000 35 6-4 17-5 1 IO 104 100 57 12-4 23-4 18500 30 5*2 12-4 120 114 8-3 15-8 23000 35 128 124 3 '9 8-25 147 23500 40 7*3 9'0 106 95 5*0 11-8 22-3 21000 35 6-0 12-4 75 _ 16-0 9000 4i 5*4 20-5 So 18-0 I2OOO 40 5*0 16-0 135 126 475 10-4 2075 20000 55 9-1 9-4 105 100 15-0 50-0 10500 ! 5 9*4 17*9 . , 125 no ' 9'5 18-0 21000 50 7-2 8-3 149 142 ~ 8-0 15-0 25OOO 54 9-1 7-8 120 5-o 10-8 20-8 21000 8.0 9-8 j . 442 AEROPLANE DESIGN Official Machines. Probably more controversy has raged round the B.E. 2 C than any other aeroplane built, nevertheless it represents the type of aeroplane that will undoubtedly be largely developed in the near future, namely, the inherently stable machine. Its inception and development was principally due to the efforts of the late Mr. E. T. Busk, of the Royal Aircraft Factory, and it shows in a striking manner the result of a sound application of theory to practice. Table I. illustrates how near the actual performance of this machine approached the calculated values. The S.E. 5 represents the most successful war product of the R.A.E. In general appearance it is similar to the Sopwith ij Strutter, and was designed as a single- seater fighting machine. It is inherently stable, a wonderful demonstration of its qualities in this direction being provided when a S.E. 5 machine returned safely to the British lines after its controls had been practically shot away over the German lines. Having considered the general trend of design with reference to complete machines, there only remains to be considered the question of detail design. CHOICE OF TYPE. It would be rash to prophesy whether the monoplane, biplane, or multiplane will be the most largely developed type in the future, since each type possesses advan- tages peculiar to itself. In comparison with the biplane, the monoplane can carry 5 per cent, more weight per square foot of wing surface, besides giving much better visibility. On the other hand, it is much weaker structurally. In the same way the triplane and quadruplane are about 5 per cent, less efficient than the biplane and triplane respectively, but if well designed should be more manoeuvrable. Generally speaking, it seems probable that the biplane will hold its own for general purposes for some considerable time to come, with the triplane as a rival in the larger sizes. Wing Design. The wind channel method of investigation has produced very efficient forms of aerofoils, and it seems pro- bable that seventeen is an optimum value of the Lift/Drag ratio for wings of practical design. Further investigation is needed as to the depth of camber and the nature of the flow in the neighbourhood of the aileron surfaces. It is probable that in the near future metal construction will replace wood for the ribs and spars of large machines at least. Optical stress analysis has shown that there is considerable divergence between the points of inflexion as calculated from 5 y 51 I ^> I I * 1 GENERAL TREND OF AEROPLANE DESIGN 443 the Theorem of Three Moments and the points obtained by loading a spar approximately as in practice, and further inquiry into this discrepancy is required. It seems at least on the safe side to use the Theory of Bending as outlined in Chapter V. For large machines the saving in weight obtained by using tapered struts is of great importance, and it is hoped that the graphical method of tackling their design, which has been fully explained in Chapter V., will enable all those whose knowledge of the