EI、Scopus 收录
中文核心期刊
Quick Search Issue Search Advanced Search
Volume 55 Issue 10
Oct.  2023
Turn off MathJax
Article Contents
Abstract
References
Related Articles
Jin Dongping. Method of polynomial vectors for nonlinear vibration systems. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2373-2380. DOI: 10.6052/0459-1879-23-331
Citation: Jin Dongping. Method of polynomial vectors for nonlinear vibration systems. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2373-2380. DOI: 10.6052/0459-1879-23-331
shu

METHOD OF POLYNOMIAL VECTORS FOR NONLINEAR VIBRATION SYSTEMS

  • State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

  • Received Date: July 25, 2023
  • Accepted Date: August 23, 2023
  • Available Online: August 24, 2023
    Graphical Abstract
    • Abstract
  • The perturbation methods for nonlinear vibration systems make it necessary to solve a set of second-order ordinary differential equations (ODEs), which are obtained by equating the like power of the perturbation solutions respectively. One of the main drawbacks of the ODEs-based methods is of low efficiency, especially for nonlinear vibration systems of multiple degrees of freedom. In this paper, a method of polynomial vectors for solving the approximate solution of nonlinear vibration systems is proposed. The second order ordinary differential equations are written in a set of state equations of the first order first, wherein the nonlinear terms of the state equations are expressed as the products of a constant matrix and a polynomial vector with the like power. By using the direct perturbation method, the linear non-homogeneous equations are obtained for the like power approximations, while the nonlinear terms are written as the products of the constant matrix and the polynomial vector with the previous approximate solutions as its element. Furthermore, the multiplication of polynomials in the polynomial vector is expressed in matrix form via Toeplitz matrix, and then all approximate analytical formulas of the state equations are determined by the first-order non-homogeneous equations. Results show that the proposed method based upon the state equations yields a concise calculation for nonlinear vibration systems of multiply degrees of freedom.
    • FullText(HTML)
    • References (15)
  • [1]
    陈予恕, 唐云, 非线性动力学中的现代分析方法, 北京: 科学出版社, 1992

    Chen Yushu, Tang Yun. Modern Analysis Methods in Nonlinear Dynamics. Beijing: Science Press, 1992 (in Chinese)
    [2]
    Nayfeh AH. Method of Normal Forms. John Wiley & Sons Inc., 1993
    [3]
    刘延柱, 陈文良, 陈立群. 振动力学. 北京: 高等教育出版社, 1998

    Liu Yanzhu, Chen Wenliang, Chen Liqun. Mechanics of Vibrations. Beijing: Higher Education Press, 1998 (in Chinese)
    [4]
    胡海岩. 应用非线性动力学. 北京: 航空工业出版社, 2000

    Hu Haiyan. Applied Nonlinear Dynamics. Beijing: Aviation Industry Press, 2000 (in Chinese)
    [5]
    陈树辉. 强非线性振动系统的定量分析方法. 北京: 科学出版社, 2007

    Chen Shuhui. Quantitative Analysis Method for Strongly Nonlinear Vibration Systems. Beijing: Science Press, 2007 (in Chinese)
    [6]
    杨绍普. 滞后非线性系统的分岔与奇异性. 北京: 科学出版社, 2003

    Yang Shaopu. Bifurcation and Singularity of Hysteretic Nonlinear Systems. Beijing: Science Press, 2003 (in Chinese)
    [7]
    杨绍普, 曹庆杰, 张伟. 非线性动力学与控制中的若干理论及应用. 北京: 科学出版社, 2011

    Yang Shaopu, Cao Qingjie, Zhang Wei. Theories and Its Applications in Nonlinear Dynamics and Control. Beijing: Science Press, 2011 (in Chines))
    [8]
    Zhang YL, Wei G, Wen H, et al. Design and analysis of a vibration isolation system with cam-roller-spring-rod mechanism. Journal of Vibration and Control, 2022, 28(13-14): 1781-1791
    [9]
    Wagg D, Neild S. Nonlinear Vibration with Control for Flexible and Adaptive Structures. Springer, 2011
    [10]
    刘福寿, 金栋平. 环形桁架结构径向振动的等效圆环模型. 力学学报, 2016, 48(5): 1184-1191 (Liu Fushou, Jin Dongping. Equivalent circular ring model for the radial vibration analysis of hoop truss structures. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5): 1184-1191 (in Chinese)

