CHAPTER XII

THE ELIMINATION OF THE QUADRANTAL ERROR

From our description of the cause of the quadrantal error it should be clear that it is of a variable erratic nature, or at least that, unlike the latitude and north steaming errors, its magnitude cannot be forecast from a knowledge of the ship’s speed, course, latitude, or other factors. Its direction is determined by the direction of the ship’s course, but its amount is settled by the violence of the rolling and pitching, and cannot therefore be calculated and tabulated in a practically useful way. It follows, therefore, that to get rid of the upsetting influence of the quadrantal error we cannot resort to “correcting” the compass readings for it, but must entirely eliminate it.

In the early (1910) Anschütz compass there were no means of eliminating the quadrantal error, for the existence of the error, or at least the importance to be attached to it, was not at first recognised. The compass, we believe, showed errors from this cause of as much as 20 deg. to 40 deg. As a result, within a very short time the design was discarded, and what amounted to an entirely different compass was substituted for it. The 1912 Anschütz compass will be briefly described later on, but here it may be said that the quadrantal error in it is eliminated by adding two more gyro-wheels to the sensitive element. The theory of this compass is not very easy to understand, so that it will be best if we explain first how the quadrantal error is eliminated in the Sperry and the Brown compasses.

The early Sperry, like the early Anschütz compass, was open to the full action of the quadrantal error. In the form now in use, however, it is prevented from taking effect by controlling automatically the position of the excentric pin connecting the weight or “bail” to the gyro casing. In the Sperry compass, as we know, the weight S (Fig. 35) is not hung from the inner supporting ring or its equivalent, but from the follow-up or phantom ring, its pendulous effect being communicated to the casing and thence to the spinning wheel through the excentric pin A. The kicks of the weight at the out positions during the rolling of the ship are also transmitted through the pin to the casing and wheel inside it.

Let us suppose, as shown in Fig. 35, that at the port out position of the compass swing the excentric pin is _east_ of the axis H J by the amount required just to bring it vertically below the centre of the spinning wheel. Then the southward kick of the weight when the vessel rolls on a north-west course is transmitted to the casing as a southward force applied at the point A. This force, acting about the axis E F as before, produces precession in the direction K, and, as before, we may resolve this movement into the components N M. The force at A has, however, in addition, a turning moment about the axis H J, for it is applied to the casing at a point lying at a distance A D from this axis, and not, as in Fig. 33, at a point virtually on it. This moment about H J would tend to make the end B of the axle turn in the direction of F, and therefore results, in accordance with the rule of the gyroscope, in an actual movement about the axis E F, the end B of the axle precessing downwards towards J, that is, in the direction R. We may resolve the precession R into two components T U.

Similarly, at the starboard out position, let us suppose that the excentric pin is this time _westwards_ of the axis H J by exactly the amount required again to bring it vertically below the centre of the wheel. As before, in the case shown in Fig. 33, the northward kick of the weight acting about E F results in a precession L, which may be resolved into the components P Q. The kick, however, as at the port out position, also has a moment about the axis H J. The kick is reversed in direction, but the point of its application to the casing is now on the other side of H J. Consequently the kick tends to rotate the casing about H J--and therefore produces precession about E F--in the same direction as does the kick at the port out position. The precession produced is represented by V, and can be resolved into the components W X.

Considering now the four pairs of component precessions, we see that they cancel out in pairs. Thus N and Q cancel, U cancels X, T wipes out M, and W does the same to P. The two components M P, which were the cause of the quadrantal error, are just balanced by the two additional components T W. The axle therefore does not rise vertically, but remains horizontal, and as a result no quadrantal error can arise.

It will be seen that the Sperry method of eliminating the quadrantal error requires the excentric pin to be movable relatively to the bail and casing. To be more precise, while the casing, the bail, the phantom ring, and all the other parts of the compass may swing round the axle of the spinning wheel, or the external gimbal axis coincident or parallel with the axle, matters have to be arranged in such a way that when the ship rolls the excentric pin shall not partake of this motion, but remain constantly in the vertical below the centre of the spinning wheel. We have to remember, however, that the Sperry method of damping the horizontal vibrations of the axle hangs upon the pin being displaced towards the east of the axis H J when the compass is in the even keel position. The two requirements are met by so controlling the position of the pin relatively to the bail and casing that when the bail, casing, etc., swing sideways under the influence of the rolling of the ship, the pin is maintained at a fixed distance eastwards of the true vertical through the centre of the spinning wheel at all points in the oscillation of the bail, casing, etc.

