Problems and corrections of classical mathematical model for piecewise linear system

Abstract

Due to the existence of gaps or backlash, many mechanical systems can be simplified into piecewise linear models. The dynamic study on mechanical systems should be based on reliable mathematical models. So that it is very important to determine the contact point and separation point between the primary system and the auxiliary spring system (ASS) in a piecewise linear system. In most existing literature, the contact point and separation point of the mathematical model are fixed at the gap. But in this paper, it is found that the contact point and separation point actually change with the system parameters when the ASS contains a damper, which implies the most existing mathematical models are incorrect. It is firstly demonstrated through numerical solution that the primary system will prematurely separate from the ASS before reaching the gap under harmonic excitation, which shows the incorrectness of the classical mathematical models. Then, based on the mechanical model and engineering practice, two corrected mathematical models are proposed. And the motions of the primary system and ASS after premature separation in the corrected models are studied. Finally, through comparisons of the contact points, separation points, amplitude-frequency curves and motion states between the corrected models and the classical mathematical model, it can be concluded that the corrected models are more reasonable. And comparisons with the experimental data imply that the corrected models can better reflect the engineering practice. These results will be helpful to the study and design of the piecewise linear system.