Combination Resonance of a Centrally Clamped Rotating Circular Disk

Abstract

The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed when the disk is subject to a space-fixed linear spring. Combination resonances are shown to occur at rotational speeds different from the classical critical speeds at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, the authors determine a set of eight nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of four interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. Using convenient variable transformations, the authors obtain an equivalent set of equations, which they use to prove that no single- mode solution is possible for this case. The stability characteristics of the trivial solution for all of the possible interactions among four of the lowest modes are studied.