DYNAMIC STABILITY OF SPINNING BEAMS WITH AN UNSYMMETRICAL CROSS-SECTION AND DISTINCT BOUNDARY-CONDITIONS SUBJECTED TO TIME-DEPENDENT SPIN SPEED

The equations of motion of a spinning beam with a rectangular cross-se ction are formulated using the Euler beam theory and the assumed mode method. The spin speed consists of steady-state and time-dependent por tions. The resulting equations of motion are not in standard Mathieu-H ill's equation form, due to the time-dependent coefficient of the gyro scopic term. These equations of motion are then reduced to a set of fi rst-order differential equations with time-dependent coefficients. The regions of instability due to parametric excitations are determined u sing the multiple scale method. Numerical results are presented for a spinning beam subjected to combinations of end conditions in the two o rthogonal planes of transverse vibration. Widths of the unstable regio ns are found to decrease as the aspect ratio of the rectangular cross- section approaches unity for spinning beams with an identical set of e nd conditions in both transverse vibration planes. These regions vanis h when the aspect ratio becomes one. However, this is not the case whe n the beam is subjected to distinct end conditions in the two planes. For a given aspect ratio, interesting changes in the unstable regions are observed as the spin speed varies within, as well as across, criti cal spin speed zones.