DAMPING THE VIBRATIONS OF THE GYRO-COMPASS
Reviewing what we have already established, we see that a gyroscopic
system possessing “three degrees of freedom” and having a pendulum
weight fixed below the wheel manifests a tendency in all latitudes
to preserve its axle pointing in the north and south direction, a
“directive force” or restoring moment being developed and applied
to the axle if the north and south position is departed from. The
magnitude of the directive force in any given latitude increases with
the deflection, from zero when the axle is pointing north and south
up to a maximum when it is aligned east and west. At any given angle
of deflection of the axle the magnitude of the directive force varies
with the latitude in which the system is stationed, being zero at the
north or south (true) pole and a maximum at the equator. Finally, at
any given angle of deflection of the axle and in any given latitude the
magnitude of the directive force is determined by (_a_) the speed of
rotation of the earth on its polar axis, (_b_) the speed of rotation of
the spinning wheel on its axle, and (_c_) the mass or moment of inertia
of the spinning wheel.
We have now to consider several important matters affecting the
practical application of the gyroscope-pendulum combination as a
substitute for the magnetic compass. The first practical consideration
which arises naturally in our minds is the question: Can a system be
devised and constructed sufficiently robust to withstand the trials and
knocks of every-day use and yet be sufficiently delicate to respond
to the feeble directive forces on the effect of which its action as
a compass depends? From the table given previously it will have been
noted that in the three chief types of gyro-compass so far developed
the directive forces developed are in two examples greater than the
corresponding directive force applied to the card of a magnetic
compass, while in the third the directive force is materially lower.
Even though they were all considerably greater than the force applied
to the needle of a magnetic compass, some doubt as to their sufficiency
to effect their work would remain, for they have to control sensitive
elements, comprising a spinning wheel, axle, supporting rings, etc.,
weighing anything from 7 lb. to about a hundredweight, whereas in the
ordinary compass the sensitive element consisting of the card and its
attached magnetic needles weighs round about ¼ oz. The actual weights
of the sensitive elements are given in the following table.
_Weight of Sensitive Element._
Anschütz compass 15 lb.
Sperry compass 100 lb.
Brown compass 7¼ lb.
Thomson magnetic compass 178 grains
Whether or not the directive force developed will be sufficient to
control the movement of the sensitive element in a gyro-compass must
clearly depend very largely upon the degree of success reached in
banishing friction from the vertical axis about which the sensitive
element moves. Without for the present describing the means actually
adopted to secure virtually frictionless support in the three types of
gyro-compass, we may say that were they other than very refined no
compass action whatever would be manifested. In the early theoretical
days of the gyro-compass before sufficiently refined practical
constructive methods had been developed, the experimental verification
of the mathematical results arrived at could not be attempted.
Granted the attainment in practice of a satisfactory frictionless
method of supporting the sensitive element, we have next to note that
the simple gyro-pendulum system which we have been considering would
be quite useless as a direction-finding device either at sea or on
land by virtue of the fact that the very absence of friction at the
vertical axis would encourage the sensitive element to oscillate
from side to side of the meridian under the least provocation. The
period of oscillation would be a prolonged one, much too prolonged, in
fact, to permit the true north to be determined by taking the mean of
the extreme positions reached by the gyro-axle in the course of its
oscillation.
We have seen that the simple gyro-pendulum system which we have so far
been considering, when placed on the equator, manifests a tendency to
set its axle north and south, that if the axle is deflected towards the
east a westerly turning directive force is developed, and that if the
axle is deflected towards the west an easterly turning directive force
is developed.
In an ordinary vertical pendulum (Fig. 13), the resting position of
the bob is at _d_. If it is swung to the position _e_--towards the
east, let us say--the weight _w_ of the bob supplies a moment about the
axis at _g_, tending to restore the pendulum to its resting position;
while if it is swung towards the position _f_--towards the west, we may
suppose--the moment is reversed and again acts to restore the pendulum
to its resting position.
[Illustration: FIG. 13. Pendulum and Compass.]
