This object is achieved, as we shall now show, by setting the axles of the gyros L M—[Fig. 41]—inclined to the axle of the gyro K. In a single-gyro compass, or in the 1912 Anschütz compass with the gyros L M suppressed, the system is very stiff against vibrations on the east-west axis E F, but is quite easily set vibrating about the north-south axis provided by the external gimbal mounting. The reason is, of course, that any vibration about the axis E F is met by the resistance of the gyro-wheel, for the vibration causes the axle to alter its direction, whereas no gyroscopic resistance is called into play by the vibration on the north and south axis, for this axis is coincident or parallel with the gyro-axle. Thus, in the 1910 Anschütz compass the period of vibration about the east-west axis was something like 70 min., whereas the period about the north and south axis was only one or two seconds. The smallness of the latter period readily resulted in the compass system getting into a swing in tune with the period of rolling of the ship—from five to twelve seconds or so. Were the axles of the gyros L M in the 1912 form set parallel with that of the gyro K, there would still be no gyroscopic resistance exerted against vibration on the north-south axis. As it is, the inclination of the axles of these two gyros, taken together, has virtually the same effect as would be obtained by the addition to the sensitive element of a separate single gyro, with its axle aligned at right angles to the axle of the gyro K.
The gyroscopic stiffness against vibration about the east-west axis E F is hardly affected by the inclination of the gyros L M at the angle chosen by the designers. The resistance, instead of being that contributed by three gyros, is equivalent to that supplied by about 2¾ gyros. The period of vibration on this axis E F is arranged to be the standard 85 min. About the north and south axis we have virtually the resistance of one gyro. The period of vibration about this axis, instead of being only one or two seconds, is about 80 sec. With this lengthened period of vibration there is practically no chance of the compass system getting into a swing in tune with the rolling of the ship, for every 2½ to 6 seconds the vibrating influence is reversed in direction, and in this interval of time the system cannot acquire any substantial degree of swinging motion. Thus, as the compass, following the rolls of the ship, moves from side to side, it suffers a pure translational movement. The axis H J remains truly vertical at all instants during the roll, and therefore the north and south kicks of the pendulous weight at the out positions of the roll precess the gyro-axle about H J in purely a horizontal plane, the precessional tendency at one out position cancelling that at the other. There is no vertical component in the precession, and therefore there can be no quadrantal error.
It is of interest and of some amusement to note that during the war the Germans applied a fourth gyro to the Anschütz compass, and that this additional gyro was carefully removed from every compass before they surrendered their submarines to us. The deceit was, however, of no avail, for the application of the fourth gyro was known to us long before the war ended, a complete compass so fitted having been recovered from a sunken submarine and repaired and carefully studied. The fourth gyro was applied to the external gimbal rings. These rings, when the submarine rolled, were found to acquire at times a violent oscillation of their own, for, of course, they received no stabilisation from the gyroscopic elements of the compass. On board submarines the violence with which the rings vibrated would occasionally threaten to wreck the compass. By adding a gyroscope to the gimbal rings, the period of vibration of the rings was lengthened to about 16 sec., so that little or no opportunity to swing was left to them. With the exception of this addition, the Anschütz 1912 compass was used by the Germans throughout the war practically without alteration.
CHAPTER XIII
CENTRIFUGAL FORCES DURING QUADRANTAL ROLLING
In addition to the “kicks” of the pendulous weight, which during quadrantal rolling, as we have seen, react gyroscopically upon the compass and tend to make the axle deviate from the north, there is a second influence at work on the compass, which, when the vessel rolls on an intercardinal course, likewise tends to deviate the axle. The two deviations are always in the same direction, so that we cannot arrange the one to reduce or eliminate the other. The second deviation arises from the centrifugal forces developed in the compass parts during quadrantal rolling. These forces, like the kicks of the pendulous weight, react gyroscopically upon the spinning wheel, and if not checked will deviate the axle away from the north.
Fig. 43. Centrifugal Forces on a Pendulum.
Let us consider the somewhat unusual type of pendulum shown in [Fig. 43], a pendulum the rod A of which is fixed rigidly to the bar B carried on knife edges, and of which the “bob” is in the form of a cylinder suspended at its mid point from the rod A in such a way as to permit the bob to turn horizontally on the bearing at C, substantially without friction. In the first instance, let the pendulum be set swinging about the knife edges with the cylindrical bob set parallel with the axis B. Each portion of the bob is thus caused to swing about that point on the axis B which is vertically over the portion when the bob is at rest. The centrifugal force developed on each portion is at all positions of the swing directed radially from such point. The centrifugal force on each portion is a maximum when the bob is passing through the mid position, and falls, with the velocity, to zero at each of the out positions. At the mid position the centrifugal force on the two extreme portions L M of the bob may be represented by D E. These two forces are equal and are directed vertically downwards. On all other similar portions the centrifugal force at the same instant is similar in magnitude and direction. The net effect of the centrifugal forces on all the portions is thus simply a tendency to bend the bob ends downwards about the central line at C. They have clearly no tendency either at the mid position of the swing or at any other to turn the bob round the axis C at the foot of the rod.
Let us now set the pendulum swinging with the bob arranged at right angles to the knife-edge axis, as shown in the second view of [Fig. 43]. Each and every portion of the bob is now swinging about the one point F on the axis. At any instant in the swing the centrifugal force on any portion is directed along the line joining the point F to the centre of that portion. Taking the two extreme end portions L M as the bob passes through the mid position, the centrifugal forces are as indicated at G H. They are not now vertical nor parallel with each other, but, again, they have no tendency to turn the bob horizontally about the axis at C. Resolving them into vertical and horizontal components, we see that the net effect of all the centrifugal forces is to try to bend the ends of the bob downwards—the effect of the vertical components—combined with the application to the bob of a horizontal stretching force—the effect of the horizontal components.