Improved approximate methods for calculating frequency response function matrix and response of MDOF systems with viscoelastic hereditary terms

As we know, it is difficult and unnecessary to obtain all the eigenpairs of a large-scaled viscoelastic (nonviscous or hysteretic) damping systems, which means that the mode truncation scheme is generally used and the mode-truncated error is therefore introduced. This study is aimed at eliminating the influence of the unavailable modes on the dynamic response of MDOF systems with viscoelastic hereditary terms. The energy dissipation terms of the system depend on the past history of motion via convolution integrals over some kernel functions. Therefore, the system is a nonviscously damped system, which has been considered as the most generalized damping model within the scope of a linear mechanical analysis. To approximate frequency response function (FRF) matrix and response without using the unavailable modes, we suggest two methods, which attempt to approximate the influence of the unavailable modes in terms of the lower modes and system matrices by using the first one or two terms of Neumann expansion of the contribution of the unavailable modes. In contrast with the FRF matrix approximated in terms of the first two terms of Neumann expansion, these procedures cannot be extended to further high order terms since all of them will be affected by the frequency-dependent variation of damping matrix from previous terms. Finally, an example is shown that the two presented methods can make the mode-truncated error reduce and may be used to approximate the influence of nonviscous modes contributed to FRF matrix due to the fact that the nonviscous modes are difficult to be obtained accurately even if a small scaled model is used for some eigensolution methods.