Eigensolution of nonclassically damped structures by complex subspace iteration

摘要

A nonclassically damped structure includes the consistent damping matrix. All structural matrices are symmetric, the stiffness matrix has to be positive definite. This leads to a quadratic eigenproblem with complex eigenvalues and eigenvectors. The common solution technique is the transformation into a linear equation system in state space form, which doubles the order of the system, and to solve it e.g. by the Lanczos method. The presented approach solves the quadratic eigenproblem directly, i.e. the order is not in- creased and the sparseness or bandedness of the system matrices is retained. The solution of the eigenproblem is obtained by the principle of subspace iteration. In comparison to undamped modes, the complex subspace iteration operates on a modal subspace of double size. The iteration sequence is based on a filter principle. This enables to utilise prefactorised system equations and starting information of eigenvectors from a previous solution, even when the structural matrices are updated. Shift techniques can be applied, which makes the algorithm useful for numerically tough problems including a large number of overdamped eigensolutions.