CHAPTER X

THE BALLISTIC DEFLECTION

Having considered the effect of the ship’s speed upon the readings of the gyro-compass, we have next to discuss the effect of the conditions which arise when the speed is changed. It is clear, of course, that if the ship’s speed is changed, say from 20 to 10 knots, the north steaming error appropriate to the course and latitude will fall to half its value when the new speed has been attained. Two questions, however, may be asked. What happens in the period during which the speed is changing, and what time elapses between the attainment of the new speed and the attainment by the compass of the new north steaming error proper to the new speed?

If an ordinary simple pendulum were hung from the roof of a railway carriage it would be found that so long as the train was travelling smoothly at _uniform_ speed the pendulum would hang vertically without oscillating just as it would do if the train were at rest. If the train increased its speed the inertia of the pendulum--that is to say, its tendency to go on moving with the old velocity--would cause the pendulum to assume an inclined position _behind_ the vertical, the deflection being proportional to the rate at which the train was gathering speed. At the instant at which the train ceased to gather speed, and again assumed a uniform velocity, the tendency for the pendulum to remain inclined to the vertical would vanish, but the deflection it possessed at that instant would result in its acquiring an oscillation which would persist until friction, etc., damped it out. If when the pendulum were again steady and hanging vertically, the train began to slow down, a similar series of events would occur, only on that occasion the inertia of the bob would cause the pendulum while the train was losing speed to assume a _forwardly_ inclined position, the angle of which would be a measure of the rate at which the speed was being reduced. On the attainment of the new uniform speed the pendulum would oscillate for a time, as before.

The gyro-compass is in part a pendulous body, and on a ship changing speed or acquiring speed from rest or coming to a stop it is open to the action of inertia forces just as is our railway carriage pendulum. As a result the compass during the change of speed may exhibit an error--the ballistic error, as it is called--on top of the existing north steaming error. Since the speed of a ship, at least of a merchant ship, cannot be changed very quickly, and since, further, the speed of a ship, at least on long voyages, is not liable to be changed very often, it might be thought that this transient ballistic error could be neglected. It probably could be safely neglected if its effect were strictly confined to the actual period of the change in the speed, but it is not so confined. We have to bear in mind the subsequent vibration of the railway carriage pendulum. A similar oscillation is set up in the gyro-compass after the new speed is attained. Since the period of oscillation of the compass is about 85 minutes, and as two or three complete swings have to be made before the damping arrangements can suppress the vibration, it follows that the compass will not settle in its resting position again until two or three hours after the new speed has been attained, although the actual change of speed may have been effected inside five minutes or less.

If the ship is sailing on an east-west course, as represented in Fig. 26, any change of speed will tend to cause the pendulum weight S to move in the direction R or T, according as the change of speed is a decrease or an increase. Such a tendency will merely throw stresses on to the journals at E F and H J, or, if the equivalent of the square outer frame is carried on gimbals, as it actually is in practice, then the tendency to rotate will be translated into an actual movement of the sensitive element on the axis of the outer mounting coincident or parallel with the axle B C of the wheel. Such a movement will have no gyroscopic effect on the compass, for the axle is not subjected by it to any non-parallel displacement. No ballistic deflection will therefore occur on that course.

If the ship changes speed while on a west to east course, the change is similarly without effect on the compass. Strictly speaking, we ought to say that the change of speed will have no effect on the compass if the ship’s course is at right angles to the direction in which the axle of the wheel is pointing. Only at the equator would this course be due east or west. In other latitudes the latitude error has to be reckoned with, so that the particular course on which changes of speed are without effect on the compass must be slightly south or north of due east or west.

If, now, the ship is steaming due north, as indicated in Fig. 27, a reduction of its speed will tend to make the pendulum weight swing forward about the east and west axis E F in the direction P. This tendency is clearly equivalent to the application of an upward force to the end B of the axle, and hence, as we know, will cause the wheel to precess, the end B of the axle going eastwards. As this precession continues it is opposed by the ever-increasing restoring moment of the directive force, so that in the end the axle assumes a position in which the directive force just balances the ballistic deflective force. This ballistic deflection remains constant all the time the ship’s speed is falling. When the speed reaches the new steady value the axle slowly oscillates back to the true resting position.

It is clear that a reduction of speed on a northerly course and an increase of speed on a southerly produce ballistic deflections in a like direction, the north end of the axle moving towards the east. An increase of speed on a northerly course or a decrease of speed on a southerly produces a westerly ballistic deflection. On intermediate courses deflections of a like kind, but of intermediate value, are caused, for the north and south _component_ of the change of speed is alone effective in tilting the wheel about the east and west axis E F.

