Numerical Analysis of Nonlinear Responses to Uncertain Excitations Modeled by Convex Sets

抄録

In this paper, the nonlinear responses to uncertain excitations modeled by convex sets are discussed.First, the nonlinear differential inclusion problem is converted into a class of optimization problems to solve the worst response.Next, it is simplified to a two-point boundary value problem of nonlinear ordinary differential equations.Finally, as an example, the Duffing equation is considered to obtain the numerical results for the two-point boundary value problem using the shooting method.The worst response and the worst excitation at a given time are discussed.The results show that the worst excitation corresponding to the worst response is a rectangular wave, and not a sine or a cosine wave for maximum-bound convex excitations.The worst excitation of a linear system is a uniform rectangular wave with natural period, while that of a nonlinear system is not a uniform one.This nonuniformity is related to the nonlinearity, the initial conditions and the excitation bound of the system.Results of further analysis are consistent with the results of a classical nonlinear theory.