Generalized Kelvin–Voigt Damping for Geometrically Nonlinear Beams

Abstract

Strain-rate-based damping is investigated in the strong form of the intrinsic equations of three-dimensional geometrically exact beams. Kelvin–Voigt damping, often limited in the literature to linear or two-dimensional beam models, is generalized to the three-dimensional case, including rigid-body motions. The result is an elegant infinite-dimensional description of geometrically exact beams that facilitates theoretical analysis and sets the baseline for any chosen numerical implementation. In particular, the dissipation rates and equilibrium points of the system are derived for the most general case and for one in which a first-order approximation of the resulting damping terms is taken. Finally, numerical examples are given that validate the resulting model against a nonlinear damped Euler–Bernoulli beam (where detail is given on how an equivalent description using our intrinsic formulation is obtained) and support the analytical results of energy decay rates and equilibrium solutions caused by damping. Throughout the paper, the relevance of damping higher-order terms, arising from the geometrically exact description, to the accurate prediction of its effect on the dynamics of highly flexible structures is highlighted.