^r, /j/y Z.^ ^-Tyt- V. ^^^/V /^/ ^/T^ University of California • Berkeley THE i^i GYROSCOPE. BY MAJOR J. G. BARNARD, A. M., CORPS OF ENGINEERS, U. S. A. m FROM Barnard's American journal of education. NEW YORK: D. VAN NOSTRAND. HARTFORD: F. C. BROWNELL. 1858. IfflfflP^^PPBfflPMffiMl Mil mfi ^^^iPiiPi'^^ ERRATA. In the first paper of this pamphlet the references to pages should, wherever met with, read instead of " 52, " " 53, " " 54, " — " 540, " " 541, " " 542. " In the second paper, for "spiral" '^' and "spiral motion, " read "helix" and helical motion. " f^O. f^^ ^-.^^^^*t^3^ cos x^ Oyrrrcos 6 cos ip sin 9-- sin V cos (p cos yj Oz=z-— sin 6 cos q> cos y^ Oy=:cos ^ cos V cos 9)-f-sin y^ sin 9 cos 2; J 0^= cos ^ cos 0j Oy^rsin cos v The differential angnlar motions, in the time dt, about the axes Ox J, Oy^j Oz^^ will be Vj;dtj Vyd% and V;2C?^. We may de- termine the values of these motions by applying the laws of composition of rotary motion to the rotations indicated by the increments of the angles <9, 9 and V- If 6 and 9 remain constant the increment d^j would indicate that amount of angular motion about the axis Oz perpendicular to the plane in which this angle is measured. In the same man- ner dcp would indicate angular motion about the axis Oz , ; while dd indicates rotation about the line of nodes ON. In using these three angles therefore, we actually refer the rotation to the three axes Oz^ Oz^, ON, of which one, Oz, is fixed in space, another, Oz , , is fixed in and moves with the body, and the third, ON, is shifting in respect to both. The angular motion produced around the axes Ox^, Oy^, Oz^, by these simultaneous increments of the angles 9, ^ and v^, will be equal to the sum of the products of these increments by the cosines of the angles of these axes, respectively, with the lines Oz, Oz, and ON The axis of Oz^ for example makes the angles ^, 0° and 90^ with these lines, hence the angular motion v^ dt is equal (taking the sum without regard to sign) to cos 6 dip-]- dtp. In the same manner (adding without regard to signs), Vxdt=QO^ Xy Oz(iv^+cos (pdd and Vy dt= cos ?/ j Oz c? v^ + cos (90° + 9) <^ G- But if we consider the motion about Oz^ indicated by dqy, posi- tive, it is plain from the directions in which 9 and v are laid off on the figure, that the motion cos OdH^ will be in the reverse di- rection and negative, and since cos 6 is positive c?V^ must be re- garded as negative, hence Vzdt=d(p— cos Odip. The first term of the value of Va;dt, cos x^Ozd^p [since cos a:;, Oz (=— sin 6 sin 9) is negative and c?v is to be taken with the negative sign] is positive. But a study of the figure will show that the rotation referred to the axis Ox , , indicated by the first term of this value, is the reverse of that measured by a positive increment of 6 in the second, and hence, (as cos 9 is positive,) dd must be considered negative. Making this change and substi- tuting the values given of cos x^ Oz, cos y, Oz, and for cos (90° -hqp),— sin (f, we have the three equations J. G. BARNARD ON THE GYROSCOPE. 641 V;rC?^=:siii 6 sin q>dip—cos will be exactly that due to the constant axial rotation ndt^ augmented by the term cos OdH^^ which is the projection on the plane of the equator of the angular motion dip of the node. This term is an increment to ndt when it is positive, and the reverse when it is negative. In the first case, the motion of the node is considered retrograde — in the second, direct. The first member of the second equation (4) being essentially positive, the difference cos ^— cos « must be always positive— that is, the axis of figure Oz , can never rise above its initial an- gle of elevation «. As a consequence -7- [in first equation (4)] must be always positive. The node N, therefore, moves always in the direction in which V is laid off positively, and the motion will be direct or retrograde, with reference to the axial rotation, according as cos^ is negative or positive — that is, as the axis of figure is above or below the horizontal plane. In either case the motion of the node in its own horizontal plane is always progressive in the same direction. If the rotation n were re- versed, so would also be the motion of the node. If this rotation n is zero, -7- must also be zero and the second equation (4) reduces at once to the equation of the compound pendulum, as it should. Eliminating ~ between the two equa- tions (4) we get . ^^dd^ 2Mgy ^.\^ C'-^n'^ , . ^-, , n \ sin2 ^ — — z=z f-^ fsm^ a — (cos — cos a) | (cos ^— cos a). The length of the simple pendulum which would make its oscillations in the same time as the body (if the rotary velocity A n were zero) is t^^.* If we call this X and make for simplicity * The length of the Bimple pendulum is (see Bartlett's Mech,, p. 252) X= — — The moment of inertia A=M(kj^ +7*); hence -,-7- =x. 644 J. G. BARNARD ON THE GYROSCOPE, — 7:r- = -^ the above equation becomes sin 2 0^ =z^ [sin 2 ^- 2 192 (cos ^-cos a)] (cos 6»-cos a) (6) and the first equation (4) becomes ^\n2 0-^—2^ f (cos^-cosa). (7.) Equation (6) would, if integrated, give the value of 6 in terms of the time ; that is, the inclination which the axis of figure makes at any moment with the vertical ; while eq. (7) (after sub- stituting the ascertained value of 0) would give the value of Knowing this fact, we may assume that the impressed velocity n is very great, and hence cos <9— cos « exceedingly minute, and on this supposition, obtain integrals of equations (6) and (7), which will express with all requisite accuracy the true gyroscopic motion. For this purpose, make 6=ia — M, ddzzi—du in which the new variable u is always extremely minute, and is the angular descent of the axis of figure below its initial eleva- tion. By developing and neglecting the powers of u superior to the square, we have siu^ 6 =. sin2 a— w sin 2a -|- u^ cos 2a * cos ^— cos oczizusmoc^^u^ cos a substituting these values in eq. 6 we get J "dt- 7 . f I V2wsina — t^2^(>QS«-[-4/^2j i |9 having been assumed very great, cos a may be neglected in comparison with 4:^^^^ and the above may be written J I y^ ^f — ^ / T\ V 2w sin a —• 4/52^2 V ) Integrating and observing that u = o, when t = o, we have * By Stirling's theorem, in which IT, U\ TJ" &c. are the values of/ (m) and its diflferent co-efficients when u is made zero. Making/ (w) =sin2 (»—«), and recollecting that sin lu =2 sin u cos u and cos 2w= cos*w — sin2 M, we get the value of sin^ ; and making /(w).