The applicability limits of the lowest-order substitute model for a cantilever beam system hard-impacting a moving base

This paper examines the circumstances under which a one-degree-of-freedom approximate system can be employed to predict the dynamics of a cantilever beam comprising an elastic element with a significant mass and a concentrated mass embedded at its end, impacting a moving rigid base. A reference model of the system was constructed using the finite element method, and an approximate lowest-order model was proposed that could be useful in engineering practice for rapidly ascertaining the dynamics of the system, particularly for predicting both periodic and chaotic motions. The number of finite elements in the reference model was determined based on the calculated values of natural frequencies, which were found to correspond to the values of natural frequencies derived from the application of analytical formulas. The precision of the parameter identification and the outcomes yielded by the substitute model were validated through the calculation of the regions of stable periodic solutions using the analytical Peterka method. Subsequently, the qualitative and quantitative limits of the substitute model's applicability were determined. The quantitative limits were delineated through the utilization of Lyapunov exponents and characteristics associated with the energy dissipation due to impacts and the average number of impacts per excitation period. These characteristics provide a foundation for the introduction of global distance measures of the dynamic behavior of diverse systems within a specified range of the control parameter.