Steady State Vibrations of a Piecewise Linear System

Abstract In this paper we examine the dynamics of a mechanical system with a piecewise linear damping and stiffness, which is characterized as a strongly nonlinear system. Searching for the exact steady state periodic response of piecewise linear systems for harmonic excitations, lead to a set of transcendental equations which should be solved numerically. The strong nonlinear characteristic of the system is the reason for failure of most traditional perturbation methods in providing an approximated frequency response. In this paper we show that the averaging method of Krylov-Bogoliobuv-Mitropolsky (or multiple time scale method) may be used to find the behavior of this type of system. Optimization of linear vibration isolators and engine mounts has a straightforward solution: the stiffness should be as soft as possible. Static deflection and maximum allowable relative mount displacement (rattle space) determine the limit of mount softness. Dividing the travel space into two (or more) stages enables us to use an isolator with a different stiffness for each stage. This is a simple method to have a more advanced passive mount isolator. However, this also makes the system piecewise linear, which has a strongly nonlinear behavior if the ratio of the two stages are far from unity. Asymmetrical bilinear systems, with different stiffness and damping values for compression and tension, are used when the relative motion in negative direction is more (or less) critical than in positive direction. Sensitivity analysis of the system shows that the difference in the damping plays a more important role than the stiffness difference in these systems. In this paper, using averaging method, we examine the nonlinear behavior of piecewise linear mounts at resonance, and provide a comprehensive analysis which may be used to better design for these kind of vibration isolators.