Chapter 18, Problem 18.158RP

The essential features of the gyrocompass are shown. The rotor spins at the rate $\stackrel{\dot{}}{\psi }$ about an axis mounted in a single gimbal, which may rotate freely about the vertical axis AB. The angle formed by the axis of the rotor and the plane of the meridian is denoted by θ, and the latitude of the position on the earth is denoted by λ. We note that line OC is parallel to the axis of the earth, and we denote by ω_e the angular velocity of the earth about its axis.

(a) Show that the equations of motion of the gyrocompass are

$\begin{array}{l}{I}^{\prime }\ddot{\theta }+I{\omega }_z{\omega }_e\cos \lambda \sin \theta -I^{\prime }{\omega }_e^2{\cos }^2\lambda \sin \theta \cos \theta =0\\ I{\stackrel{\dot{}}{\omega }}_z=0\end{array}$

where ω_z is the rectangular component of the total angular velocity ω along the axis of the rotor, and I and I′ are the moments of inertia of the rotor with respect to its axis of symmetry and a transverse axis through O, respectively.

(b) Neglecting the term containing ${\omega }_e^2$, show that for small values of θ, we have

$\ddot{\theta }+\frac{I{\omega }_z{\omega }_e\cos \lambda }{I^{\prime }}\theta =0$

and that the axis of the gyrocompass oscillates about the north–south direction.

Fig. P18.158