RLC电路弹簧耦合系统非线性动力学研究
摘要
随着电子技术的发展,大量的电、磁、机械耦合系统得到广泛应用,这类机电耦合系统存在着丰富的非线性动力学现象,对其进行深入地研究,掌握系统中电路子系统和机械子系统及全系统的动力学规律,可以提高系统运行的稳定性及安全性。
应用Lagrange-Maxwell方程建立了一个RLC电路与弹簧耦合系统的多自由度动力学模型,充分考虑了系统的动能、势能、磁能和电场能,记入了系统的耗散做功和非保守的广义力的影响。运用非线性振动理论对该系统的单自由度模型、双自由度模型和三自由度模型分别进行动力学分析。发现在单自由度系统中,系统的固有频率随极板间距增大而增大,随极板面积和线性电感系数的增大而减小;系统的响应曲线存在着“跳跃”现象,电阻和电感的非线性,能改变系统响应曲线的软硬特性;用级数法得到的结果与龙格库塔法得到的结论相吻合;采用Lindstedt-Poincare方法可以迅速而准确地分析系统的稳定边界,极板间距的取值决定了阻尼和弹簧刚度的取值范围。对于具有2:1内共振关系的双自由度系统,两个模态之间出现了能量传递,在满足系统的双重共振ω2≈2ω1且?≈ω2的情况下,幅频响应曲线出现了“饱和”现象;电感的非线性,在双重共振ω2≈2ω1且?≈ω1的情况下,能在共振区的一侧激起新的振动,在双重共振ω2≈2ω1且?≈ω2的情况下,改变了响应曲线的拓扑结构;无论是电阻非线性还是电感非线性,都会抑制两个模态的振动;改变系统参数,振幅和共振区的大小会发生相应的变化,但两个模态对参数变化的响应程度是不一样的,可以通过一个模态控制另一个模态。对于三自由度系统,改变外激励和系统参数,能使三个模态的振幅发生变化。电路系统的固有频率随极板间距的增大而增大,随极板面积和电感的增大而减小。机械系统的固有频率随极板重量的增大而减小,随弹簧刚度的增大而增大。改变调谐值,能使系统响应的拓扑结构发生变化,在一些情况下,三自由度系统出现了与二自由度系统相近的动力学现象。
在本系统中,电极板的振幅和电量的大小都可以被控制在安全的范围之内,从而保证系统的安全性,也可以利用其特有的“饱和”现象,开发出新型的减振器。图139;参58
With the development of electronic technology, more and more coupled systems of electricity-magnetism-mechanism are used widely. These coupled systems have abundant non-linear dynamics phenomena. In order to improve stability and security of operation of these systems, we ought to study these systems in detail.
Multi-degrees of freedom dynamics model of coupled RLC circuit and spring system is established by means of the Lagrange-Maxwell equation, considering the kinetic energy, the potential energy, the magnetic energy and electrical energy, the dissipation function and the influence of the non-conservative generalized force. Using the nonlinear oscillation theory to analyze single degree of freedom model, double degrees of freedom model and three degrees of freedom model. For single degree of freedom systems, with the increasing of plate distance, nature frequency of the system enlarge; with the increasing of plate area and linear inductance coefficient, nature frequency of the system decreases. The jump phenomenon is found in these systems. The non-linearity of resistance and inductance can change the nonlinear stiffness characteristic of response curves, and the results of the series method are in good agreement with the results of Runge-Kutta method. Based on the method of Lindstedt-Poincare, the stability of the system is analyzed, the values of damping and stiffness of spring can be effect by the value of plate distance. For double degree of freedom system, two coupled modals are all excited and vibrated; energy is transformed between two modals. The saturation phenomenon appears when the system meets double resonances conditionω2≈2ω1and ?≈ω2. In nonlinear inductance system, new vibrations are excited in one side of the response curves when the system meets double resonances conditionω2≈2ω1 and ?≈ω1. In double resonances conditionω2≈2ω1and ?≈ω2, with the changing of parameters of the system, amplitudes of certain modal can be changed. Changing the excitation, a similar saturation phenomenon can be found. When the parameters and resonant conditions are equal, nonlinear inductance can change the topology structure of response curves. Changing the nonlinear resistance and the nonlinear inductance can control peak value of response curves. Changing other parameters, amplitudes of response curves will change correspondingly, but the responses are different, one modal could be controlled by the other. For three degrees of freedom system, with the increasing of plate distance, the frequency of circuit systemω1 increases.
