An efficient scheme for stability analysis of finite element asymmetric rotor models in a rotating frame

Abstract

The present work deals with efficient stability analysis of finite element models of asymmetric rotors in a rotating frame, which rotates about the undeformed centre-line of the rotor with a speed equal to shaft spin speed. In asymmetric rotors the shaft is non-axisymmetric and the bearings are non-isotropic. For stability analysis of finite element models of such rotors one requires to deal with a large set of homogeneous linear differential equations with periodic coefficients having a period π / Ω , where Ω denotes the shaft spin speed in rad/s. Stability analysis of above problems can be attempted using the standard method of assumed solution in the state-space. If the number of equations is n in state-space and the number of terms considered in the truncated periodic part of the assumed solution is m, for stability analysis one has to solve an eigenproblem of size nm × nm , when full matrices are considered. To avoid this costly approach, a specific arrangement of terms is proposed here in the otherwise standard method of assumed solution in state space, which would easily exploit the inherent sparsity of the finite element matrices and all other matrices generated in the process. The efficiency of the proposed method lies in the fact that here only the non-zero terms and their row-column locations of all the associated matrices are stored and processed. The sparse finite element matrices are directly generated in a sparse finite element assembler. Finally first few eigenvalues of a sparse eigenvalue problem are computed using a standard available sparse eigensolver routine to check for the stability of an asymmetric rotor at a given spin speed.