Analyzing Nonlinear Structures with Random Excitation Using Integral Quadratic Constraints

Modeling the response of nonlinear structures due to random excitation is crucial for the design of mechanical systems, including the estimation of loading on mechanical joints and the fatigue life of nonlinear components. This chapter presents a method for bounding the maximum variance of the output response of a nonlinear system under random excitation of known power spectral density. The proposed approach leverages integral quadratic constraints (IQCs) that enclose the relationship between inputs and outputs of the nonlinearity sufficiently for analysis. While IQCs have traditionally been employed in robust control to analyze stability and performance, recent advancements have extended its applications to analyzing optimization algorithm rate of convergence and stability of transitional flows. In this chapter, we explore an optimization-based algorithm that harnesses different IQCs to bound the nonlinearities in the system. To validate the efficacy of the proposed algorithm, we apply it to the analysis of the Duffing equation, a well-known nonlinear oscillator. Results demonstrate the effectiveness of the algorithm in bounding the maximum variance of the system’s response and its potential for application in the design and analysis of nonlinear structures subject to random vibration.