A coefficient of restitution based approach to model spatial vibro-impact dynamics in continuous systems

Abstract

This work proposes an efficient numerical methodology to simulate the non-smooth vibro-impact dynamics between two one-dimensional continuums, more specifically, two taut strings. At the outset, we invoke Galerkin’s projection to render the partial differential equation (PDE) governing the dynamics of the infinite-dimensional continuum to a finite-dimensional system of ordinary differential equations (ODEs). Initially, we address the complex impact interactions between the systems through a transformation from modal-to-physical coordinates on a spatially discretized grid. Subsequently, the impact forces and the post-impact velocities are obtained using the impulse-momentum principle and the coefficient of restitution. Notably, the methodology eliminates the necessity of detecting the time instant of the impact event, thereby rendering the procedure computationally efficient. The temporal evolution is handled using a modified Moreau’s midpoint scheme, where the impact force at a given time step is found explicitly. The considered strings, being non-dispersive, admit closed-form solutions for specific cases like point obstacles and smooth sinusoidal and flat obstacles. The solutions were obtained by Cabannes et al. (H. Cabannes, “Presentation of software for movies of vibrating strings with obstacles”, Applied Mathematics Letters 10 (5) (1997) 79-84.) in terms of travelling waves, invoking the method of characteristics. We compare the results from the presented methodology with these closed-form solutions and show that they match extremely well. Furthermore, to show that the proposed methodology is computationally more efficient than other methods, we compare the results with the Galerkin−Ivanov numerical method. We once again observe extremely good correspondence between the results and exhibit significant computational advantages of the proposed approach for several test cases considered herein. The numerical investigations provide evidence to support the method’s efficacy in predicting precise and energy-conserving responses for different parameter ranges.