1902 Encyclopedia > Gyroscope

## Gyroscope

GYROSCOPE, GYROSTAT, are names given to instruments which are used to demonstrate certain properties of rigid bodies, when made to rotate rapidly about the axis round which they are kinetically symmetrical.

In some of its forms the gyroscope has been known for a very long time, and is, in all probability, of French or German invention. Almost the first instrument of the kind that we hear of, and of which the present gyroscope is merely a modification, is that of Bohnenberger, which was constructed as early as 1810, and is described in Gilbert's Annalen for 1818 (lx. p. 60). It consisted of a heavy spheroid which could rotate inside a circular ring round its shorter axis,—the axis running on pivots situated at opposite ends of the ring's diameter. This ring, with its contained spheroid, was similarly made movable inside a second ring, and round an axis at right angles to the axis of the spheroid. In the same way this second ring, with its contents, could rotate inside a third ring, and round an axis at right angles to each of the others. From this it will be seen that the spheroid had all degrees of free rota-I tion,—one point only within it being fixed, namely, the intersection of the three axes.

The construction of the gyroscope will be readily understood from fig. 1, which is engraved from a photograph of a large instrument made by Mr Sang.

It consists of a fly-wheel, with a heavy rim accurately turned and balanced, which can rotate round an axis GC forming a diameter of the ring K. This ring also can rotate about the axis AF which is at right angles to GC, and is a diameter of the ring L. Similarly this ring L can rotate about the vertical axis BH, which is perpendicular both to GC and AF, and forms a diameter of the ring M, which is screwed to the heavy sole-plate N. In order to suit certain experiments the ring M, by means of a clamp arrangement seen at X, can be turned round to make the axis BBC inclined at any angle with the vertical. With this instru-ment the following among many interesting experiments may be made.

1. Either by a large fly-wheel and band, or by running out a long eord coiled several times round the V-grooved wheel on the end of the axis, let a very rapid rotation be given to the fly-wheel round the axis OC, and in the positive direction (that is, counter-clock-wise as seen from C). Such a rotation can readily be given so rapid as to last a long time. While so revolving let a weight be placed at G so as to cause a rotation round OA, also in the positive direction, and it will then be observed that a slow rotation of the whole mass takes place about the vertical axis OB, but in the nega-tive direction. If the weight be put at C, the wheel rotating as before, the rotation round OB will be in the positive direction. Reversing the rotation of the wheel will in both cases produce a reversal of the rotation round OB.
2. The fly-wheel still rotating in the positive direction, let a positive rotation be given to the ring L round OB, and immediately the axis of the wheel OC will tilt up, rotating in the positive direc-tion round OA till it reach the vertical position, where it will remain, and then it will be observed that the fly-wheel is rotating in the positive direction round OB. Should the axis OC, in eon-sequence of inertia, be carried a little past the vertical, and the direction of rotation round OB be then suddenly reversed, the axis OC will continue to revolve in the positive direction round OA, till it reaches the vertical pointing downwards, when it will again be stationary, and the wheel will now be rotating in the negative direction round OB. By gently oscillating the ring L at the instant the axis of the fly-wheel is passing the vertical, a continuous rapid rotation round OA can be given to the ring K carrying the rotating fiy-wheef with it.

3. Let the fly-wheel be now made stationary, and, while in this position, let simultaneous rotations in the positive direction be given round both OA and OB ; the fly-wheel will then be observed to begin to move in the negative direction round OC.

The explanation of all these experiments follows readily from the theory of the composition of rotations round different axes. Let the angular velocities about OA, OB, OC, be OT, p, a respectively. If simultaneous angular velocities a and -a be given about OC and OA in the positive direction, these can be compounded into an angular velocity about a new axis 01, which divides the angle AOC into two parts such that
_a sin COI sin COI
a =sin AOI = co7COI = fem «01 = tan 9 (suppose). 01 is, therefore, the position of the instantaneous axis, and as OC will always tend to coincide with it, OC will move round the axis OB in the negative direction. It will be noticed also that, -a remaining the same, tan fl will be less as <r is greater. This shows that the rotation round OB, or the motion in azimuth, will be slower as the speed of the fly-wheel is greater. This accords with experi-ment. In precisely the same way by considering simultaneous angular velocities about OC and OB and OA and OB, can experi-ments 2 and 3 be explained.

