A data-driven technique for discovering the dynamical system with rigid impact characteristic

We propose a data-driven technique for discovering the equation of motion of the dynamical system with rigid impact. The method first discovers a dynamical system close to impact by the Fourier series with a high rate of convergence, known as the double-even extended series. Then, we use the system to construct an impact mapping, which maps the data close to impact to an estimated impact instant. By minimizing the error of impact mapping, we find the location of impact surface and energy lost during impact that generally satisfies the data close to impact. Finally, we discover the equation of motion without impact by the double-even extended series The analyzed data can be collected at equal time intervals with measurement error, and there is no need to deliberately collect data at the impact instant. The technique is able to capture the impact characteristic when there is a lack of knowledge about the critical changes of the dynamics at the impact instant and the non-linear dynamical behaviors without impact. We test the identification ability of the new technique using impact dynamical systems connected with cubic damping term and strong non-linear damping, respectively. The identified systems accurately capture impact dynamics such as the long-time prediction with impact, multistable dynamical phenomenon, and chattering dynamics.