Calculus is limited, or even non-existent, to apply this theory in practice. Internal bracing is generally effected by either plain or stranded wire in machines of all countries, the Fokker biplane and triplane being notable exceptions. A great improvement in the design of the wings will be the development of a section with a stationary centre of pressure over the range of flying angles. Further improvements likely to follow are : (i) A practical design for a variable camber of surface, in which the mechanism is simple and reliable, and does not add appreciably to the weight of the machine. (ii) The elimination of the major portion of the external bracing of the wing structure. Fuselages The design of the fuselage is largely governed by the type of engine employed and the particular purpose for which the machine is intended. Recent investigations tend to show that the circular (or elliptical) body does not possess any material advantage over the square section. Constructionally,, either wooden formers, suitably lightened out and of the required cross-sectional shape, support the longerons at regular intervals ; or the strut and cross-bracing wire method is used. It may be remarked in passing that enclosing the rear portion of the body of several well-known war machines has led to a reduced overall resistance and consequent improvement in performance. The monocoque method of fuselage construction, which dis- penses with the longerons and employs a moulded three-ply method of construction, offers considerable advantages from the commercial point of view, since internal bracing is not needed,, and consequently the space inside the fuselage is left clear for passengers, luggage, and cargo. Control Surfaces As shown in Chapter XL, the attain- ment of stability by means of a correct disposition of the various control surfaces in relation to the fixed surface areas of the machine itself is now well within the compass of the aero- 444 AEROPLANE DESIGN plane designer. It is also possible to achieve stability by means of external stabilising devices, such as, for example, the use of -a gyroscope, but the success of the inherently stable machine has obviated the need for developing such methods. The Airscrew. Rapid development has taken place in the design of the airscrew during the war period, and it is now stated that the limit to the speed of the airscrew is fixed by the circum- ferential velocity of the tip, which must not exceed the velocity of sound (iioo feet per second). Airscrews must therefore be geared down so that the maximum tip speed under no circum- stances is greater than 1000 feet per second. Metal airscrews have been manufactured, and will probably be developed for countries where the climatic conditions do not permit of a continued use of a wooden airscrew. There are also several experimental designs of airscrews with variable pitch under trial, of which more will doubtless be heard in the future. Performance. With the passing of the special conditions imposed by the War, the need for very rapid climb will dis- appear, and aeroplanes will cease to be required to operate at 20,000 feet, and to be capable of reaching that height in the minimum time. The engine employed is a vital factor in the performance of any machine, and it is quite -a truism that in all far-reaching developments the aeroplane designer has to wait upon the engine designer. ' FIG. 306. Scout. Reproduced by courtesy of Messrs Boulton & Paul. FIG. 307. Passenger. Boulton & Paul Machines. Facing page 444. GENERAL TREND OF AEROPLANE DESIGN 445 ^!Sv LAN DING CHASSIS Axle (OR UNDERCARRIAGE) FIG. 308. Aeroplane Nomenclature. 