A hybrid expansion method for frequency response functions of non-proportionally damped systems

• This study considers non-proportionally viscously damped systems.
• Frequency response functions are computed using complex modal superposition method.
• A general relationship satisfied by eigensolutions and system matrices is established.
• An explicit expression on the contribution of the higher-order modes is presented.
• A correction method to non-proportionally viscously damped systems is presented.

This study is aimed at eliminating the influence of the higher-order modes on the frequency response functions (FRFs) of non-proportionally viscously damped systems. Based on the Neumann expansion theorem, two power-series expansions in terms of eigenpairs and system matrices are derived to obtain the FRF matrix. The relationships satisfied by eigensolutions and system matrices are established by combining the two power-series expansions. By using the relationships, an explicit expression on the contribution of the higher-order modes to FRF matrix can be obtained by expressing it as a sum of the lower-order modes and system matrices. A hybrid expansion method (HEM) is then presented by expressing FRFs as the explicit expression of the contribution of the higher-order modes and the modal superposition of the lower-order modes. The HEM maintains original-space without having to use the state-space equation of motion such that it is efficient in computational effort and storage capacity. Finally, a two-stage floating raft isolation system is used to illustrate the effectiveness of the derived results.