Perhaps the most common form of gyroscope is that which has been largely sold under the name of the gyroscope top. It is a modification of the gyroscope introduced by Fessel. In it the ring containing the rotating wheel is either supported on a vertical pivot at the end of the axis, or simply held suspended by a string attached to the same point, as represented in fig. 2. In this case, when the wheel is rapidly rota-ted, and its axis initially inclined at an angle with the vertical, a slow motion round the vertical axis is observed to take place. This rotation is accom-panied by a feeble nutation of the axis, which is unobserv-able by the eye so long as the wheel rotates rapidly. The rotation in azimuth would continue uni-form if the velocity of the wheel re-mained constant, but it grad ually increases in speed, the nuta-tion at the same time becoming more apparent as the velocity dies away, till the gyroscope finally settles with the axis of the wheel in the direction of the prolongation of the supporting string. A short analytical investigation of this particular case may serve to give some idea of how problems on the gyroscope generally may be attacked. For clearness in the woodcut (fig. 3) we suppose the ring containing the wheel removed.

Let K represent the wheel of mass M; h the distance of the centre of gravity G from 0 ; OC the axis of the wheel, supposed of unit length ; OA, OB, two other axes through the fixed point 0 at right angles to 00 and to each other ; OX, OY, OZ, three rect-angular axes fixed in space, with which OA, OB, OC (axes movable with the body) initially coincide ; if/, <p, 8, angles on the unit sphere which define the point C.

If A, B, C be the moments of inertia, and -ar, p, <r the angular velocities about OA, OB, OC respectively, the equation of kinetic energy T gives
T = 4(A*r2 + Bp2-rCo-2) ;
but » = 8 sin <f> - iff sin 8 cos <p ;
p \$ cos (p + ij/ sin 8 sin * ; <r = if> cos 8 + <j>.
Substituting and making A = B, as is clearly the case, we have
T = iAfl'2 + iAif-2 sin H + |C(iJ/ cos 8 + <j>)\ Applying Lagrange's equations of motion, which are for this case
d_ dT_*T_0
dt ' dtj/ difi '
d_ dT_dT = 0
dt
we get (1)
dt „ dq> dcp '
A if* sin 20 + C(i|/ cos 8 + <j>) cos 8 I =0 ; I C(i£cos0 + <p) ( -0;
d8 d8 y '
(2)4.
d_ dt d_ dt
(3) AS - Ai|/2 sin 8 cos 8 + C sin 8\$\$ cos 8 + tp) = Mgh sin 8. Integrating (2), we get
4< cos 8 + <p = a constant = n, which represents the angular velocity about 00.
Now if the motion be steady we must have both 8 and >(/ constants. Let them be represented by a and fj. respectively. Then equation (3) gives
- A,u2 sin a cos a + Cp.n sin a = Mgh sin a _ Mgh + A,us cos a C»

By this formula we can calculate, for anyparticular instrument, the angular velocity in azimuth from having given the angular velocity of rotation of the fly-wheel and vice versa.
An important case of motion, and one also very interesting both mathematically and physically, is got by including a gyroscope in a pendulum bob and supporting the rod from a universal flexure joint. This constitutes the gyroscopic pendulum. The joint is usually' got by attaching a short length of fine steel wire rigidly to the end of the rod, and suspending the whole by means of the other end of this wire firmly clamped into a fixed support. When the gyroscope is rapidly rotated, and the pendulum drawn a little aside from the vertical and then let go, its lower end is observed to describe a beautiful curve consisting of a series of equal closed loops, all equally near each other and radiating from a centre point. This curve, which is a species of hypotrochoid, is figured in Thomson and Tait's Natural Philosophy, where also the whole theory of the gyroscopic pendulum will be found.

As has been already mentioned, a remarkable apparent effect is produced by the earth's diurnat rotation upon a rapidly rotating gyroscope. This arises from what Foucault called the "fixity of the plane of rotation," and what Thomson and Tait have recently called " gyrostatic domination." In virtue of this principle, the axis of the rotating fly-wheel tends always to preserve a fixed direction in space, and, in consequence, will appear to move in a direction opposite to that in which the earth's rotation is at each instant actually carrying it.

