A vibrating pendulum ([Fig. 15]) will be satisfactorily damped if we can apply to the bob in all positions of its swing a force proportional to the velocity with which the bob is moving at each given instant and directed always so as to oppose the motion. At e, the position of release after deflection, the bob has no velocity; the damping force should therefore be zero. As it travels from e to d the bob is gaining velocity; the damping force should therefore grow from zero to a maximum at d and be directed at each instant tangentially to the curve of swing and act from d towards e. During this portion of its swing the bob is thus robbed of some of its velocity, so that it fails to rise to the position f and comes to rest at some point g. In travelling from d to g the velocity of the bob is decreasing naturally, and is still further decreased by the damping force which, acting in the same direction as before, falls from a maximum at d to zero at g. As it moves from g to d on the return stroke the bob gains velocity; the damping force should therefore increase from zero at g to a maximum at d—of a lower value than the maximum at the same point during the first stroke—and should be directed along the curve in the reversed sense, namely, from d towards g. During the swing from d to h the damping force, still reversed, should decrease from the second maximum value once more to zero. At the start of the second swing the damping force should again act from d towards e and should rise from zero to a third maximum value at d—less than either the second or first maximum value, and so on until the amplitude of the swing is reduced to the required degree. It will be noticed that in a damped vibration the mean position of the bob on any one swing is not coincident with the resting position d, but lies somewhere between the resting position and the position from which the bob commences the swing.

With each passage of the bob through the resting position d the value of the damping force rises to a maximum, the value of which becomes less and less on each successive swing. Ultimately, when the bob settles in the resting position, the maximum value becomes zero. In other words, the damping force, having completed its work by bringing the bob to rest, entirely disappears and leaves the pendulum exactly in the same condition as it would be under in the resting position if no damping force had ever been in action. The pendulum is thus as free as formerly to respond, in the resting position, to a vibrating influence, but as soon as it acquires motion the damping force is again called into existence to a degree directly dependent on the strength of the vibrating influence, with the result that the motion is first checked, and then finally stopped.

The damping force required is, as we have said, one which at all times is proportional in magnitude to the velocity of the bob—or what is the same thing, to the angular velocity of the pendulum as a whole—and which at all times acts to oppose the motion of the bob. Metallic friction—say, at the supporting axis of the pendulum—it would bring the motion to rest sooner or later, would not provide a satisfactory damping force, for solid friction is independent of the rubbing velocity, at least at low speeds such as we are here concerned with. The damping force provided by it being constant, would not be automatically adjusted to the velocity of the bob. It would vanish, it is true, when the bob was at rest, but as soon as the slightest vibration set in it would spring up to its full value straight away and would preserve the same value throughout a large swing as throughout a small one. In any event the presence of metallic or other solid friction at the point in the gyro-compass corresponding to the axis of the pendulum—namely, at the bearings of the vertical axis H J—cannot be permitted, and must be eliminated to the utmost possible degree if the directive force is to be sufficient to control the movement of the sensitive element.

Fluid friction, on the other hand, would provide a satisfactory damping force, for fluid friction is proportional to the velocity, at least at low speeds. A pendulum vibrating with its bob in a vessel of water or the floating card of an ordinary magnetic compass is satisfactorily damped by fluid friction. In the gyro-compass, however, the motion to be damped is, as we have seen, an exceedingly slow one, slower in fact than the small hand of a watch if the deflection of the axle from the meridian is initially less than 11½ deg. east or west. A fluid damping force would be proportionately low, so that without making the damping elements of enormous size the force derived would be insignificant and next to useless for practical purposes. As an illustration of this statement it may be remarked that in the early Anschütz compass the sensitive element was virtually floated in a bowl of mercury. Yet the drag of the mercury, the velocity of the vibration being so small, did not measurably reduce the amplitudes of the vibration during observation extending over several hours. This example is not quite a good one, however, for the friction at the surface of a body immersed in mercury would appear to be not of the fluid description, but of the solid type.

Solid and fluid friction being thus ruled out, at least as direct means of providing the required damping force, we have to find some other method of applying it. It is, or should be, clear that in whatever way the damping force is applied it should originate within the sensitive element itself. If it originates outside, then its transmission to the sensitive element cannot, in view of the fact that its origination, growth, and decay are to be controlled by the motion of the element, be effected in any conceivable way without the introduction of some material connection between the element and the outside source of the force. Such a connection can only be made frictionless if the outside source moves in exact unison with the sensitive element. If it does so move it clearly ceases to be an outside source and becomes really part of the sensitive element itself. This consideration suggests generating and applying the damping force gyroscopically by the exertion of some suitable action on the spinning wheel itself.

CHAPTER V
THE DAMPING SYSTEM OF THE ANSCHÜTZ (1910) COMPASS

In the preceding chapter we demonstrated the necessity for damping the horizontal oscillatory movement of the gyro-compass axle and discussed the nature of damped vibrations in general. We now turn to describe the damping means provided in each of the practical forms of gyro-compass so far evolved.

Turning to the early (1910) form of Anschütz compass, we find that, as shown in the purely diagrammatic sketch given in [Fig. 16], the spinning wheel is enclosed within a metallic case formed with tubular journals B C for the axle, and provided with trunnions E F, whereby it is supported within the vertical ring, which, as before, is free to turn on a vertical axis H J. The casing, it will be seen is, so far as the support of the wheel is concerned, exactly equivalent to the horizontal inner ring of our preceding illustrations. In the actual instrument the method of supporting the casing so as to permit it to turn about a horizontal and a vertical axis is not in the least like that shown in our sketch.

The wheel, running at 20,000 revolutions per minute, although it is quite plain, has a powerful blower-like action. On one side of the casing an orifice D is formed for the inlet of air, and on the periphery below an outlet duct K, directing the air blast tangentially away from the casing, is provided. The exact value of the pressure of the blast in the early Anschütz compass is not known to us, but in the Brown compass, wherein a similar blast is developed, the wheel, running at 15,000 revolutions per minute, gives an air pressure equal to some 3 in. of water.