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258   GYROSCOPIC STABILIZATION

(b) If the inner frame and spinning wheel of a gyroscope be statically unstable with respect to a horizontal axis perpendicular to the spin-axle, and if at the same time the second frame be also statically unstable with respect to a horizontal axis perpendicular to the first, then the system may be dynamically stable, and when it is dynamically stable the two axes will precess in the direction of the spin (Art. 155, b).

(c) Precession of a statically stable gyroscopic system is suppressed by retarding the precession till the system is in its equilibrium position.

(d) Precession of a statically unstable gyroscopic system is suppressed by accelerating the precession till the system is in its equilibrium position.

(e) Hurrying the precession of a top causes the top to rise against the torque due to gravity. Hurrying the precession of a statically unstable gyroscope causes the spin-axle to rotate about the axis of torque in the direction opposite that of the torque (Art. 40).

(f) Retarding the precession of a top causes the top to fall. Retarding the precession of a statically unstable gyroscope causes the spin-axle to rotate about the axis of torque in the direction of the torque (Art. 40).

(g) A gyro-wheel incapable of precessing produces no effect on the stability of a body to which it may be attached (Art. 34).

(h) A body that is statically stable and subject to periodic oscillations can be rendered dynamically stable by retarding the precessional velocity of the vertical gyro-axle of an attached gyroscope that is either statically stable or neutral (Art. 88).*

(i) A body that is statically unstable and subject to oscillations can be rendered dynamically stable by accelerating the precessional velocity of the vertical gyro-axle of an attached unstable gyroscope.*

(j) A body consisting of two gyroscopically coupled systems capable of oscillating about horizontal axes cannot be dynamically stable unless both are either statically stable or statically unstable. t

(k) The overloading of one side of a system that is statically unstable, but that has been rendered dynamically stable by means of a spinning gyroscope, causes the overloaded side to rise and the center of gravity of the system to oscillate across a vertical line through the point of support of the system.

* Bogaert, L'Effet gyrostatique et ses applications (1912), Art. 68.

t Deimel, Mechanics of the Gyroscope (1929), Art. 101.

GYROSCOPICALLY STABILIZED MONORAIL CARS 259

(1) When a statically unstable body rendered dynamically stable by a statically unstable gyroscope with vertical spin-axle is moved around a curve in the direction of the spin of the gyro-wheel, the system becomes more stable; when revolved in the direction opposite the spin, the system becomes less stable (Art. 159).

(m) When a statically unstable body to which is attached a gyroscope of neutral stability and with vertical spin-axle goes around a curve in the direction of the spin of the gyro-wheel, the entire system becomes both statically and dynamically unstable. When the system goes around the curve in the direction opposite the spin, the system oscillates about an equilibrium position.

(n) When a statically stable body to which is attached a gyroscope with vertical spin-axle and positive static stability is rotated in the direction of the gyro-wheel, the stability of the system is increased; whereas if the rotation be in the direction opposite the spin, then the stability of the system is decreased.

§2. Gyroscopically Stabilized Monorail Cars

157. The Economy of Monorail Cars. - Cars that would safely run at high speed on a single rail on the ground would effect considerable economies over birail cars, not only in original cost of installation but also in maintenance of way and in operation. The ability of a birail locomotive to go up a grade is limited by the coefficient of static friction between the edge of the wheels and the top of the rails. Grades are seldom over two per cent. Five per cent is considered very high. Wheels with double flanges, slightly tapered, would give a greater frictional force but could not be used on a curve with birail cars. They could, however, be used on even a sharp curve with monorail cars, thereby enabling steeper grades to be traversed by monorail cars than by birail cars. A monorailway could be built in a terrain requiring curves and grades too sharp for an ordinary birail railway. There would be many places where a monorailway would be used if we were certain as to the dynamic stability of the statically unstable cars. A considerable amount of thought has been devoted to the dynamic stabilization of monorail cars but at the present time there is no commercial line in operation.

158. The Principles upon Which Depend the Dynamic Stabilization of Monorail Cars. - The dynamic stabilization of a monorail car requires that a torque acting upon the car from out-