= cos(a— w)-cosa the value in text of cos B — cos a is obtained. f Eq. 6 may be written •K d9^ , (cos Q - cos a)2 -^=2(cos0-cosa)-4/3^ ,i„. ^ \ By substituting the values just found, of dB, sin^ Q and cos 9 — cos a and per- forming the operations indicated, neglecting the higher powers of ii, (by which fcOS ^ — " COS Ot) I '^Q reduces simply to w^) and deducing the value ^ (Z<, the expres- sion in the text, is obtained. No. 9.— [Vol. Ill, No. 2.]— 35. 546 J. G. BARNARD ON THE GYROSCOPE. 17 , 1 r , 4.s^"l. ^^.^=-.arc|-cos=l--^_J. or, (since cos 2a = 1—2 sin^ a) ?* = ^sinasin2^ l^-.^j (9.) Putting a— t* in place of (equat. 7) neglecting square of u, we get from which, observing that V = 0, when ^=0 These three expressions (9), (10), (11), represent the vertical angular depression — the horizontal angular velocity — and the sin a . du , . , ^ 2/3 4«2 „ / - may be put m the form -: — . * V2m sin a -4/32^2 '' ^ sma J sin a sin a Call -T^ =R, and the integral of the 2d factor of the above is the arc whose radius is R and versed sine is u ; or whose cosine is R — ^i ; or it is R times the arc whose u cosine 1 — ^ with radius unity. Substituting the value of R in the integral and 23 1"^ multiplying by the factor 7 — we get the value of -^ t, of the text. f In eq. (7) if we divide both members by sin^ 9, and, in reducing the fraction cos 9 - cos a r---r — > use the values already found and neglect the square, as well as higher powers u, (which may be done without sensible error owing to the minuteness of w, though it could not be done in the foregoing values of dt and t, since the co-efficient 4^2 in those values, is reciprocally great, as u is small) the quotient will be simply u sin a Substituting the value of u and dividing out sin a w^e get the value of — in the text. The integral of sin 2 13 | 5^ < ,:-'f.e:i iXi »^ ,' riPiV: -^ X\ ^^^:j' : -A M"^-' 547 contd. v/e see then the revolving body does not in fact main- tain a uniform unchanging elevation, and move about its point of support at a uniform rate, (as it appears to do) But the axis of figure generates what may be called s- corrugated cone , and any * The assumption that-y^-O iriien t. is zero supposes that the initial position of the node coincides with the fixed axis of x. In my subsequent illustrations and analysis I suppose the initial position to be at 90° therefrom, which would require to the above value of vA, the constant -'TT to be added. The horizontal ^ Z angular motion of the axis of figure is the same as that of the node. 548 J. G. Barnard on the Gyroscope point it would describe an undulating curve (fig«2) whose superior culminations a, a , a , &o, , are CUSPS lying in the same horizontal plane, eind whose Fig. 2 sagittae cb, ^y , &c. , are to the amplitudes aajfl.a* If the initial elevations is 90^, this ratio is as the diajneter to the circumference of the circle : a property v/hich indicates the cycloid * Assuming X »90° and sin« =1, equations (9) and (10 ) will give, by elimination of sin^^ ^^ f. dLi- -x^ x/3;i«. substituting this value in eq. ( d) we get the differential equation of the cycloid generated by the circle whose diameter is— 4— - In this position of the axis, both the angles u and 1/^ are arcs of great circles described by a point of the axis of figure at a units distance from 0^, and owing to their minuteness may be considered as rectilinear co-ordinates. If c{ is not 900, the sagittae b£=/7isin<\; but then-, while the angular motion is the same, the arc described by the same point of the axis v/ill be that tVA "' A v^ ',* • ! ":j": 7 ■ T'.i ; "^^^'^ ;"77 ™r ' . --i J J- o . ; -c i ' 0- " 'C/ • i>ri *: C x J" ! 548 contd. of a small circle « whose actual length will likewise be reduced in the ratio of 1 : since* The curve is therefore a cycloid in all circumstances; and the axis of figure moves as if it were attached to the circumference of a minute circle whose dieimeter is •r^ sin ^ , which rolled along the horizontal circle, aa^a',' about the verticeuL throughthe point of support. The centre je of this little circle moves with uniform velocity. The first term of the value of y (equation ll) is due to this uniform motion : it may be called the mean precession * i^o :.'^''Ci:o .>5:. •-.; .;..: J-. G. BARNARD ON THE GYROSCOPE. 549 The second term is due to the circular motion of the axis about this centre, and, combined with the corresponding values of u^ constitutes what may be called the nutation. These cycloidal undulations are so minute — succeed each other with such rapidity, (with the high degrees of velocity usually given to the gyroscope,) that they are entirely lost to the eye, and the axis seems to maintain an unvarying elevation and move around the vertical with a uniform slow motion. It is in omitting to take into account these minute undulations that nearly all popular explanations fail. They fail, in the first place, because they substitute, in the place of the real phenome- non, one which is purely imaginary and inexplicable^ since it is in direct variance with fact and the laws of nature ; — and they fail, because these undulations — (great or small, according as the impressed rotation is small or great) furnish the only true clue to an understanding of the subject. ^The fact is, that the phenomenon exhibited by the gyroscope which* is so striking, and for which explanations are so much sought, is only a particular and extreme phase of the motion ex- pressed by equations (6) and (7) — that the self-sustaining power is not absolute^ but one of degree — ^that however minute the axial rotation may be, the body never will fall quite to the vertical ; — however great, it cannot sustain itself without any depression. I have exhibited the undulations, as they exist with high veloci- ties, — when they become minute and nearly true cycloids ; with low velocities, they would occupy (horizontally) a larger portion of the arc of a semi-circle, and reach downward approximating, more or less nearly, to contact with the vertical : and, finally^ when the rotary velocity is zero, their cusps are in diametrically opposite points of the horizontal circle, while the curves resolve themselves into vertical circular arcs which coincide with each other, and the vibration of the pendulum is exhibited. All these varieties of motion, of which that of the pendulum is one extreme phase and the gyroscopic another, are embraced in equations (6) and (7) and exhibited by varying § from to high values, though, (wanting general integrals to these equations) we cannot determine, except in these extreme cases, the exact - elements of the undulations. The minimum value of may however always be determined by equation (8). If we scrutinize the meaning of equations (6) and (7), it will be found that they represent, the first, the horizontal angular component of the velocity of a point at units distance from 0, and the second, the actual velocity gf such point.