MATHEMATICAL-ANALYSIS OF STABILITY OF A SPINNING DISK UNDER ROTATING,ARBITRARILY LARGE DAMPING FORCES

Stability of a rotating disk render rotating, arbitrarily large dampin g forces is investigated analytically. Points possibly residing on the stability boundary are located exactly in parameter space based on th e criterion that at least one nontrivial periodic solution is necessar y at every boundary point. A perturbation technique and the Galerkin m ethod are used to predict whether these points of periodic solution re side on the stability boundary, and to identify the stable region in p arameter space. A nontrivial periodic solution is shown to exist only when the damping does not generate forces with respect to that solutio n. Instability occurs when the wave speed of a made in the uncoupled d isk, when observed on the disk, is exceeded by the rotation speed of t he damping force relative to the disk. The instability is independent of the magnitude of the force and the type of positive-definite dampin g operator in the applied region. For a single dashpot, nontrivial per iodic solutions exist at the points where the uncoupled disk has repea ted eigenfrequencies on a frame rotating with the dashpot and the dash pot neither damps nor energizes these modes substantially around these points.