An Average Vector Field Method for Nonlinear Vibration Analysis

BAO Siyuan; DENG Zichen

doi:10.21656/1000-0887.390178

Volume 40 Issue 1

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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

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

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An Average Vector Field Method for Nonlinear Vibration Analysis

doi: 10.21656/1000-0887.390178

BAO Siyuan¹,
DENG Zichen²

¹School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China;²School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Natural Science Foundation of China（11202146）

Received Date: 2018-06-26
Rev Recd Date: 2018-11-07
Publish Date: 2019-01-01

Abstract

Abstract

Through construction of differential equations in the vector form, the differential iteration form of the vibration response was obtained according to the average vector field (AVF) method. This discrete form is energy-preserving for the Hamiltonian system, and has the characteristics of 2ndorder accuracy. The detailed steps of the AVF method were given. To establish the AVF scheme, the mapping forms were deduced directly for several common items in the differential equations. The pendulum problem and the Kepler problem were studied with the AVF method. The numerical results demonstrate the advantages of the AVF method in solving nonlinear vibration problems, i.e. the conservation of energy and the longterm solution stability.
- average vector field method,
- nonlinear vibration,
- energy-preserving,
- pendulum problem,
- Kepler problem

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References(24)

References

[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))