* * In more general terms equations (4) express, the first, that the moment of the quantity of motion about the fixed vertical axis Oz remains always constant : the second that the living forces generated in the body (over and above the impressed axial rotation) are exactly what is due to gravity through the height, h. Both are expressions of truths that might have been anticipated ; for gravity 660 J. G. BARNARD ON THE GYROSCOPE. For sin ^ -77 is the horizontal, and -7- the vertical, component of this velocity. Calling the first t'A, and the second v^, and the resultant Vg^ and calling cos ^— cos «, (which is the true height of fall) A, those equations may be written Cn h ''= A ^6 <^) This velocity Vs (as a function of the height of fall) is exactly that of the compound pendulum, and is entirely independent of the axial rotation n. Hence, (as we might reasonably suppose) ro- tary motion has no power to impair the work of gravity through a given height, in generating velocity ; but it does have power to change tJie direction of that velocity. Its effect is precisely that of a material undulatory curve, which, deflecting the body's path from vertical descent, finally directs it upward, and causes its velocity to be destroyed by the same forces which generated it. And it may be remarked, that, were the cycloid, we have de- scribed, ^wc/i a material curve, on which the axis of the gyroscope rested, without friction and without rotation, it would travel along this curve by the effect of gravity alone, (the velocity of descent on the downward branch carrying it up the ascending one,) with exactly the same velocity that the rotating disk does, through the combined effects of gravity and rotation. Equation {a) expresses the horizontal velocity produced by the rotation. If we substitute its value in the second, we may deduce de__ \2g C-^n^ h^ If we take this value at the commencement of descent, and before any horizontal velocity is acquired, (making h indefinitely small), the second term under the radical may be neglected, and the first increment of descending velocity becomes ^ h, pre- cisely what is due to gravity, and what it would he were there no rotation. Hence the popular idea that a rotating body offer s any direct resistance to a change of its plane, is unfounded. It requires as little exertion of force (in the direction of motion) to move it cannot increase the moment of the quantity of motion about an axis parallel to itself; while its power of generating living force by working through a given height, cannot be impaired. Had we considered ourselves at liberty to assume them, however, the equations might have been got without the tedious analysis by which we have reached them. J. G. BARNARD ON THE GYROSCOPE. 551 from one plane to another, as if no rotation existed ; and (as a corollary) as little expenditure of work. But deflecting forces are developed, by angular motion given to the axis, and normal to its direction, which are very sensible, and are mistaken for direct resistances. If the extremity of the axis of rotation were confined in a vertical circular groove, in which it could move without friction ; or if any similar fixed re- sistance, as a material vertical plane, were opposed to the de- flecting force, the rotating disk would vibrate in the vertical plane, as if no rotation existed. Its equation of motion would become that of the compound pendulum, — -= V^h, What then is the resistance to a change of plane of rotation so often alluded to and described'/ A mzinomS^lentirely . The above may be otherwise establisEeJ Ifin equations (3) we introduce in the second member an indeterminate horizontal force, g\ applied to the centre of gravity, parallel to the fixed axis of ?/, and contrary to the direction in which, in our figure, we suppose the angle v to increase, the projections of this force on the axes Ox^^ Oy^^ will be a'g' and h' g' and the last two of these equations will become, (calhng cosines x^Oy and y^Oy^ a' and 6',) Advy—{C—A)nVxdt^^yM(ag-ira'g')dt AdVa:+\C—A)nVydt=-yM(bg-{-h'g')dt Multiplying the first by Vy and the second by v^ and adding A (vydvy -^Vxdvx )=zyM\g (a Vy^hvj;)d t-\-g' (a' Vy—h'vj;)d t~\. But (avy—bva;)dt}iSiS been shown (p. 53) to be =d.cosO^ — and by a similar process it may be shown that {a'Vy — 'b'Vx)dt= =d. (sin cos V'). (For values of a' and h\ see p. 52.) Let us suppose now that the force g' is such that the axis of the disk may be always maintained in the plane of its initial po- sition xz. The angle v^ would always be 90°, dip=0, and c?.(sin^ cos y^)=0. That is, the co-efficient of the new force g' becomes zero; and the integral of the above equation is as before (p. 54), A(Vy2+Va;^)=2YMg cosd-^h. , But the value of Vy^+v^^ likewise reduces (since -^=0) to -7-^ and the above becomes the equation of the compound pendulum. dO^ 2Yj}fg 2, a (g) 7772^^ — A — ^^^ ^"^~^^^X ^^^^ ^— cos a), (h being determined.) This is the principle just before announced, that, with a force so applied as to prevent any deflection from the plane in which gravity tends to cause the axis to vibrate, the motion would be precisely as if no axial rotation existed. 