Critical damping in nonviscously damped linear systems

•Viscoelastic structures are characterized by a frequency dependent damping matrix.•Critically damped modes of nonviscous systems are characterized.•Mathematical conditions to find critical surfaces in nonviscous systems are presented.•A new numerical method to compute critical curves in nonviscous systems is developed.•Critical surfaces of single and multiple degrees of freedom systems are presented. In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are named critical damping surfaces (or critical manifolds, since several parameters can be involved). In this paper the theoretical foundations to determine critical damping surfaces in nonviscously damped systems are established. In addition, a numerical method to obtain critical curves is developed. The approach is based on the transformation of the algebraic equations, which define implicitly the critical curves, into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.