The stabilisation of the excentric pin in this manner is effected gyroscopically by means of the attachment shown in Fig. 36. This device consists of a small high-speed electrically driven gyroscope running inside a casing which is mounted rotatably on a vertical axis inside a stirrup frame. This frame, as shown in Fig. 37, is hung pendulum-wise on the north side of the main gyro casing, the axis of its suspension being collinear with the axle of the main gyro-wheel. The axle of the small gyro-wheel is thus aligned in the east and west direction. The stirrup bracket is turned up horizontally below the casing of the small gyro, and at its end is fitted with guides, carrying a pair of rollers. These rollers constitute the excentric pin and, as shown in Fig. 37, engage within two curved channel-sectioned tracks attached one to the bail and one to the main gyro casing. If when the bail, casing, etc., swing under the influence of the ship’s rolling the excentric pin should attempt to follow suit--either by reason of friction at the axis of suspension of the stirrup bracket or at the track rollers--the wheel and casing of the small gyro will start processing round the vertical axis inside the stirrup, for the attempt is equivalent to an endeavour to tilt the small gyro-axle in an east and west vertical plane, and therefore calls forth the usual gyroscopic reaction. The precession of the small gyro on its vertical axis is made to react on the stirrup bracket by means of a spring connection between the bracket and the casing, as shown in Fig. 36. The direction of spin of the small gyro-wheel is such that the force thus applied to the stirrup bracket opposes and just balances the frictional or other force trying to make it swing with the bail, casing, etc., of the main gyro. In this way the excentric pin as the vessel rolls is caused to maintain its original distance from the vertical line through the centre of the spinning wheel.

The suppression, or rather the avoidance of the quadrantal error in the Brown compass, is achieved in a manner which is mechanically very distinct from that adopted for the same purpose in the Sperry compass.

An elementary diagram of the Brown compass is given in Fig. 38. In this sketch A is the casing, inside which the gyro-wheel mounted on the axle B (C) rotates in the direction of the dotted arrow--B, as before, representing the north-seeking end of the axle. The casing is supported on an east and west horizontal axis E F inside a vertical ring, this ring being journalled at H J inside a frame D, the equivalent of the square frame in our simple model. To obtain complete freedom for the gyro the frame D is mounted on an athwartship axis G K, which is itself carried inside a ring journalled within the binnacle on an axis L parallel with the ship’s longitudinal centre line. The pendulum weight S is fixed to the lowest point of the frame D. If the weight S and frame D be set swinging on the axis G K, the swinging movement will, of course, be directly communicated through the journals H J to the vertical supporting ring, but, as the trunnions E F of the casing are really supported on knife edges, the swinging movement of the frame D cannot be communicated from the vertical ring through the trunnions E F to the casing A, and thence to the wheel. Yet it is essential that the weight S should be able to act pendulum-wise on the casing and wheel, for otherwise, as we know, the system would be without directive force.

The connection between the weight and the casing is not a mechanical one, but is effected by making use of the air blast created by the fan-like action of the wheel inside the casing. As we have explained in connection with the damping system adopted in this compass, the trunnion F is hollow, and delivers the air blast through a nozzle M, fixed relatively to the vertical supporting ring, into a divided box N. From this box pipes are led to two oil bottles--one of which is shown at P--fixed to the casing on the east side of the axle, one bottle being on the north face of the casing and the other on the south. The pressure of the air blast acting unequally upon the oil in these two bottles when the casing tilts about the axis E F results, as we have already explained, in any horizontal oscillation of the sensitive element about the axis H J being damped. In a similar way two bottles Q R half-filled with oil are fixed to the north and south faces of the casing on the west side of the axle of the spinning wheel. These bottles are also connected to the box N, but the connecting pipes are crossed so as to join each bottle Q R to the division of the box remote from it, and not to the adjacent division, as in the case of the damping bottles P. When, therefore, the weight S and frame D are swung slowly pendulum-wise on the axis G K, as shown in the first view in Fig. 39, the nozzle M, being fixed to the vertical ring, delivers more air into one division of the box N than the other, and therefore a greater air pressure is exerted inside one bottle than inside the other. Consequently oil flows through the connecting pipe T from the former bottle to the latter until the columns of oil are sufficiently unequal to balance the difference of pressure of the air inside the bottles. It will be noticed that the crossing of the pipes connecting the bottles with the box N results in the oil being accumulated in that bottle, which lies away from the side to which the weight S has been swung. Hence, although there is no mechanical connection between the weight and the casing of the spinning wheel, the effect when the weight S is swung slowly is thus substantially the same as it would be if there were, for the weight of the extra oil forced into the bottle R exerts a turning moment on the casing about the axis E F, and tends therefore to make the casing follow the deflection of the weight S and frame D.

