Primary resonance of coupled RLC circuit and spring system with internal resonance considering inductance nonlinearity

With the development of electronic technology, more and more coupled systems of electricity-magnetism-mechanism are used in industrial field. RLC (Resistance-Inductance-Capacitance) circuit and spring system can simulate these coupled systems. In order to analyze the vibration of the RLC circuit and spring system, a mathematical model considering inductance nonlinearity and external harmonic excitation is established by the Lagrange-Maxwell equation. Based on the nonlinear vibration theory, the vibration of the system is analyzed; the first approximation solutions and corresponding to steady state solutions of the resonance system are obtained by multiple scales method. Numerical results show that the two coupled modals are excited and vibrated when the system meets the double resonances condition. Energy transforming between two modals is found. With the increasing of voltage, amplitudes of the response curves increase. With the increasing of resistance, amplitudes of the response curves decrease. With the increasing of nonlinear coefficient of inductance, amplitudes of the response curves increase first, and then decrease. With the increasing of damping, amplitude of electric charge increases and amplitude of plate decreases. It has also been found nonlinear inductance can excite new vibrations in one side of the response curves.