THE QUADRANTAL ERROR
Let a stone be tied to the end of a string and be flung forward while
the free end of the string is held in the hand. Let the hand holding
the free end be moved forward at first with the same speed as the
stone in its flight, and then let the hand be drawn back. We know
that as soon as the string becomes taut--if it were not already fully
extended--the hand would feel a pull exerted on it in the direction
in which the stone was initially projected. This pull arises from
the fact that the stone has been given momentum in the direction of
its flight, and while this momentum is being taken out of it by the
backward pull of the hand and new momentum in the backward direction is
being communicated to it the stone reacts forcibly on the string and
therefore on the hand.
In the gyro-compass the pendulum weight may be likened to the stone,
the stirrup carrying the weight to the string, and the spinning wheel
to the hand. While the ship is steaming at a uniform speed, the system
is in the condition existing just after the stone has been projected
from the hand, and while the hand is following it with equal speed.
A change in the ship’s speed--a reduction of its speed to be quite
correct in our analogy--is comparable with the drawing back of the
hand. The pull of the stone exerted on the hand at this instant is
represented by the tendency of the weight to continue moving at the
former speed, and the resultant “kick” which by its attempt to do so
it applies to the spinning wheel. This kick is communicated to the
spinning wheel through stiff members and rigid connections, and not
through a flexible string. It really acts at the centre of gravity of
the pendulum weight, and is therefore felt by the spinning wheel, not
as a straight pull, but as a force tending to turn the wheel about the
horizontal axis E F.
The kick of the weight when the ship changes speed on any course having
a north or south component produces, as we have seen, a transient
ballistic error which may influence the correctness of the compass
readings for some hours after the change of speed has been completed.
We have now to show that similar kicks occur under other conditions,
and may similarly affect the accuracy of the readings.
The subject which we are about to discuss is the so-called quadrantal
error which, unless steps are taken to eliminate it, appears in the
compass readings when the ship is sailing in a rough sea on any course
other than due north, south, east, or west. It is caused by the rolling
and pitching of the ship. The efforts made to eliminate it have
influenced the evolution of the gyro-compass to a greater extent than
probably those directed towards overcoming or allowing for all the
other errors combined to which the device is open.
[Illustration: FIG. 29. Effect of Rolling on due North Course.]
The gyro-compass on board ship is usually, although in no way
necessarily, mounted above the vessel’s metacentres--transverse and
longitudinal--about which rolling and pitching take place. In Fig.
29 we show a section, looking aft, of a vessel steaming due north
with a gyro-compass mounted on its deck. It is clear that, so far as
the weight S is concerned, it may be regarded when the ship rolls as
the bob of an inverted pendulum vibrating in an east and west plane,
through an angle equal to the angle of the ship’s roll. From what we
have already said, it will be seen that at the end of a roll to port
the hand holding the string attached to the flying stone, hitherto
following it, begins its withdrawal to the east. The weight checked in
its movement to the west communicates a kick to the sensitive element.
This kick, a westward force, is applied at the centre of gravity of
the weight, and clearly does nothing more than throw a stress on to
the journals at E F and H J, and through the square frame on to the
deck. It does not therefore affect the direction in which the gyro-axle
is pointing. If, as is actually the case in practice, the equivalent
of the square frame is not fixed directly to the deck, but is mounted
inside the binnacle on athwartship and longitudinal gimbals, the square
frame will swing on the longitudinal gimbal axis. The weight S will,
however, still act during the ship’s roll as an inverted pendulum,
as at Z Z, in which the bob remains more or less parallel with its
original direction. Under these conditions the kick of the weight at
the end of a roll to port will rotate the frame in the direction of
the curved arrow about the longitudinal gimbal axis. As this axis is
coincident or parallel with the gyro-axle, the gyro-axle is not moved
otherwise than parallel with itself either by the actual roll of the
ship or by the kick of the weight at either out position. Thus a ship
steaming north may roll with impunity without affecting the accuracy of
the direction in which the gyro-axle is pointing.
[Illustration: FIG. 30. Effect of Rolling on due West Course.]
It is clear that rolling on a due south course or _pitching_ on a due
east or west course is similarly without effect on the gyro-compass.
Now consider the case of the ship _rolling_ when on a due west course
(Fig. 30). It is clear that here again the weight S may by itself be
considered as the bob of an inverted pendulum vibrating through the
angle of the ship’s roll, this time in a north and south plane. The
weight S has, in addition, its own pendulum action on the horizontal
axis E F, and tends to keep the gyro-axle horizontal throughout the
roll.
