三自由度模态分析解析求解策略研究  

Research on Analytic Solution Strategies for 3-DOF Modal Analysis

作　　者：张国智[1] 宋徽[2] ZHANG Guozhi;SONG Hui(School of Mechanical and Electrical Engineering,Xinxiang University,Xinxiang 453003,China;Department of Architecture and Civil Engineering,Xinxiang Vocational and Technical College,Xinxiang 453000,China)

机构地区：[1]新乡学院机电工程学院,河南新乡453003 [2]新乡职业技术学院建筑学院,河南新乡453003

出　　处：《新乡学院学报》2024年第12期44-47,共4页Journal of Xinxiang University

基　　金：新乡市重大科技专项(21ZD009);2021年新乡学院课程思政示范课程。

摘　　要：针对复杂结构的三自由度模态分析问题的速解,研究了基于解析方法的求解策略。在分析模态问题求解难点基础上,提出了三自由度模态分析的解析求解思路。同时,系统梳理了三次方程的三角代换和多项式代换的解析求解方法,讨论了其各自的适用范围。通过对典型的计算算例解析可知,三角代换方法和多项式代换方法对于3个不同实数解的求解结果相同,三角代换的结果形式更为简洁,另外,多项式代换和因式分解的方法可对四次和五次方程进行求解。针对某超声波加工机床的配重机构的模态分析,基于三角代换方法可得到解析解,与二分法的数值解的最大误差不超过1.5%。在保证计算精度的前提下解析求解方法为模态分析的数字孪生提供了方法。A solution strategy based on analytical methods was studied for the fast solution of 3-DOF modal analysis problem of complex structures.On the basis of analyzing the difficulties in solving modal problems,an analytical solution approach for 3-DOF modal analysis was proposed.At the same time,the analytical solution methods for polynomial and trigonometric transformations of cubic equations were systematically reviewed,and their respective applicability was discussed.Through the analysis of typical calculation examples,it can be seen that polynomial substitution and trigonometric substitution have the same solution results for three different real solutions,and the result form of trigonometric substitution is more concise.In addition,for fourth and fifth degree equations,polynomial substitution and factorization methods can be used to solve.For the modal analysis of the counterweight mechanism of an ultrasonic machining machine tool,an analytical solution can be obtained based on the triangular substitution method,and a maximum error is 1.5%compared to the binary numerical solution.The analytical solution method provides a method for the digital twinning of modal analysis while ensuring computational accuracy.

关 键 词：三自由度 模态 三角代换 多项式代换 超声波加工机床 配重机构 

分 类 号：TH12[机械工程—机械设计及理论]