For time-stepping algorithms that exhibit numerical dissipation in high-frequency range, the authors present a formulation in the general context of nonlinear dynamics. They examine existing and new methods for nonlinear elastic spring/mass systems and consider a simplified model of thin elastic beams and the fully nonlinear problem of elastodynamics. The new dissipative schemes consist of a modified stress formula together with a modified dynamic equation relating displacements and velocities. Representative numerical simulations in nonlinear three-dimensional elastodynamics are performed by using the finite element method.