Abstract
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 closedform
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.