If we suppose the gyroscope represented at fig. 1 to have all degrees of free rotation round ths point 0, and to be in every way exactly balanced about that point, and also to have all its pivots nearly void of friction, then, at whatever part of the earth's surface it may be placed, the fly-wheel while rotating rapidly will le observed to move gradually and finally take up such a position that its axis OC is parallel to the earth's axis, and also that its direction of rotation round OC is the same as that of the earth round its axis. Should the ring K be fixed so that the axis GC can only move in a horizontal plane, then the ring L will move in azimuth till it has placed GC in the direction north and south and such that the direction of rotation of the fly-wheel coincides with that of the earth. Further, should the ring L be fixed in the plane of the meridian, so that GC can only move in altitude, then GC will be observed to tilt up till it is parallel to the earth's axis, the direction of rotation of the wheel being, as before, the same as that of the earth. These effects may be explained as follows :—_

Let A be the latitude of the place, and a the angular velocity of the earth on its axis ; also, at starting, let the ring K be in the horizontal plane, and let OC make an angle a, in azimuth, with a horizontal line ON drawn from 0 northwards ; further, let the rotation of the wheel round OC be positive. Now a can be resolved into three angular velocities at right angles to each other, viz.:—_ a sin A round the vertical line, ai cos A cos a round a line parallel to OC, and a cos A sin a round a line parallel to ON.

Of these o> sin A gives the velocity in azimuth, a cos A cos a can only affect the velocity of the wheel, while » cos A sin a gives the velocity of OC in altitude. These being the actual component velocities communicated to the gyroscope by the earth, the apparent velocities will be equal and opposite to them.

The apparatus actually employed by Foucault to demonstrate the rotation of the earth differed somewhat from that represented in fig. 1. In it the corresponding ring to L was suspended from a fixed stand by a thread without torsion, and rested at its lowest point by a pivot in an agate cup. Also the ring corresponding to K, which carried the fly-wheel, rested on knife edges within L, and could be removed at pleasure, in order that the rotation might be given to the fly-wheel. Great care was also taken to have every part thoroughly well balanced. It is stated that the experiment was several times successfully performed, and Foucault by means of his apparatus was thus able, without astronomical observation, to find out the latitude of the place, the east and west points, and the rate of the earth's motion. The same experiment has lately, it is said, been successfully performed by Mr G. M. Hopkins. His fly-wheel, being driven by electricity, has the advantage of rotating at a uniform rate for any length of time. By attaching a small mirror to the frame which carries the revolving wheel, and using a spot of light reflected from it as an index, he has been able to make manifest the earth's rotation in a very short time.

An ingenious practical application of the gyroscope principle was suggested and carried into effect about the year 1856 by Professor Piazzi Smyth. His aim was to devise a telescope-stand which would always remain level on board ship, notwithstanding the pitching and rolling, and so facilitate the taking of astronomical observations at sea. For this purpose the stand was supported on gimbals, and underneath it were placed on fine pivots several heavy fly-wheels which could be put in rapid rotation, some on vertical and some on horizontal axes. The complete apparatus, involving many ingenious details as to driving the flywheels, &c, was tested by Professor Smyth on board the yacht "Titania" during a voyage to Teneriff'e, and found to work with perfect satisfaction. A full account of the method will be found in Trans. Royal Scottish Society of Arts, vol. iv.

