Approximate complex eigensolution of proportionally damped linear systems supplemented with a passive damper

Abstract

An approximate and numerically efficient method is developed to complex eigenproblem associated with the discrete proportionally damped systems equipped with a passive damper. The presence of the damper changes the dissipative character of the complete system and makes it non-proportionally damped. The eigenanalysis of such systems is conventionally performed in a space of twice the system’s dimension. This makes analysis costly, particularly for large systems. The proposed method avoids using this numerically demanding state-space formulation. The determination of complex eigenvalues is based on the approximate solution of the characteristic equation that is derived in the modal space. The perturbation approach is adopted to reflect the differences in the eigenvalues of proportionally and non-proportionally damped systems. The complex eigenvectors are calculated afterwards with the use of a significantly reduced modal system and obtained eigenvalues. The proposed procedure is easily programmable and enables the calculation of the individual complex eigenvalues and eigenmodes separately, which significantly reduces computational time.