552 J. G. BARNARD ON THE GYROSCOPE. To determine the force of g' ; multiply tlie first of preceding equations by J, and the second by a, and add the two, and add likewise A{vydh+Vxda)=—AndQOQd (see p. 54) and we shall get Ad{hVy-\-aVx)-^Cndco^d=:yMg'{a'h — ah')dt. By referring to the values of a, a', h, h\ and performing the operations indicated and making cos ^=o, sin v=l, the above becomes, Ad{bvy-\-aVa:)-\-Ond cos 0=YMg^ sin 6 dt. But the value of {hvy+av:^) (p. 54) becomes zero when --^^=0. TT / OndcosO Cn dd ^ Hence g :=—% ^^ — = — * yM&mOdt yMdt The second factor — is the angular velocity with which the axis of rotation is moving. Hence calling Vs that angular velocity, the value of the deflect- ing force^ g' may be written (irrespective of signs), ^-^^^- (^) that is, it is directly proportional to the axial rotation n, and to the angular velocity of the axis of that rotation. By putting for 0, Mk^ (in which h is the distance from the axis at which the mass M^ if concentrated, would have the moment of inertia, (7,) the above takes the simple form In the case we have been considering above, in which g' is sup- posed to counteract the deflecting force of axial rotation, the angu- lar velocity Vs ,ot—j- (equation g) is equal to hr (^^^ ^ ~ ^^^ ")• But in the case of the free motion of the gyroscope, this de- flecting force combines with gravity to produce the observed movements of the axis of figure. If, therefore, we disregard the axial rotation and consider the body simply as fixed at the point 0, and acted upon, at the cen- ter of gravity, by two forces — one of gravity, constant in inten- sity and direction — the other, the deflecting force due to an axial C rotational, whose variable intensity is represented hj—=rz.nvsj * The effect of gravity is to diminish 9 and the increment dd is negative in the case we are considering. Hence the negative sign to the value of ^', indicating that the force is in the direction of the positive axis of y, as it should, since the tendency of the node is to move in the reverse direction. J. G. BARNARD ON THE GYROSCOPE. 553 and whose direction is always normal to the plane of motion of the axis ; we ought, introducing these forces, and making the axial rotation n zero, in general equations (8), to be able to de- duce therefrom the identical equations (4) which express the mo- tion of the gyroscope. • a This I have done ; but as it is only a Verification of what has previously been said, I omit in the text the introduction of the somewhat difficult analysis.* Equation (5) becomes (in the case we consider), by integration, (p■=^nt^\^^J cos a which, with the values of u and V already obtained, determines completely the position of the body at any instant of time. Knowing now not only the exact nature of the motion of the gyroscope, but the direction and intensity of the forces which * To introduce these forces in eq, (3) I observe, first, that as both are applied at O (in the axis Oz^ the moment L^ is still zero and the first eq. becomes, as before, CdVg = or Vg=: const. And as we disregard the impressed axial rotation, we make this constant (or v^ ) zero. Cn The deflecting force — ^ Vg (taken with contrary sign to the counteracting force Cn d9 Cn d-^ just obtained) resolves itself into two components — t> -jt and — —^ -jr sin 9, the first in a horizontal, the second in a vertical plane, and both normal to the axis of figure. The second is opposed to gravity, whose component normal to the axis of figure, is g sin 9. Hence we have the two component forces (in the directions above indicated), ^ Cnd^ , / Cn c?^^ \ 'Wf'dl ^iff--^7irr-^] sine. / Cnd^\ . ^[^-WdF) ''' I Cn dA,\ , ,^ / Cn d-^\ , ^ . ^^Cn d^ These moments with reference to the axes of y j and x j will be . " ^ / Cn dA,\ , ^ ,^ C'n (f9 -sm(P7Jf \9-;7M-ir\ 8me-cos(P7if ^ ^, and Hence equations (3) (making v^ zero, and putting for M^ and Ni the above values, and recollecting the values of a and b, (p. 53) become d-^ d^ 1 Advy = a*^^ ^e^ /ssy^, ^^ XVI. EDUCATIONAL MISCELLANY. ON THE MOTION OF THE GYROSCOPE AS MODIFIED BY THE RETARDING FORCES OF FRICTION, AND THE RESISTANCE OF THE AIR*. WITH A BRIEF ANALYSIS OF THE TOP. O^^^c^lJi^C " BY MAJOE J. G. BAENAED, A. M. Corps of Engineers U. S. A. In a previous paper (see article in this Joumal for June, 1857, to wMcli this paper is intended to be supplementary,) I have investigated the ''Self-sustaining power of the Gyro- scope" in the light of analysis. From the general equations of "Eotary motion" I have deduced the laws of motion for the particular case of a solid of revolution moving about a fixed point in its axis of figure, (or the prolongation thereof). I have shown that such a body, having its axis placed in any degree of inclination to the vertical, and having a high rotary motion about that axis, will not, under the influence of grav- ity, sensibly fall ; but that any point in the. axis will describe "an undulating curve whose superior culminations are cusps lying in the same horizontal plane ;" that this curve approaches more and more nearly to the cycloid, as the velocity of axial rotation is greater ; that when this velocity is very great the undulations become very minute and " the axis of figure per- forming undulations too rapid and too minute to be perceived, moves slowly about its point of support." I have shown how the direction and velocity of this gyration are determined by the direction and velocity of axial rotation and the distance of the center of gravity of the figure from the point of support, and that the remarkable phenomenon exhibited by the gyroscope is but a particular case due to a very high velocity of axial rotation, of the general laws of motion of such a body as described, which embrace the motion of the pendulum in one extreme and that of the gyroscope in the other, and that intermediate between these two extreme cases (for moderate rotary velocities) the un- dulations of the axis, will be large and sensible. I have likewise shown that whenever, to the axis of a rotating solid, an angular velocity is imparted, a force which I have called " the deflecting force^"* acting perpendicular to the plane of motion of that axis, is developed, whose intensity is proportional, to this angular velocity, and likewise to the rotary velocity of the body ; and that it is this deflecting force which is the imme- diate sustaining agent, in the gyroscope. In the above deductions of analysis is found the full and com- plete solution of the " self-sustaining power of the gyroscope." To make the character of the motion indicated by analysis, No. 11.— [IV., No. 2.]— 34. 