Approach To The Vibration Response Of Multi-Degree-of-Freedom Parametric System With Matrix Trigonometric Series

Parametric vibration caused by periodic time-varying parameters is widespread in engineering area.Vibration response prediction,stability analysis and vibration control strategy are three basic problems of parametric system vibration.This paper focuses on vibration response resolution,and develop an approximation method to solve free and forced response of multi-degree-of-freedom(MDOF)system,utilizing modulation feedback and harmonic balance.The parametric vibration equation can be transformed into an infinite set of homogeneous linear equations by matrix trigonometric series.Based on this homogeneous equation set,the characteristic equation can be obtained in the condition of its non-zero solution existing.Then,the principal oscillation frequencies can be computed.The approximative expression of the free response is a linear combination of the principal oscillation frequencies and the parametric excitation frequency,and that of forced response is a linear combination of the external excitation frequency and the parametric frequency.While,the combined frequencies may cause the combined harmonic resonance of the parametric system.In the homogeneous equation,normalized modal method is introduced to solve the coefficient matrix and the modal matrix,and obtain the general solution of the equation.The undetermined constants in the general solution are determined by the initial conditions,and the closed solution of the vibration response is given.Taking the unit impulse excitation into account,the closed solution of the unit impulse response is obtained in the same manner.In general,the main mode of parametric vibration is no longer the main mode of linear vibration due to the periodic time-varying parameters,whether in the real mode or the complex mode.At the same time,the essence of the coefficient matrix is the combined mode matrix,which corresponds to the mode of the combined harmonic resonance.To analyze the precision of the matrix trigonometric series approximation,an approach error function is defined to calculate the computational error in the proposed approach and make comparison with the Runge-Kutta method.When the number of terms of the approximation series is greater than a certain number,the error will be much smaller than that of Runge-Kutta method.The results show that the matrix trigonometric series method proposed in this paper provides a high-precision mathematical approximation method for the vibration response analysis of MDOF parametric system,with important application value in vibration theory and engineering.Finally,the coupled inverted pendulum system is established through the rotor wing of helicopter,and the prediction of its free response and forced response are completed with the proposed method.The result proves that the presented method is practicable and valid,and provides an effective calculation method for future research of parametric vibration problem.