At the end of a roll to port the kick of the weight S, a southerly
force acting at the centre of gravity of the weight, tends to turn the
spinning wheel in the direction R about the horizontal east and west
axis E F. This kick is clearly equivalent to the application of an
upward force at the end C of the axle or a _downward_ force at the end
B, and therefore, as we know, will cause the axle not to turn about
E F, but to precess the north end B towards the _west_ about the axis
H J.
The roll to starboard now takes place, and at its end the kick of the
weight occurs in the northerly direction. This kick will tend to turn
the spinning wheel about the east and west axis E F in the direction T,
and is clearly equivalent to the application of an _upward_ force at
the end B of the axle. Such a force, as we know, will cause the end B
of the axle to precess towards the _east_.
It will thus be seen that the western deflection of the axle produced
by the precession which occurs at the end of the roll to port is
counterbalanced and automatically eliminated by the eastern deflection
produced by the precession occurring at the end of the roll to
starboard. The only effect on the compass caused by the rolling of the
ship is therefore a vibration of the axle about the north and south
direction. We should really say that the only effect is the application
to the sensitive element of a vibratory influence in tune with the
rolling of the ship. As the ship’s period is very small compared with
that of the compass about the axis H J--about 5 to 12 seconds as
compared with about 85 minutes--this vibratory influence practically
fails to disturb the steadiness with which the axle points to the north.
It is clear that pitching of the ship when on a due west course has
the same effect--or lack of effect--on the compass as rolling on a due
north course, and that pitching on a due north course has the same
effect as rolling on a due west course. It is further clear that due
south and due east courses are similar in this respect to due north and
due west courses. Thus when the ship is on a cardinal course neither
pitching nor rolling disturbs the axle of the spinning wheel from its
north resting position.
Matters are quite otherwise, however, when the course is an
inter-cardinal, or quadrantal, one. The discovery of this unlikely fact
was in large part due to the investigations of the Compass Department
of the British Admiralty. It does not appear to have been known, or at
least its full significance does not seem to have been appreciated,
in the earlier days of gyro-compass construction. At least one early
design of compass--the Anschütz of 1910--did not include means of
eliminating or allowing for the “quadrantal error,” and soon became
obsolete as a result primarily of the omission. That the later design
of Anschütz compass successfully overcomes the difficulties introduced
by this error is shown by the admitted excellence of the navigation
of the German submarines, which vessels were universally fitted with
compasses of this design, and which, like all their class, suffer much
from rolling and pitching.
[Illustration: FIG. 31. External Gimbal Mounting.]
So far we have been able to use a very simple model to demonstrate
the properties and errors of the gyro-compass. We have now reached a
subject which, if it is to be explained correctly, requires us to adopt
a model of a more elaborate nature, one in which the outer square frame
of our simple model is itself mounted on a pair of gimbal axes. We
have already referred to this type of mounting in our second chapter
and above in connection with Fig. 29. In Fig. 31 we show it with the
pendulous weight added to the sensitive element. We may take it that
the axis T U, about which the entire compass system may swing, is
parallel with the longitudinal centre line of the ship, and that the
frame Y is fixed to the deck.
[Illustration: FIG. 32. Effect of Rolling on a Due North Course (Simple
Mounting).]
[Illustration: FIG. 33. Effect of Rolling on a Due North Course
(External Gimbal Mounting).]
In order to make quite clear the difference between the two forms of
mounting, we give in Figs. 32 and 33 two corresponding sets of views
showing the compass system on a ship while steaming due north and
rolling. With the simple mounting (Fig. 32) the weight S constantly
remains radially below the centre of the spinning wheel, and the
kicks which it applies at the out port and out starboard positions
merely stress the compass mountings. With the more elaborate mounting
(Fig. 33) the weight tends to remain constantly in the vertical below
the centre of the spinning wheel, but the kicks occurring at the out
positions, if the rolling is continued for any length of time, cause
it to swing beyond the vertical, so that as the ship rolls the whole
compass system acquires an oscillation on the axis T U (Fig. 31) in
tune with the rolls. It is to be noted that the period of vibration of
the compass system about the axis T U is very much less than that of
the sensitive element about the axis H J. The latter, as we have seen,
is about 85 minutes, and is determined, in part at least, by the high
speed of the spinning wheel. About T U the vibration, however, does not
call any gyroscopic force into play, for it takes place without causing
the axle of the spinning wheel to alter its direction. The period of
this vibration is therefore not affected by the speed of the spinning
wheel. It may be quite small--from one to two seconds--and therefore
comparable with the ship’s rolling period. If, then, no steps are taken
to prevent it, the compass system as a whole will acquire a swing on
the axis T U (Fig. 31) when the ship is steaming north--or south--and
rolling. Its period of vibration on this axis is not strictly that
which it would have were it set oscillating about the axis T U with
this axis mounted on a fixed frame, for the vibration is not a free
one, but is a forced vibration under the impulses communicated by the
rolling of the ship. Whatever may be the exact free period of swing of
the compass system on the axis T U, so long as it is somewhere near
that of the ship’s roll or a small sub-multiple of the rolling period,
the tendency is for the compass swings to settle down in such a way
that, as represented in Fig. 33, the pendulum weight S reaches its
extreme out positions simultaneously with the ship’s arrival at its
extreme port and starboard heels. In the course of one complete roll of
the ship, however, the compass system may make more than one complete
vibration about T U.