Imagine a ship steaming due north at 20 knots and changing its speed to 10 knots during a period of five minutes. As it is steaming north there will be a north steaming error, the axle pointing westwards of true north by an amount dependent upon the speed of the ship and the latitude in which it is sailing. Let O N (Fig. 28) be the direction of true north, and let O A be the direction in which the axle of the gyro-compass aligns itself when the speed is 20 knots, the angle N O A being the combined latitude and north steaming errors. Let O B be the resting position for the axle when the ship’s speed is 10 knots, the angle N O B being the latitude error--which has not altered--plus the north steaming error--which is now less because of the reduced speed. As the speed is being reduced, the ballistic action of the pendulum weight causes the axle temporarily to turn eastwards from the true resting position and to align itself in some direction O C. When the ship has settled down to 10 knots, the axle leaving the position O C vibrates about O B with amplitudes which are being continually reduced by the action of the damping system.

The point of importance to notice is that the temporary ballistic deflection O C is eastward of O A, just as is O B the new resting position for the axle. This result is a general one. Had the speed been increased instead of decreased, the ballistic deflection would have been westward of O A, just as would be the new resting position for the axle at the increased speed. Similarly, on due southerly courses or on quadrantal courses the ballistic deflection produced by any change of speed is always in the same direction as that in which the axle moves in passing from the old to the new north steaming error. This being so, it is conceivable that at least, under certain conditions, the ballistic deflection position O C may coincide with the new position O B for the axle under the new north steaming error. Should such a result be obtained, the ballistic action will, on a change of speed occurring, swing the axle straight away into the new resting position. Thus O C and O B being coincident, there will be no tendency for the axle to oscillate about O B when the speed assumes the new steady value, so that the axle will swing into the new resting position in a “dead-beat” manner. The effect of the ballistic action will thus be confined to the period during which the speed is being changed--five minutes in our example--and will not influence the readings of the compass for a period of anything up to two or three hours after the new speed is reached.

The ballistic deflection of a pendulum hung from the roof of a railway carriage is dependent upon the length of the pendulum, and therefore upon its period of natural vibration. So, too, the ballistic deflection of the gyro-compass is dependent upon the period of its vibration about the north and south direction. Once the characteristics of the compass are determined, therefore, the angle A O C (Fig. 28) of the ballistic deflection is settled by (1) the direction of the ship’s course, and (2) by the rate at which the ship’s speed towards the north or south is being changed. It is not affected by the latitude in which the ship is sailing, the deflection produced by a change of 10 knots in five minutes being the same at the equator as in 60 deg. or any other latitude.

Coming to the difference between the old and the new north steaming errors, it will be seen from our previous explanation of the errors themselves that the magnitude of the angle A O B is independent of the design of the compass in use. It will vary with (1) the direction of the ship’s course, and (2) the extent by which the ship’s speed is changed. In addition, it will also vary with (3) the latitude in which the ship is sailing.

From these considerations it is argued that by suitably selecting the period of vibration the ballistic deflection can be made _for one particular latitude_ just equal to the difference between the north steaming errors for any initial and any new speed on any course. In practice the latitude selected is 40 deg. north or south. It is found by calculation that the period of vibration which must be given to the compass to secure this dead-beat ballistic deflection is such that in this latitude the compass should oscillate in the same period as would a simple pendulum, the length of which was equal to the radius of the earth. It is for this reason that all modern gyro-compasses have a period of vibration of approximately 85 minutes.[5]

In latitudes other than 40 deg. north or south the ballistic deflection will not be dead-beat, but by taking this latitude as the mean the subsequent oscillation after a change of speed is--in mercantile vessels at least--sufficiently small to introduce no error of great importance, except for purposes of very accurate observation, in which case the observation, if possible, would not be made until the vessel had been running for two or three hours after the last considerable change of speed.

It is to be noticed that the ballistic error may arise when the ship is altering course without altering its speed. Thus, if it is sailing north and makes a sharp turn eastwards or westwards, the speed northwards during the turn falls progressively from a maximum at the beginning to zero at the end of the turn, even although the actual linear speed of the ship remains constant throughout. Turning in either direction is thus equivalent to a deceleration of the ship’s northerly speed, so that an easterly ballistic deflection may be expected in both cases. On the other hand, a turn towards the north from a due east or west course is equivalent to an acceleration of the ship’s northerly speed, and, as a result, a westerly ballistic deflection is produced. In latitude 40 deg. north or south both these ballistic deflections would be dead-beat in a modern gyro-compass, being just sufficient in the one case to eliminate and in the other to apply the appropriate north steaming error.

In a very recent improvement, it is understood, it has been found possible to provide means whereby the ballistic deflection is made dead-beat in _all_ latitudes.