»n^ />ii 'iu v^. .0^. \^ IMAGE EVALUATiON TEST TARGET (MT-3) // ^ A.-^ / z 1.0 I.I 1.25 2.5 2.0 1.4 1.6 V] m # ^/ W '^ /A Photographic Sciences Corporation 4 23 WEST MAIN STREET WEBSTER, N.Y. )4;80 (716) 872-4503 :\^' \ % v ■^\ <^\^\ Oi'' CIHM/ICMH Microfiche CIHM/ICiVIH Collection de microfiches. Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions historiques \ * (', and draw the line A E; consider A E to represent a spring pivotally sup- ported at A and pivotally connected at /f, and of such force as to counterbalance the centrifugal force of B, or of its resultant weight at E. Then, if the zero of tension of this spring is at yl, it will counterbalance correctly the centrifugal force of the weight at all other positions, as at /ii, for the spring lies in the line of action of centrifugal force, and its leverage I) (■ about C is the same as that of the centrifugid force. From which it is clear that a spring pivotally adjusted at A, and pivotally con- nected to the weight-arm (d i' n ^e line B (\ is cor- rectly located to produce exact isoj- In Fig. 5 let A, li, 7>'„ and C pivot, as before. Draw lines yl 6* a C through any point U on the lin>. From the point E drxw AM,, parallel to ^> .. to El. The figure ^1, E Ei is exactly similar to the figuv^ A B Bf, and corresponding sides of the figures are parallel to each other. It will be clear from the foregoing and from inspec- tion, that if a spring be pivotally connetited from Ai to E, and has its zero of tension at Ai, and is adjusted to balance the centrifugal moment of the weiglit, it will balance it in all other positions, as at Ei. But the arc B Bi need not have been drawn through the centre of gravity B, for, from the previous demon- stration (see Fig. 3), it could have been drawn through any other point of line B C ; therefore, the line B A might have had any direction between that of the direction of the line A to > hft, weight, and 'aw an arc from ing Bi C at El- and connect Ai 8 NOTKS ON THE TIIKOltY OF HlIAFl' GOVEIINORS. tlmt of the (lirnction of a lino A /?.., parallel to />' f. SiK-h un iudofinito nutnlx^r of coiiHtruetioiiH would brin^' tlio poir' /I, at any ponition on tho lino A (\ or its eontiuuatioii tlirouj^h A. It liaH therefore beeTi shown that a spring pivola/ly sicung at any jxnnt on the line (J A, or it,s nontinnation through A, and pivotally connected to any point on C B or its continuation throwjh B, arid Fig. a \/ ye having its zero of extension at the first-named point, is correctly placed to produce exact isochronism. Referring to Fig. 6, it will be clear without demonstration NOTEiJ ON THE THLOBY OP 8HAFT QOVERNOHH. 9 HII I at 7 at ally and ,tly ion timt t'lio point of connoction to wei^lit-iirin nned not be ay tlie lino /> « '. It may bi* anywhoro or the woiglit-arm, as at / vidnl a now znra lino (I (' bo drawn anguh;;ly tUo same arfiauoe and (lik'ocrtion from J (' ah tho line /''('from A't'. it follows that tlu! point of conno(!tion of Hprin*^ to woi^lit- arm, and tho dinuition of action of H})ring, ininj tk svhvttil cnliirfy at rdUihnn, or for convenioneo, provided only that tho length betwaon pivots and tho tension of isprin<^ bo fixed according to tho principles laid down. The ctinclusions arrived at by tho preceding reasoning may be expressed in tho form of a second Theoreu) : The combined zero and fixed pivotal point of a spring, anangotl to act isochrononsly on any point of tho line of weight-arm from weight pivot through centre of gravity of weight, may be taken at any j)oiut on the line from weight pivot through centre of shaft. Moreover, the spring force required will be inversely -s the distance of the fixed pivot from the weight-arm pivot. In Fig. 7, ^otters A A^ B C E represent the stn :. parta as in Drop porpendiculars C G and CD from (J on lines A K Fi.'. 