The gyro-pendulum system as set up at the equator is, it will be
seen, subjected to an exactly analogous set of forces when its axle
is deflected east or west. The system, in fact, constitutes virtually
a horizontal pendulum, the vertical axis H J being identified with
the axis at _g_ in the ordinary pendulum. Now we know that if we
deflect an ordinary pendulum to some such position as _e_ and let it
go it will swing through the resting position _d_ to a position _f_
equidistant on the other side, and will continue to vibrate until
friction at the axis _g_, air resistance, etc., sap its original stock
of energy communicated to it by the initial deflection. The period of
vibration--the time elapsing between two successive passages of the bob
in the same direction through the resting position--is determined by
the length of the pendulum and remains constant throughout, even when
the amplitude of the swing has fallen off virtually to nothing.
An exactly analogous state of affairs exists with the gyro-pendulum
system. If the axle is deflected towards the east and then let go
it will swing back under the action of the directive or restoring
force through the north and south position over to an equal angle on
the western side, and will thereafter vibrate back and forth with a
constant period, until frictional and other losses cause the motion
to die away. The period of vibration is determined by a complication
of factors, among which are the speed at which the wheel is spinning
on its axle, the speed of rotation of the earth, and the mass of
the sensitive element. If the sensitive element can be regarded as
consisting solely of the wheel, then, no matter what may be the size
of the wheel, so long as it is in the form of a circular disc, the
period of vibration is determined solely by the speed of the wheel
and the speed of rotation of the earth. For a wheel at the equator
running at 20,000 revolutions per minute the period of vibration would
be about eleven seconds. In practice, however, the weight of the
axle, the inner and outer supporting rings--or their equivalents--the
pendulum bob and various other fittings and adjuncts of a secondary
nature have to be added to the weight of the wheel in assessing the
influence of the sensitive element upon the period of vibration. The
greater the mass--or more correctly, the moment of inertia--of the
sensitive element the longer will be the period of vibration. In the
early--1910--Anschütz gyro-compass the sensitive element had a moment
of inertia such that the period of vibration at the equator was just
over 61 minutes; that is to say, 334 times as long as it would have
been if the sensitive element had consisted of nothing but the spinning
wheel.
This very prolonged period, were nothing done to rectify matters, would
be a very serious objection in practice to the use of the gyro-compass.
The axle, if deflected, would take about half an hour to reach an
equal position on the opposite side of the meridian. Hence, if, when
a compass reading was desired, the axle were found to be vibrating,
at least half an hour would be required to determine the north and
south direction by observing the two extreme positions of the axle and
taking the mean. The alternative would be to wait until the vibration
died away. This course would involve, however, a very much greater
delay, for the virtual absence of friction at the vertical axis of the
system--an essential, as we have seen, if the directive force is to
be allowed to come into play at all--results in the vibration being
practically unchecked, so that, once started, it would continue almost
indefinitely.
Some means of damping the vibration analogous to the damping action of
the liquid in a magnetic compass must clearly then be provided. Ideally
the means should be such that if the axle is deflected through any
angle it will return to the north and south position in a “dead-beat”
manner and not swing across the meridian over to the opposite side.
This ideal cannot be realised in practice.
Returning to the simple pendulum illustrated in Fig. 13 we have to
notice that the influence at work causing the vibration is the weight
of the bob acting about the axis at _g_. This influence is a maximum
when the bob is at the extreme positions _e f_ and is zero when the
bob is at _d_. On the other hand, the velocity of the bob is zero at
the two extreme positions _e f_ and is a maximum at _d_. During the
swing from _e_ to _d_ the vibrating influence is helping the motion of
the bob and the velocity consequently increases. At _d_ the vibrating
influence disappears, while during the swing from _d_ to _f_ it
reappears and this time opposes the motion of the bob, the velocity of
which consequently becomes less and less. The movement of the bob from
_d_ to _f_ in opposition to the vibrating influence is achieved by the
momentum of the bob arising from the velocity which it gathers during
its swing from _e_ to _d_. For the angle _d g f_ to be equal to the
angle _d g e_ the velocity of the bob as it passes through the position
_d_ must just be a certain amount, no more and no less, namely, the
velocity which a body would acquire in falling from rest at the level
of the bob at _e_ vertically downwards to the level of the bob at _d_.
