Let a wheel A (Fig.1) be mounted on an axle BC journalled within a horizontal ring D. Let this ring in turn be mounted on journals EF within a vertical ring G and, further, let this vertical ring be carried on journals HJ within a vertical frame K. This arrangement constitutes a gyroscopic system having three degrees of freedom, because relatively to the frame K the wheel may turn about three axes BC, EF, and HJ mutually at right angles to each other and because, if the wheel is set spinning on its axle, gyroscopic properties will be manifested.

The following is a brief statement of the gyroscopic properties manifested when the wheel is spun on its axle:

(a) Let the wheel be spinning in the direction of the arrow L and let a weight W be hung on the horizontal ring at the end B of the axle. The movement produced by this weight is not a rotation of the horizontal ring, and the wheel within it, about the axis EF. Instead, the horizontal ring remains horizontal and the whole system inside the square frame sets off rotating at a uniform speed about the axis HJ in the direction of the arrow marked M on the horizontal ring. This rotation or precession, as it is called, will be maintained so long as the weight W remains in action. There is here no question of perpetual motion. The work expended in overcoming the friction at the vertical journals is derived from the energy of the spinning wheel, and when this energy is exhausted the phenomenon ceases. The phenomenon can, in fact, only be maintained indefinitely by expending power to drive the wheel against the leakage of energy through friction at the journals of the axle and the vertical axis HJ. A closer examination of the phenomenon would show that there is a slight rocking motion of the horizontal ring on its axis EF, and therefore an additional leakage of energy at the journals of this axis. This rocking motion can be neglected for our present purposes. It is sufficiently accurate to say that the horizontal ring remains horizontal.

(b) The speed of the precession is proportional to the weight W and to the speed of rotation of the wheel on its axle. For instance, doubling the weight doubles the speed of precession.

(c) If the direction of spin of the wheel is reversed the direction of the precession is also reversed.

(d) If the spin of the wheel is in the direction L, and if instead of attaching a weight at the end B of the axle we exert an upward force at this point the precession developed will be opposed to the direction of the arrow M.

(e) If instead of trying to rotate the wheel about the axis EF by means of a weight or force applied at B we attempt to turn it about the vertical axis HJ by applying a horizontal force V to the outer ring, the wheel will not turn about the vertical axis HJ, but about the horizontal axis EF, the end B of the axle rising up towards H.

(f) As before, the direction of this movement is reversed by reversing either the direction of spin of the wheel or action of the force V. If both are reversed simultaneously the direction of the movement produced by the applied force is not altered.

The behaviour set forth above can be summarised in a general rule as follows:—If to a spinning wheel possessing three degrees of freedom a force be applied tending to turn the wheel about some axis XX, the actual motion produced will not be about XX but about some other axis YY; this axis YY will be such that rotation about it will tend to bring the axle of the spinning wheel into coincidence with or parallel with the axis XX; the direction of the rotation produced about YY will be such that when the condition of coincidence or parallelism is reached the spin of the wheel will coincide in direction with the rotation we are attempting to produce about the axis XX.

Taking case (a) (Fig.1), it will be seen that the axis EF about which we are attempting to produce rotation by means of the weight W, together with the weight W itself, is of necessity carried round by the precession in the direction M at the same rate as the axle of the spinning wheel. The axle in this case cannot therefore place itself in coincidence with the axis of the applied force. But it does its best to do so. The precession persists and is an expression of the fruitless chase of the axis EF by the axle BC.

If, however, the weight is attached by some kind of sliding connection on the horizontal ring in such a way that its line of action remains stationary in space, then the axis about which we are attempting to produce rotation will also remain stationary in the position occupied by the axis EF before precession commences. In this case it is quite possible for the axle of the wheel to place itself in coincidence with the axis of the applied force. Precession about HJ through 90 deg. will accomplish this result, as indicated in Fig.2. The weight W is now acting at a point on the horizontal ring where it ceases to have any tendency to turn the wheel about the axis EF. When, therefore, the position of coincidence is reached precession ceases and the system comes to rest in this position.

If the experiment were actually made it would be found that the momentum acquired by the system during the 90 deg. turn would carry the axle through the position of coincidence with the axis of the applied force. But immediately the axle passes to the opposite side the force W is exerted on a point of the horizontal ring between F and C. The action of the force passing on to this, the opposite, segment of the ring reverses the conditions under which the system started its movement and as a result precession in the direction opposed to the arrow M is set up. The axle thus tends to recover its position of coincidence and in the end settles down to a vibratory motion from side to side of the axis of the applied weight. Friction at the vertical journals will “damp” this vibratory motion, the amplitudes of the swings will decrease, and the axle will ultimately settle in steady coincidence with the axis of the applied force. In this condition the force will have no further effect on the system beyond throwing a bending moment on to the vertical axis.