5 Fig. 7. and 7s' vl, respectively; also from ^drop the perpendicular EF on line A C. Then similar triangles yl, EFn,n(\. A^ CD, as well as A F E and A C (S, are formed, from which proportions may be made as follows : AE A,G EF. CD. AC A,E CG, EF. 10 NOTES ON THE THEORY OF SHAFT GOVERNORS. Multiplying together, equating products of extremes and means, and canceling K F, we Lave AEx CGy. A,C=AyE^ CD -K A(\ or AE^ CG^AE-x CI) x~ AC AiC If we consider A C to be the unit of force oi the spring when the fixed pivot is at A^, and J, the unit of force when the pivot is at A, then this becomes intelligible. It shows that the linear extensions of the springs A E and AJ^., multiplied by their leverages G C ■axiOi CD and by the units of force of the springs, are equal. The points A' and A^ were taken at random, which makes the demonstration general. A further consideration of Fig. 3 Avill show tliat when A is the fixed pivot the unit of spring force is inversely as the distance of the point of connection on line B C from C. If, therefore, we have computed the centrifugal force of the weight />*, we have merely to multiply this centrifugal force by the ^. B C A C r T^' r , ■■,. ratio j^ X -^— ^ (see b ig. 5 for illustration) to find the corre- sponding balancing spring force ; or to multiply the centrifugal force per inch of radius by this ratio to find the correspond- ing spring force per inch of extension. The linear extension of the spring was before shown to be EAi. 4. Af proximate isochronism. In Fig. 8 let AD C be the centres t.i shaft, of gravity of weight, and of pivot, as before. Draw a line from B through A to any point O. Let C be the fixed point of a spring pivotally attached to //, and having its zero of extension at A. It is clear that the arrangement may be made isochronous for the two positions A and B. For the moment consider O to be infinitely removed from A, and investigate the mid-position of B at />'i. Draw />', O^ parallel to B O, and A A^ parallel to i?i C; also A B^, and B C at right angles to A B^ through C. Lat A C be represented by 11 and the angle A C By by a. Then the spring moment at By is R- sin. a, and the centrifugal moment is 2 /?'sin. - x cosin. ^, which expressions are equal to each other, by trigonometry. Therefore, wiuh a spring so located NOTES ON THE THEORY OF SHAFT GOVERNORS. 11 and adjusted, a third point, i?i, is isochronous. If the point O is a finite distance from A, as shown, it will be found that there will still be a point Bx, near the middle of the arc A B, which will be isochronously balanced ; other points, however, between A and Z?i will have their spring moment too small, and points between B^ and B will have their spring moment too large. Suppose the spring to be arranged as in Fig. 9, the point B^ being the inner or initial position of the weight B ; then clearly, from previous demonstrations — being the pivotal jooint of the spring — to produce isochronism at points Z>\ and B, A^ must be the zero point of the spring. It will be found as before that a point nearly midway between B and B^ is also isochronous ; also, if the angle B C By is not large, the approach to complete isochronism is very close. This corresponds to the arrangement commonly used in prac- tice. Clearly, the arc Ji By may be drawn in any other place 12 NOTES ON THE THEORY OP SHAFT GOVERNOES. from C as a centre, as at D D„ the line of the spring being made to pass through these two points, and their angular distance apart being the same as that of U and /?,. \ 5. Influence of the weight of the spring. Let the spring be applied, as in Fig. 10, and let G be its centre of gravity. Determine its moment m of centrifugal force about its pivot 0, and divide bj the length of the spring I), O, which we will call I. Owing to the weight G being constant, and the direction of / practically always the same, — is very nearly con- stant— ^/i being the moment of centrifugal force and I tlie length of the spring— for any degree of extension between I)^ and 1), and the quantity -^ is the tangential force at /), due to the cen- trifugal force of the spring. Extend the arc D Z>, across the line A C, and lay off each side of the line to F and J'\ one-half the arc JJ D,. Draw radial lines from ^'to 7'\and /;. Draw a circle through O whose diameter is the cho-d I) />,. From A NOTES ON THE THEORY OF SHAFT GOVERNORS. 