446 AEROPLANE DESIGN TABLE LXXIV. SAFE LOADS IN Outside diam. Area. Sq. ins. Weight. Lbs. run per ft. LENGTHS 1O 20 30 40 50 60 SAFE LOADS Inches. GAUGE 22. THICKNESS 4 ... 0415 141 1300 650 4OO ! 25O 150 ' too | ... 0635 216 2800 1800 1000 650 400 500 I .- 0745 253 3400 2500 1500 looo 700 | 5^0 I 0855 '291 4000 3250 2200 1450 1000 I 700 1* - .0965 328 4600 3850 2850 1900 1350 1000 It ... 1075 366 5100 4500 3550 2500 1800 1300 i .. Il8 5 '403 5700 5100 4250 3150 2300 1700 1295 440 6300 5750 4950 3800 2900 2150 if ... 1405 478 6850 6300 5550 4500 ; 3500 2650 if ... 1515 515 7400 6900 6200 5200 4100 3200 i-g .- 1625 553 8000 7500 6900 5900 ! 4750 3800 2 1735 590 8500 8050 7500 6550 5500 4400 GAUGE 20. THICKNESS 4- -. 0525 -178 2OOO 1200 700 450 300 200 . '0807 '275 3900 2500 1450 850 500 350 1 ... 0949 "323 4800 3550 2200 1400 900 650 1090 571 5600 4500 3000 2COO 1400 IOOO 1232 419 6500 5500 4OOO 2750 1950 1350 i ... '373 ^7 7300 6400 5000 3600 2500 1800 1 ... I5I4 '515 8200 7300 5950 450 33oo 2400 h ... 1656 .563 9000 8200 6900 5400 4000 3000 1 ... 1797 -611 9700 9050 7900 6300 5000 3800 1938 -659 10600 9850 8800 7300 5800 4600 15 ... 2080 .707 I[400 10700 9700 8400 6700 5400 2 2221 .755 12200 II500 10650 9400 7800 6400 GAUGE 17. THICKNESS \ ... 0781 '401 2000 1000 400 250 200 170 f 1221 647 5900 3500 1900 I 100 800 500 1441 "769 7200 5600 3150 1900 1150 800 ... 1661 .892 ,3700 7000 4500 2900 1950 1350 i ... 1881 015 9800 8250 6000 4000 2800 2OOO i ... 2101 I 3 8 III50 9850 7500 5000 3700 2750 f '2321 26l 12600 1 1200 9OOO t6oo 475 3500 4 2540 384 13700 12500 10600 8000 6000 4500 1 2760 507 14900 13800 12100 9700 7400 5600 f ... 2980 '629 16300 15200 13700 11250 8900 6900 I 3200 752 17600 16600 14900 12750 10150 8000 2 3420 875 18800 17850 16400 14250 11800 9500 GENERAL TREND OF AEROPLANE DESIGN 447 LBS. FOR TUBULAR STEEL STRUTS. IN INCHES. 70 80 90 1OO 110 120 130 140 15O IN LBS. OF METAL, 0*028" 80 70 60 5 40 1 200 I 150 120 100 80 35o 300 230 2OO 150 120 100 500 400 330 280 200 1 60 130 110 750 550 440 350 280 24O 1 80 1 60 150 1000 8co 600 460 360 3 20 260 220 2OO 1300 1000 800 650 520 450 340 280 250 i 700 i 300 1080 830 700 600 500 420 35-3 2100 1650 1350 1 100 870 750 650 5 60 500 2550 2000 1650 1350 1130 920 800 750 650 3000 2450 2000 1700 I4OO 1200 1000 880 800 3600 29CO 2400 2000 1650 1400 1200 IIOO IOOO OF METAL, 0-036' | 150 100 90 80 70 . 250 200 200 170 150 130 500 350 300 280 230 200 180 700 550 450 4OO 330 280 250 1 80 IOOO 800 650 550 450 370 320 280 250 1400 IIOO 850 700 600 500 420 380 300 1800 1400 1150 900 800 650 550 500 450 2300 1800 1500 1200 IOOO 850 800 700 550 2800 2300 1850 1500 1300 IIOO 920 800 ! 700 3600 2800 2300 1850 1550 1300 1150 I ooo 900 4300 3400 2950 2300 1950 1650 1400 1300 1150 5200 4100 3350 2750 2353 2000 1700 1500 1350 OF METAL, 0*056' 150 120 100 70 _ __ 350 250 200 120 100 ' too 500 4OO 320 300 260 950 800 650 550 500 350 320 1500 1200 950 800 700 550 500 450 2000 I600 1300 IOOO 850 750 650 $70 5 00 2700 2100 J700 1400 1200 IOOO 850 750 6 5 3450 2700 22OO 1800 1500 1250 IOOO 875 800 4300 3500 2800 2300 1900 1650 1350 TI50 IOOO 5300 4300 3500 2850 2400 2000 1700 1500 1250 6500 5100 4250 3550 2000 2500 2100 1850 1600 7700 6200 5000 4200 3600 3050 2600 2300 2000 449 LIST OF TABLES. I. Comparison of Calculated and Actual