Fig. 6. Lost Degrees of Freedom Restored.
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 T U and V W being at right angles to each other. In the position shown in [Fig. 4] the new axis T U would restore the lost third degree of freedom, while the second new axis V W 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 E F ([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].
CHAPTER III
THE GYROSCOPE AND THE ROTATION OF THE EARTH
Let us now suppose that the gyroscopic system shown in [Fig. 1]—without the weight W—is placed on the equator as represented at P ([Fig. 7]), and that the axle is set pointing due east and west as at B C. At the end of an interval of time, say two hours, the earth will have rotated through some angle α, say 30 deg., and will have carried the gyroscope with it to the position Q. The square frame has thus clearly been inclined relatively to its original position. It has, in fact, suffered the exact equivalent of a direct translation R together with a pure rotation about a horizontal axis through E of amount α. The translation leaves the system unaffected, but the rotation of the frame results in the frame moving relatively to the axle, wheel, and inner ring. The axle, in fact, remains parallel with its original position at P. It is still pointing east and west, but the frame is now inclined to it and, relatively to the horizontal surface of the earth at Q, the axle is dipping at an angle β which is equal to α—or 30 deg. Actually, if we fixed a disc to the square frame and a hand to the inner ring, as indicated in Fig. 8, the system as erected at the equator would form a twenty-four hour clock indicating strictly accurate sidereal time as distinct from mean solar time. To an observer on the earth the hand would appear to travel clockwise round the disc once in twenty-four hours. Actually, however, the hand would not rotate, but would remain constantly parallel with its original position, while the disc would travel anti-clockwise relatively to the hand and would make one complete turn in twenty-four hours. The hand would remain parallel with its original position by virtue of the fact already stated, namely, that the force applied to the inner ring through its all-but-frictionless supports is very small, and in any event does not turn the hand clockwise, but causes the wheel, inner ring, and hand to precess about the vertical axis H J. The rate at which this precession took place would be a measure of the success with which we had eliminated friction at the horizontal axis E F.
Fig. 7. Elementary Gyroscope at Equator.
Fig. 8. Gyroscopic Clock.