5.1 Introduction

Occurrence of spinning systems is common in aerospace and mechanical engineering, among other disciplines. Relevant structures may be made of a combination of nonspinning and spinning components; the spinning components may rotate around any arbitrary axes with differing speeds of rotation. An accurate free vibration analysis is a vital prerequisite for subsequent dynamic response and aeroservoelastic analysis of such structural systems. Such an analysis is especially complicated for the case of spinning structures. Examples for the relevant structures include helicopters, spacecraft, and rotating machinery.

5.2 Derivation of Equation of Motion

Figure 5.1 depicts the elastic deformation u _j of a point j in a flexible structure rotating at a constant angular velocity ? in an arbitrary axis having components ? _X, ? _Y, ? _Z along the reference coordinate system axes. Then the position, velocity, and acceleration vectors are defined as

in which the second term in the right-hand side of Eq. (5.3) is zero for steady spin rate and the third and fourth terms relate to Coriolis and centripetal accelerations, respectively; also

Figure 5.1: Flexible structure subjected to arbitrary spinning motion.

Further, the first term of Eq. (5.3) relates to effects of in-plane stretching on out-of-plane deformation. In the absence of damping, the governing equations of motion for the entire structure can be derived using the preceding equation in formulating strain and kinetic energies and using Hamilton's principle as

or

with r representing all r _j, the term on...