Performance. II. Weights of Structural Components expressed as Percentages of the Total Weight. III. Diminution in Weight per H.P. of Aero Engines. IV. Percentage Resistance of Aeroplane Components. V. Strength and Weight of Timbers. VI. Properties of Duralumin. VII. Specific Tenacity of Different Materials. VIII. Steels to Standard Specifications. IX. Brinell Hardness Numbers. X. Factors of Safety. XI. Wind Pressures. XII. Table of Forces. XIII. Table of Forces. XIV. Influence of Aspect Ratio on the Normal Pressure of a Flat Plate (Eiffel). XV. R.A.F.6. Coefficients. XVI. Influence of Aspect Ratio. XVII. Camber. XVIII. Reduction Coefficients due to Biplane Effects. XIX. Comparison of Lift Coefficients. XX. Comparison of the Wings of a Triplane. XXI. Calculations of V/V. XXII. Moments of Inertia Geometrical Sections. XXIII. Load, Shear Force, Bending Moment and Deflection. XXIV. Shear, Bending Moment, Slope, and Deflection by Tabular Integration. XXV. Wing Loading of Modern Machines. XXVI. Bracing Wires and Tie-rods. XXVII. Particulars of Strands for Aircraft Purposes. XXVIII. Strainers. XXIX. Resistances of Struts. XXX. Resistance Coefficients for Fuselage Shapes. XXXI. Monocoque Fuselages without Airscrews. XXXII. Monocoque Fuselages with Airscrews. XXXIII. Comparison of Four Fuselage Bodies. XXXIV. Values of K with Increase of dV. XXXV. Resistance of Inclined Struts and Wires. HH 45 List of Tables. XXXVI. Resistances of Landing Wheels. XXXVII. Estimate of Body Resistance of B.E.2 at 60 M.P.H. XXXVIII. Resistance of Aeroplane Components. XXXIX. Skin Frictional Resistances. XL. Percentage Weights of Aeroplane Components. XLI. Weights and Particulars of Leading Aero Engines. XLII. Elongation of Shock Absorber. XLI 1 1. Stresses due to Centrifrugal Force. XLIV. Stresses due to Bending. XLV. Maximum Stresses in the Airscrew Blade. XLVI. Stability Nomenclature. XLVII. (Case I.) Speed 79 m.p.h. Angle of Attack (i) i XLVIII. (Case II.) Speed 45 m.p.h. Angle of Attack (i) 12. XLIX. Wing Characteristics. L. Determination of Angle of Downwash. LI. & LI I. Angle of Tailplane tq Body. LI 1 1. Elevator Moment. LIV. K y ' LV. Lateral Force of Aeroplane Components. LVI. Determination of Wing Resistance. LVII. H.P. available. LVI 1 1. Mean Atmospheric Pressure, Temperature, and Density at Various Heights above Sea Level. LIX. Percentage of Standard Density due to Change of Altitude. LX. Calibration of an Airspeed Indicator by Direct Test. LXI. Airspeed at Heights. LXII. Rate of Climb Test. LXI 1 1. Determination of the Position of the Centre of Gravity. LXIV. Balancing. LXV. Body Resistance (Outside Slip Stream). LXVI. Body Resistance (In Slip Stream). LXVII. Calculation of Slip Stream Coefficient. LXVI 1 1. Body Resistance of Machine at Different Speeds. LXIX. Calculation of Total Resistance. LXX. Horse-power Required and Available, and Rate of Climb. LXXI. Progress of Civil Aviation in England, May I5th to October 31 st, 1919. LXXI I. Data relating to Successful Aeroplanes. LXXI Hi .. ,, ,, ;, LXXIV. Safe Loads in Ibs. for Tubular Steel Struts. .45 1 INDEX. A CCELEROMETER, 145. 