非线性电容LC电路的位移小参数摄动法分析
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摘要
为了将位移小参数摄动法拓展到电路领域,用于非线性LC电路方程的求解,为非线性LC电路的研究提供一种分析方法,从系统的拉格朗日函数出发,将变分式进行有限元离散得到时段刚度阵,对电路的特征方程进行一次摄动,用一次摄动后的近似刚度阵代替原时段刚度阵,利用刚度阵与辛传递矩阵的关系将问题转换成辛传递矩阵求解的问题,从而求出非线性电容LC电路中电容的电荷及电感的磁通链随时间变化的关系图.将算例与四阶龙格库塔法进行比较,验证了本文方法的正确性.将本文方法与传递辛矩阵加法摄动进行比较,结果表明:本文方法具有一定的精度、效率及稳定性.
In this paper,the perturbation based on displacement method was extended to the field of circuit,which is used to solve the nonlinear LC circuit equation,and an analytical method for the research of nonlinear LC circuits was provided. Deducing from the Lagrange function of the system,the variational equation was discretized by the finite element to obtain the time stiffness matrix. The characteristic equation of the circuit was perturbed for the first time,and the original time stiffness matrix was replaced by a perturbed approximate stiffness matrix. On the basis of the relationship between the stiffness matrix and the transfer matrix,the problem was transformed into the problem of symplectic transfer matrix,and the oscillation characteristics of the LC circuit can be solved. The comparison between the perturbation based on displacement method and the fourth order Runge Kutta method demonstrated the correctness of the proposed method. Compared with the transfer symplectic matrix method,it is proved that the proposed method has definite precision,efficiency and stability.
In this paper,the perturbation based on displacement method was extended to the field of circuit,which is used to solve the nonlinear LC circuit equation,and an analytical method for the research of nonlinear LC circuits was provided. Deducing from the Lagrange function of the system,the variational equation was discretized by the finite element to obtain the time stiffness matrix. The characteristic equation of the circuit was perturbed for the first time,and the original time stiffness matrix was replaced by a perturbed approximate stiffness matrix. On the basis of the relationship between the stiffness matrix and the transfer matrix,the problem was transformed into the problem of symplectic transfer matrix,and the oscillation characteristics of the LC circuit can be solved. The comparison between the perturbation based on displacement method and the fourth order Runge Kutta method demonstrated the correctness of the proposed method. Compared with the transfer symplectic matrix method,it is proved that the proposed method has definite precision,efficiency and stability.
引文
[1] TANG Q,CHEN C M,LIU L H. Energy conservation and symplectic properties of continuous finite element methods for Hamiltonian systems[J]. Applied Mathematics and Computation,2006,181(2):1357-1368.
[2]蒋宪宏,邓子辰,张凯,等.线性Hamilton系统边值问题的保辛数值方法[J].应用数学和力学,2017,38(9):988-998.JIANG X H,DENG Z C,ZHANG K,et al. A symplectic approach for boundary-value problems of linear Hamiltonian systems[J]. Applied Mathematics and Mechanics,2017,38(9):988-998.(in Chinese)
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[4] ZI-XIANG X U,ZHOU D Y,DENG Z C. Numerical method based on Hamilton system and symplectic algorithm to differential games[J]. Applied Mathematics and Mechanics(English Edition),2006,27(3):341-346.
[5]侯平兰,邓子辰.高层建筑结构地震响应的辛精细积分方法[J].应用数学和力学,2018,39(4):442-451.HOU P L,DENG Z C. A symplectic precise integration method for seismic responses of tall buildings[J]. Applied Mathematics and Mechanics,2018,39(4):442-451.(in Chinese)
[6] PAN H,YANG Y,LI X,et al. Symplectic geometry mode decomposition and its application to rotating machinery compound fault diagnosis[J]. Mechanical Systems and Signal Processing,2019,114:189-211.
[7] TAN S J,ZHONG W X. Numerical solution of differential Riccati equation with variable coefficients via symplectic conservative perturbation method[J]. Journal of Dalian University of Technology,2007,28(3):277-287.
[8] USHIDA A,TANJI Y,NISHIO Y. Analysis of nonlinear traveling waves by frequency-domain perturbation method[C]∥IEEE International Symposium on Circuits&Systems. Piscataway:IEEE,1996:150-153.
[9] GILMORE R J,STEER M B. Nonlinear circuit analysis using the method of harmonic balance—a review of the art.II. advanced concepts[J]. International Journal of Microwave and Millimeter-Wave, Computer-Aided Engineering,1991,1(2):159-180.
