Computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments

In this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non-proportional and non-viscous systems are presented. These methods are specially conceived for practical engineering applications making use of the finite element analysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural systems. Considering the solution of the undamped problem, the complex eigenpair is estimated by finite increments using the eigenvector derivatives. For non-proportional viscous systems with low and medium damping, a simple single-step technique is presented whose rapidity and accuracy is verified by means of numerical applications. For higher damped systems an incremental approach is proposed that keeps the accuracy without significantly increasing the computational time. For non-viscously damped systems a fast iterative modality is suggested, which allows to approximate, in an efficient and simple way, the complex eigenpair. As numerical applications, the study of a metallic beam with free layer damping treatment is completed using finite element procedures, where the damping material is modelized by an exponential model whose parameters are obtained from curve fitting to experimental data.