非线性振动分析的均向量场法
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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.
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 2 nd-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 longterm solution stability.
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 2 nd-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 longterm solution stability.
引文
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[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 A pplied Mechanics,2007,39(5):668-671.(in Chinese))
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[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,SU NWeipeng,WU Baisheng.Analytical approximate solutions to large amplitude oscillation of a simple pendulum[J].Journal of V ibration 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.(LZ hongrong,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.(Z HO U Kaihong,WANG Yuanxun,LI Chunzhi.T he application of differential quadrature method in nonlinear vibration analysis of simple pendulum[J].Mechanics in Engineering,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,Z HAO Dingbai.A simple treatment of the Kepler problem[J].College Physics,2000,19(1):45-47.(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.(QINMengzhao,WANG Yushun.Structure-Preserving A lgorithms for Partial Differential Equation[M].Hangzhou:Z hejiang Science and T echnolog 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 Z ichen.The Infinite Dimensional Dynamical System Structure A nalysis Method[M].Xi’an:Northwestern Polytechnical U niversity Press,2015.(in Chinese))
[5]HU WP,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 Q iang,Z HO NG Wanxie.T he 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.(CHENLu,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 T hesis.Yichang:China T hree Gorges U niversity,2015.(in Chinese))
[10]QUISPEL G R W,MCLACHLAND I.A new class of energy-preserving numerical integration methods[J].Journal of Physics A:Mathematical and Theoretical,2008,41(4):045206.DO I:10.1088/1751-8113/41/4/045206.
[11]CELLEDONI E,MCLACHLAND I,OWRENB,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 A pplied Mathematics,2014,259:233-243.
[13]CAI J X,WANG Y S,GONG Y Z.Numerical analysis of AVF methods for three-dimensional timedomain 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,SU NJianqiang,LU O Siyu.An averaged vector field method for the nonlinear Schr9dinger equation[J].Mathematica Numerica Sinica,2013,35(1):60-66.(in Chinese))
[15]HAIRER E,LUBICH C,WANNER G.G eometric Numerical Integration:Structure-Preserving Algorithms for O rdinary D ifferential 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.(CHENLu.Phase error analysis and correction of structure preserving algorithms[D].Master Thesis.Nanjing:Nanjing Normal U niversity,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 A pplied 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,Z HO U Gang,WANG Yonghong,et al.R ectifying 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 A pplied 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.(Q INYuyue,DENG Z ichen,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,SU NWeipeng,WU Baisheng.Analytical approximate solutions to large amplitude oscillation of a simple pendulum[J].Journal of V ibration 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.(LZ hongrong,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.(Z HO U Kaihong,WANG Yuanxun,LI Chunzhi.T he application of differential quadrature method in nonlinear vibration analysis of simple pendulum[J].Mechanics in Engineering,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,Z HAO Dingbai.A simple treatment of the Kepler problem[J].College Physics,2000,19(1):45-47.(in Chinese))