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.
[Illustration: 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.
Let us, however, rotate the bob about the axis at C through something
less than 90 deg. from the first position--say, through 45 deg., as
indicated in the third sketch. Consider again the two extreme end
portions as the bob is passing through the mid position of its swing.
Let J K be a line through the centre of the bob parallel with the
knife-edge axis. From the centres of the end portions, let lines L N
and M P be drawn at right angles to J K. From N and P draw lines
upwards parallel with the rod A to meet the knife-edge axis at Q and R.
Then Q and R are the points about which the two extreme portions of the
bob are swinging. The centrifugal forces on these portions are directed
along the lines Q L and R M, and are indicated at S and T. These
forces may, as shown, be resolved into vertical components--parallel
with Q N and R P--and horizontal components--parallel with N L and P M.
Treating the other portions of the bob similarly, it will be seen that
while the vertical components of the centrifugal forces, as before,
merely tend to bend the ends of the bob downwards, the horizontal
components are no longer parallel with the length of the bob, as in the
second sketch. They are inclined to the bob and clearly apply to it a
moment which will turn it about the axis at C in the direction of the
arrow U until the bob reaches the position shown in the second sketch.
It is obvious that any rotation of the bob on the axis at C away from
the position shown in the first sketch, even a slight one, will call
into action horizontal components of centrifugal force after the manner
shown in the third sketch. The equilibrium of the bob in the position
shown in the first sketch is therefore unstable. With the bob in the
second position a movement of the end L to the _left_ will, as we know,
call into action horizontal components of centrifugal force, tending
to turn the bob about C in the direction of the arrow U. If the end L
is moved to the _right_ the horizontal components, as can easily be
verified by repeating the argument, will tend to turn the bob about C
in the direction opposed to the arrow U. Thus deflection to either side
of the second position calls forth forces which lead the bob to recover
that position. We conclude, therefore, that the second position is one
of stable equilibrium.
It may further be shown by repeating the argument that if the bob
is rotated about C from the first position in such a way as to send
the end L to the _rear_ the horizontal components called into action
tend to set the bob into the second position with the end L pointing
_backwards_, and that when the bob is so aligned relatively to the
knife-edge axis the equilibrium is again stable. Thus, altogether,
if the bob is parallel with the knife-edge axis, the equilibrium is
unstable. If it is at right angles the equilibrium is stable. If it is
set into any intermediate position, forces are called into play which
endeavour to align it at right angles to the knife-edge axis.
Let a second cylindrical bob of the same size and weight be attached
to the first at right angles to it and in the same plane. Then in all
positions of the bobs relatively to the knife-edge axis the equilibrium
will be stable, for at any setting of the bobs the rotary force about C
developed by the one bob will be equal and opposed in direction to that
developed by the other bob.
The above discussion covers a principle of considerable general
importance in mathematical physics. Briefly put, it shows us that a
body suspended pendulum-wise in the manner sketched will develop a
tendency when swung to set its longer axis parallel with the plane
of the swing, and that to avoid this tendency in a swinging body
mounted in the manner we have indicated, the mass of the body must
be distributed symmetrically and equally about the pendulum rod, so
that there shall be no “longer axis.”[8] So far as the gyro-compass is
concerned, the circumstances described necessitate--at least in the
Sperry and Brown compasses--the addition of compensator weights to
represent the second cylindrical bob to which we have referred above.
The masses composing the gyro-compass are not disposed symmetrically
and equally about the vertical axis--the axis which we have throughout
called H J. As seen in plan, the masses are concentrated towards the
east and west plane containing the spinning wheel, and are deficient
towards the north and south. By virtue of the presence of the
pendulous weight, or its equivalent, below the spinning wheel, the
whole mass of the compass is capable of swinging pendulum-wise, the
axis of this swing being one or other of the axes of the external
gimbal mounting. Further, the swinging mass is free--if the wheel is
not spinning--to turn about the vertical axis which we have called H J.
Reference to Fig. 31 will make these statements quite clear.
Thus the compass and its system of mounting inevitably reproduce the
essential features of the pendulum shown in Fig. 43. Taking the second
sketch in that engraving, the bob represents the masses of the compass
concentrated in the east and west plane. The gyro-axle being at right
angles to this plane is parallel with the knife-edge axis B. This
axis B may be regarded as representing the longitudinal axis of the
external gimbal mounting, so that in the conditions imagined the ship
is supposed to be sailing due north or south. It is clear that when the
ship rolls on either of these courses, the fact that the mass of the
compass is concentrated in the east and west plane will not tend to
deviate the axle, for the “pendulum bob” is in its position of stable
equilibrium.
The first sketch similarly represents the condition of matters existing
when the ship rolls on an east or west course, the knife-edge axis
B again being identified with the longitudinal axis of the external
gimbal mounting. The pendulum bob is now in its position of unstable
equilibrium, so that if the mass distribution of the compass is not
corrected by the addition of compensator weights, a tendency for the
axle to deviate away from the north _may_ occur.
If we identify the knife-edge axis B with the athwartship axis of the
external gimbal mounting, the second sketch in Fig. 43 illustrates the
condition of matters existing when the ship _pitches_ on an east or
west course, while the first will represent pitching on a due north or
south course.
On quadrantal courses, failure to correct the inequality of the mass
distribution, if that distribution is not uniform, will _inevitably_
result in the axle deviating when the ship rolls. Thus, the third
view in Fig. 43 represents the conditions prevailing when the ship
rolls on a due north-west--or south-east--course. Looking at the
plan, the line J K represents the direction of the longitudinal axis
of the external gimbal mounting and therefore the direction of the
ship’s course, while a line at right angles to the pendulum bob would
represent the gyro-axle, and therefore the north and south direction.
When the ship rolls the compass masses oscillate pendulum-wise about
the external longitudinal gimbal axis--represented by the bar B--but
the directive force of the compass tends to preserve the general plane
of the oscillating masses inclined at 45 deg. to the direction of the
bar B. As a result, the horizontal components of the centrifugal forces
developed in the compass masses apply to the wheel, casing, etc., a
turning moment in the direction of the arrow U. This moment, it will
be seen, tends to turn the sensitive element in the same direction as
that in which the “kicks” of the weight at the out positions of the
swing endeavour to deviate the axle on the same course. The centrifugal
moment would, however, cause an actual deviation in the direction U in
a direct manner only if the gyro-wheel were not spinning. As it is, the
centrifugal moment causes the north end of the gyro-axle to precess
upwards and therefore leads to a deflection of the pendulous weight
of the north. This deflection throws a turning moment on to the wheel
about the horizontal axis--the axis we have called E F throughout--and,
finally, this turning moment precesses the axle horizontally in
the direction of the arrow U. Thus, with the wheel spinning, the
centrifugal moment actually does produce rotation in the direction U,
but its action is not direct.
As we have said, the tendency of the axle to deviate during rolling on
quadrantal courses as a result of the action of centrifugal forces,
is corrected in those compasses requiring it by the addition to the
compass masses of compensator weights on the north and south sides of
the sensitive element. These weights are adjusted for mass and position
in such a way as to make the mass distribution of the sensitive element
substantially the same in the north and south direction as in the east
and west. In this way the centrifugal forces tending to cause deviation
are neutralised on all courses, both during rolling and pitching.