If the velocity of the bob when it swings through _d_ is greater than
this amount the bob will swing beyond the position _f_. If it is less
the bob will fail to reach _f_.
[Illustration: FIG. 14. Damped and Undamped Vibrations.]
The analogue of the problem to be solved in connection with the
gyro-compass is to devise some means that will rob the pendulum bob on
each successive swing of some percentage of the velocity with which it
passed through the resting position _d_ during the preceding swing. By
so doing we shall obviously decrease continuously the angle to which
the pendulum swings on each side of the position _d_. Thus instead of
the swings as traced out on a piece of paper moving below the bob being
as shown at A (Fig. 14), they will be of the form represented at B.
The amplitudes, instead of remaining of uniform amount practically
indefinitely, will diminish with each swing until they become so small
as to be invisible. It is to be noted that theoretically the vibrations
cannot be completely suppressed even after an indefinite number of
swings, for if the velocity at the resting position is at each swing,
say, 50 per cent. less than at the previous passage, it will always be
something and never become zero. It will, however, in quite a small
number of swings become so low that the motion of the pendulum will
be practically undiscernible. Thus with a 50 per cent. decrement the
velocity at the eighth passage of the bob through the resting position
will be less than 1 per cent. of what it was at the first passage.
It is also to be noted that while the amplitudes are decreased in the
manner indicated the periods of the swings are not being made less. In
an ordinary pendulum the period, as we have said, depends solely upon
the length and--within quite wide limits at least--remains the same
whatever be the angle to which we originally deflect the bob. We should
therefore expect that if the swings are “damped” in the way shown at
B (Fig. 14), the period of each swing would be the same and equal to
that of the undamped swings represented at A. Actually the period of
a damped vibration is always somewhat greater than that of the same
system vibrating freely, for by robbing the pendulum of some of its
velocity at each swing we are virtually causing the bob to pass through
the resting position with the velocity of a free swinging pendulum of
greater length and therefore of increased period. The increase in the
period of the damped pendulum over the same pendulum when undamped is
determined by the strength of the damping means employed, or, in other
words, by the percentage by which we reduce the velocity at each swing.
In the early (1910) Anschütz compass the period of vibration at the
equator without damping was, as we have stated already, about 61
minutes. With its damping device in action the period of the compass
at the equator became approximately 70 minutes. In later designs of
gyro-compasses the period of the damped vibration is deliberately made
85 minutes or thereabouts. A practical advantage--to be explained
later--is secured by adopting this particular value. It is the period
which a simple pendulum would have if its length were equal to the
radius of the earth--4000 miles or so.
[Illustration: FIG. 15. Damped Pendulum.]
A vibrating pendulum (Fig. 15) will be satisfactorily damped if we can
apply to the bob in all positions of its swing a force proportional to
the velocity with which the bob is moving at each given instant and
directed always so as to oppose the motion. At _e_, the position of
release after deflection, the bob has no velocity; the damping force
should therefore be zero. As it travels from _e_ to _d_ the bob is
gaining velocity; the damping force should therefore grow from zero to
a maximum at _d_ and be directed at each instant tangentially to the
curve of swing and act from _d_ towards _e_. During this portion of
its swing the bob is thus robbed of some of its velocity, so that it
fails to rise to the position _f_ and comes to rest at some point _g_.
