A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances $Zαβ,α,β={x,y}$ had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for $Zxx$⁠, $Zxy$⁠, $Zyx$⁠, and $Zyy$⁠. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.