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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分段线性系统的经典数学模型的问题和修正


由于存在间隙或反冲，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，在分段线性系统中确定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。在大多数现有文献中，数学模型的接触点和分离点固定在间隙处。但在本文中发现，当 ASS 包含阻尼器时，接触点和分离点实际上会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都是不正确的。首先通过数值解证明，在谐波激励下，初级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。并研究了校正模型中早逝分离后初级系统和 ASS 的运动。最后，通过对修正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合理。与实验数据的比较表明，校正后的模型可以更好地反映工程实践。这些结果将有助于分段线性系统的研究和设计。