GYROSTAT.—This is a modification of the gyroscope, devised by Sir William Thomson, which has been used by him as well as by Professor Tait for a number of years to illustrate the dynamics of rotating rigid bodies. It consists essentially of a fly-wheel, with a massive rim, fixed on the middle of an axis which can rotate on fine steel pivots inside a rigid, case. The rigid case exactly resembles a similarly-shaped, but hollow, fly-wheel and axis closely surrounding the other but still leaving it freedom to move. Slits are made in the containing case whereby a cord can be coiled several times round the axis for the purpose of setting the fly-wheel in motion. There is also attached to the rigid case, in the plane
passing through the i
centre of gravity of A §
the wheel at right angles to its axis, a thin flange of metal, which is called the bearing edge. The circumference of this flange is not a circle but a curvilinear polygon of sixteen or more equal sides. The object of mak-ing it so is to pre-vent the instrument from rolling like a wheel on the bearing edge when the fly-wheel is rotating rapidly. A draw-ing of the gyrostat, with a portion of the case removed to show the inside, is given in fig. 4. A represents the bearing edge upon which the whole balances. To suit certain experiments, a sharp conical steel point can be fixed on to the end B by means of a bayonet joint.

The gyrostat is a most instructive instrument, and with it many interesting experiments can be performed. We can only mention a few taken from Thomson and Tait's Natural Philosophy, vol. i. Part I., where the mathematics of the whole subject will be found fully given.

1. Let the gyrostat be placed on a fiat sheet of glass with its bearing edge in a vertical plane as represented in the figure. Neglecting translations, it has in that position clearly two free-doms, one in azimuth and the other inclinational. Of these, the first is neutral and the second unstable when the fly-wheel is still; but when it rotates rapidly the second will become stable, the first remaining neutral as before. When the fly-wheef is spinning rapidly, the persistence with which the gyrostat keeps the erect position is very remarkable. A blow from the fist on the side of the case is met by a strong resistance, the instrument being thrown into a state of violent tremor, which subsides, however, after a few seconds. If, while the fly-wheel is still rapidly spinning, a weight be hung on at B, the whole apparatus will, like the gyroscope, immediately begin to move round a vertical axis.

2. Let the gyrostat be supported on knife-edge gimbals at its lower end, and with the axis vertical. In this position it has two freedoms each unstable, without rotation of the fly-wheel, but with it both stable. A similar result is obtained by supporting the gyrostat on a universal flexure joint, constituting an inverted gyroscopic pendulum.

3. Let the gyrostat be supported on two equally long stilts tangential to the bearing edge. In this position the two freedoms, one azimuthal the other inclinational, are both unstable without, but both stable with, rapid rotation of the fly-wheel.

4. Let the gyrostat be attached to a bar of wood by thrusting the bearing edge through a narrow slot in the middle of the bar, and then let it be suspended by strings attached to the ends of the bar, the bar being horizontal. By this we have the means of slinging the gyrostat bifilarly in four different ways, two with the strings parallel and two with them crossed, the gyrostat being alternately above and below the wooden bar. Each of these ways has four freedoms, which can be reduced to three by a third string in each ease. A little consideration will show in each ease the stable and unstable modes without and with rotation of the fly-wheel.

It only remains to mention a very interesting and intricate gyrostatic problem which has been lately suggested by Sir William Thomson. He supposes a string of gyrostats to be formed with their axes all in the same straight line, and each attached to the other by a small universal flexure joint of thin steel wire. If this string be disturbed after the fly-wheels have all been put into rapid rotation, a remarkable cork-screw motion passes along the wire. For particulars we must refer to a paper announced by Sir William Thomson in the R. S. E. Proceedings.

In addition to the references in the text the following will be found useful :—Ast. Notices, vol. i. ; Comptes Rendus, Sept. 1852 ; Paper by Professor Magnus translated in Taylor's Foreign Scientific Jferaoi'rs, new ser., part3, p. 210: Ast. Notices, vol. xiii. pp. 221-248; Theory of Foucault's Gyroscope Experiments, by the Rev. Baden Powell, F.R.S. ; Ast. Notices, vol. xv. ; articles by Major J. G. Barnard in Silliman's Journal, 2d ser., vols. xxiv. and xxv. ; E. Hunt on " Rotatory Motion," Proc. Phil. Soc. Glasgow, vol. iv. ; J. Clerk Maxwell, "On a Dynamical Top," Trans. R.S.E., vol. xxi.; Phil. Mag., 4th ser., vols. 7, 13, 14 ; Proc. Royal Irish Academy, vol. viii.; Sir William Thomson on "Gyrostat," Nature, vol. xv. p. 297 ; Price's Infinitesimal Calculus, vol. iv. (J. BL.)