530 ^- ^' BARNARD ON THE GYROSCOPE. sensible to the eye, it is only necessary to attach to the ordinary gyroscope, in the prolongation of the axis, an arm of five or six inches in length, and having an universal joint at its extremity, and to swing the instrument as a pendulum ; or, the extremity of an arm of such a length may be rested in the usual way, upon the point of the standard, when, with the centre of gyra- tion removed at so great a distance from the point of support, the undulatory motion becomes very evident. But it cannot fail to be observed that the motion preserves this peculiar feature but for a very short period. The undula- tions speedily disappear; instead of periodical moments of rest (which the theory requires at each cusp) the gyratory velocity becomes continuous^ and nearly uniform and horizontal; audit increases as the axis (owing to the retarding influences of friction and the resistance of the air) slowly falls. In short, the axis soon seems to move upon a descending spiral described about a vertical through the point of support. The experimental gyroscope, in its simplest form consists of two distinct masses, the rotating disk, and the mounting (or ring in which the disk turns). The point of support in the latter, though it gives free motion about a vertical axis, constrains more or less, the motion of the combined mass about any other. The rotating disk turns at the extremities of its axle, upon points or surfaces in the mass of the mounting, with friction ; it is rare, too, that the point of support, of the mounting, is ad- justed in the exact prolongation of the axis of the disk. Without attempting to subject to analysis causes so difficult to grasp as these, I shall first attempt to show, by general con- siderations, what would be the immediate influence of the re- tarding forces of friction and the resistance of the air upon our theoretical solid ; and then point out the further effect due to the discrepancies of figure, above indicated. Leaving out of con- sideration the minute effect of friction at the point of support, these forces exert their influence, mainly in retarding the rotary velocity of the disk. Friction — at the extremities of the axle of the disk, and the resistance of the air, at its surface, are power- ful enough to destroy entirely in a Yerj few minutes, the high velocity originally given to it. It is in this way, mainly, that they modify the motion indicated by analysis. If the rotary velocity remained co?25^ar2 Awhile the axis made one of the little cycloidal curves aba', (fig. 1) the deflecting force would be just sufficient, as I have shown (p. 556 of the article cited) to lift the axis back to its original elevation a', and to destroy, entirely, the velocity it had acquired through its fall cb. If, at a', the rotary velocity n underwent an instantaneous dimi- nution, and remained constant through another undulation, a curve, of larger amplitude and sagitta a' b' a" would be described, and the axis would again rise to its original elevation a", and again be brought to rest. We might then, on casual considera- J, G. BARNARD ON THE GYROSCOPK 531 tion of the subject, expect to see the undula- tions become more and more sensible as the rotary velocity decreased. The reverse, how- ever, is the case, as I have already stated. In fact, the above supposition would require the rotary velocity n to be a discontinuous decreas- ing function of the time ; whereas it is, really a continuous decreasing function. It is under- going a gradual diminution between a and a'. The deflecting force, which is constantly pro- portional to it, is therefore insufficient to keep the axis up to the theoretical curve aha', but a bluer Guive ah^a^ is described; and when the culmination a , is reached, it is helow the original elevation a\ But the 2d of our general equations for the gyroscope (4), [afterwards put under the sim- ple form jeq. (f)\vs^ =—h'\ which is inde- pendent of n, shows that the angular velocity of the axis will always be that due to its actual fall h below the initial elevation. On reaching the culmination a , therefore, the axis will not come to rest, but will have a horizontal veloc- ity due to the fall a'a^ and the curve will not form a cusp but an inflexion at a^. The axis will commence its second descent, therefore, with an initial horizontal velocity. It will not descend as much as it would have done had it started from rest with its dimin- ished value of n ; and, for the same reason as before, will not be able as again to rise high as its starting point a^ but to a some- what lower point a^ and with an increased horizontal velocity. These increments of hori- zontal velocity will constantly ensue as the culminations become lower and lower, while on the other hand, the undulations become less and less marked, as indicated by the ligure. I have stated in my former paper (p. 559) that a certain initial horizontal angular velocity such as would " make its corresponding deflect- ing force equal to the component of gravity, g sin <5, would cause a horizontal motion without undulation." This horizontal velocity is rapidly attained through the agencies just described : or, at least, nearly approximated to, and the axis, as observation shows, soon acquires a continuous and uniform hori- zontal motion. On the other hand, this sustaining power being directly pro- 532 J- ^- BARNARD OJM THE GYROSCOPE. portional to the rotary velocity of the disk, as well as to the an- gular velocity of the axis, diminishes with the former, and as it diminishes, the axis must descend, acquiring angular velocity due to the height of fall : hence the rapid gyration and the descend- ing spiral motion which accompanies the loss of rotary velocity. A more curious and puzzling effect of the friction of the axle is presented, when we come to take into consideration, instead of our theoretical solid, the discrepancies of figure presented by the actual gyroscope. If, with a high initial rotation, the com- mon gyroscope be placed on its point of support with its axis somewhat inclined above a horizontal position, it will soon be observed to rise. In my analytical examination (p. 543) I have stated as a deduction from the second equation (4), that " the axis of figure can never rise above its initial angle of elevation." That equation supposes that the rotary velocity n remains unim- paired, and is the expression of a fundamental principle of dy- namics — that of "living forces" (so-called), which requires that the living force generated by gravity be directly proportional to the height of fall, and involves as a corollary that through the agency of its own gravity alone, the centre of gravity of a body can never rise above its initial height.