Abstract
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.
