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非线性振动分析的均向量场法

鲍四元 邓子辰

鲍四元, 邓子辰. 非线性振动分析的均向量场法[J]. 应用数学和力学, 2019, 40(1): 47-57. doi: 10.21656/1000-0887.390178
引用本文: 鲍四元, 邓子辰. 非线性振动分析的均向量场法[J]. 应用数学和力学, 2019, 40(1): 47-57. doi: 10.21656/1000-0887.390178
BAO Siyuan, DENG Zichen. An Average Vector Field Method for Nonlinear Vibration Analysis[J]. Applied Mathematics and Mechanics, 2019, 40(1): 47-57. doi: 10.21656/1000-0887.390178
Citation: BAO Siyuan, DENG Zichen. An Average Vector Field Method for Nonlinear Vibration Analysis[J]. Applied Mathematics and Mechanics, 2019, 40(1): 47-57. doi: 10.21656/1000-0887.390178

非线性振动分析的均向量场法

doi: 10.21656/1000-0887.390178
基金项目: 国家自然科学基金(11202146)
详细信息
    作者简介:

    鲍四元(1980—),男,副教授(E-mail: bsiyuan@126.com);邓子辰(1964—),男,教授,博士生导师(通讯作者. E-mail: dweifen@nwpu.edu.cn).