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[11]孙雁,陈晓辉,钟万勰.非线性方程的保辛近似算法[J].力学季刊,2006,27(3):365-370.SUN Y, CHEN X H, ZHONG W X. Symplectic conservative integration for nonlinear differential equation[J]. Chinese Quarterly of Mechanics,2006,27(3):365-370.(in Chinese)
[12]陆克浪,富明慧,李纬华,等.一种基于加权残值法的高阶辛算法[J].中山大学学报(自然科学版),2015,54(4):8-12.LU K L, FU M H, LI W H, et al. A high order symplectic algorithm based on weighted residual method[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni,2015,54(4):8-12.(in Chinese)
[13]高强,彭海军,吴志刚,等.非线性动力学系统最优控制问题的保辛求解方法[J].动力学与控制学报,2010,8(1):1-7.GAO Q,PENG H J,WU Z G,et al. Symplectic method for solving optimal control problem of nonlinear dymamical systems[J]. Journal of Dynamics and Control,2010,8(1):1-7.(in Chinese)
[14]吴锋,高强,钟万勰. FPU方程的多尺度保辛摄动积分[J].计算力学学报,2015,32(5):587-594.WU F,GAO Q,ZHONG W X. Multiscale symplectic perturbation method for FPU equation[J]. Chinese Journal of Computational Mechanics,2015,32(5):587-594.(in Chinese)
[15] FU M H,LAN L H. High order symplectic conservative perturbation method for time-varying Hamiltonian system[J]. Acta Mechanica Sinica,2012,28(3):885-890.
[16] CHANDIA K J,BOLOGNA M,TELLINI B. Multiple scale approach to dynamics of an LC circuit with a chargecontrolled memristor[J]. IEEE Transactions on Circuits&Systems II Express Briefs,2018,66(1):120-124.
[17] ZHIAN Y,YIHUI C. Double resonances of coupled RLC circuit and spring system with internal resonance considering inductance nonlinearity[C]∥International Conference on Electronic Measurement&Instruments.Piscataway:IEEE,2009:16-19.
[18]安利强,周邢银,张颖,等.基于节点位移法的变耦合区域风力机叶片弯扭耦合效应分析[J].科学技术与工程,2018,18(8):209-213.AN L Q,ZHOU X Y,ZHANG Y,et al. Analyses on bend-twist coupling characteristic of wind turbine blade with varied coupling region based on the nodal displacement method[J]. Science Technology and Engineering,2018,18(8):209-213.(in Chinese)
[19]钟万勰,陈晓辉.浅水波的位移法求解[J].水动力学研究与进展(A辑),2006,21(4):486-493.ZHONG W X,CHEN X H. Solving shallow water with the displacement method[J]. Journal of Hydrodynamics(Ser A),2006,21(4):486-493.(in Chinese)
[20]姚征.辛体系算法在波的传播与振动问题中的应用[D].大连:大连理工大学,2007.YAO Z. Applications of symplectic algorithm in wave propagation and vibration problems[D]. Dalian:Dalian University of Technology,2007.(in Chinese)
[21]钟万勰,孙雁.小参数摄动法与保辛[J].动力学与控制学报,2005,3(1):3-8.ZHONG W X, SUN Y. Small parameter perturbation method and symplectic conservation[J]. Journal of Dynamics and Control,2005,3(1):3-8.(in Chinese)
[22]钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2013:282-284.
[23] CHEN J, LEE J H, LIU Q H. A high-precision integration scheme for the spectral-element time-domain method in electromagnetic simulation[J]. IEEE Transactions on Antennas&Propagation,2009,57(10):3223-3231.
[2]蒋宪宏,邓子辰,张凯,等.线性Hamilton系统边值问题的保辛数值方法[J].应用数学和力学,2017,38(9):988-998.JIANG X H,DENG Z C,ZHANG K,et al. A symplectic approach for boundary-value problems of linear Hamiltonian systems[J]. Applied Mathematics and Mechanics,2017,38(9):988-998.(in Chinese)
[3] ZHU B Q,ZHUO J S,ZHOU J F. Conservation law of Hamiltonian function in symplectic system of polar coordinate elasticity[J]. Engineering Mechanics,2006,23(12):63-68.
[4] ZI-XIANG X U,ZHOU D Y,DENG Z C. Numerical method based on Hamilton system and symplectic algorithm to differential games[J]. Applied Mathematics and Mechanics(English Edition),2006,27(3):341-346.
[5]侯平兰,邓子辰.高层建筑结构地震响应的辛精细积分方法[J].应用数学和力学,2018,39(4):442-451.HOU P L,DENG Z C. A symplectic precise integration method for seismic responses of tall buildings[J]. Applied Mathematics and Mechanics,2018,39(4):442-451.(in Chinese)
[6] PAN H,YANG Y,LI X,et al. Symplectic geometry mode decomposition and its application to rotating machinery compound fault diagnosis[J]. Mechanical Systems and Signal Processing,2019,114:189-211.
