Forced Vibrations of Rotating Annular Discs Under Space-Fixed Point-Force Excitation

This study investigates the forced transverse vibration of a thin rotating annular disc subjected to time-varying point forces that are fixed in space and act perpendicular to the disc surface. Earlier analytical treatments of this problem have almost always been restricted to a single support condition, most often the clamped–free disc of a hard-disk drive. The boundary conditions of the disc have not been treated as a design variable of the forced response. The contribution of this work is to remove that restriction: the same generalised Galerkin formulation is applied to clamped–clamped, clamped–free and free–clamped rotating annular discs, and the sensitivity of the forced response to the excitation parameters is compared across all three. The governing differential equation, which includes gyroscopic coupling and the membrane stresses induced by rotation, is nondimensionalised and solved by the Galerkin method with polynomial radial trial functions. The modal equations are then integrated in state-space form with light modal damping. The physical transverse response at a fixed observation point is characterised by power spectral density diagrams assembled into waterfall plots, and the corresponding steady-state harmonic responses are obtained in closed form from the frequency-domain resolvent of the same state-space model. The formulation is verified against an independent radial finite-element model for all three boundary conditions and against published rotating clamped–free natural frequencies. The central finding concerns how the excitation parameters act on the response. The excitation frequency, the radial position of a force, and the angular separation and phase of a pair of forces act as largely independent levers. The quantitative sensitivity to each lever, however, is set by the boundary conditions, as is the force placement that minimises a chosen travelling-wave family. The radial position that minimises the excitation of a chosen radial family is governed by the interior node of that mode, which lies at ≈52 , 68 and 44 mm for the clamped–clamped, clamped–free and free–clamped discs, respectively, and does not in general coincide with a free edge. Angular separation, by contrast, suppresses a nodal-diameter family in a manner that is essentially boundary-condition independent. The parameter dependences are shown to be steady-state properties: the driven spectral line of the finite-duration records reproduces the resolvent solution to within 0.09 dB.