Free Linear Vibration of Self-Pressurized Foil Bearings

Abstract

A foil bearing is formed when a flexible medium travels across a stationary rigid surface and entrains a thin layer of fluid that lubricates the relative sliding motion. Such bearings are used in magnetic tape drives to prevent excessive wear of the recording head and tape interface. In the one-dimensional model considered here, the tape is approximated as an axially-moving Euler-Bernoulli beam under tension, and the air pressure in the bearing region satisfies Reynolds equation for unsteady compressible flow. To the extent that transverse deformation of the tape couples with the air pressure, the “foil bearing problem” falls within the discipline of elastohydrodynamic lubrication. The governing equations for the tape and recording head are linearized about the equilibrium displacement and pressure fields, and the two resulting coupled partial differential equations with nonconstant coefficients describe the linear response. Following global discretization through Galerkin’s method, the natural frequencies, damping, and mode shapes of the tape and recording head system are determined through numerical solution of the generalized matrix eigenvalue problem. The coupled displacement and pressure modes depend on the transport speed, and they are complex because of viscous dissipation of the air and convection of the tape. For the illustrative case of a semicircular recording head, the dependence of the system’s eigenvalues on the transport speed, and on the location of the recording head within the tape’s span, is discussed.