Instead of trying to make the wheel rotate about the axis EF by applying a weight to the inner ring as in Fig.1, let us, as shown in Fig.3, mount the square frame K on a horizontal axis NP and attach the weight W to an arm fixed on the frame. The axes NP and EF being—at least initially—collinear, the effect of this arrangement is to throw a turning moment on to the wheel about the axis EF just as does the weight W in Fig.1. It is to be noticed, however, that the moment of the weight W in Fig.3 about the axis NP is transmitted to the inner ring as a moment about the axis EF solely because of the friction existing at the journals of the axis EF. This friction may be very small, so that the turning moment received by the wheel is only a very small fraction of the turning moment exerted by the weight W about NP. The effect of the arrangement is thus exactly the same as would be produced in the arrangement Fig.1 if we reduced the weight W to a hundredth or a thousandth of its value. In other words, precession about the vertical axis HJ will set in in the direction of the arrow M just as before, but the speed of this precession will be only a hundredth or a thousandth of the previous value.

It is not very important to trace out the behaviour of the system shown in Fig.3 beyond a very brief period immediately after the weight W is applied. The point of importance is that the precession produced by the weight is very slow, and therefore that in a given interval of time the amount precessed is very small. Further, the rate of the precession depends solely upon the friction at the journals of the axis EF and not upon the weight W or the movement of the frame K except in so far as these factors affect the friction. The less the friction the less will be the rate of precession and the amount precessed in a given time. Thus by mounting the axis EF on knife edges the friction can be made so small that the precession produced by the weight W becomes immeasurable. Hence we deduce that if friction is substantially absent at the axis EF the frame K might be violently rocked on the axis NP or even set into continuous rotation without causing the axle of the wheel either to dip or to precess.

Continuing the argument, we might mount the square frame on a vertical axis and attempt to produce rotation of the wheel about the axis HJ by applying a horizontal force to one side of the square frame instead of a force V on the outer ring as shown in Fig.1. A similar result would be obtained. Granted an all but total absence of friction at the journals of the vertical axis HJ, the precession produced about the horizontal axis EF would be immeasurably small. Thus the frame might be set into violent motion about its vertical axis without causing the axle either to rotate in a horizontal plane or to precess in a vertical one.

Finally, if the square frame were mounted on a horizontal axis collinear with the axle BC it might obviously be rotated about this axis without affecting the system otherwise than by increasing or reducing the rubbing speed of the axle BC in its bearings.

Since pure translation of the frame in any direction cannot apply a turning moment to the system about any axis, and as rotation of the frame about any one of the three principal axes has no effect which is measurable on the orientation of the axle, it follows that, given substantial absence of friction at the axes EF and HJ, the axle of the wheel will remain constantly pointing parallel with its original position, no matter how the frame K may be moved or turned about.

The gyroscopic system shown in Fig.1 has, as we have said, “three degrees of freedom,” because its wheel is free to spin about three different axes mutually at right angles. It is to be carefully noted that it can only truly be said to have three degrees of freedom so long as the inner ring and the parts inside it are not rotated on the axis EF away from the position which in Fig.1 they are shown as occupying relatively to the outer ring. Thus rotation of the wheel on its axle or of the whole system inside the square frame on the axis HJ leaves the three axes BC, EF, HJ undisturbed at right angles to each other. But rotation of the inner ring and the parts inside it on the axis EF tends to destroy one of the degrees of freedom. If, for instance, the inner ring is rotated through 90 deg., as shown in Fig.4, the axle BC and the axis HJ will coincide in direction. In this position the wheel cannot be rotated about a horizontal axis at right angles to EF and has therefore virtually only two degrees of freedom, namely, about the axis EF and about the axis HBCJ. Again, if with the inner ring in the position shown in Fig.4 the outer ring is turned through 90 deg. relatively to the square frame, the system assumes the configuration shown in Fig.5 and the wheel loses the power of rotating about a horizontal axis in the plane of the square frame.

If, then, in any application of the gyroscope it is necessary to guarantee that the system shall have three degrees of freedom in all possible configurations, the simple mounting shown in Fig.1 will not serve the purpose. It can be made to do so in the manner shown in Fig.6, namely, by mounting the square frame inside a gimbal ring X, which in turn is supported by a frame Y, the two new axes TU and VW being at right angles to each other. In the position shown in Fig.4 the new axis TU would restore the lost third degree of freedom, while the second new axis VW would restore the degree of freedom lost when the system assumed the configuration shown in Fig.5.

In the gyro-compass it is necessary to guarantee that the spinning wheel in all possible configurations shall have three degrees of freedom, and accordingly we find the wheel mounted in a manner reproducing the features of Fig.6. On the other hand, the majority of the movements which the compass system is called upon to make do not entail anything except very small degrees of rotation of the inner ring and wheel about the axis EF (Fig.1), and therefore for most purposes the simple mounting there shown reproduces the required three degrees of freedom sufficiently closely to permit us to use it for demonstration purposes. In one very important portion of our subsequent discussion, however—namely, that dealing with the effect on a marine gyro-compass produced by rolling and pitching of the vessel—it will be necessary for us to take cognisance of the fact that the square frame shown in Fig.1 is not fixed directly to the ship’s deck, but is really carried in gimbals as shown in Fig.6.