    Liu Fushou, Jin Dongping. Equivalent circular ring model for the radial vibration analysis of hoop truss structures. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5):1184-1191 (in Chinese)
    [11]
    Liu FS, Wang LB, Jin DP, et al. Equivalent beam model for spatial repetitive lattice structures with hysteretic nonlinear joints. International Journal of Mechanical Sciences, 2021, 200: 106449
    [12]
    Li XY, Wei G, Guo JJ, et al. Multi-harmonic equivalent modeling for a planner repetitive structure with polynomial-nonlinear joints. Acta Mechanica Sinica, 2022, 38: 122020
    [13]
    Li XY, Wei G, Guo JJ, et al. Nonlinear response of a planar repetitive space truss with joints based on multi-harmonic equivalent model. Meccanica, 2023, 58: 1495-1510
    [14]
    Jin DP, Wen H, Chen H. Nonlinear resonance of a subsatellite on a short constant tether. Nonlinear Dynamics, 2013, 71(3): 479-488
    [15]
    Gao XM, Jin DP, Hu HY. Nonlinear resonances and their bifurcations of a rigid-flexible space antenna. International Journal of Nonlinear Mechanics, 2017, 94: 160-173
    • Related Articles

  • Related Articles

    [1]Minghui Fu, Jinghua Lin. An exponential matrix function and its application[J]. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(5): 808-814. DOI: 10.6052/0459-1879-2009-5-2008-052
    [2]Chaos control of form iterative algorithm in structural reliability analysis[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(5): 647-654. DOI: 10.6052/0459-1879-2007-5-2006-501
    [3]Jinyang Liu, Jiazhen Hong. Nonlinear formulation for flexible multibody system with large deformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(1): 111-119. DOI: 10.6052/0459-1879-2007-1-2006-113
    [4]Transfer matrix method for 2-dimension system[J]. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(5): 712-720. DOI: 10.6052/0459-1879-2006-5-2005-429
    [5]FORM in structural reilianbility analysis and chaotic dynamics[J]. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(6): 799-804. DOI: 10.6052/0459-1879-2005-6-2004-244
    [7]CALCULATING NORMAL FORMS FOR NONSEMI SIMPLE BIFURCATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1998, 30(4): 423-433. DOI: 10.6052/0459-1879-1998-4-1995-145
    [8]MATRIX PERTURBATION FOR LINEAR VIBRATION DEFECTIVE SYSTEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1992, 24(6): 747-753. DOI: 10.6052/0459-1879-1992-6-1995-799
    [9]APPLICATION OF CAPACITANCE MATRIX METHOD IN VORTEX METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 1992, 24(1): 19-25. DOI: 10.6052/0459-1879-1992-1-1995-706
    [10]A NEW METHOD OF CALCULATING THE ASYMPTOTIC SOLUTION OF NONLINEAR VIBRATION SYSTEMS——A SIMPLE METHOD OF CALCULATING THECOEFFICIENTS OF NORMAL FORM[J]. Chinese Journal of Theoretical and Applied Mechanics, 1990, 22(4): 413-419. DOI: 10.6052/0459-1879-1990-4-1995-964

Catalog

    Get Citation
    PDF
    XML

    Article Metrics

    Article views (456) PDF downloads (117) Cited by()
    Related

    Download Center

    Author Center

    Copyright © Editorial Office of the Chinese Journal of Mechanics   京ICP备05039218号-1

    Address：15 Beishihuan Xi Lu, Haidian District, Beijing, China   China Pos：100190

    Tel：010-62536271

    Email：lxxb@cstam.org.cn

    微信公众号
    RSS

    Supported by: Beijing Renhe Information Technology Co., Ltd.  Baidu Analytics

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return