Similarly, should the axle B C dip, as shown in the second view in Fig. 39, extra oil will accumulate in the bottle which has been elevated by the dipping, and as a result a restoring moment about the axis E F will be applied to the casing, just as it would be if the weight S had been directly connected to the casing and had been deflected by the dipping movement.

The second view in Fig. 39 illustrates the generation of the directive force in the Brown compass. Let us suppose that the compass is at the equator, and that by some agency the axle is turned so that its end B points due west. The rotation of the earth will cause the axle to dip slowly into some such position as that shown. During this slow dipping movement oil will flow slowly from the bottle Q into the bottle R. The unbalanced weight of oil will apply a turning moment to the sensitive element about the axis E F and will therefore produce actual motion about the axis H J, which will precess the end B of the axle towards the north. Any tendency for the axle to vibrate about the north and south line as a result of the momentum acquired by the sensitive element while coming up from the west will be damped by the air blast acting upon the oil in the two other bottles P fixed, as shown in Fig. 38, on the east side of the axle.

The point we require here to notice especially about this action is that, as the tilting of the axle produced by the rotation of the earth takes place very slowly--it cannot exceed the speed of rotation of the earth on its polar axis, namely, 0.0007 of a revolution per minute--the flow of oil from the bottle Q to the bottle R takes place very slowly also. The oil therefore acquires practically no momentum, and rises in the bottle R in strict accordance with the tilt acquired by the axle. If the tilting motion be stopped or reversed, the oil would remain in the bottle R at just the level it had reached, or would immediately begin to flow back again, for its momentum being negligible, it has not sufficient kinetic energy to rise higher in the bottle R after the upward motion of the bottle has ceased.

On the other hand, when the ship rolls, say, on a due west course, as represented in Fig. 40, the outer gimbal ring supporting the athwartship axis G K, the frame D carrying the weight S, and the vertical supporting ring will acquire an oscillation about the fore and aft axis L in tune with the rolls of the ship. The air blast will thus be directed into the two divisions of the box alternately, and therefore oil will flow from one bottle to the other. It is to be noticed, however, that the rolling motion of the ship induces in this manner a flow of oil from one bottle to the other at a very much greater rate than does the tilting action of the earth’s rotation dealt with above.

A ship rolling through 45 deg., out to out, in a complete period of 10 seconds rotates about its rolling centre with an average velocity equivalent to 1.5 revolutions per minute--or over 2000 times as fast as the speed of rotation of the earth on its polar axis--and at the mid point of its roll it will move with an actual velocity of about twice the average. The momentum acquired by the oil in flowing from bottle to bottle is therefore in this case not negligible. In fact, when the ship reaches one of its out positions and starts to return, the oil does not immediately begin to flow back into the other bottle, but is carried by its kinetic energy to a still higher level in the bottle in which it has been rising. As a result, the oscillation of the oil between the two bottles lags behind the oscillation of the pendulum weight S, and therefore that of the ship itself. The lag acquired is such that the oil is just level in the two bottles when the ship is at either of its out positions, and when the ship is at the mid point, or even keel position of its roll, although the air blast is for the moment evenly distributed between the two compartments of the divided box, the oil is standing at its maximum level in the bottle on that side of the wheel from which the ship is recovering herself. The action of the oil in the bottles during a rapid oscillation of the compass system is, in fact, quite analogous to that of the water in the Frahm system of anti-rolling tanks.

