A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

Abstract

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,â€‌ ASME J. Tribol., 134(1), p. 011704). The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by secondorder rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor leads to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However, the method (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,â€‌ ASME J. Tribol., 134(1), p. 011704) requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,â€‌ ASME Paper No. GT201394802) showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,â€‌ ASME Paper No. GT201394802) was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach also enhances the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.