Transverse vibrations of a centrally clamped rotating circular disk

The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed. The results specialized to the linear case of disks clamped at the center and free at the periphery are in good agreement with those reported in the literature. The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, we determine a set of four nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of two interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. They are reduced to an equivalent set of two first-order equations whose equilibrium solutions are determined analytically. The stability characteristics of these solutions is studied; the qualitative behavior of the response is independent of the mode being considered.