It is clear that whether the bob S remains radially below the centre
of the wheel, as in Fig. 32, or remains vertically below it, or swings
past the vertical, as in Fig. 33, the effect of rolling when the ship
is on a due north course is as established in connection with Fig. 29.
Either a tendency to rotate the wheel about an axis coincident with its
axle or an actual rotation of the wheel about this axis results from
the kicks of the weight S at the out positions. Whichever it is, there
is no gyroscopic effect called into play, and the axle is not subjected
to any influence causing it to move away from the due north direction.
We have now to discuss what happens if the vessel rolls or pitches when
sailing on an inter-cardinal or quadrantal course.
In Fig. 34 we show in plan a vessel steaming on a north-west course.
When the ship rolls the compass oscillates in the path A D, a path,
that is to say, at right angles to the ship’s centre line and curved
upwardly by reason of the fact that the compass is mounted above the
rolling centre of the ship. The axle of the compass during this
movement tends to point steadily towards the north under the action of
the directive force. The wheel therefore oscillates from A to D with
its axle askew relatively to the path of oscillation, and not at right
angles to it, as in Fig. 29, or parallel with it, as in Fig. 30. The
oscillation A D can, however, be resolved into two components, namely,
a north and south oscillation D K, in which the axle is parallel with
the path, and an east and west oscillation K A, in which it is at right
angles to the path, both paths being curved upwardly.
[Illustration: FIG. 34. Ship Rolling on N.W. Course.]
The oscillation D K, taken by itself, is clearly exactly equivalent to
that shown in Fig. 30. The southward kick of the weight at the “out
position” D (Fig. 34) just neutralises the disturbing effect of the
northward kick at the “out position” K.
Similarly, the oscillation K A, taken by itself, is exactly equivalent
to that shown in Fig. 29 or its modification, Fig. 33. The eastward
kick of the weight at A and the westward kick at K cannot do more than
turn the mounting round the axle of the spinning wheel. They do not
tend to change the direction of the axle.
Hence, taken separately, each component oscillation D K and K A is
without disturbing effect upon the direction in which the axle is
pointing. Taken together, however, their united effect is not the sum
of their separate effects, as we might hastily assume. We must not
suppose that the north and south kicks proper to the north and south
component of the oscillation are without effect upon the east and
west component or vice versa. At the out position D of the component
oscillation K D the weight is not only kicked southwards, but also
receives a westward kick from the other component oscillation A K. At
K it receives both a northward kick and an eastward kick. Similarly,
at the out position K of the component oscillation A K the weight is
kicked not only to the west but also to the south, by virtue of the
oscillation from K to D. At the other out position A of this component
the weight is kicked eastwardly, and also to the north. If we take
cognisance of these additional kicks, we may treat the component
oscillations separately and add the separate results to get the true
effect of the actual oscillation A D.
Returning, then, to Fig. 30, let us suppose that in the oscillation
there shown, the equivalent of the component K D, we apply a westward
kick to the weight S when the ship reaches its port out position--that
is to say, a kick out of the paper--and that at the starboard out
position it receives an eastward kick into the plane of the paper.
It is clear that these additional kicks do nothing more than tend to
rotate the wheel about its axle or an axis coincident or parallel
with its axle. They cannot therefore call any gyroscopic effect into
action, and as a result do not disturb the direction in which the axle
is pointing. The component oscillation D K, even with the extra kicks
added, is thus harmless.
