Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization

WANG Deliang; LIU Jike; LIU Guang

doi:10.13471/j.cnki.acta.snus.2023A001

您当前的位置：

首页 >

文章列表页 >

Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization

Research articles | 更新时间：2023-11-07

- Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 62, Issue 5, Pages: 78-84(2023)
- 作者机构：
  
  1.中山大学航空航天学院，广东深圳 518107
  2.深圳市智能微小卫星星座技术与应用重点实验室，广东深圳 518107
- 作者简介：
- 基金信息：
- DOI：10.13471/j.cnki.acta.snus.2023A001
  CLC： V21
- Published：25 September 2023，
  
  Published Online：13 July 2023，
  
  Received：13 January 2023，
  
  Accepted：27 February 2023
扫描看全文
王德亮,刘济科,刘广.基于Tikhonov正则化改进的IHB法求解Mathieu-Duffing系统多重解[J].中山大学学报(自然科学版),2023,62(05):78-84.

WANG Deliang,LIU Jike,LIU Guang.Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(05):78-84.
王德亮,刘济科,刘广.基于Tikhonov正则化改进的IHB法求解Mathieu-Duffing系统多重解[J].中山大学学报(自然科学版),2023,62(05):78-84. DOI： 10.13471/j.cnki.acta.snus.2023A001.

WANG Deliang,LIU Jike,LIU Guang.Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(05):78-84. DOI： 10.13471/j.cnki.acta.snus.2023A001.

摘要

增量谐波平衡法（IHB法）是研究强非线性振动系统的一种半数值半解析方法，然而已有研究表明，在求解含多重解的系统时该方法的收敛性强烈地依赖于初值的选择。Tikhonov正则化常被用于优化问题中来解决可能出现的病态问题。文章通过在原始的IHB法中引入Tikhonov正则化，提出一种改进的IHB法（TIHB法）来求解具有多重解的Mathieu-Duffing系统。结果表明，改进的TIHB法可以快速、高效地获得系统的多个稳定或不稳定解，且算法的收敛性能要远远优于原始的IHB法。

Abstract

The incremental harmonic balance method （IHB method） is a semi-numerical and semi-analytical method for strongly nonlinear dynamic systems. However， previous studies have shown that the convergence performance of the original IHB method in solving systems with multiple solutions strongly depends on the selection of initial values. The Tikhonov regularization is often used in optimization problems to solve potential ill-posed problems. In this paper， by incorporating the Tikhonov regularization into the original IHB method， an improved IHB method （TIHB method） is proposed to obtain the multiple solutions of the Mathieu-Duffing system. The results show that the improved TIHB method can obtain the stable and unstable solutions of the Mathieu-Duffing system quickly and efficiently， and the convergence performance of the TIHB method is much better than the original IHB method.

关键词

非线性振动IHB法Tikhonov正则化多重解

Keywords

nonlinear vibrationIHB methodTikhonov regularizationmultiple solution

references

陈树辉， 2007. 强非线性振动系统的定量分析方法［M］. 北京：科学出版社.

陈予恕， 1992. 非线性振动、分叉和混沌理论及其应用［J］. 振动工程学报， 5（3）： 235-250.

富明慧，林敬华， 2008. 精细积分法在非线性动力学问题中的应用［J］. 中山大学学报（自然科学版）， 47（3）： 1-5.

黄建亮，王腾，陈树辉， 2021. 含外激励van der Pol-Mathieu方程的非线性动力学特性分析［J］. 力学学报， 53（2）：496-510.

刘广，刘济科，陈衍茂， 2017. 基于Wilson-θ和Newmark-β法的非线性动力学方程改进算法［J］. 计算力学学报， 34（4）： 433-439.

钱征文，程礼，陈卫，等， 2011. 非线性振动方程多重解求解方法［J］. 振动与冲击， 30（10）： 14-18.

徐加初，方焕斌， 2008. 冲击载荷作用下夹层扁锥壳的非线性动力屈曲［J］. 中山大学学报（自然科学版）， S2： 62-66.

张丹伟，刘济科，黄建亮， 2018. 非线性系统准周期振动的多时间尺度IHB法［J］. 中山大学学报（自然科学版）， 57（6）： 63-70.

