An approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix

This paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials, characterised by the complex structural damping matrix. This new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the Rayleigh quotient. It is implemented in three different approaches: single-step, incremental and iterative schemes. The single-step technique is presented for systems with low and medium damping. From numerical examples, it can be verified that the errors committed by the approximate single-step technique with respect to the exact solutions, obtained by the IRAM method, are less than 0.2% when the loss factor of the material damping is lower than 1; this is a considerable improvement on other approximate methods. For higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time. The complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach, whose validity is proved using a four-parameter fractional derivative model.