一种改进的KBM法求解非线性振动方程

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振动与冲击 ›› 2018, Vol. 37 ›› Issue (13) : 165-170.

论文

一种改进的KBM法求解非线性振动方程

王磊佳,张鹄志,胡辉,祝明桥

作者信息 +

An improved KBM method for solving nonlinear vibration equations

WANG Leijia，ZHANG Huzhi，HU Hui，ZHU Mingqiao

Author information +

文章历史 +

摘要

经典KBM法可以较好地解决含有小参数的非线性振动方程的求解问题，但这个小参数很大程度限制了经典KBM法的应用。本文将经典KBM法中频率表达式的基本方程进行平方修正，使得该方法的求解范围不受小参数的限制；应用改进后的方法求解Duffing方程，并在计算中对振动系统的频率方程进行简化，求得方程的二次近似解和频率解分别与其它方法的解进行对比，验证了本文改进的KBM法的可行性，且证实其计算结果的精度较其它方法高。

Abstract

The classical KBM method can be used to solve nonlinear vibration equations with small parameters, but these small parameters greatly restrict the application of the classical KBM method. Here, the square modification was done for the basic equation of frequency expression in the classical KBM method so that the solving range of the method could not be restrained by small parameters. The improved KBM method was used to solve Duffing equation. In calculation, the frequency equation of the vibration system was simplified. The equation’s quadratic approximate solution and frequency solution were compared with solutions of other methods. It was shown that the feasibility of the improved KBM method is verified; the accuracy of its calculation results is higher than that of other methods.

导出引用

王磊佳,张鹄志,胡辉,祝明桥. 一种改进的KBM法求解非线性振动方程[J]. 振动与冲击, 2018, 37(13): 165-170

WANG Leijia，ZHANG Huzhi，HU Hui，ZHU Mingqiao. An improved KBM method for solving nonlinear vibration equations[J]. Journal of Vibration and Shock, 2018, 37(13): 165-170

参考文献

[1] 张建生,赵登峰.动力学中的复杂现象分岔、混动及在冲击消振系统中的应用[J].四川:现代机械, 2006,(1):60-63.
Zhang JianSheng,zhaoDengFeng.Bifurcation and Chaos in Dynamics Complex Phenomenon and Their Applications in Impact Damper System[J].GuiZhou:Modern Machinery,2006(1):60-63.
[2] 陈立群,刘延柱.振动力学发展历史概述[J].上海:上海交通大学学报,1997,31(7):132-136.
Chen LiQun,LiuYanZhu. Survey of the Historical Developments of Vibrat ion Mechanics [J]. Shang Hai: Journal of Shanghai Jiaotong University, 199 7,31(7):132-136.
[3] 陈树辉.强非线性振动系统的定量分析方法[M].北京:科学出版社,2007,1-2.
Cheng ShuHui. Quantitative Analysis Method for Strongly Nonlinear Vibration System[M].BeiJin: Science Press,2007,1-2.
[4] 高书磊. 波浪激励下非线性柔性隔振系统的动力学分析[D]. 山东大学, 2007.
GaoShuLei. Dynamic Analysis of the Nonlinear Flexible Isolation System by the Wave[D],Shandong University,2007.
[5] 李骊.强非线性振动系统的定性理论与定量方法[M].北京:科学出版社,1997:155-156.
LI Li. Qualitative Theory and Quantitative Method for Strongly Nonlinear Vibration Systems[M].BeiJin: Science Press,1997:155-156.
[6] 王海波.Duffing方程非线性振动特性的计算与分析[D]. 西安建筑科技大学, 2009.
Wang HaiBo.The Calculation and Analysis of Nonlinear vibration characteristics with Duffing Equation. Xi`an University of Architecture and Technology.2009.
[7] Amore P, Aranda A. Improved Lindstedt–Poincaré method for the solution of nonlinear problems[J]. Journal of Sound and Vibration, 2005, 283(3): 1115-1136.
[8] 胡庆泉, 陈立群. 多尺度法在阻尼多自由度系统中的应用[J]. 振动与冲击, 2006, 25(4):61-63.
Hu QingQuan.ApplicationOfmultiscale Method to Dampedmulti-degree Systems[J].Journa Of Vibration And Shock,2006,25(4):61-63.
[9] 张宝善. 求解一类非线性方程的改进的平均法[J]. 应用数学和力学, 1994, 15(12):1119-1126.
Zhang BaoShan,A Modified Method of Averaging for Solving a Class of Nonlinear Equations[J].Applied Mathematics and Mechanics, 1994 ,15(12):1119-11126.
[10] KakutaniT,Sugimoto N. Krylov‐Bogoliubov‐Mitropolsky method for nonlinear wave modulation[J]. 1974, 17(8):1617-1625.
[11] Cheung Y K, Chen S H, Lau S L. A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators[J]. International Journal of Non-Linear Mechanics, 1991, 26(3-4): 367-378.
[12] 杨志安, 李熙, 孟佳佳. 改进多尺度法求解环形极板机电耦合强非线性系统主共振的研究[J]. 振动与冲击, 2015, 34(19): 208-212.
Yang zhiAn,LIXi,MengJiaJia..Improved multi-scale method for primary resonance of an annular plate electromechanical coupled strong nonlinear system[J].Journa Of Vibration And Shock, 2015,34(19):208-212.
[13] 陈立群, 吴哲民. 多自由度非线性振动分析的平均法[J]. 振动与冲击, 2002, 21(3): 63-64.
Chen LiQun,WuZheMing.Averaging Method For Analyzing a Multi_Degrees Of Freedom Nonlinear Oscillation[j]. Journa Of Vibration And Shock, 2002,21(3):63-64.
[14] C.W. Lim, B.S. Wu. A modified Mickens procedure for certain non-linearoscillators [J]. Journal of Sound andVibration,2002,257:202–206.
[15] J.H. He.A new perturbation technique which is also valid for large parameters [J]. Journal of Sound and Vibration,2000,229: 1257–126.
[16] Nayfeh A H. Introduction to perturbation techniques[M]. John Wiley & Sons, 2011.
[17] 刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:95-100.
Liu YanZhu,ChengLiQun.The Nonlinear Vibration [M].BeiJin:Higher Education press, 2001:95-100.
[18] Meirovitch, L. Elements of Vibration Analysis [J]. New York: McGraw-Hill, 1975:364-368.
[19] Ahmed W K. Advantages and disadvantages of using MATLAB/ode45 for solving differential equations in engineering applications[J]. International Journal of Engineering, 2013, 7(1): 25-31