In travelling from _d_ to _g_ the velocity of the bob is decreasing
naturally, and is still further decreased by the damping force which,
acting in the same direction as before, falls from a maximum at _d_
to zero at _g_. As it moves from _g_ to _d_ on the return stroke the
bob gains velocity; the damping force should therefore increase from
zero at _g_ to a maximum at _d_--of a lower value than the maximum at
the same point during the first stroke--and should be directed along
the curve in the reversed sense, namely, from _d_ towards _g_. During
the swing from _d_ to _h_ the damping force, still reversed, should
decrease from the second maximum value once more to zero. At the start
of the second swing the damping force should again act from _d_ towards
_e_ and should rise from zero to a third maximum value at _d_--less
than either the second or first maximum value, and so on until the
amplitude of the swing is reduced to the required degree. It will be
noticed that in a damped vibration the mean position of the bob on any
one swing is not coincident with the resting position _d_, but lies
somewhere between the resting position and the position from which the
bob commences the swing.
With each passage of the bob through the resting position _d_ the value
of the damping force rises to a maximum, the value of which becomes
less and less on each successive swing. Ultimately, when the bob
settles in the resting position, the maximum value becomes zero. In
other words, the damping force, having completed its work by bringing
the bob to rest, entirely disappears and leaves the pendulum exactly in
the same condition as it would be under in the resting position if no
damping force had ever been in action. The pendulum is thus as free as
formerly to respond, in the resting position, to a vibrating influence,
but as soon as it acquires motion the damping force is again called
into existence to a degree directly dependent on the strength of the
vibrating influence, with the result that the motion is first checked,
and then finally stopped.
The damping force required is, as we have said, one which at all
times is proportional in magnitude to the velocity of the bob--or
what is the same thing, to the angular velocity of the pendulum as a
whole--and which at all times acts to oppose the motion of the bob.
Metallic friction--say, at the supporting axis of the pendulum--it
would bring the motion to rest sooner or later, would not provide a
satisfactory damping force, for solid friction is independent of the
rubbing velocity, at least at low speeds such as we are here concerned
with. The damping force provided by it being constant, would not be
automatically adjusted to the velocity of the bob. It would vanish,
it is true, when the bob was at rest, but as soon as the slightest
vibration set in it would spring up to its full value straight away and
would preserve the same value throughout a large swing as throughout
a small one. In any event the presence of metallic or other solid
friction at the point in the gyro-compass corresponding to the axis of
the pendulum--namely, at the bearings of the vertical axis H J--cannot
be permitted, and must be eliminated to the utmost possible degree if
the directive force is to be sufficient to control the movement of the
sensitive element.
Fluid friction, on the other hand, would provide a satisfactory damping
force, for fluid friction is proportional to the velocity, at least
at low speeds. A pendulum vibrating with its bob in a vessel of water
or the floating card of an ordinary magnetic compass is satisfactorily
damped by fluid friction. In the gyro-compass, however, the motion to
be damped is, as we have seen, an exceedingly slow one, slower in fact
than the small hand of a watch if the deflection of the axle from the
meridian is initially less than 11½ deg. east or west. A fluid damping
force would be proportionately low, so that without making the damping
elements of enormous size the force derived would be insignificant and
next to useless for practical purposes. As an illustration of this
statement it may be remarked that in the early Anschütz compass the
sensitive element was virtually floated in a bowl of mercury. Yet the
drag of the mercury, the velocity of the vibration being so small,
did not measurably reduce the amplitudes of the vibration during
observation extending over several hours. This example is not quite a
good one, however, for the friction at the surface of a body immersed
in mercury would appear to be not of the fluid description, but of the
solid type.
Solid and fluid friction being thus ruled out, at least as direct means
of providing the required damping force, we have to find some other
method of applying it. It is, or should be, clear that in whatever way
the damping force is applied it should originate within the sensitive
element itself. If it originates outside, then its transmission to the
sensitive element cannot, in view of the fact that its origination,
growth, and decay are to be controlled by the motion of the element,
be effected in any conceivable way without the introduction of some
material connection between the element and the outside source of the
force. Such a connection can only be made frictionless if the outside
source moves in exact unison with the sensitive element. If it does so
move it clearly ceases to be an outside source and becomes really part
of the sensitive element itself. This consideration suggests generating
and applying the damping force gyroscopically by the exertion of some
suitable action on the spinning wheel itself.