Eigenvalues of linear viscoelastic systems

The calculation of eigenvalues of single- and multiple-degree-of-freedom linear viscoelastic systems is considered. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the size of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here an approximate non-state-space based approach is proposed for this type of problem. The proposed approximations are based on certain physical assumptions which simplify the underlying characteristic equation to be solved. Closed-form approximate expressions of the complex and real eigenvalues of the system are derived. These approximate expressions are obtained as functions of the elastic eigenvalues only. This enables one to approximately calculate the eigenvalues of complex viscoelastic systems by simple post-processing of the elastic (undamped) eigenvalues. Representative numerical examples are given to verify the accuracy of the derived expressions.