ELEMENTARY GYROSCOPIC PHENOMENA
Let a wheel A (Fig. 1) be mounted on an axle B C journalled within a
horizontal ring D. Let this ring in turn be mounted on journals E F
within a vertical ring G and, further, let this vertical ring be
carried on journals H J within a vertical frame K. This arrangement
constitutes a gyroscopic system having three degrees of freedom,
because relatively to the frame K the wheel may turn about three axes
B C, E F, and H J mutually at right angles to each other and because,
if the wheel is set spinning on its axle, gyroscopic properties will be
manifested.
[Illustration: FIG. 1. Model Gyroscope, with Three Degrees of Freedom.]
The following is a brief statement of the gyroscopic properties
manifested when the wheel is spun on its axle:
(_a_) Let the wheel be spinning in the direction of the arrow L and
let a weight W be hung on the horizontal ring at the end B of the
axle. The movement produced by this weight is not a rotation of the
horizontal ring, and the wheel within it, about the axis E F. Instead,
the horizontal ring remains horizontal and the whole system inside the
square frame sets off rotating at a uniform speed about the axis H J
in the direction of the arrow marked M on the horizontal ring. This
rotation or precession, as it is called, will be maintained so long as
the weight W remains in action. There is here no question of perpetual
motion. The work expended in overcoming the friction at the vertical
journals is derived from the energy of the spinning wheel, and when
this energy is exhausted the phenomenon ceases. The phenomenon can, in
fact, only be maintained indefinitely by expending power to drive the
wheel against the leakage of energy through friction at the journals
of the axle and the vertical axis H J. A closer examination of the
phenomenon would show that there is a slight rocking motion of the
horizontal ring on its axis E F, and therefore an additional leakage
of energy at the journals of this axis. This rocking motion can be
neglected for our present purposes. It is sufficiently accurate to say
that the horizontal ring remains horizontal.
(_b_) The speed of the precession is proportional to the weight W
and to the speed of rotation of the wheel on its axle. For instance,
doubling the weight doubles the speed of precession.
(_c_) If the direction of spin of the wheel is reversed the direction
of the precession is also reversed.
(_d_) If the spin of the wheel is in the direction L, and if instead
of attaching a weight at the end B of the axle we exert an _upward_
force at this point the precession developed will be _opposed_ to the
direction of the arrow M.
(_e_) If instead of trying to rotate the wheel about the axis E F by
means of a weight or force applied at B we attempt to turn it about the
vertical axis H J by applying a horizontal force V to the outer ring,
the wheel will not turn about the vertical axis H J, but about the
horizontal axis E F, the end B of the axle rising up towards H.
(_f_) As before, the direction of this movement is reversed by
reversing either the direction of spin of the wheel or action of the
force V. If both are reversed simultaneously the direction of the
movement produced by the applied force is not altered.
The behaviour set forth above can be summarised in a general rule as
follows:--If to a spinning wheel possessing three degrees of freedom
a force be applied tending to turn the wheel about some axis X X, the
actual motion produced will not be about X X but about some other axis
Y Y; this axis Y Y will be such that rotation about it will tend to
bring the axle of the spinning wheel into coincidence with or parallel
with the axis X X; the direction of the rotation produced about Y Y
will be such that when the condition of coincidence or parallelism
is reached the spin of the wheel will coincide in direction with the
rotation we are attempting to produce about the axis X X.
Taking case (_a_) (Fig. 1), it will be seen that the axis E F about
which we are attempting to produce rotation by means of the weight W,
together with the weight W itself, is of necessity carried round by
the precession in the direction M at the same rate as the axle of the
spinning wheel. The axle in this case cannot therefore place itself in
coincidence with the axis of the applied force. But it does its best to
do so. The precession persists and is an expression of the fruitless
chase of the axis E F by the axle B C.
[Illustration: FIG. 2. Model Gyroscope, with Three Degrees of Freedom.]
If, however, the weight is attached by some kind of sliding connection
on the horizontal ring in such a way that its line of action remains
stationary in space, then the axis about which we are attempting to
produce rotation will also remain stationary in the position occupied
by the axis E F before precession commences. In this case it is quite
possible for the axle of the wheel to place itself in coincidence with
the axis of the applied force. Precession about H J through 90 deg.
will accomplish this result, as indicated in Fig. 2. The weight W is
now acting at a point on the horizontal ring where it ceases to have
any tendency to turn the wheel about the axis E F. When, therefore, the
position of coincidence is reached precession ceases and the system
comes to rest in this position.
If the experiment were actually made it would be found that the
momentum acquired by the system during the 90 deg. turn would carry
the axle through the position of coincidence with the axis of the
applied force. But immediately the axle passes to the opposite side
the force W is exerted on a point of the horizontal ring between F and
C. The action of the force passing on to this, the opposite, segment
of the ring reverses the conditions under which the system started
its movement and as a result precession in the direction opposed to
the arrow M is set up. The axle thus tends to recover its position of
coincidence and in the end settles down to a vibratory motion from side
to side of the axis of the applied weight. Friction at the vertical
journals will “damp” this vibratory motion, the amplitudes of the
swings will decrease, and the axle will ultimately settle in steady
coincidence with the axis of the applied force. In this condition the
force will have no further effect on the system beyond throwing a
bending moment on to the vertical axis.
[Illustration: FIG. 3. Frictional Transmission of Turning Moment.]
