The fact, then, that the axle is prevented from aligning itself completely parallel with the earth’s polar axis thus apparently results, once it has found the north, in making it wander, in northern latitudes, towards the west. This is not so. Once the axle has found the north a steady uniform precession towards the west is required to maintain it on the north. Thus in [Fig. 12] let A be the wheel and axle of the compass when it finds the north. If the axle maintained this alignment then some time later it would assume the position shown dotted at D; that is to say, it would be pointing towards east of north. To maintain it on the north it must rotate westwardly through the angle ϕ in the interval between A and D. As this angle ϕ grows with the interval the required rotation is really equivalent to a steady uniform precession towards the west.
Fig. 12. Compass at Equator and near North Pole.
If the compass in [Fig. 12] is practically at the north pole it is clear that to hold the end B of the axle on the north the axle has to precess about the vertical axis H J of the compass mounting at the rate of practically one complete revolution per twenty-four hours—that is, 0.0007 revolution per minute. At the equator the required rate of precession about H J is zero, for any movement about this axis will carry the axle away from the north. At intermediate latitudes the precession required to hold the axle on the north has an intermediate value. Its value for any latitude θ is, in fact, 0.0007 × sin θ in revolutions per minute.
Thus as the latitude is changed the required rate of precession also changes. So, too, does the angle θ ([Fig. 11]), by which the axle fails to reach parallelism with the earth’s polar axis, and consequently so does the strength with which the axle desires to reach this alignment. As the equator is left farther and farther behind, then, the axle comes to rest pointing north with a greater and greater upward tilt from the horizontal. The applied moment of the weight S thus increases. It increases just at the rate necessary to give the required rate of westerly precession for the particular latitude in which the compass is at any moment stationed. Should anything prevent the axle from acquiring the tilt appropriate to the latitude, or should the westerly precession on the vertical axis caused by this tilt be opposed and reduced in any way, the axle will fail to keep on the north and will lag behind the meridian with an easterly deviation. We shall see later on that the precession about the vertical axis is in some designs of gyro-compass unavoidably opposed, and that as a consequence these compasses exhibit a latitude error.
We have thus shown that the effort of the compass to set its axle parallel with the earth’s polar axis, combined with the action of gravity on the pendulum weight, is necessary to the compass if the axle once having found the north is to remain on it, and that this effort of the axle increases in strength the farther north—or south—the compass is moved from the equator. What, however, this effort gains in strength as the angle of latitude increases the effective directive force on the compass loses. Thus in [Fig. 11] the directive force may be represented as at f by a line parallel with the earth’s polar axis. This line represents the magnitude of the compass’s effort to set its axle parallel with the polar axis. The speed of the spinning wheel and its moment of inertia have not been altered by moving the compass away from the equator, nor has the angular speed of the compass round the earth’s axis, for although the compass is moving in a circle of reduced radius T B, and therefore is travelling with less linear velocity than at the equator, it is still making one turn per twenty-four hours round the polar axis of the earth. Thus the three factors fixing the magnitude of the “directive force” are unaltered. The force f is thus the same as that exerted on the compass at the equator. It does not, however, act as before, purely about the vertical axis H J, but partly about H J and partly about the horizontal axis E F. It may be regarded as a force applied at the end B of the axle and therefore as tending to turn the wheel about an imaginary axis a b. We may resolve it into two components p and q, p being at right angles to the axis H J and q at right angles to E F. The component q represents the magnitude of the upwardly tilting effect applied to the axle by the rotation of the earth. The component p represents the effective directive force tending to restore the axle from the deflected position represented towards the north in the horizontal plane. The angle between this effective component and the full force f is θ, the angle of latitude of the station at which the compass is set up. The effective directive force is thus f cos θ, and therefore diminishes from the value f at the equator towards zero as the north—or south—pole is approached.
CHAPTER IV
DAMPING THE VIBRATIONS OF THE GYRO-COMPASS
Reviewing what we have already established, we see that a gyroscopic system possessing “three degrees of freedom” and having a pendulum weight fixed below the wheel manifests a tendency in all latitudes to preserve its axle pointing in the north and south direction, a “directive force” or restoring moment being developed and applied to the axle if the north and south position is departed from. The magnitude of the directive force in any given latitude increases with the deflection, from zero when the axle is pointing north and south up to a maximum when it is aligned east and west. At any given angle of deflection of the axle the magnitude of the directive force varies with the latitude in which the system is stationed, being zero at the north or south (true) pole and a maximum at the equator. Finally, at any given angle of deflection of the axle and in any given latitude the magnitude of the directive force is determined by (a) the speed of rotation of the earth on its polar axis, (b) the speed of rotation of the spinning wheel on its axle, and (c) the mass or moment of inertia of the spinning wheel.
We have now to consider several important matters affecting the practical application of the gyroscope-pendulum combination as a substitute for the magnetic compass. The first practical consideration which arises naturally in our minds is the question: Can a system be devised and constructed sufficiently robust to withstand the trials and knocks of every-day use and yet be sufficiently delicate to respond to the feeble directive forces on the effect of which its action as a compass depends? From the table given previously it will have been noted that in the three chief types of gyro-compass so far developed the directive forces developed are in two examples greater than the corresponding directive force applied to the card of a magnetic compass, while in the third the directive force is materially lower. Even though they were all considerably greater than the force applied to the needle of a magnetic compass, some doubt as to their sufficiency to effect their work would remain, for they have to control sensitive elements, comprising a spinning wheel, axle, supporting rings, etc., weighing anything from 7 lb. to about a hundredweight, whereas in the ordinary compass the sensitive element consisting of the card and its attached magnetic needles weighs round about ¼ oz. The actual weights of the sensitive elements are given in the following table.