[7] TAN S J,ZHONG W X. Numerical solution of differential Riccati equation with variable coefficients via symplectic conservative perturbation method[J]. Journal of Dalian University of Technology,2007,28(3):277-287.
[8] USHIDA A,TANJI Y,NISHIO Y. Analysis of nonlinear traveling waves by frequency-domain perturbation method[C]∥IEEE International Symposium on Circuits&Systems. Piscataway:IEEE,1996:150-153.
[9] GILMORE R J,STEER M B. Nonlinear circuit analysis using the method of harmonic balance—a review of the art.II. advanced concepts[J]. International Journal of Microwave and Millimeter-Wave, Computer-Aided Engineering,1991,1(2):159-180.
[10]钟万勰,姚征.时间有限元与保辛[J].机械强度,2005,27(2):178-183.ZHONG W X,YAO Z. Time domain fem and symplectic conservation[J]. Journal of Mechanical Strength,2005,27(2):178-183.(in Chinese)
[11]孙雁,陈晓辉,钟万勰.非线性方程的保辛近似算法[J].力学季刊,2006,27(3):365-370.SUN Y, CHEN X H, ZHONG W X. Symplectic conservative integration for nonlinear differential equation[J]. Chinese Quarterly of Mechanics,2006,27(3):365-370.(in Chinese)
[12]陆克浪,富明慧,李纬华,等.一种基于加权残值法的高阶辛算法[J].中山大学学报(自然科学版),2015,54(4):8-12.LU K L, FU M H, LI W H, et al. A high order symplectic algorithm based on weighted residual method[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni,2015,54(4):8-12.(in Chinese)
[13]高强,彭海军,吴志刚,等.非线性动力学系统最优控制问题的保辛求解方法[J].动力学与控制学报,2010,8(1):1-7.GAO Q,PENG H J,WU Z G,et al. Symplectic method for solving optimal control problem of nonlinear dymamical systems[J]. Journal of Dynamics and Control,2010,8(1):1-7.(in Chinese)
[14]吴锋,高强,钟万勰. FPU方程的多尺度保辛摄动积分[J].计算力学学报,2015,32(5):587-594.WU F,GAO Q,ZHONG W X. Multiscale symplectic perturbation method for FPU equation[J]. Chinese Journal of Computational Mechanics,2015,32(5):587-594.(in Chinese)
[15] FU M H,LAN L H. High order symplectic conservative perturbation method for time-varying Hamiltonian system[J]. Acta Mechanica Sinica,2012,28(3):885-890.
[16] CHANDIA K J,BOLOGNA M,TELLINI B. Multiple scale approach to dynamics of an LC circuit with a chargecontrolled memristor[J]. IEEE Transactions on Circuits&Systems II Express Briefs,2018,66(1):120-124.
[17] ZHIAN Y,YIHUI C. Double resonances of coupled RLC circuit and spring system with internal resonance considering inductance nonlinearity[C]∥International Conference on Electronic Measurement&Instruments.Piscataway:IEEE,2009:16-19.
[18]安利强,周邢银,张颖,等.基于节点位移法的变耦合区域风力机叶片弯扭耦合效应分析[J].科学技术与工程,2018,18(8):209-213.AN L Q,ZHOU X Y,ZHANG Y,et al. Analyses on bend-twist coupling characteristic of wind turbine blade with varied coupling region based on the nodal displacement method[J]. Science Technology and Engineering,2018,18(8):209-213.(in Chinese)
[19]钟万勰,陈晓辉.浅水波的位移法求解[J].水动力学研究与进展(A辑),2006,21(4):486-493.ZHONG W X,CHEN X H. Solving shallow water with the displacement method[J]. Journal of Hydrodynamics(Ser A),2006,21(4):486-493.(in Chinese)
[20]姚征.辛体系算法在波的传播与振动问题中的应用[D].大连:大连理工大学,2007.YAO Z. Applications of symplectic algorithm in wave propagation and vibration problems[D]. Dalian:Dalian University of Technology,2007.(in Chinese)
[21]钟万勰,孙雁.小参数摄动法与保辛[J].动力学与控制学报,2005,3(1):3-8.ZHONG W X, SUN Y. Small parameter perturbation method and symplectic conservation[J]. Journal of Dynamics and Control,2005,3(1):3-8.(in Chinese)
[22]钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2013:282-284.
[23] CHEN J, LEE J H, LIU Q H. A high-precision integration scheme for the spectral-element time-domain method in electromagnetic simulation[J]. IEEE Transactions on Antennas&Propagation,2009,57(10):3223-3231.