* The anomaly observed, therefore, either requires the action of some foreign force ; or^ that the living force lost by the rotating disk, shall, through some hidden agency, be expended in performing this work of lifting the mass. The discrepancy here exhibited between the motion proper to our theoretical solid of revolution and the experimental gyro- scope is due to the division of the latter into two distinct masses, one of which rotates, loith friction, upon points or surfaces in the other ; and to the fact that at the point of support (in the latter) there is r^oi perfectly free motion in all directions. The friction at the extremities of the axle of the disk, tends to impress on the mass which constitutes the "mounting," a ro- tation in the same direction. Were the motion of the latter upon its fixed point of support perfectly free, the mounting and disk would soon acquire a common rotatory velocity about the axis of the disk. But the mounting is perfectly free to turn about the vertical axis through the point of support, though not about any other. If we decompose, therefore, the rotation which would be impressed upon the mounting into two components, one about this vertical, and the other about a horizontal axis — the first X2k.QQfull effect, and the latter is destroyed at the point fo support. If the axis of the instrument is above the horizon- tal, this component of rotation is in the same direction as the gyration due to gravity, and adds to it ; if the axis is below the horizontal, the component is the reverse of the natural gyration, and diminishes it. * The first of these equations (as I have remarked in a note to p. 547) is the expres- sion of another fundamental principle — more usually called the " principle of areas." J. G. BARNARD ON THE GYROSCOPE. 533 But I have shown that the axis soon acquires, independent of this cause, a gyration whose deflecting or sustaining force is just equivalent to the downward component of gravity. The addi- tion to this gyratory velocity caused by friction when the axis is inclined upwards puts the deflecting force in eoccess^ and the axis is raised ; it is raised, as in all other cases in which work is done through acquired velocity — viz., by an expenditure of living force ; but in this instance, through a most curious and compli- cated series of agencies. The phenomenon may be best illustrated in the following man- ner. Let the outer extremity of the common gyroscope, having its axis inclined above the horizontal, be supported by a thread attached to some fixed point vertically above the point of support, so that gyration shall be free. Here gravity is eliminated, and the axis of our theoretical solid of revolution would remain per- fectly motionless ; but the gyroscope starts off, of itself, to gy- rate in the same direction that it would were its extremity ^ree. This gyration increases (if the rotary velocity is great) until the deflecting force due to it, lifts the outer extremity from its sup- port on the thread, and it continues indefinitely to rise. Try the same experiment with the axis helow the horizontal. The gyration will commence spontaneously as before, but in the reverse direction : it will increase until the inner extremity is lifted from the point of support^ (the action of the deflecting force being here reversed,) the instrument supporting itself on the thread alone. If the experiment is tried with the axis perfectly hori- zontal, no gyration takes place, for the component of rotation, due to friction, is, in this position, zero. The foregoing reasoning accounts, I believe, for all the ob- served phenomena of the experimental gyroscope, and shows how, from the theory of oar imaginary solid of revolution, a consideration of the effects of the discrepancies of form, and of the actual disturbing forces, leads to their satisfactory explanation. The great similarity between the phenomena of the top and gyroscope, renders it not uninteresting to compare the laws of motion of the two. If we conceive a solid of revolution ter- minated at its lower extremity by & point (the ordinary form of the top), resting upon a horizontal plane without fi-iction, and having a rotary motion about its axis of figure, such a body will be subject to the action of two forces; its weight, acting at the centre of gravity, and the resistance of the plane, acting at the point vertically upwards. According to the fundamental principles of dynamics, the centre of gravity will move as if the mass and forces were con- centrated at that point, while the mass will turn about this cen- tre as if it were fixed. Calling E the resistance of the plane, if the mass, and Mg the weight of the top, and z the height of 634 J. G. BARNARD ON THE GYROSCOPE. the centre of gravity above the plane, we shall have for the equation of motion of the centre of gravity* ^^£=^-^^ (!•) As the angular motion of the body is the same as if the centre of gravity was fixed, and as R is the only force which operates to produce rotation about that centre, if we call G the moment of inertia of the top about its axis of figure, and A its moment with reference to a perpendicular axis through the centre of gravity, and / the distance, GK (fig. 2) of the point of support from that centre; the equations of rotary motion will become identical with equations (3) (p. 541), substituting B for Mg Cdv^=0 ') Advy-^{C—A)v^Vxdt = yaRdt > (2.) Advx-\-{C^A)vyVzdt=:—yhRdt ) The first of equations (2) gives us Vz as for the gyroscope, equal a constant n. Multiplying the 2d and 8d of equations (2) by Vy and v^ re- spectively, and adding and making the same reduction as on p. 63, we shall get A{yydvy-\-Vxdvx)=:Ry d ,co^d. But z (the height of the centre of gravity above the fixed plane) = — y cos (9 ; hence yd.Q>o&d =—dz; and equation (1) gives -r-^ -h-g ). Substituting these values of E and yd.aosO in the preceding equation, and integrating, we have A{vy2+Vx2)^M{^^ + 2gz'^=k (3.) From the 2d and 3d of equations (2) the equation (c) (of the gyroscope, p. 542) is deduced by an identical process. A(bVy-\-aVa;)-\- On co& 6:=:l, and a substitution in the two foregoing equations of the values of the cosines a and h, and of the angular velocities v^ and Vy, in terms of the angles (p, and ip (see pp. 540, 541), and for z and — - their values, — / cos (9, and '/sin (9— and a determination of the dt d t constants, on the supposition of an initial inclination of the axis a, and of initial velocity of axial rotation w, will give us for the equations of motion of the top : hm^d-j-zzz—r- (cos (9 — cos a) Ai^^m2e^-^^^J1^ * As there are no horizontal forces in action, there can be no horizontal motion of the centre of gravity except from initial impulse, which I here exclude. J. G. BARNARD ON THE GYROSCOPE. 