The Gyroscopic Compass: A Non-Mathematical Treatment

by T. W. Chalmers

Fig. 16. Air Blast Damping System of Anschütz (1910) Compass.

The mouth of the outlet duct K is partially closed by a plate L fixed at the end of a pendulum arm suspended frictionlessly, or practically so, from some convenient point on the casing, so that when the casing turns on the vertical axis H J the arm and plate turn with it. The pendulum arm is carefully balanced in such a way that when the axle of the spinning wheel is horizontal the plate L exactly divides the orifice K, leaving equal passages for the air blast on each side. In this condition the two passages M N being equal in area, the air blast is divided by the plate into two streams of equal volume and momentum, so that if their free discharge is not influenced by surrounding objects their reactions on the casing, one on the one side of the vertical axis H J, the other on the opposite side, will be equal.

Let the compass be deflected until its axle points east and west, the end B being towards the east. Then, as we have seen, the tendency of the axle to remain parallel with its original position, combined with the rotation of the earth, will cause the axle to assume an inclined position relatively to the earth’s horizontal surface. Gravity acting on the pendulous weight S as thus displaced from the plumb line will, as we know, set up a precession about the vertical axis H J, so as to cause the end B of the axle to move towards the north. When, however, the axle tilts in this manner about the horizontal axis E F, the pendulum plate L, hanging freely, remains in the plumb line. Consequently the equal areas M N become unequal, as at P Q, the larger P being towards that end of the axle which has tilted upwards, namely, the north-seeking end B. The reaction on the casing of the portion of the air blast issuing from P is now greater than that of the portion emitted through Q. This inequality results in the application to the sensitive element of a force acting about the vertical axis H J. The reaction of a jet of air or water or other fluid being opposed to the direction in which the jet is issuing, the force applied to the casing is such as to drive P into the plane of the paper and to bring Q out of it—that is to say, to tend to rotate the casing on the vertical axis in the direction of the arrow R.

The force thus applied clearly tends to oppose the return of the axle from the east towards the north. On the other hand, if the axle succeeds in turning round until the end B points to the west, the rotation of the earth will result in the axle tilting as before, but with the end B downwards. Consequently the area M diminishes and the area N increases, and the reaction applied to the casing is such as to tend to turn it in the reverse direction to R. The applied force in this case therefore opposes the return of the axle from the west towards the north. The exact manner in which the opposition acts is to be noted. A force applied to a simple gyroscope, so as to tend to make it turn about the vertical axis H J in the direction R ([Fig. 16]) will, as was stated in our second chapter, cause the wheel to precess about the horizontal axis E F, the end B of the axle going down and the end C up. If the force about H J is reversed, the movement produced will be a precession of the wheel on E F such that the end B of the axle rises and the end C falls. The opposition between the moment applied to the sensitive element by the air blast and the moment applied to it by the directive force is therefore not quite direct. When the end B of the axle is deflected towards the east a directive force is called into being by virtue of the tendency of this end to rise. The air blast reaction about H J induces precession on the axis E F, causing the end B to fall. By the amount by which the air blast reaction succeeds in lowering the end B of the axle, by that amount will it reduce the normal magnitude of the directive force. When the axle swings through the meridian towards the west a reversed directive force is called into play by virtue of the tendency of the end B to fall. The opposition of the air blast reaction arises from the fact that its tendency now is to make the end B rise. Thus on both sides of the meridian the opposition of the air blast reaction is effective because it tends to precess the wheel on the axis E F in the direction opposed to that in which the rotation of the earth is trying to tilt the axle. The tilt moves the pendulous weight S away from the vertical, so producing the directive force; the air blast reaction reduces the tilt and so opposes the directive force.

The reaction applied to the sensitive element by the air blast thus fulfils one requirement of a satisfactory damping force; its effect at all times is opposed in direction to the direction in which the axle is moving. The second requirement is that the magnitude of its effect should always be proportional to the velocity with which the axle is moving.

