Closed-form solutions for oscillators with inelastic impacts

• Non-smooth time transformations are generalized on inelastic impact models.
• Analytical tool is developed to solve the related boundary-value problems.
• Closed-form solutions for vibro-impact dynamics with energy loss are obtained.
• Possible area of application is outlined.

A class of vibrating systems with perfectly stiff amplitude limiters is considered by means of non-smooth time substitutions. The motion is represented as a combination of oscillating component, which is due to cyclic collisions with the limiters, and a slow decay caused by the gradual energy loss at collision times. A specific modification of the two variable expansions is used, where the non-smooth (triangle wave) temporal argument is viewed as a fast time while the energy decay is described in a slow time scale. As a result, closed-form analytical solutions are obtained that automatically satisfy collision conditions with the energy loss. Three qualitatively different basic types of vibrations are considered to cover periodic, frequency modulated, and amplitude–frequency modulated motions.