535 from wliicli the angular motions of the top can be determined. The first is identical with the first equation (4) for the gyroscope. The second differs from the jsecond gyroscopic equation only in containing in its first member the term My^ ^in^d-—^ or its equivalent M-j-^ , expressing the living force of vertical transla- tion of the whole mass. The second member (as in the corresponding equation for the gyroscope) expresses the work of^ gravity^ and the first term of the first member expresses the living force due to the angular motion of the axis. Instead therefore of the work of gravity being expended (as in the gyroscope) ivhoUy in producing angu- lar motion, part of it is expended in vertical translation of the centre of gravity. The angular motion takes place not (as in the gyroscope) about the point of support (which in this case is not fixed\ but about the centre of gravity (to which the moments of inertia A and B refer) ; and that centre, motionless horizon- tally, moves vertically up and down, coincident with the small angular undulations of the axis through a space which will be more and more minute as the rotary velocity n is greater. An elimination of -r- between the two equations (4) and a study of the resulting equation, would lead us to the same gen- eral results, as the similar process, p. 544, for the gyroscope. The vertical angular motion, expressed by the variation which the angle undergoes, becomes exceedingly minute (the maxi- mum and minimum values of 6 approximating each other) when n is great, and the axis gyrates with slow undulatory motion about a vertical through the centre of gravity. It would be easy, likewise, to show by substituting for another variable, u=ct—dj always (in case of high values of n) extremely small, and whose higher powers may therefore be neglected, that the co-ordinates of angular motion, u and V, approximate more and more nearly to the relation expressed by the equation of the cycloid as n increases ; though the approximation is not so rapid as in the gyroscope. All the results and conclusions flowing from the similar process for the gyroscope (see pp. 545, 546, 547, 548) would be deduced. As, however, the centre of gravity, to which these angular motions are referred, is not a fixed ])oint^ but is itself constantly rising and falling as d increases or di- minishes, the actual motion of the axis is of a more complicated character. If OK" (see fig. 2) is the initial position of the axis of the top, the motion of the centre of gravity will consist ina vertical falling and rising through the distance GG'— GK"{q,o^z^ G'G"— coaZiG G") = y (cos ^ , — cos «) (in which ^ is the minimum value of ^) 536 J. G. BARNARD ON THE GYROSCOPE. while the extremity of the axis or pointy K^ describes on the supporting surface and about the projection G" of the cen- tre of gravity, an undulating curve a, Z>, a', h'^ a'\ &c., hav- ing cusps a, a\ ko,.^ in the circle described about G" with the radius G"K"—y sina, and tangent, externally, to the circle described with a radius G" K'=y sin^,. But, as in the case of the gyroscope, these little undulations speedi- ly disappear through the re- tarding influence of friction and resistance of the air, and the point of the top describes about G". a circle, more or less perfect. The rationale of the self-sustaining power of the top is identi- cal with that of the gyroscope ; the deflecting force due to the angular motion of the axis plays the same part as the sustaining agent, and has the same analytical expression. Owing io friction^ the top likewise rises, and soon attains a vertical position ; but the agency by which this effect is produced is not exactly the same as for the gyroscope. If the extremity of the top is rounded, or is not a perfect mathematical point, it will roll^ by friction, on the supporting surface along the circular track just described. This rolling speedily imparts an angular motion to the axis greater than the horizontal gyration due to gravity, and the deflecting force be- comes in excess, (as explained in the case of the gyroscope,) and the axis rises until the top assumes a vertical position. Even though the extremity of the top is a very perfect point, yet if it happens to be slightly out of the axis of figure (and rotation) the same result will, in a less degree, ensue : for the point, instead of resting permanenthj on the surface, will strike it, at each revo- lution, and in so doing, propel the extremity along. The condi- tions of a perfect point, perfectly centered in the axis of figure, are rarely combined, or rather ixre practically impossible; but it is easy to ascertain by experiment that the more nearly they are fulfilled, and the harder and more highly polished the support- ing surface, the less tendency to rise is exhibited ; while the great stiffness (or tendency to assume a vertical position) of tops with rounded points, is a fact well known and made use of in the construction of these toys. C^f"The references throughout this paper are to my paper on the gyroscope in the June number of the Am. Journal of Education. ^££^9'^^ , /SO^ - "99 XVIII. EDUCATIONAL MISCELLANY AND INTELLIGENCE. 0^ THE EFFECTS OF INITIAL GYRATORY VELOCITIES, AND OF RETARDDfG FORCES, ON THE MOTION OF THE GYROSCOPE. BY MAJOR J. G. BARNARD, A. M Corps of Engineers, U. S. A.* In one of the concluding paragraphs' of my first paper on the Gyro- scope (Am. Journal of Education, June, 1857,) I stated that " an initial im- pulse may be applied to the rotating disk in such a way that the horizon- tal motion shall be absolutely without undulation. An initial angular velocity such as would make its corresponding deflective force equal to the component of gravity g sin ^, would cause a horizontal motion without undulation." The statement contained in the last sentence quoted, is not rigidly true ; for besides the component of gravity, there is another force to be consid- ered, viz., the centrifugal force due to the gyratory velocity, which acts either in conjunction with, or in opposition to, the component of gravity, according as the axis of the disk is above or below a horizontal. In this last position this force is null (as regards its effects in sustaining or depressing the axis), and to this angular elevation of the axis the statement quoted is true without qualification. The assumption of