13 made tance draw lines tangent to this circle, crossing the radial lines C F and CFi at K and J\\. Let d represent the distance A K. Then, since j is the force, . is the weight, which, concen- trated at Kov /ri— according to the position of the weiglit-arm /> ("— Avill produce a centrifugal moment about 6' almost exactly equivalent, at the three points K, K^, and (">,, to the centrifugal moment due to the spring weight. Between the extreme posi- tions K and /li and the central position the action of such a 'ntre bout hich [the con- iigth \D, cen- the half iw a n A / ; io weight is not exactly equivalent to that of the spring, but with a n. /derate arc the error is extremely small. Tlie object of determining the location and amount of the equivalent weight A' is to find the influence of the spring on the location of a correct centre of gravity line B C. G. Influence of the ivcight of the link. Theorem : If a governor link be constrained to move at one end in a circular path about the centre of the driving shaft, and u NOTES ON THE THEOBY OP SHAFT GOVERNORS. at the other end in a circular path about the centre of the weight pivot, then is the centrifugal e£fect of the link the same as If that portion of the weight of the link were concentrated at Its weiglit-arm pivot, which would rest on its support if the link were placed in a horizontal position on two end supports rnf'^'}^ ^^^ '^ ^^ ^^^ ^^'^^^ ^e"<^re, 6' the weight-arm pivot I^ O the link, and G the centre of gravity of the link. G A = 1 1 "^7 ^^ *''^^'' *° represent the centrifugal force of the link. Clearly, this force may be resolved into two forces, Om and Z », the sum of which is equal to G A, while their ratio is as 6^ ^ to G 0, and their direction of action parallel to G A. Eesolye Om into the components Oj> and Oq, Op having a radial direction from A, and q lying in the line of the link. Lay off |> y from L to q„ in the direction of the link, and combine ^^, with Ln by the parallelogram of forces, which gives the force Z. as the total resultant force of the link tending to rotate tne weight-arm about its pivot. DrawZ^, also Gt parallel to A, and tv parallel to ^ Z By geometry G A is divided at ^•, and Z A at U in the same ratio as Z O at G; therefore, since Om was made equal io Gv m amount and direction, the triangles m Oq and Gvt are equal Ln was made parallel to and equal iovA, and ns is parallel I NOTES ON THE THEORY OF SHAFT GOVERNORS. 15 to and equal to v t by construction ; therefore the triangle nsL is equal to the triangle vf A, and Z« is equal to J. iJ in amount and direction. But At AL OG OL or, At = AL -x OG T)L Ir other words, when the centrifugal force acting at G is rep- resented by the radius A G, that acting at L may be represented by the radius .1 Z, multiplied by a fraction which is the ratio of the weight G, which would be supported at L, provided the link were to rest in a horizontal position on two supports at Z and 0; which was to be proved. If the centre of gravity of the link were at its centre, as is common, then it would be exactly right to consider one-half the link concentrated at Z. 7. FricHonal effect of valve. In Fig. 12 let A C represent the maximum tension of spring and B C the tension to inner position of weight arm. Let A D and ^Z' represent the spring force corresponding to positions A and B of weight-arm. Assuming perfect isochronism between weight and spring, then .1 D and BE also represent the bal- anced centrifugal force, and this force for any intermediate posi- tion of weight-arm is the corresponding height from the line A B to the line 1) K Up to this time we have neglected the effect of valve-gear friction. Supposing this effect to be a constant force acting in the same direction as the centrifugal force of the weight, then it may bo 16 NOTES ON THE THEORY OF SHAFT QOVERNORS. represented bj a line parallel to and above I) C, as FO^. If the constant friction of the valve-gear acts against centrifngal force, or with the spring, then F, (I, parallel to I) (\ may represent its Fig. 13. effect. In the former of these cases the maximum spring force becomes .1 /; and the maximum spring tension A C^, while^in the latter case these quantities become A F^ and A d. In some constructions the connection between the governor and the valve-gear is such as to produce a variable effect of fric- Fi2. 14, tion. This is the case with the " Buckeye " governor, shown in Fig. 13, in which, moreover, this friction has a centripetal effect. . If the ;al force, esent its iig force e in the overnor 1 of fric- N0TE3 ON THE THEORY OF HIIAFT G0VERN0H8. 17 In Fig. 14 this variable resistance is illustrated by the curved line F, A',. By reference to Fig. 13 an auxiliary spring P will be seen, which is designed to act through a little more than half the range of the weight-arm, and to produce an effect illustrated by the shaded portion of Fig, 14. The result is that a line F, 6\ approximately straight, illustrates the centripetal action of the main spring, B C\ being its initial tension, and B Ei its total force at first stop, or initial position. 