APPLICATION OF k-STEP LANCZOS METHOD TO EXTRACT THE INTERIOR EIGENVALUES OF SKELETAL SYSTEMS

Unlike analyses performed to determine structural response to environmental loadings (i.e., wind or earthquake), where only a few of the lowest natural modes are assumed to participate in the overall response, mechanically induced loadings can excite higher-order modes and require solution of the interior eigenvalue problem. In such applications, care must be taken to ensure that the finite element model of the vibrating structure effectively models the higher-order modal response and that the eigensolver used to extract the natural frequencies does not encounter convergence problems due to the close spacing of the interior eigenvalues. This paper formulates and applies an implicitly restarted Lanczos-based eigensolver for computing the natural frequencies of skeletal systems from the interior of the eigenvalue spectrum. This interior eigenvalue problem is formulated using a finite element method which utilizes both the polynomialand frequency-dependent shape functions in representing beam deformation; thus, the resulting eigenvalue problem is nonlinear, or eigenvalue dependent. Planar frame examples are given which illustrate the effectiveness of the proposed modelling technique and eigensolver.