On the distribution of real eigenvalues in linear viscoelastic oscillators

Abstract

In this paper a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via convolution integral over exponentially decaying kernel function. The free-motion equation of this nonviscous system yields a nonlinear eigenvalue problem where it has a certain number of real eigenvalues corresponding to the non-oscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related with 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 level of damping and a theorem which 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 an insight into the behavior of the real eigenvalues while the damping level is changing.