8. Ttmrtia in a shaft gcvernor. In Fig. 15 let A and C be the centres of shaft and weight pivot, respectively, and consider the total effective weight of the governor weight and arm centred at B. Inertia acts on the weight B at right angles to the line A B. From analytical mechanics (see Weisbach) the force of inertia may be represented by the expression P = M B .^, while cen- trifugal force is F — of MR; in both expressions oo is the angu- lar velocity, J/ the mass, A' the radius A B, and d t the small interval of time in which a change of velocity occurs. Substituting for co its equivalent value 2 tt T, in which T is the number of revolutions per second, and differentiating the equation for centrifugal force, since it is only the difference of force due to change of speed which is effective, we have 1 own in P r= 2 I effec!;. and a dF = ^e MBTdT. 18 NOTES ON THE THKORY OF SHAFT GOVERNORS. li B L or Ji li, accordiiip; to the direction of motion, repre- sents the force P, and // ^1, the force il F, tlien li A, or /> Ifi is the resultant of these forces, and the tangent of the angle .li li Z„ or Ai B Ai, which angle we designate by n\ is P tangent rr = ^-p , or, substituting above values, tangent a =-. ^^-^^ t We see from this that the effect of inertia to increase or decrease (according to the direction of motion) the moment of Fig. 16. force about the weight pivot is less the greater the number of revolutions per unit of time, and is greater the less the interval of time in which the change of speed takes place. Let us assume that the weight B is no longer concentrated in a point, but is spread out into a disk of considerable size, as in Fig. 16, whose radius we call r ; then the force of inertia relative to the axis A is greater than before. By a well-known law of inertia, the radius of gyration of the — , therefore the force of inertia acting at B is V'2 B -,-!)' V'2 / , r d GO B '^^ dt ' weight is i? + I NOTES ON THE THEORY OF SHAFT 00VERN0R8. 19 Substituting Irr Tior ro, and dividing by d F as before, we have ri + ---V tangent a ^n TdT Suppose r to be \ of R, T to be three revolutions per sec- ond, and d t to be one second ; then tangent (^ becomes 0.0406, and a is less than 2i°. If o?< is ' of one second, then (x becomes about 22', and if dt is rJn of one second, a is about 70''. The extremes of these three cases are shown grapnically in Fig. 16 for both right and left hand motion. This illustrates to how great an extent, when changes of speed are sudden, iner- tia force may be useful to assist centrifugal force ; also to what a slight extent inert i;i acts when changes are not sudden. It also shows that if the direction of motion be badly chosen, Fig. 17. the combined forces may produce an instantaneous moment about the weight pivot in the wrong direction, thus interfering with sensitive governing. As to the actual value of d t in practice, it may often be a very 20 NOTES ON THE THEORY OF SHAFT QOVERNORS. small quantity, for in an engine having dead points the velocity changes a number of times, to a greater or less extent, during each revolution. These changes are loss the heavier tl.o fly-wheel, therefore with a light fly-wheel an inertia governor should be specially efficient. In one of tlie papers referred to at the beginning of this ar- ticle, Mr. Armstrong advocates the use of inertia in the way wliich wouLl reduce the efl'ectivc moment about (\ for the sake of " stability." The fiict seems to be, however, that stability and sensitive- ness are best arrived at by using the force of inertia to aid cen- trifugal force, as in the left-hand motion of Fig. 16. Fig. 17 illustrates a governor for a single-valve engine — de- signed by Mr. J. W. Thompson — which is said to have performed so perfectly that no perceptible variation of spoed in the range of the governor could be detected by careful test, and there was no trouble from racing. Fig. 18. It will be noted that the arc through which the weight-arm moves is so small that isochronism could be practically perfect, while inertia was utilized to a great degree. In Fig. 18 is represented a very ingenious method of combin- ing a separate inertia weight with a shaft governor. This was NOTES ON Till". THKOnV OF flnAPT GOVERNOIIP. 21 applie.l l.y Mo88r8 Buiicn^ft ct Lewin. of Plnla