Chapter 18.3, Problem 18.137P

The top shown is supported at the fixed point O. Denoting by ϕ, θ, and ψ the Eulerian angles defining the position of the top with respect to a fixed frame of reference, consider the general motion of the top in which all Eulerian angles vary.

(a) Observing that ΣM_Z = 0 and ΣM_z = 0, and denoting by I and I′, respectively, the moments of inertia of the top about its axis of symmetry and about a transverse axis through O, derive the two first-order differential equations of motion

$I^{\prime }\stackrel{\dot{}}{\varphi }{\text{ sin}}^2\theta +I(\stackrel{\dot{}}{\psi }+\stackrel{\dot{}}{\varphi }\text{ cos}\theta )\text{cos}\theta =\alpha$    (1)

$I(\stackrel{\dot{}}{\psi }+\stackrel{\dot{}}{\varphi }\text{ cos}\theta )=\beta$    (2)

where α and β are constants depending upon the initial conditions.

These equations express that the angular momentum of the top is conserved about both the Z and z axes; that is, that the rectangular component of H_O along each of these axes is constant.

(b) Use Eqs. (1) and (2) to show that the rectangular component ω_z of the angular velocity of the top is constant and that the rate of precession $\stackrel{\dot{}}{\varphi }$ depends upon the value of the angle of nutation θ.

Fig. P18.137 and P18.138