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.

中文翻译：

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分段线性系统经典数学模型的问题及修正


由于间隙或间隙的存在，许多机械系统可以简化为分段线性模型。机械系统的动力学研究应基于可靠的数学模型。因此确定分段线性系统中主系统和辅助弹簧系统(ASS)之间的接触点和分离点非常重要。现有文献大多将数学模型的接触点和分离点固定在间隙处。但本文发现，当ASS包含阻尼器时，接触点和分离点实际上随着系统参数的变化而变化，这意味着大多数现有的数学模型是不正确的。首次通过数值求解证明了在谐波激励下主系统在到达间隙之前会过早地与ASS分离，这表明了经典数学模型的不正确性。然后，基于力学模型和工程实践，提出了两种修正的数学模型。并对校正模型中过早分离后主系统和ASS的运动进行了研究。最后，通过修正后的模型与经典数学模型的接触点、分离点、幅频曲线和运动状态的比较，可以得出修正后的模型更加合理。与实验数据的比较表明，修正后的模型更能反映工程实际。这些结果将有助于分段线性系统的研究和设计。