It will thus be seen that the net effect of transmitting the “kicks” derived during rolling from the pendulous weight of the wheel through the Brown oil bottle arrangement is simply to delay the application of the kicks to the gyro-wheel by a constant amount, namely, by the time taken for the ship to roll from either out position to the mid position or a quarter of a complete period. Thus as the ship sailing due west rolls through the mid position from starboard to port the compass system experiences the turning moment about the axis E F, which with a rigidly fixed pendulum it would receive at the starboard out position from the northwards kick of the weight. Similarly, the equivalent of the southwards kick of the weight at the port out position is felt by the wheel when the compass is passing through the even keel condition on the subsequent roll from port to starboard. It is clear that with the ship sailing due west--or east--this delaying of the kick does not affect the result established previously for a rigidly connected pendulous weight. Any tendency for the axle to precess towards the west when the compass is passing through the even keel position in one direction is completely annulled by the tendency to precess towards the east at the succeeding passage through the even keel position in the opposite direction.

On quadrantal courses, however, the delay in the application of the kick is most important, for when the ship rolls it results in the elimination of the quadrantal error. That it does so can easily be understood by reference to Fig. 33. The kicks, being received when the compass is passing through the even keel position, and not at the out positions, precess the wheel about the axis H J at a time when this axis is truly vertical, and not when it is inclined. The precessional movements have therefore no vertical components M P. They are represented completely by the horizontal components N Q. The axle thus does not depart from the horizontal plane, and any movements in this plane arising from the tendency to precess in the directions N Q cancel each other at successive passages of the compass through the mid position.

The early form of Anschütz compass was followed by the 1912 pattern in which the quadrantal error was successfully eliminated. An example of the 1910 form was obtained by Messrs. Elliott Brothers, of Lewisham, from Anschütz and Co., of Kiel, and was, we believe, fitted on board H.M.S. _New Zealand_. Its defects becoming apparent, its manufacture in this country was not proceeded with, but upon the appearance of the improved type in 1912, Messrs. Elliott took up its construction and supplied several to the Admiralty. The Sperry compass, however, secured the preference in the British Navy, and with the outbreak of the war Messrs. Elliott ceased practically to make Anschütz compasses. On the other hand, the German Navy continued to use the 1912 type of Anschütz compass, with very little alteration or addition, right throughout the war. In view of the fact that every German submarine was fitted with a compass of this form, and bearing in mind the high degree of excellence attained in the navigation of the enemy’s underwater craft, there can be no doubt that the modern Anschütz compass is a very satisfactory device.

In Fig. 41 we give a purely diagrammatic representation of the compass. The outer square frame may, as usual, be regarded as mounted within external gimbal rings, providing a fore-and-aft axis and an athwartship axis. The square frame contains a vertical ring free to turn about the axis H J and itself containing a horizontal ring mounted on an east and west axis F E. The pendulum weight S is attached, as usual, to the inner horizontal ring. The essential difference between the 1912 and the 1910 forms of Anschütz compass lies in the fact that, as shown in our diagram, the inner ring, or its equivalent, does not surround a single gyro-wheel, but has attached to it the casings of three distinct gyros.

As shown in the first plan--in Fig. 42--the three gyros are situated at the corners of an equilateral triangle. One gyro K is placed at the south end of the meridional diameter of the horizontal ring, with its axle pointing towards the centre of the ring. The two other gyros L M are placed at 60 deg. east and 60 deg. west of north, with their axes pointing towards the centre of the gyro K--_not_ towards the centre of the ring. The wheels of all three gyros rotate anti-clockwise, as seen from the north, looking south--that is, they rotate in the same direction as do the wheels of all single-gyro compasses as seen from the same standpoint.

If we neglect the two gyros L M for the moment, imagining them to be replaced by simple deadweights to counterbalance the weight of the gyro K, it will be seen that this system differs from that of the early Anschütz compass only in the fact that the gyro-wheel has been reduced in size, and has been displaced from the centre of the horizontal ring to the southern edge. The displacement of the wheel in this manner in no way affects the essential working of the compass. In a single-gyro compass we might with some advantage similarly displace the spinning wheel, for by so doing and by counterbalancing the displaced weight we should, at least partially, get rid of the concentration of mass about the east-west axis, which, as we shall see, introduces, if not corrected, an additional source of error into the compass readings when the ship rolls on quadrantal courses.