2~\. Aeroplane nomenclature, 445. Aerodynamic balance, 41-45. Aero engines, 239. Aerofoil, 38, et seq. characteristics, 3, 50, 59. choice, 84-86. comparison with model, 64. effect of thickening L. E., 73. T. E., 74- full-scale pressure distribution, 62, 63- pressure distribution, 55, 57, 60, 61. superimposed, 79, 80. upper and lower surfaces compared, 65,66. variation of maximum ordinate, 71, 72. Ailerons (wing flaps), 372. balanced, 377. Airco machines, 429, 430. Airscrew, 7, 280, et seq. airscrew balance, 302. angle of attack, 9. construction, 299. design, 285, 286, 295, 296. efficiency, 8, 140, 287, 289. slip ratio, 282, 283. stresses, 295-98. testing, 302. variable pitch, 9. Airspeed indicator, 390, 392. calibration, 401. Altimeter, 398, 399. Alloys, light, 15. steel, 17. Anemometer, 390. Aneroid barometer, 397, 398. Angular accelerations, 308. Armstrong - Whitworth machines, 137, 428, 431. Aspect ratio, 5, 47-49, 63-65, 382, 383. Atlantic flight, 425. Australian flight, 425. 120, 121, Avro biplane, 421. Avro machines, 426, et seq. Avro triplane, 426. BANKING, 143. Bairstow, 304, 313, 333. Bending moment, 110-13, IJ 8, 198. B.E. 2 biplane, 349. B.E. 2 fuselage. B.E. 3 fuselage, 222-27. Bernouilli's assumption, 144, 155. Biplane, 5, 80. strutless, 135. trusses, 132. wireless, 134. Blade element theory, 280, 490, et seq. Bleriot, 131, 425, 426. stability model, 315, 321, 323, 335. Body resistance, 213, 383, 386, 417,419. Boulton & Paul machines, 437, 444. Bow's notation, 28. Brinell hardness, 22, 23. Bristol fighter, 146, 147. machines, 431, 432. monoplane, 424. triplane, 433. Bryan, 304. Bulk modulus, 21. /CAMBER, upper service variable, \^ 67-69. lower service variable, 70. Ceiling, 8. Centre of gravity, 75, 132, 257, 260, 410. Centre of pressure, 16, 74, 75, 149, 160, 162, 3^6, 374. Centre of resistance, 414, 417, 418. Chanute, I, 132. Chassis, 260. design, 262, et seq. location, 261, 262. resistance, 230. 452 Index. Chassis, stresses, 269-71. types, 264, 269. Civil aviation, 425. Clerk Maxwell, 26. Controllability, 5, 9, n, 303, 346. Control surfaces, 346, et seq., 443. balanced, 375. Controls, 407. Cowling, 255. Curtiss, 253. stability model, 339-44. DECALAGE, 83. Deflexion, 118, 120, 121. De Havilland, 429. Density, standard, 390. Derivatives, resistance, 301, 314. rotary, 304, 314. Dihedral, 407. Dines, 389. Down-loading stresses, 151, 155-57, 165, 197. Drag, 4, 7, 49-52, 54, 58. Drag-bracing, 137, 138, 163, 177, 191, 194. Drag-struts, 177, 178, 195. Drawings, general scheme, 411. Dual control, 378. Duplication, 161. Duralumin, 15. T^ CCENTRIC loading, 126. j"j Eddy motion, 59. Eiffel, 38, 47, 231. Eiffel laboratory, 45, 46. Elasticity, 20. Elevator, 346, 356, 379. settings, 359, et seq. End effect, 62, 287. End correction factor, 198. Engine mountings, 250-53. Equations of motion, 307. Equivalent chord, 357. Euler's formula, 125, 126. FABRIC, 20. Factor of safety, 23, 24, 140, 141. Farman, H., 132. Fin, 368, 379. Fineness ratio, 129. Flat plate, force on, 47. inclined, 48. edgewise, 50. resistance, 230. Forces, 10. Fuselage design, 237, et seq. monocoque, 223, 238, 245, 443. resistance, 213, 215-20, 222. stressing, 247, 249. AP, 77, 407- Grieve, 425. Gyroscopic action, 257-59. Handley Page, 424, 425, 433, 434. fuselage, 245. Hanging on the prop, 8. Hawker, 433, 435. Hooke's law, 21, 114. Horse-power, 7. available, 384, 424, 487. required, 384, 487, 420. T NCIDENCE bracing, 137, 138, 162. 