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.
[Illustration: FIG. 26. Ballistic Force on Compass when Ship’s Speed
Changes.]
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.
[Illustration: FIG. 27. Ballistic Force on Compass when Ship’s Speed
Changes.]
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.
[Illustration: FIG. 28. Ballistic Deflection.]
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.