The reduction in the size of the gyro-wheel would, of course, reduce the magnitude of the directive force. In the 1912 design the wheels of all three gyros run at 20,000 revolutions per minute--that is, at the same speed as the wheel in the 1910 design. But each wheel is only 5 in. in diameter instead of 6 in., and weighs, with its axle, 5 lb. 2 oz., or only half as much as the wheel in the earlier form. The gyro K considered alone would therefore supply a directive force of but half the former amount.[6]

It follows, obviously, that if a second gyro of the same speed and size as that at K be fixed in accurate alignment at the north end of the meridional diameter of the horizontal ring the directive force at any angle of horizontal deflection away from the north will be doubled, that is, will be made equal to that developed in the 1910 design. Such a compass might be constructed, but it would exhibit the quadrantal error--or at least the inertia force portion of that error--just as badly as did the early Anschütz.

The essence of the 1912 design lies in the fact that not one but two gyros are attached to the north side of the horizontal ring, and in the additional fact that the axles of these two gyros are not parallel with but are inclined to the axle of the south gyro K.

When the sensitive element of this compass is in the north resting position, the axle of the gyro K is aligned with the meridian and therefore the rotation of the earth--the compass being supposed at the equator--merely moves the axle parallel with itself. On the other hand, with the sensitive element in the north resting position, the axles of the gyros L M are inclined to the meridian at 30 deg. The gyro L is virtually in the condition of a single-gyro compass, with the north end of its axle partially turned towards the east. Under this condition, as we know, the rotation of the earth will tend to make the north end B of the axle rise above the horizontal plane. Conversely, the gyro M is in the condition of a single-gyro compass, with the north end of its axle partially turned towards the west. The earth’s rotation in this case tends to make the north end of the axle dip below the horizontal plane. Thus, in the 1912 Anschütz compass, when the sensitive element is in the north resting position, the gyro K under the rotation of the earth is without effect on the pendulous weight S, the gyro L is striving to swing it towards the north, and the gyro M is trying with an equal effort to swing it towards the south. The weight, therefore, remains in the plumb line and applies no turning moment to any of the gyros. As there is no turning moment, there is no precessional tendency. The sensitive element remains directed towards the north. This alignment, as in a single-gyro system, is the true resting position, and under it no directive force is applied to the sensitive element.

To study how the three gyros act together to restore the sensitive element to the north resting position should it be deflected therefrom, let us suppose that the deflection suffered is one of 30 deg. towards the east. As shown in the second plan in Fig. 42, the effect of such a deflection is to place the three gyros K L M in the condition respectively of a single-gyro system when the axle is (_k_) deflected 30 deg. to the east, (_l_) deflected 60 deg. to the east, and (_m_) aligned on the north. At this deflection, then, the gyro M contributes no restoring force. The directive force contributed by the gyro K is half that contributed by the single gyro of the 1910 design when the deflection is 30 deg., for the mass--or moment of inertia--of the wheel is half the earlier value. The directive force contributed by the gyro L is something greater than that contributed by the gyro K, for the virtual deflection eastwards is 60 deg. instead of 30 deg., and, as we know, the directive force increases with the deflection. Thus, the three gyros taken together supply a directive force, when the sensitive element is deflected through 30 deg., which is somewhat greater than that supplied by the single double-sized wheel of the 1910 design.[7] This result is a general one. Whatever the deflection may be, the directive force supplied by the three-gyro compass is always about one-third greater than the directive force of the 1910 design at the same deflection. This increased force is developed even if the deflection be less than 30 deg. In such cases the gyro M will, of course, supply a force, a non-restoring force, to the sensitive element. Only when the deflection exceeds 30 deg. eastwards does the gyro M assist the gyros K and L to restore the sensitive element to the north resting position. If the deflection be to the west, however, the gyro M is the chief assistant of the gyro K, a laggard’s part being played by the gyro L until 30 deg. of westerly deflection is reached.

