Symplectic Eigenspace Expansion for the Random Vibration Analysis of Gyroscopic Systems

The random dynamic responses of the damped gyroscopic system were investigated under random loads. The pseudo-excitation method, as a highly efficient and accurate method for random vibration analysis, had been widely used in the fields of structural seismic and wind engineering. In the Lagrange framework based on a single physics variable the method of modal superposition is effective to reduce the degrees of freedom for complex structures in the numerical random vibration analysis. However, for the random analysis of gyroscopic systems, given the existing gyroscopic effects, application of the modal superposition method based on the Rayleigh quotient eigenvalues will be quite limited. Therefore, the general description of the symplectic eigenvalue problem was introduced firstly. Furthermore, for the damped gyroscopic system subjected to stationary random loads, the pseudo-excitation method was used and the solution formulae were derived based on the symplectic eigenspace expansion. For the conservative gyroscopic system, the solution expression was in an explicit form. In the numerical examples, the stationary random responses of a gyroscopic system were computed with the present method, of which the accuracy and efficiency were verified through comparison of the results with those out of other methods. The present method is of significance for the random vibration problems about mechanical engineering equipments with gyroscopic systems.