UBC Theses and Dissertations
The motion of a self-excited rigid body Lee, Richard Way Mah
This thesis discusses the motion of a rigid body about a fixed point subject to a torque arising from internal reaction. Such a body is called self-excited, and its motion is governed by Euler’s dynamical equations. First, in Section 2 we consider the case of a torque vector which is fixed in direction along the largest or smallest principal axis of the body, and has a component in the chosen axis equal to a constant plus a perturbation term that is proportional to the square of the modulus of the spin vector [character omitted] (p,q,r). It is shown that Euler's equations can be integrated in terms of a variable φ, introduced by means of a differential relation. Further quadrature and inversion yield p,q, and r as functions of the time t. Using the method of phase-plane analysis, we show that the spin vector can perform a variety of motions with respect to the body-fixed trihedral. In particular, when the perturbation is zero, we infer from the corresponding phase-plane trajectories that the spin vector can perform asymptotic motions of the first and second kinds and periodic motions about permanent axes lying in the principal plane perpendicular to the torque vector. Some of the results for this case were also obtained by Grammel, using different method. In the general case, when the perturbation is not zero, these motions are preserved. However, a second type of periodic motion exists; it occurs about the principal axis containing the torque vector, the principal axis itself being a direction of stable permanent rotation. In Section 3 we consider the same problem with the torque vector acting along the middle principal axis. Using the methods of the previous Section, we show that [character omitted] can assume periodic motions as well as asymptotic motions of various kinds. The periodic motions established in these two Sections are then computed in Section 4 as power series in a small parameter. Finally, in Section 5 the motion of a symmetric rigid body moving in a viscous medium subject to a time-dependent torque is studied. Its motion is compared with that in a vacuum. We show first that p,q, and r can be expressed in terms of certain integrals. For the special case where the self-excitement is time-independent and fixed in direction within the body, these integrals can be reduced to the generalized sine and cosine integrals. Their values can be computed from asymptotic and power series which are developed in the same Section. The asymptotic behavior of the spin vector is then discussed, yielding qualitative results which are summarized in three theorems.
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