Yongjun Shen , Ruiliang Zhang , Dong Han , Xiaoyan Liu
{"title":"Problems and corrections of classical mathematical model for piecewise linear system","authors":"Yongjun Shen , Ruiliang Zhang , Dong Han , Xiaoyan Liu","doi":"10.1016/j.cnsns.2024.108300","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1007570424004854/pdfft?md5=d9cb27071fecb3155437befee4393b3d&pid=1-s2.0-S1007570424004854-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1007570424004854","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 0
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.
由于间隙或反冲的存在,许多机械系统可以简化为片断线性模型。机械系统的动态研究应基于可靠的数学模型。因此,确定分片线性系统中主系统与辅助弹簧系统(ASS)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都固定在间隙处。但本文发现,当辅助弹簧系统包含阻尼器时,接触点和分离点实际上会随着系统参数的变化而变化,这意味着现有的大多数数学模型是不正确的。本文首先通过数值求解证明,在谐波激励下,主系统在到达间隙之前会提前与 ASS 分离,这说明经典数学模型是不正确的。然后,根据机械模型和工程实践,提出了两个修正的数学模型。并研究了修正模型中主系统和 ASS 过早分离后的运动。最后,通过比较修正模型和经典数学模型的接触点、分离点、幅频曲线和运动状态,得出修正模型更为合理的结论。而与实验数据的比较则表明,修正后的模型能更好地反映工程实践。这些结果将有助于片线性系统的研究和设计。