Instead of trying to make the wheel rotate about the axis E F by
applying a weight to the inner ring as in Fig. 1, let us, as shown
in Fig. 3, mount the square frame K on a horizontal axis N P and
attach the weight W to an arm fixed on the frame. The axes N P and E F
being--at least initially--collinear, the effect of this arrangement
is to throw a turning moment on to the wheel about the axis E F just
as does the weight W in Fig. 1. It is to be noticed, however, that the
moment of the weight W in Fig. 3 about the axis N P is transmitted to
the inner ring as a moment about the axis E F solely because of the
friction existing at the journals of the axis E F. This friction may be
very small, so that the turning moment received by the wheel is only a
very small fraction of the turning moment exerted by the weight W about
N P. The effect of the arrangement is thus exactly the same as would
be produced in the arrangement Fig. 1 if we reduced the weight W to
a hundredth or a thousandth of its value. In other words, precession
about the vertical axis H J will set in in the direction of the arrow
M just as before, but the speed of this precession will be only a
hundredth or a thousandth of the previous value.
It is not very important to trace out the behaviour of the system shown
in Fig. 3 beyond a very brief period immediately after the weight W is
applied. The point of importance is that the precession produced by the
weight is very slow, and therefore that in a given interval of time the
amount precessed is very small. Further, the rate of the precession
depends solely upon the friction at the journals of the axis E F and
not upon the weight W or the movement of the frame K except in so far
as these factors affect the friction. The less the friction the less
will be the rate of precession and the amount precessed in a given
time. Thus by mounting the axis E F on knife edges the friction can
be made so small that the precession produced by the weight W becomes
immeasurable. Hence we deduce that if friction is substantially absent
at the axis E F the frame K might be violently rocked on the axis N P
or even set into continuous rotation without causing the axle of the
wheel either to dip or to precess.
Continuing the argument, we might mount the square frame on a vertical
axis and attempt to produce rotation of the wheel about the axis H J
by applying a horizontal force to one side of the square frame instead
of a force V on the outer ring as shown in Fig. 1. A similar result
would be obtained. Granted an all but total absence of friction at the
journals of the vertical axis H J, the precession produced about the
horizontal axis E F would be immeasurably small. Thus the frame might
be set into violent motion about its vertical axis without causing the
axle either to rotate in a horizontal plane or to precess in a vertical
one.
[Illustration: FIG. 4. One Degree of Freedom Lost.]
Finally, if the square frame were mounted on a horizontal axis
collinear with the axle B C it might obviously be rotated about this
axis without affecting the system otherwise than by increasing or
reducing the rubbing speed of the axle B C in its bearings.
Since pure translation of the frame in any direction cannot apply a
turning moment to the system about any axis, and as rotation of the
frame about any one of the three principal axes has no effect which
is measurable on the orientation of the axle, it follows that, given
substantial absence of friction at the axes E F and H J, the axle of
the wheel will remain constantly pointing parallel with its original
position, no matter how the frame K may be moved or turned about.
[Illustration: FIG. 5. Second Degree of Freedom Lost.]
The gyroscopic system shown in Fig. 1 has, as we have said, “three
degrees of freedom,” because its wheel is free to spin about three
different axes mutually at right angles. It is to be carefully noted
that it can only truly be said to have three degrees of freedom so long
as the inner ring and the parts inside it are not rotated on the axis
E F away from the position which in Fig. 1 they are shown as occupying
relatively to the outer ring. Thus rotation of the wheel on its axle
or of the whole system inside the square frame on the axis H J leaves
the three axes B C, E F, H J undisturbed at right angles to each other.
But rotation of the inner ring and the parts inside it on the axis E F
tends to destroy one of the degrees of freedom. If, for instance, the
inner ring is rotated through 90 deg., as shown in Fig. 4, the axle B C
and the axis H J will coincide in direction. In this position the wheel
cannot be rotated about a horizontal axis at right angles to E F and
has therefore virtually only two degrees of freedom, namely, about the
axis E F and about the axis H B C J. Again, if with the inner ring in
the position shown in Fig. 4 the outer ring is turned through 90 deg.
relatively to the square frame, the system assumes the configuration
shown in Fig. 5 and the wheel loses the power of rotating about a
horizontal axis in the plane of the square frame.
[Illustration: FIG. 6. Lost Degrees of Freedom Restored.]
If, then, in any application of the gyroscope it is necessary to
guarantee that the system shall have three degrees of freedom in all
possible configurations, the simple mounting shown in Fig. 1 will not
serve the purpose. It can be made to do so in the manner shown in Fig.
6, namely, by mounting the square frame inside a gimbal ring X, which
in turn is supported by a frame Y, the two new axes T U and V W being
at right angles to each other. In the position shown in Fig. 4 the new
axis T U would restore the lost third degree of freedom, while the
second new axis V W would restore the degree of freedom lost when the
system assumed the configuration shown in Fig. 5.
In the gyro-compass it is necessary to guarantee that the spinning
wheel in all possible configurations shall have three degrees of
freedom, and accordingly we find the wheel mounted in a manner
reproducing the features of Fig. 6. On the other hand, the majority
of the movements which the compass system is called upon to make do
not entail anything except very small degrees of rotation of the inner
ring and wheel about the axis E F (Fig. 1), and therefore for most
purposes the simple mounting there shown reproduces the required three
degrees of freedom sufficiently closely to permit us to use it for
demonstration purposes. In one very important portion of our subsequent
discussion, however--namely, that dealing with the effect on a marine
gyro-compass produced by rolling and pitching of the vessel--it will be
necessary for us to take cognisance of the fact that the square frame
shown in Fig. 1 is not fixed directly to the ship’s deck, but is really
carried in gimbals as shown in Fig. 6.