电感非线性RLC电路弹簧耦合系统3次超谐共振研究

Study on 3RD Superharmonic Resonance of Coupled RLC Circuit and Spring System Considering Inductance Nonlinearity

研究电感非线性RLC电路弹簧耦合系统的非线性振动,应用拉格朗日—麦克斯韦方程,建立受简谐激励的具有电感非线性RLC电路弹簧耦合系统的数学模型。根据非线性振动的多尺度法,得到系统满足3次超谐共振条件的一次近似解以及对应的定常解。对其进行数值计算,分析系统参数对幅频响应曲线的影响。当系统3次超谐共振调谐值等于零时,幅频响应曲线的振幅最大。增大电压、极板面积和非线性电感系数,幅频响应曲线的振幅和共振区增大。增大极板间距、电阻和线性电感系数,幅频响应曲线的振幅和共振区减小。系统的固有频率随极板间距的增大而增大,随极板面积和线性电感系数的增大而减小。

In order to study on nonlinear vibration of coupled RLC circuit and spring system, a mathematical model of coupled RLC circuit and spring system considering inductance nonlinearity and harmonic excitation is established combined with Lagrange-Maxwell equation. Based on multiple scales method for nonlinear vibration analysis, the first approximation solutions and corresponding to steady state solutions of the 3 superharmonic resonance system are obtained. Numerical analysis results show that when detuning of the 3 superharmonic resonance system equals zero, amplitude of amplitude frequency response curve reaches maximum, the amplitude and resonant region of the system increase with increasing of voltage, plate area and nonlinear inductance coefficient, while they decrease with increasing plate distance, resistance and linear inductance coefficient. It has also been found the nature frequency of the system increases with the increasing plate distance but it decreases when the plate area and linear inductance coefficient increase.