an initial horizontal velocity requires only a new determination of constants for equations (a) and (c) (pp. 541, 542, June No.). If we make, in those equations ^=a, ()d:zi90°, V=90°, «=-sina, v^zzim, Vy=0, VzZ=:.n, (in which m is the assumed initial velocity) and determine the constants h and I therefrom, the equations of motion will become sm 2 o — =z — (cos o — cos a) -|- m sm a . Bm^^^ + — =-^(cos6-cos«) + m^ J and from them we get . ^^dO^ r^Mgy . ^^ "iCmn . C^n^ , ^ sm2 d — — -=z| i^-^ sm^ 0^ — sm a — — (cos c^— cos a) dt^ \^ A A ^2 V / — m2 (cos 6 -|- cos a) I (cos ^— cos a) (2) dd dip From this we ffet -;-= when cos ^ — cos a =0 : and as -^ is not zero ^ dt ' dt for this initial elevation, it indicates, instead of a cusp, a tangency to the horizontal here. This paper is intended to give a more rigidly mathematical demonstration of the effects of " retarding forces" than is given in (December No. p. 529,) of this Jour- nal >■ and to give the theory of the " motions " of the Gyroscope a more general form, by the introduction of " Initial Gyratory Velocities." 300 J. G. BARNARD ON THE GYROSCOPE. If the curve described is horizontal without undulation, the other fac- tor of the second member of eq. (2) should likewise become zero with dz=za : an effect which may ensue from a suitable value given to m. The value of the deflecting force due to a given angular velocity m is (J (p. 552, June number) -—mn, and if we suppose this equal to the com- ponent of gravity g sin «, we shall have m j= --f^sin a. Cn If we substitute this value of m in the second member of equation (2) and assume a =r 90° the factor in question becomes zero for ^z= a, and the maximum and minimum values of d are the same, indicating a hori- zontal motion without undulation. For every other initial elevation than 90° a different value of m is re- quired to produce this result, in consequence of the influence of the cen- trifugal force of gyration at other elevations. With «= 90°, equation (2) becomes 'dt^~\ —J-^^'"^^ -J ^^cos<9-.m2cos^ cos(9 (3) Placing the first factor of the second member equal to zero and solving with reference to cos^ we get (recollecting the value given to § in our former article) sin2 e - 52-4?- + 4:Mgy ^V^4.Mgy] + 1- Cmn Mgy' (4) For ^/^ = 0, equation (3) expresses the cycloidal curve with cusps a, a', a'\ &c., as has been already shown in our former investigation. For M g'^ m > but <^ —^ — the minimum value of 6 derived from equation (4) is greater than when m is zero, while instead of a cusp (there is as has already been observed) a tangency at a, and the curve has the wave form a6j a'h\ (the points b^b^'b^", &c. being higher than bb'b").* Mgy When mr=-— — the curve unites with the horizontal a a' a" a'" and Cn there is no undulation ; equation (4) giving cos ^ =: 0, or ^ = 90°. * In reality, the amplitudes, a a', a' a", of the undulations become increased, at the same time that the sagittic are diminished, but, for the sake of comparison, I have represented them the same for each variety of curve. J. G. BAENARD ON THE GYROSCOPE. 301 When m >► --^, -r- becomes still zero with ^z=a = 90°; but this Cn at instead of a njaximum is now a minimum value of ^, for the value of which satisfies equation (4) is greater than 90°, and the curve ah^ a'h^', &c., undulates above the plane a a' a". 2 Mo Y 1 Finally when m= ^ , equation (4) will give cos5=- -^ and a substitution of this in the first equation (1) (making a = 90°), will give — r= : showing that the curve makes cusps at its superior culminations, and that the common cycloidal motion is resumed. In fact the value of —- = -3 {^ (p. 547, June number) at the lowest point h of the cycloid, is, 2 Mo y (substituting the values of ^ and i) exactly equal to —^ , and the value of the sagitta u corresponding to e 6 is what we have just found for cos<9, or e 63, viz. — -. If now, retaining m constant at this value to which we have brought it, we increase the rotary velocity, w, or vice versa, a curve with loops, (fig. 2,) may be described, as it can be shown that, for the maximum value d ip of dj — becomes negative.* 2. In my supplementary paper in the December number of this Journal I have endeavored to show how the theoretical cycloidal motion of a sim- ple solid of revolution is modified by the retarding forces of friction and the resistance of the air, and that the theory explains all the phenomena observed in the ordinary gyroscope. It may be objected however that the nature of the curve given in Fig. 1, (p. 531,) is in some degree assumed, and I therefore" wish to show that it can be confirmed by mathematical demonstration. The rotary velocity n of the disk is supposed to be gradually destroyed through the retarding forces of friction at the extremities of the axle, and of the resistance of the air at the surface. Without attempting to give analytical expressions for the retarding forces, it is sufliicient to say that the rotary velocity, at the end of any *If m is made negative and small (i. e., a backward initial velocity given) a looped curve like the above, but lying below the plane a a' a", results. All these curves (n being always supposed very great) are but the different forms of the " cycloid " known as prolate, common, and curtate cycloids ; the common — a particular case of the curve — corresponding to the particular case of the problem in which the initial gyratory velocity is either zero or has the particular value —^ • 302 J. G. BARNARD ON THE GYROSCOPE. time i, counting from the commencement of motion, may be expressed thus in which n is the initial rotary velocity of the disk. If we substitute this expression for v^ in the last two equations (3) (p. 541, June No.,) and follow a similar process to that by which equations (4) of that paper are deduced, we shall get, for the equations of motion sin2|9— =:— - (cos(9-cosa)-— / f{t)d.co&d 7 at A Af/ ( /k\ For the sake of simplicity suppose the initial position of the axis be hori- zontal, or a z=90 and the above become sin2^^ = ^cos^^-^ rf(t)d.coB,e ) dt A Aj o'^^ ( ,_. dt ^ dt^ A . J « If aff a' represents the cycloidal curve, and aee' e" g' the curve in question, it will be observed that the angular velocity of the axis given by the 2nd equation (6) is the same for both, for equal values of ^, while the value of i^e horizontal component oi \h2ii velocity, sin^— , is less r^ than for the cycloidal curve, by the term - / f{t)d. cos (9. A sm C