Taking Fig. 33 rather than Fig. 29 as the equivalent of the component
oscillation A K, let us suppose that at the port out position the
weight is subjected to an additional kick to the south--that is,
into the plane of the paper--and that at the starboard out position
it receives an extra kick to the north, or out of the plane of the
paper. The southward kick on the weight at the port out position will
tend to cause the end B of the axle to turn down towards J, but, in
accordance with the fundamental gyroscopic rule, the actual motion
of the sensitive element will not be a rotation about the axis E F,
but a precession about the axis H J, the end B of the axle moving in
the direction K. Similarly, the northward kick at the starboard out
position will tend to cause the end B of the axle to rise towards H,
and will therefore produce precession in the direction L. Now these
two precessional movements, K and L, can be resolved into horizontal
and vertical components N M and Q P respectively, as shown separately,
and of these components it is clear that the movement N cancels the
movement Q. On the other hand, the two components M and P are in the
same direction, and therefore do not cancel each other. Their effect
is cumulative, and with each succeeding roll the end B of the axle
tends to rise higher and higher in the true vertical plane B R. It may
be explained, perhaps, that this motion is possible, in spite of the
fact that there is no actual axis in the mounting of the compass at
right angles to the true vertical plane, except at the instant when
the compass is passing through the even keel position in the course of
its swing from side to side. As the rolling of the ship starting from
zero increases up to a more or less steady value, the end B of the axle
rises slowly at first and then at a constant rate, as exaggerated at
T. Such of this upward movement as occurs while the compass is passing
through the even keel position can be effected by means of a rotary
motion purely about the axis E F. Elsewhere in the swing of the compass
from side to side it is accommodated by a partial rotation on E F and a
partial rotation about H J. These two axes between them permit the end
B of the axle to rise in a vertical plane at any point of the compass
swing just as effectively as would an axis which was at right angles to
the vertical plane B R.
To appreciate the significance of this vertical rise of the north end
B of the axle, let us consider the condition of the compass as it is
passing through the even keel position. The elevation of the end B out
of the horizontal plane causes the weight S to swing forward towards
the north, and therefore to apply a turning moment to the spinning
wheel about the axis E F. This turning moment will tend to bring
the end B down again to the horizontal, but, as we know, the actual
motion produced will be a precession about the axis H J, the end B
moving _westwards_. Looking at Fig. 34, it will thus be seen that the
net effect of the rolling of the ship is to deviate the axle of the
compass in the direction shown at V--that is to say, in the direction
required to set the plane of the spinning wheel parallel with the
plane of the rolling by the shortest possible course--in our example
by a rotation westward through 45 deg. With the axle so deflected
the earth’s rotation, of course, calls a directive force into play,
tending to restore the axle to the north and south line, but if the
ship is rolling violently the deviating force will be much stronger
than the directive force until a very considerable angle of deflection
is reached. When the balance is struck, the axle settles down with a
steady deviation towards the west, which will remain constant as long
as the rolling is maintained. In practice the violence of the rolling
varies almost from roll to roll, for it represents a conflict between
the natural period of rolling of the ship and the period of the waves.
As a consequence, the deviation of the compass varies somewhat during
the rolling, but it is always westwardly if the course is in the
north-west quadrant.
A little consideration will show that if the ship is steaming towards
the north-east the additional kick on the weight S (Fig. 33) at the
port out position will be towards the north and at the starboard
out position towards the south. The end B of the axle is therefore
precessed vertically downwards instead of upwards, and as a result
the deviation of the axle is _eastwardly_. In general, if the ship
is steaming on any course in the north-west or south-east quadrant,
the deviation caused by her rolling will be towards the west. It will
be towards the east if the course is in the north-east or south-west
quadrant. The effect of pitching on a quadrantal course is to cause a
deviation of the axle in the direction opposed to that of the deviation
caused by rolling, so that if the vessel be both rolling and pitching,
the deviation is somewhat less than it would be if rolling only had to
be considered.
It may perhaps be thought that the quadrantal error is an effect
produced by the double gimbal mounting of the compass, and that had
we adhered to our simple model, as shown in Fig. 32, it would not
arise. This is not so. If Fig. 32 be taken as representing, after the
manner of Fig. 33, the north view of the compass on a ship steaming
due north-west, it will be seen that the southern kick at the port
out position causes the axle to precess in the direction K and the
northern kick at the starboard out position in the direction L.
Resolving these movements as before, we see that, while the horizontal
components N Q again cancel, the vertical components M P are, as in
Fig. 33, cumulative in effect, but this time they result in the end B
of the axle moving downwards, and therefore finally cause it to precess
eastwards instead of westwards, as before. The quadrantal error is
therefore not eliminated, but merely reversed in direction by adopting
the simpler mounting for the compass.