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254   NAVIGATIONAL COMPASSES

applicable only to a compass that spins in the same direction as the rotation of the earth.

8. Why are the spinning wheels of gyro-compasses always of great angular momentum?

8. Deduce an expression for the magnitude of the directive force acting upon a gyro-compass. Where, upon the earth's surface, does the gyro-compass possess its maximum and where its minimum directive force?

10. Suppose that, in assembling a gyro-compass, the gyro-wheel were mounted slightly too far toward the south. What would be the effect upon the direction of the settling position of the spin-axle of the gyro?

11. Show that if the spin-velocity of a Sperry gyro-compass diminishes considerably from the correct value, the compass will deflect westward of the meridian plane when the compass is in northern latitudes.

12. In going from the equator to latitude 60°, the spin-velocity of both the Anschutz and the Arma gyro-compass is reduced 50 per cent. Why? State the reason why this change is not accompanied by a deflection of the spinaxle out of the meridian plane.

13. Describe the cause of the " latitude error." Give the names of the gyro-compasses in which it exists and show how it is compensated.

14. Describe the cause of the " meridian-steaming " or " north-steaming " error and show how it is compensated.

15. Describe the cause of the " ballistic deflection " error and show that this error will not occur if the period of vibration of the spin-axle has the proper value.

16. Find the period of a gyro-compass that will have zero ballistic deflection error when the compass is at a given latitude.

17. Show what is meant by the " ballistic damping error " and state how it can be prevented.

18. Describe the cause of the " quadrantal " or "rolling " error and give two methods by which it is suppressed.

19. The mercury ballistic of the Sperry gyro-compass is supported by a " phantom element." Why could it not be attached directly to the gyrocasing?

20. The mercury ballistic of the Sperry gyro-compass is loosely connected to the gyro-casing by a pin that is a little to the east of the vertical line through the center of the gyro. Show what would be the effect upon the compass if this eccentric connection were offset to the west of the vertical axis rather than to the east.

21. Describe the methods used in three different makes of gyro-compass for causing the position in azimuth of the compass card of the master compass to remain in fixed relation with respect to the direction of the spin-axle of the gyro.

22. Name a natural error to which gyro-compasses are subject, that is avoided in each of the following types: (a) pendulous sensitive element, (b) non-pendulous element provided with a mercury ballistic, (c) two-gyro sensitive element. Explain how each type avoids the error named.

CHAPTER Vl
GYROSCOPIC STABILIZATIU1
§1. General Principles

154. Static and Kinetic Stability.-If a body, after suffering a slight angular displacement, recovers its former position, the body is said to be stable. If, after suffering a slight angular displacement, the body departs further and further from its former position, the body is said to be unstable. If, however, after suffering a slight angular displacement, there is no tendency of the body either to recover its former condition or to depart further from it, the body is said to be in neutral or indifferent stability. A body that is stable is also said to have positive stability and one that is unstable is said to have negative stability. The degree of stability of a body is measured by the amount of work necessary to effect a permanent change in the position of the body.

When a whipping-top is standing on its flat end it is statically stable; when standing on its peg and not spinning, it is unstable; when lying on its side, it is neutral. If, however, the top be given an angular displacement when spinning with constant speed, the top will not tumble over as it would if not spinning but will oscillate and eventually regain very nearly the condition it had before it was disturbed. A top, when spinning, is said to be kinetically or dynamically stable. Since, in this case, the top is acted upon by an unbalanced torque, the top is not in equilibrium.

A steady motion is said to be kinetically or dynamically stable if, when slightly disturbed, it oscillates in such a manner that the vibratory deviation from steady motion approaches steady motion as a limit when the disturbance approaches zero.

155. The Stability of a System Consisting of a Body Capable of Oscillation and an Attached Precessing Gyro-Wheel. - Represent the mass of the system by m, the radius of gyration with respect to the axis of oscillation by k, and the distance from the center of mass to the point of support by H. Represent by 8 the angle at any instant between any line fixed in the system and the position of this line when the system is in its equilibrium position.

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