In connection with this second requirement it is assumed that the momenta of the air jets issuing from the openings P Q ([Fig. 16]) are at all angles of tilt equivalent to the momentum of a single jet issuing from an imaginary orifice U situated at some distance d from the axis H J on the side of the larger opening P, the area of the imaginary orifice R and the velocity of the air through it being constant at all angles of tilt. If this assumption is correct, then the moment about H J applied by the reaction of the air blast is proportional to the distance d—that is to say, to the angle of the tilt.

The assumption here made is, we believe, substantially justified if the angle of tilt is never very great. In actual practice it is always small. Various considerations, however, suggest that the reactions of the two jets P Q are not equivalent strictly to the reaction of a single jet through an orifice R of constant area. Thus from geometrical considerations we can show that the sum of the areas P Q is not equal to the sum of the areas M N. Again, the total weight of air drawn in per minute through the orifice D may be constant, and therefore the total weight of air delivered per minute through the combined openings P Q may be unaffected by the tilt. But the ratio in which the total volume divides itself between the two openings P Q and the velocity through each certainly vary with the tilt. A peculiar practical phenomenon also has to be considered in this connection. In the 1910 form of Anschütz compass the peripheral speed of the spinning wheel was 500 ft. per second, or 340 miles an hour. The air friction at this speed was so very great that after the wheel had been run a few thousand hours its surface was found to be noticeably smoother than it was when the wheel left the grinding machine on which it was finished. As a result of this polishing effect, we should expect that even though the speed of the wheel remained perfectly constant, its blower-like action would decrease somewhat until the compass had been in use for a certain length of time. If the blower action does so decrease the magnitude of the air blast reaction on the sensitive element at any given angle of tilt must diminish with time. We do not know whether the diminution would be sufficiently great to introduce a serious error in the reading of the compass.

Taking the assumption to be correct, at least for small angles of tilt, we have next to study how the angle of tilt varies as the axle swings from the east side of the meridian over to the west and back again.

When the axle is pointed due east the rotation of the earth, as we have seen, tends to make the end B rise. If it is pointed due west, the end B tends to fall. If it is lying dead on the meridian, the earth’s rotation has no tilting effect at all on the axle. If the axle is pointed in some direction between due east and due north, the tilting effect is less than it would be if it were pointed due east, but it is in the same direction; the end B tends to rise. Similarly, in any position in the north-west quadrant the end B tends to tilt downwards under a tilting influence which is somewhat less than that experienced in the due west position. Let, then, the axle be turned to point due east. The end B begins at once to rise, but immediately the axle thus leaves the horizontal position it begins to feel the directive force, and it commences to turn towards the north. Until it reaches the north the tilting influence of the earth’s rotation continues so that during the whole time the axle is swinging through the north-east quadrant the tilt is increasing and the end B is rising, higher and higher. When the north direction is reached the tilting influence vanishes and the end B of the axle tends to travel horizontally in its elevated position over into the north-west quadrant under the influence of the momentum acquired in that direction during its movement through the north-east quadrant. But on passing to the west side of the meridian it again comes under the tilting influence of the earth’s rotation, which this time tends to make the end B fall. When the axle reaches the due west position, the tilting influence being equal to that acting over the north-east quadrant and reversed in direction, and having acted for the same length of time, will just have wiped out the elevation of the end B, and the axle will again be horizontal. On the return journey through the north-west quadrant the tilting influence is still reversed, so that on reaching the north position again the end B is as far below the horizontal as it was above it on the swing from east to west. Passing to the east side of the meridian, the tilting action assumes its former direction, the end B begins to rise, and when the due east position is reached again the axle is once more horizontal. The path traced out by the end B during a complete vibration from east to west and back again is thus in the nature of an ellipse—or, in extreme cases, a circle—such as that shown at a b c d e in [Fig. 17].