The theory of calculating the response of forced vibration in time-varying systems is still imperfect, and the numerical solution method cannot accurately reflect the analytical nature of the vibration response. The Legendre Polynomial Approximation Method (LPAM) can be used to obtain a continuous approximate function solution of the response of the time-varying system, but the computational efficiency of this method is low. In this paper, the time-domain response solving theory of time-invariant systems is transplanted to time-varying systems. An improved Legendre series algorithm based on Duhamel integral is proposed based on the superposition principle of linear systems. The Legendre polynomial approximation of Dirac function and Duhamel integral are used to solve the response of time-varying systems. A system of first-order non-homogeneous coefficient ordinary differential equations is used to illustrate the effectiveness of the improved algorithm. Simulation examples of a single-degree-of-freedom system with uniform linear and nonlinear variation of both stiffness and damping are designed. The displacement response of the system under transient excitation and simple harmonic excitation obtained by the methods are compared with the calculation results of the fourth-order Runge-Kutta numerical method, which illustrates the divergence of the improved algorithm and the enhancement of calculating speed.
Key words
time-varying system /
Legendre polynomial approximation /
linear superposition /
time domain response
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