This very prolonged period, were nothing done to rectify matters, would be a very serious objection in practice to the use of the gyro-compass. The axle, if deflected, would take about half an hour to reach an equal position on the opposite side of the meridian. Hence, if, when a compass reading was desired, the axle were found to be vibrating, at least half an hour would be required to determine the north and south direction by observing the two extreme positions of the axle and taking the mean. The alternative would be to wait until the vibration died away. This course would involve, however, a very much greater delay, for the virtual absence of friction at the vertical axis of the system—an essential, as we have seen, if the directive force is to be allowed to come into play at all—results in the vibration being practically unchecked, so that, once started, it would continue almost indefinitely.
Some means of damping the vibration analogous to the damping action of the liquid in a magnetic compass must clearly then be provided. Ideally the means should be such that if the axle is deflected through any angle it will return to the north and south position in a “dead-beat” manner and not swing across the meridian over to the opposite side. This ideal cannot be realised in practice.
Returning to the simple pendulum illustrated in [Fig. 13] we have to notice that the influence at work causing the vibration is the weight of the bob acting about the axis at g. This influence is a maximum when the bob is at the extreme positions e f and is zero when the bob is at d. On the other hand, the velocity of the bob is zero at the two extreme positions e f and is a maximum at d. During the swing from e to d the vibrating influence is helping the motion of the bob and the velocity consequently increases. At d the vibrating influence disappears, while during the swing from d to f it reappears and this time opposes the motion of the bob, the velocity of which consequently becomes less and less. The movement of the bob from d to f in opposition to the vibrating influence is achieved by the momentum of the bob arising from the velocity which it gathers during its swing from e to d. For the angle d g f to be equal to the angle d g e the velocity of the bob as it passes through the position d must just be a certain amount, no more and no less, namely, the velocity which a body would acquire in falling from rest at the level of the bob at e vertically downwards to the level of the bob at d. If the velocity of the bob when it swings through d is greater than this amount the bob will swing beyond the position f. If it is less the bob will fail to reach f.
Fig. 14. Damped and Undamped Vibrations.
The analogue of the problem to be solved in connection with the gyro-compass is to devise some means that will rob the pendulum bob on each successive swing of some percentage of the velocity with which it passed through the resting position d during the preceding swing. By so doing we shall obviously decrease continuously the angle to which the pendulum swings on each side of the position d. Thus instead of the swings as traced out on a piece of paper moving below the bob being as shown at A ([Fig. 14]), they will be of the form represented at B. The amplitudes, instead of remaining of uniform amount practically indefinitely, will diminish with each swing until they become so small as to be invisible. It is to be noted that theoretically the vibrations cannot be completely suppressed even after an indefinite number of swings, for if the velocity at the resting position is at each swing, say, 50 per cent. less than at the previous passage, it will always be something and never become zero. It will, however, in quite a small number of swings become so low that the motion of the pendulum will be practically undiscernible. Thus with a 50 per cent. decrement the velocity at the eighth passage of the bob through the resting position will be less than 1 per cent. of what it was at the first passage.
It is also to be noted that while the amplitudes are decreased in the manner indicated the periods of the swings are not being made less. In an ordinary pendulum the period, as we have said, depends solely upon the length and—within quite wide limits at least—remains the same whatever be the angle to which we originally deflect the bob. We should therefore expect that if the swings are “damped” in the way shown at B ([Fig. 14]), the period of each swing would be the same and equal to that of the undamped swings represented at A. Actually the period of a damped vibration is always somewhat greater than that of the same system vibrating freely, for by robbing the pendulum of some of its velocity at each swing we are virtually causing the bob to pass through the resting position with the velocity of a free swinging pendulum of greater length and therefore of increased period. The increase in the period of the damped pendulum over the same pendulum when undamped is determined by the strength of the damping means employed, or, in other words, by the percentage by which we reduce the velocity at each swing.
In the early (1910) Anschütz compass the period of vibration at the equator without damping was, as we have stated already, about 61 minutes. With its damping device in action the period of the compass at the equator became approximately 70 minutes. In later designs of gyro-compasses the period of the damped vibration is deliberately made 85 minutes or thereabouts. A practical advantage—to be explained later—is secured by adopting this particular value. It is the period which a simple pendulum would have if its length were equal to the radius of the earth—4000 miles or so.
Fig. 15. Damped Pendulum.