应用Lagrange-Maxwell方程建立了一个RLC电路与弹簧耦合系统的多自由度动力学模型,充分考虑了系统的动能、势能、磁能和电场能,记入了系统的耗散做功和非保守的广义力的影响。运用非线性振动理论对该系统的单自由度模型、双自由度模型和三自由度模型分别进行动力学分析。发现在单自由度系统中,系统的固有频率随极板间距增大而增大,随极板面积和线性电感系数的增大而减小;系统的响应曲线存在着“跳跃”现象,电阻和电感的非线性,能改变系统响应曲线的软硬特性;用级数法得到的结果与龙格库塔法得到的结论相吻合;采用Lindstedt-Poincare方法可以迅速而准确地分析系统的稳定边界,极板间距的取值决定了阻尼和弹簧刚度的取值范围。对于具有2:1内共振关系的双自由度系统,两个模态之间出现了能量传递,在满足系统的双重共振ω2≈2ω1且?≈ω2的情况下,幅频响应曲线出现了“饱和”现象;电感的非线性,在双重共振ω2≈2ω1且?≈ω1的情况下,能在共振区的一侧激起新的振动,在双重共振ω2≈2ω1且?≈ω2的情况下,改变了响应曲线的拓扑结构;无论是电阻非线性还是电感非线性,都会抑制两个模态的振动;改变系统参数,振幅和共振区的大小会发生相应的变化,但两个模态对参数变化的响应程度是不一样的,可以通过一个模态控制另一个模态。对于三自由度系统,改变外激励和系统参数,能使三个模态的振幅发生变化。电路系统的固有频率随极板间距的增大而增大,随极板面积和电感的增大而减小。机械系统的固有频率随极板重量的增大而减小,随弹簧刚度的增大而增大。改变调谐值,能使系统响应的拓扑结构发生变化,在一些情况下,三自由度系统出现了与二自由度系统相近的动力学现象。
在本系统中,电极板的振幅和电量的大小都可以被控制在安全的范围之内,从而保证系统的安全性,也可以利用其特有的“饱和”现象,开发出新型的减振器。图139;参58
With the development of electronic technology, more and more coupled systems of electricity-magnetism-mechanism are used widely. These coupled systems have abundant non-linear dynamics phenomena. In order to improve stability and security of operation of these systems, we ought to study these systems in detail.
Multi-degrees of freedom dynamics model of coupled RLC circuit and spring system is established by means of the Lagrange-Maxwell equation, considering the kinetic energy, the potential energy, the magnetic energy and electrical energy, the dissipation function and the influence of the non-conservative generalized force. Using the nonlinear oscillation theory to analyze single degree of freedom model, double degrees of freedom model and three degrees of freedom model. For single degree of freedom systems, with the increasing of plate distance, nature frequency of the system enlarge; with the increasing of plate area and linear inductance coefficient, nature frequency of the system decreases. The jump phenomenon is found in these systems. The non-linearity of resistance and inductance can change the nonlinear stiffness characteristic of response curves, and the results of the series method are in good agreement with the results of Runge-Kutta method. Based on the method of Lindstedt-Poincare, the stability of the system is analyzed, the values of damping and stiffness of spring can be effect by the value of plate distance. For double degree of freedom system, two coupled modals are all excited and vibrated; energy is transformed between two modals. The saturation phenomenon appears when the system meets double resonances conditionω2≈2ω1and ?≈ω2. In nonlinear inductance system, new vibrations are excited in one side of the response curves when the system meets double resonances conditionω2≈2ω1 and ?≈ω1. In double resonances conditionω2≈2ω1and ?≈ω2, with the changing of parameters of the system, amplitudes of certain modal can be changed. Changing the excitation, a similar saturation phenomenon can be found. When the parameters and resonant conditions are equal, nonlinear inductance can change the topology structure of response curves. Changing the nonlinear resistance and the nonlinear inductance can control peak value of response curves. Changing other parameters, amplitudes of response curves will change correspondingly, but the responses are different, one modal could be controlled by the other. For three degrees of freedom system, with the increasing of plate distance, the frequency of circuit systemω1 increases.
