Analytical investigations of stable periodic solutions in a two-degree-of-freedom kinematically forced impacting cantilever beam.

Abstract

This paper presents a comprehensive analytical study of a two-degree-of-freedom vibrating system with impacts, which can model a kinematically forced cantilever beam with a substantial mass and a concentrated mass at its end that impacts a rigid base during motion. An analytical method, based on Peterka's approach and tailored to the specific features of the system, is developed to analyze periodic motions, with particular emphasis on their occurrence and stability. The influence of system parameters, including clearance, mass distribution, and excitation frequency, on the system behavior is investigated, and parameter ranges are identified that lead to stable periodic solutions. The analytical results are then compared with numerical simulations in which Lyapunov exponents are calculated using an adapted Müller approach for numerical verification of stability. The two methods yield consistent results, confirming the effectiveness and precision of the approaches employed. It is demonstrated that the location and extent of regions of stable periodic solutions are significantly influenced by the relationships between the excitation frequency and the system eigenvalues. These results provide important insights for the design of kinematically forced vibro-impact systems with significant masses of elastic elements.