1 Incidence gears, 363, 379. Inherently stable, 303. Interference, 77, 383. Interplane struts, 165, 171-76, 178, 184-91. Tx-INEMATIC viscosity, 89. LANCHESTER, 304. Langley, 48, 424. Lateral force, 319, 325, 327, 370. in airscrew, 372. Lay-out, 403, et seq. Leading edge, 73. Lift, 4, 7, 49-52, 54, 58. effect of variable camber on, 68-71. bracing, 165. Lift/Drag, 51, 53, 54. superimposed aerofoils, 83. Lilienthal, i. Linear accelerations, 307. Load, 118, 122, 27. diagram, 122. Longitudinal force, 317, 353, 354. MATERIALS, 13, et seq. Maxim, 424. Meridian curve (tapered strut), 174. Method of sections, 35-37. Moments of inertia, 101, 102, 105-108^ Monoplane, 5, 80. trusses, 130. Multiplane, 6. NIEUPORT, 6, 133, 139. Nomograms, 1 08, 109. Normal force, 318, 352, 355. Nose diving, 144. National Physical Laboratory, 38. OLEO gear, 276. Oscillations, longitudinal, 315. lateral, 313. Index. 453 TT)ARALLEL axes, 102. \^ Parasol type, 132. Performance, 2, 381, et seq., 389, 444. curves, 421. test, full scale, 395. Perry formula, 129. Pfalz, 248. Pitching, 305. moment, 319, 254, 352, 355. Pitot tube, 56, 392, 393. Poisson's ratio, 21, 22. Polar moment of inertia, 105. Polygon offerees, 26. Pratt truss, 131, 132. Pusher, 139, 176, 254. ^vUADRUPLANE trusses, 137. T~) ADIATORS, 255-57. \\_ R.A.F. 6, characteristics, 54, 164. R.A.F. wires, 167 Rankine-Gordon formula, 126, 127, 249, 250. Rate of climb, 386, 3)6, 400, 402, 421. Rayleigh, 88. Resistance, 6, 208, et seq, aeroplane components, 233. complete machine, 232. wires, 228, 229, 382, 420. Ribs, 177, 203-206. Rigidity modulus, 21. Rolling, 305. moment, 320, 325, 327. Ross-Smith, 425. Routh's discriminant, 313, 330, 343. Rudder, 368, 379, 408. SECTION modulus, 116. S. E. 4. 369- S. E. 5, 145, 146, 442. Shear force, 110-13, I 9%- Shear stress, 116, 118. Shock absorbers, 272-76. Similitude, 88. Skin friction, 209, 234-37, 244. Slip stream, air-screw, 231., coefficient, 234, 419. Slope, 118, 120, 121. Sopwith machines, 433, 435. Spars, 179, 181, 195, 196, 199, 200, 201. Specific tenacity, 15. Spiral instability, 334. Stability, 9, 1 1, 303, et seq., 346, 408. nomenclature, 305. - lateral, 319-21, 323, 333. longitudinal, 315-17, 330, 338. Stable machine, 303. Stagger, 83, 84, 164, 383, 407. Stanton, Dr., 47. Statoscope, 400. Steel, 16. Strain, 20, 101, et seq. Strainers, 169, 170. Stream lining, 208, et seq. chassis, 279. Stress, 20, 101, et seq., 148. airscrew, 295. beams, 1 14. chassis, 269- 71. fuselage, 247, 249. Stress diagrams, 26, 34. reciprocal figures, 35. Struts, 121, 123, 124. tapered, 121, 174-76, 184-87, 189, 191. stream line, 129. interplane, 165, 171-74. inclined, 213, 229. Strut resistance, 213, 214. Sturtevant, 246, 253. Superimposed aerofoils, 79, 80. TABULAR integration, 120. Tail-plane incidence gears, 363, 379- Tail plane, 80, 346, 379. design, 357. interference, 80. loading, 347. skid, 277, 278. Theorem of three moments, 113, 154, 155. Tie rods, 167, 168. Timber, 13, 14. seasoning, 13. Torque, 286. Tractive power, 289. airscrew, 231. Tractor, 139, 176. Trailing edge, 74. reflexure, 76, 77. Triangle of forces, 26. Triplane, 5, 80. Trusses, 136. Thrust, 288, 293. u NITS, 87, VENTURI meter, 393, 394. Vickers Vimy-Rolls, 425, 432, 436, 437- Vimy bomber, 240. commercial, 244, 245, 248, 432. V. squared law, 210. 454 WASH from main planes, 349. Weight, 2, 238. Wind pressure, 24, 25. tunnel, 38-40. Wing assembly, 206, 207. design, 130, et seq., 160, 183, 184, 443- folding, 8, 208. resistance, 384, 419. sections and characteristics, 90-100. stresses, 148, 152, 153. structures, 130. tip losses, 287. Index. Wings, loading, 149, 150. - pressure distribution, 62, 63. - weights, 149. Wires resistance, 228, 229. Wright brothers, 424, 425. glider, 8. AWING, 305- JL - moment, 324, 326, 328, 369. Young's modulus, 21, 114. 235. 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