赵倩，刘子良，姚红良，等， 2018. IHB法在多自由度Bouc-Wen滞回非线性系统响应特性研究中的应用［J］. 振动与冲击， 37（10）： 57-62.

AMABILI M， 2008. Nonlinear vibrations and stability of shells and plates［M］. Cambridge：Cambridge University Press.

CHEN Y M， LIU J K， 2009. Improving convergence of incremental harmonic balance method using homotopy analysis method［J］. Acta Mech Sin， 25： 707-712.

CHEUNG Y K， CHEN S H， LAU S L， 1990. Application of the incremental harmonic balance method to cubic non-linearity systems［J］. J Sound Vib， 140（2）： 273-286.

DEVORE R A， 1998. Nonlinear approximation［J］. Acta numerica， 7： 51-150.

DOWELL E H， ILGAMOV M， 1988. Studies in nonlinear aeroelasticity［M］. New York：Springer Science & Business Media.

FERRI A A， 1986. On the equivalence of the incremental harmonic balance method and the harmonic balance-newton raphson method［J］. J Appl Mech， 53（2）： 455-457.

LIAO S J， SHERIF S A， 2004. Beyond perturbation： Introduction to the homotopy analysis method［J］. Appl Mech Rev， 57（5）： B25-B26.

LIU G， LIU J K， WANG L， et al， 2022. Time-domain minimum residual method combined with energy balance for nonlinear conservative systems［J］. Mech Syst Signal Processing， 170： 108818.

LIU G， LV Z R， LIU J K， et al， 2018. Quasi-periodic aeroelastic response analysis of an airfoil with external store by incremental harmonic balance method［J］. Int J NonLinear Mech， 100： 10-19.

LIU G， LU Z R， WANG L， et al， 2021a. A new semi-analytical technique for nonlinear systems based on response sensitivity analysis［J］. Nonlinear Dyn， 103（2）： 1529-1551.

LIU G， LIU J K， WANG L， et al， 2021b. A new semi-analytical approach for quasi-periodic vibrations of nonlinear systems［J］. Commun Nonlinear Sci Numer Simul， 103： 105999.

JU R， FAN W， ZHU W D， 2020. An efficient Galerkin averaging-incremental harmonic balance method based on the fast Fourier transform and tensor contraction［C］// American Society of Mechanical Engineers.

SHEN J H， LIN K C， CHEN S H， et al， 2008. Bifurcation and route-to-chaos analyses for Mathieu-Duffing oscillator by the incremental harmonic balance method ［J］. Nonlinear Dyn， 52（4）： 403-414.

ZHENG Z C， CHEN Y M， LU Z R， et al， 2022a. Residual-tuned analytical approximation for the limit cycle of aeroelastic systems with hysteresis nonlinearity［J］. J Fluids Struct， 108： 103440.

ZHENG Z C ， LU Z R ， CHEN Y M， et al， 2022b. A modified incremental harmonic balance method combined with Tikhonov regularization for periodic motion of nonlinear system［J］. J Appl Mech， 89（2）： 021001.

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Study on Nonlinear Vibration of Axially Moving Beams with Varying Velocities

The IHB method with multiple time scales for quasi-periodic motions of nonlinear systems

Semi-analytical solution of nonlinear aeroelastic systems with freeplay based on the time-domain minimum residual method

Electrical impedance imaging for cavity detection

Related Author

CHEN Shuhui

HUANG Jianliang

HUANG Jianliang

LIU Jike

ZHANG Danwei

LIU Zuoqiu

QIN Yingquan

WU Zeyu

Related Institution

Department of Applied Mechanics and Engineering, Sun Yatsen University

Department of Applied Mechanics and Engineering, School of Engineering, Sun Yat-sen University

Postal code：510275
Tel：020-84112585，84113223 Email：xuebaozr@mail.sysu.edu.cn
Technical support is provided by Beijing Founder electronics co., LTD 京ICP备09064830号-19 京公网安备11010802024621
It is recommended to read the content of this site in Chrome&IE9+. Please switch to extreme mode in browser 360.
Cookies We use cookies to help provide and enhance our service and tailor content. By continuing, you agree to the use of cookies.

⁰