Dynamic stability of spinning beams with an unsymmetrical cross-section and distinct boundary conditions subjected to time-dependent spin speed

Abstract

The equations of motion of a spinning beam with a rectangular cross-section are formulated using the Euler beam theory and the assumed mode method. The spin speed consists of steady-state and time-dependent portions. The resulting equations of motion are not in standard Mathieu-Hill's equation form, due to the time-dependent coefficient of the gyroscopic term. These equations of motion are then reduced to a set of first-order differential equations with time-dependent coefficients. The regions of instability due to parametric excitations are determined using the multiple scale method. Numerical results are presented for a spinning beam subjected to combinations of end conditions in the two orthogonal planes of transverse vibration. Widths of the unstable regions are found to decrease as the aspect ratio of the rectangular cross-section approaches unity for spinning beams with an identical set of end conditions in both transverse vibration planes.