RLC电路弹簧耦合系统的级数解

THE SERIES SOLUTIONS TO COUPLED RLC CIRCUIT AND SPRING SYSTEM

为了研究RLC电路弹簧耦合系统的非线性振动,用统一的能量法考虑机电耦合系统的电场能、磁场能和机械能,应用拉格朗日-麦克斯韦方程建立起一个受到简谐激励的RLC电路弹簧耦合系统的数学模型,该机电耦合系统具有平方非线性。根据线性振动理论对系统运动微分方程组进行分析,得到了一个受简谐激励的M ath ieu方程,通过积分变换,得到了M ath ieu方程的级数形式解。分别用龙格库塔法和级数法计算了在无外激励的情况下,有阻尼和无阻尼时系统分别对应的时间响应,通过M atlab软件进行模拟分析,发现二者得到的响应曲线吻合,证明了级数法对分析类似系统是个很有效的手段。

In order to study nonlinear vibration of a coupled RLC circuit and spring sytem,its energies including electric field energy,magnetic field energy and mechanical energy are obtained.Applying Lagrange-Maxwell equation,a mathematical model of the coupled RLC circuit and spring system subject to harmonic excitation is established.The coupled RLC circuit and spring system has square nonlinearity.A typical Mathieu equation subject to harmonic excitation is acquired by means of the theory of linear vibration.By using integral transformation,the series solutions to the Mathieu equation are obtained.The dynamic responses of the system with damping and without damping in the case of no external excitation are computed by using Runge-Kutta technique and series solutions in MATLAB software.The numerical results of the series solution are in good agreement with those of Runge-Kutta method.The method of series is available to other similar system.