The manner in which this three-gyro system avoids the quadrantal error can now be discussed. The quadrantal error, it may be recalled, arises when the ship rolls on an intercardinal course, and is primarily caused by the fact that the whole compass system can swing on its external gimbal axis in tune with the rolls of the ship. If we could so arrange matters that during the roll of the ship from side to side the compass system would swing on the external gimbals, no more and no less than the amount required just to keep the axis H J truly vertical at all points of the roll, then the north and south “kicks” received on the weight S at the out positions of the roll would cause the sensitive element to precess first one way, then the other, but always in a horizontal plane. There would be no vertical component in the precession, the cumulative effect of which, as we have seen, produces the quadrantal error.

In the Brown compass, the quadrantal error is eliminated by delaying the effect of the “kicks” on the weight S until the axis H J is truly vertical--that is to say, the “kicks” on the weight are not transmitted to the spinning wheel until the compass system is passing through the even keel position. In the Sperry compass the weight S is virtually shifted back and forth from east to west of the axis H J in tune with the rolling of the ship in such a way as to introduce a second component of vertical precession, which in amount and direction is just sufficient to nullify the vertical component causing the quadrantal error. In the 1912 Anschütz compass, the object aimed at is the maintenance of the axis H J truly vertical at all times during the rolling condition.

This object is achieved, as we shall now show, by setting the axles of the gyros L M--Fig. 41--inclined to the axle of the gyro K. In a single-gyro compass, or in the 1912 Anschütz compass with the gyros L M suppressed, the system is very stiff against vibrations on the east-west axis E F, but is quite easily set vibrating about the north-south axis provided by the external gimbal mounting. The reason is, of course, that any vibration about the axis E F is met by the resistance of the gyro-wheel, for the vibration causes the axle to alter its direction, whereas no gyroscopic resistance is called into play by the vibration on the north and south axis, for this axis is coincident or parallel with the gyro-axle. Thus, in the 1910 Anschütz compass the period of vibration about the east-west axis was something like 70 min., whereas the period about the north and south axis was only one or two seconds. The smallness of the latter period readily resulted in the compass system getting into a swing in tune with the period of rolling of the ship--from five to twelve seconds or so. Were the axles of the gyros L M in the 1912 form set parallel with that of the gyro K, there would still be no gyroscopic resistance exerted against vibration on the north-south axis. As it is, the inclination of the axles of these two gyros, taken together, has virtually the same effect as would be obtained by the addition to the sensitive element of a separate single gyro, with its axle aligned at right angles to the axle of the gyro K.

The gyroscopic stiffness against vibration about the east-west axis E F is hardly affected by the inclination of the gyros L M at the angle chosen by the designers. The resistance, instead of being that contributed by three gyros, is equivalent to that supplied by about 2¾ gyros. The period of vibration on this axis E F is arranged to be the standard 85 min. About the north and south axis we have virtually the resistance of one gyro. The period of vibration about this axis, instead of being only one or two seconds, is about 80 sec. With this lengthened period of vibration there is practically no chance of the compass system getting into a swing in tune with the rolling of the ship, for every 2½ to 6 seconds the vibrating influence is reversed in direction, and in this interval of time the system cannot acquire any substantial degree of swinging motion. Thus, as the compass, following the rolls of the ship, moves from side to side, it suffers a pure translational movement. The axis H J remains truly vertical at all instants during the roll, and therefore the north and south kicks of the pendulous weight at the out positions of the roll precess the gyro-axle about H J in purely a horizontal plane, the precessional tendency at one out position cancelling that at the other. There is no vertical component in the precession, and therefore there can be no quadrantal error.

It is of interest and of some amusement to note that during the war the Germans applied a fourth gyro to the Anschütz compass, and that this additional gyro was carefully removed from every compass before they surrendered their submarines to us. The deceit was, however, of no avail, for the application of the fourth gyro was known to us long before the war ended, a complete compass so fitted having been recovered from a sunken submarine and repaired and carefully studied. The fourth gyro was applied to the external gimbal rings. These rings, when the submarine rolled, were found to acquire at times a violent oscillation of their own, for, of course, they received no stabilisation from the gyroscopic elements of the compass. On board submarines the violence with which the rings vibrated would occasionally threaten to wreck the compass. By adding a gyroscope to the gimbal rings, the period of vibration of the rings was lengthened to about 16 sec., so that little or no opportunity to swing was left to them. With the exception of this addition, the Anschütz 1912 compass was used by the Germans throughout the war practically without alteration.