引文
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[2] 文成秀,赵常宽,闻邦椿. 分段线性振动机械关于外激励频率的分岔与混沌[J]. 东北大学学报,2001,22(2):200-202
[3] 李鹤,姚红良,闻邦椿等. 振动锤的周期运动稳定性与分岔[J]. 东北大学学报, 2004,25(1):66-69
[4] 李鸿光,闻邦椿. 一类具有可动边界的机械振动系统的混沌行为[J]. 东北大学学报,1999,20(4):376-379
[5] 罗跃纲,曾海泉,李振平,闻邦椿. 基础松动—碰摩转子系统的混沌特性研究[J]. 振动工程学报,2003,16(2):184-188
[6] 孙保苍,周传荣. 不平衡转子—轴承系统非线性行为研究[J]. 振动与冲击,2003,22(2):85-90
[7] 侯书军,陈予恕. 柔性机械臂的非线性振动分析[J]. 振动工程学报,1999,12(4):492-498
[8] 杨永锋,任兴民,秦卫阳. 松动-裂纹耦合故障转子系统的非线性响应[J]. 机械科学与技术,2005,24(8): 985-987
[9] 刘占生,崔颖,黄文虎等.转子弯扭耦合振动非线性动力学特性研究[J]. 机械工程, 2003,14(7):603-606
[10] 牛武.基于非线性电阻的 RLC 混沌电路实验分析[J]. 物理实验 ,2001,21(11):11-12
[11] 詹士昌,梁方束. RLC 电路非线性现象产生机制的研究[J]. 杭州师范学院学报,2002,1(1):31-38
[12] 孙海宁.一个非线性电路的梅尔尼科夫方法分析[J]. 上海大学学报,2003,9(3):232-237
[13] Ali Oksasoglu and Dimitry Vavriv. Interaction of Low-and High-Frequency Oscillations in a Nonlinear RLC Circuit[J] . Fundamental Theory and Applications. 1994,41(10):669-672
[14] S.K.Chakravarthy. Nonlinear Oscillations Due To Spurious Energisation of Transformers[J],IEE Proc-Electr.Power Appl.1998,145(6):585-592
[15] S.K.Chakravarthy, C.V.Nayar. Parallel (Quasi-Periodic) Ferroresonant Oscillations in Electrical Power Systems[J]. Fundamental Theory and Applications. 1995,42(9):530-534
[16] S.K.Chakravarthy, C.V.Nayar.Frequency-locked and Quasi-Periodic(QP) Oscillations in Power Systems[J]. IEEE Transations on Power Delivery, 1998,13(2):560-569
[17] C.Kieny. Application of the Bifurcation theory in studying and understangding the global behaviour of a ferroresonant electric power circuit[J]. IEEE Trans on PWRD.1991(2):866-872
[18] P.Bornard,V.Collet Billon ,C.Kieny. Protection of EHV power systems against ferroresonance[J]. CIGRE Paper,1990(8):34-103
[19] N.Germay, S.Mastero, and J.Vroman . Review of ferroresonance phenomena in high voltage power system and presentation of a voltage transformer model for predetermining them[J]. CIGRE Paper,1974,No22:33-18
[20] J.R.Marti and A.C.Soudack. Ferroresonance in power systems:Fundamental solutions[J]. IEE Proc.Pt-C,1991:321-329
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