Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, which is mathematically represented by convolution integrals involving hereditary functions. The widespread Biot damping model assumes that such functions are exponential kernels, which modify the eigenvalues' set so that as many real eigenvalues (named nonviscous eigenvalues) as kernels are added to the system. This paper is focused on the study of a mathematical characterization of the nonviscous eigenvalues. The theoretical results allow the bounding of a set belonging to the real negative numbers, called the nonviscous set, constructed as the union of closed intervals. Exact analytical solutions of the nonviscous set for one and two exponential kernels and approximated solutions for the general case of N kernels are developed. In addition, the nonviscous set is used to build closed-form expressions to compute the nonviscous eigenvalues. The results are validated with numerical examples covering single and multiple degree-of-freedom systems where the proposed method is compared with other existing one-step approaches available in the literature.