  • 中图分类号: O322;O326

An Average Vector Field Method for Nonlinear Vibration Analysis

Funds: The National Natural Science Foundation of China(11202146)
  • 摘要: 通过构造向量形式的振动微分方程组,利用均向量场(AVF)法得到振动响应的向量差分迭代格式.该离散格式能够保能量,同时具有二阶精度的特征,从而给出非线性振动问题的均向量场法.介绍了均向量场法的基本步骤.在建立AVF格式时,对于微分方程中若干常见的项,直接给出相应的映射项.应用均向量场法研究了非线性单摆问题和Kepler(开普勒)问题,数值结果说明了该方法保能量和具有长时间求解能力的特性.
  • [1] 冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M]. 杭州: 浙江科学技术出版社, 2003.(FENG Kang, QIN Mengzhao. Symplectic Geometric Algorithm for Hamilton System [M]. Hangzhou: Zhejiang Science and Technolog Press. 2003.(in Chinese))
    [2] 秦孟兆, 王雨顺. 偏微分方程中的保结构算法[M]. 杭州: 浙江科技出版社, 2010.(QIN Mengzhao, WANG Yushun. Structure-Preserving Algorithms for Partial Differential Equation [M]. Hangzhou: Zhejiang Science and Technolog Press, 2010.(in Chinese))
    [3] FENG K. Difference schemes for Hamiltonian formalism and symplectic geometry[J]. Journal of Computational Mathematics,1986,4(3): 279-289.
    [4] 胡伟鹏, 邓子辰. 无限维动力学系统的保结构分析方法[M]. 西安: 西北工业大学出版社, 2015.(HU Weipeng, DENG Zichen. The Infinite Dimensional Dynamical System Structure Analysis Method [M]. Xi’an: Northwestern Polytechnical University Press, 2015.(in Chinese))
    [5] HU W P, DENG Z C, ZHANG Y. Multi-symplectic method for peakon-antipeakon collision of quasi-Degasperis-Procesi equation[J]. Computer Physics Communications,2014,185(7): 2020-2028.
    [6] 高强, 钟万勰. Hamilton系统的保辛-守恒积分算法[J]. 动力学与控制学报, 2009,7(3): 193-199.(GAO Qiang, ZHONG Wanxie. The symplectic and energy preserving method for the integration of Hamilton system[J]. Journal of Dynamics and Control,2009,7(3): 193-199.(in Chinese))
    [7] BRUGNANO L, IAVERNARO F, TRGIANTE D. A two-step, fourth-order method with energy preserving properties[J]. Computer Physics Communications,2012,183(9): 1860-1868.
    [8] 陈璐, 王雨顺. 保结构算法的相位误差分析及其修正[J]. 计算数学, 2014,36(3): 271-290.(CHEN Lu, WANG Yushun. Phase error analysis and correction of structure preserving algorithms[J]. Mathematica Numerica Sinica,2014,36(3): 271-290.(in Chinese))
    [9] 叶霄霄. 基于平均向量场方法的暂态稳定计算[D]. 硕士学位论文. 宜昌: 三峡大学, 2015.(YE Xiaoxiao. Transient stability calculation based on the average vector field method[D]. Master Thesis. Yichang: China Three Gorges University, 2015.(in Chinese))
    [10] QUISPEL G R W, MCLACHLAN D I. A new class of energy-preserving numerical integration methods[J]. Journal of Physics A: Mathematical and Theoretical,2008,41(4): 045206. DOI: 10.1088/1751-8113/41/4/045206.
    [11] CELLEDONI E, MCLACHLAND I, OWREN B, et al. Energy-preserving integrators and the structure of B-series[J]. Foundations of Computational Mathematics,2010,10(6): 673-693.
    [12] CIESLINSKI J L. Improving the accuracy of the AVF method[J]. Journal of Computational and Applied Mathematics,2014,259: 233-243.
    [13] CAI J X, WANG Y S, GONG Y Z. Numerical analysis of AVF methods for three-dimensional time-domain Maxwell’s equations[J]. Journal of Scientific Computing,2016,66(1): 141-176.
    [14] 李昊辰, 孙建强, 骆思宇. 非线性薛定谔方程的平均向量场方法[J]. 计算数学, 2013,35(1): 60-66.(LI Haochen, SUN Jianqiang, LUO Siyu. An averaged vector field method for the nonlinear Schrdinger equation[J]. Mathematica Numerica Sinica,2013,35(1): 60-66.(in Chinese))
    [15] HAIRER E, LUBICH C, WANNER G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations [M]. Berlin: Springer, 2006.
    [16] 陈璐. 保结构算法的相位误差分析及其修正[D]. 硕士学位论文. 南京: 南京师范大学, 2014.(CHEN Lu. Phase error analysis and correction of structure preserving algorithms[D]. Master Thesis. Nanjing: Nanjing Normal University, 2014.(in Chinese))
    [17] 邢誉峰, 杨蓉. 动力学平衡方程的中点辛差分求解格式[J]. 力学学报, 2007,39(1): 100-105.(XING Yufeng, YANG Rong. Application of Euler midpoint symplectic integration method for the solution of dynamic equilibrium equations[J]. Chinese Journal of Theoretical and Applied Mechanics,2007,39(1): 100-105.(in Chinese))
    [18] 刘晓梅, 周钢, 王永泓, 等. 辛算法的纠飘研究[J]. 北京航空航天大学学报, 2013,39(1): 22-26.(LIU Xiaomei, ZHOU Gang, WANG Yonghong, et al. Rectifying drifts of symplectic algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics,2013,39(1): 22-26.(in Chinese))
    [19] 邢誉峰, 杨蓉. 单步辛算法的相位误差分析及修正[J]. 力学学报, 2007,39(5): 668-671.(XING Yufeng, YANG Rong. Phase errors and their correction in symplectic implicit single-step algorithm[J]. Chinese Journal of Theoretical and Applied Mechanics,2007,39(5): 668-671.(in Chinese))
    [20] 秦于越, 邓子辰, 胡伟鹏. 谐振子的辛欧拉分析方法[J]. 动力学与控制学报, 2014,12(1): 9-12.(QIN Yuyue, DENG Zichen, HU Weipeng. Symplectic Euler method for harmonic oscillator[J]. Journal of Dynamics and Control,2014,12(1): 9-12.(in Chinese))
    [21] 李鹏松, 孙维鹏, 吴柏生. 单摆大振幅振动的解析逼近解[J]. 振动与冲击, 2008,27(2): 72-74.(LI Pengsong, SUN Weipeng, WU Baisheng. Analytical approximate solutions to large amplitude oscillation of a simple pendulum[J]. Journal of Vibration and Shock,2008,27(2): 72-74.(in Chinese))
    [22] 吕中荣, 刘济科. 摆的振动分析[J]. 暨南大学学报(自然科学版), 1999,20(1): 42-45.(L Zhongrong, LIU Jike. Vibration analysis of a pendulum[J]. Journal of Jinan University(Natural Science),1999,20(1): 42-45.
    [23] 周凯红, 王元勋, 李春植. 微分求积法在单摆非线性振动分析中的应用[J]. 力学与实践, 2003,25(3): 50-52.(ZHOU Kaihong, WANG Yuanxun, LI Chunzhi. The application of differential quadrature method in nonlinear vibration analysis of simple pendulum[J]. Mechanics in Engineerin g, 2003,25(3): 50-52.(in Chinese))
    [24] 李文博, 赵定柏. 开普勒问题的一种简单处理[J]. 大学物理, 2000,19(1): 45-47.(LI Wenbo, ZHAO Dingbai. A simple treatment of the Kepler problem[J]. College Physics,2000,19(1): 45-47.(in Chinese))
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出版历程
  • 收稿日期:  2018-06-26
  • 修回日期:  2018-11-07
  • 刊出日期:  2019-01-01

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