The Stability of Flexible Spinning Disks Supported by Incompressible Hydrodynamic Lubrication

This paper uses Lyapunov’s method to examine the stability of a flexible spinning disk that is hydrodynamically coupled to the thin films of air surrounding it. A hydrodynamically coupled Lyapunov functional that defines the stable operational speed range of the disk is presented here for the first time. The stability boundary or, equivalently, the maximum stable speed, defined by this functional is independent of the strength of the coupling and allows significantly higher rotation speeds than allowed in the absence of hydrodynamic coupling. The hydrodynamically coupled critical speed is an order of magnitude higher than the uncoupled critical speed for clamping ratios less than 0.3. Several vibration modes of the disk which travel forwards at one half the rotation speed are stable but not asymptotically stable despite the large hydrodynamic damping present in the system. Unlike many elastohydrodynamic stability problems, calculation of the stability boundary using Lyapunov’s method is linear and tractable. The form of the Lyapunov functional suggests several design features that may be helpful in the design of improved, high speed configurations.