This paper proposes a new approach for computing the real eigenvalues of a multiple-degrees-of-freedom viscoelastic system in which we assume an exponentially decaying damping. The free-motion equations lead to a nonlinear eigenvalue problem. If the system matrices are symmetric, the eigenvalues allow for a variational characterization of maxmin type, and the eigenvalues and eigenvectors can be determined very efficiently by the safeguarded iteration, which converges quadratically and, for extreme eigenvalues, monotonically. Numerical methods demonstrate the performance and the reliability of the approach. The method succeeds where some current approaches, with restrictive physical assumptions, fail.

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Summary In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.

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