調制白噪聲激勵下的單自由度強非線性系統的近似瞬態響應概率密度

Abstract

研究調制白噪聲激勵下，包含弱非線性阻尼及強非線性剛度的單自由度系統的近似瞬態響應概率密度．應用基於廣義諧和函數的隨機平均法，導出關於幅值瞬態概率密度的平均Fokker－Planck－Kolmogorov 方程．該方程的解可近似表示為適當的正交基函數的級數和，其中系數是隨時間變化的．應用Galerkin方法，這些系數可由一階線性微分方程組解得，從而可得幅值響應的瞬態概率密度的半解析表達式及系統狀態響應的瞬態概率密度和幅值的統計矩．以受調制白噪聲激勵的van der Pol-Duffing振子為例驗證其求解過程，並討論了線性阻尼系數及非線性剛度系數等系統參數對系統響應的影響．

The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to dulated white noise excitation were studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude was derived.The solution of the equation was approximated by a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients.According to the Galerkin method,the time-dependent coefficients can be solved from a set of first-order linear differential equations.Then the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained.A van der Pol-Duffing oscillator subject to modulated white noise was given as an example to illustrate the proposed procedures.The effects of the system parameters,such as linear damping coefficient and nonlinear stiffness coefficient,on the system response were discussed.