On the dynamic response of non-linear systems with parameter uncertainties

This paper presents a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties. Consideration is given to systems with polynomial non-linearity subjected to deterministic excitation. The uncertain parameters are modeled as time-independent random variables. The set of orthogonal polynomials associated with the probability density function is used as the solution basis, and the response variables are expanded in terms of a finite sum of these polynomials. A set of deterministic non-linear differential equations is derived using the weighted residual method. The discrete-time solutions to the equation set are evaluated numerically using a step-by-step time-integration scheme and the response statistics are determined. Application of the proposed method is illustrated through the analysis of non-linear single-degree-of-freedom structural systems exhibiting uncertain stiffnesses. Both hardening and softening stiffness characteristics are examined. The accuracy of the results is validated by direct numerical integration.

[1]  P. C. Jennings.  Periodic Response of a General Yielding Structure , 1964 .

[2]  Hector A. Jensen,et al.  Response variability in structural dynamics , 1991 .

[3]  R. Ibrahim.  Structural Dynamics with Parameter Uncertainties , 1987 .

[4]  Wing Kam Liu,et al.  Probabilistic finite elements for nonlinear structural dynamics , 1986 .

[5]  Stochastic Finite Element Analysis for the Transport of Trichloroethylene Vapors , 1994 .

[6]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[7]  R. Ghanem,et al.  Stochastic Finite Element Expansion for Random Media , 1989 .

[8]  Haym Benaroya,et al.  Finite Element Methods in Probabilistic Structural Analysis: A Selective Review , 1988 .

[9]  R. Ghanem,et al.  Polynomial Chaos in Stochastic Finite Elements , 1990 .

[10]  Pol D. Spanos,et al.  A stochastic Galerkin expansion for nonlinear random vibration analysis , 1993 .

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  Wing Kam Liu,et al.  Random field finite elements , 1986 .

[13]  Pol D. Spanos,et al.  Galerkin Sampling Method for Stochastic Mechanics Problems , 1994 .

[14]  Rajendra Singh,et al.  Frequency response of linear systems with parameter uncertainties , 1993 .

[15]  C. E. Brenner,et al.  Stochastic response of uncertain systems , 1992, Archive of Applied Mechanics.

[16]  Ted Belytschko,et al.  Transient probabilistic systems , 1988 .

[17]  W. W. Soroka,et al.  Impulse response of a dynamic system with statistical properties , 1973 .

[18]  Masanobu Shinozuka,et al.  Monte Carlo solution of structural dynamics , 1972 .