Efficient explicit method of acquiring frequency-response curves for nonlinear vibrating systems

Abstract

The characteristics of nonlinear vibrating systems are universally investigated by accurately describing the amplitude of the response across a range of frequencies, which is referred to as the frequency-response curve. These curves are conventionally obtained by the arc-length continuation method, where a set of nonlinear algebraic equations is implicitly solved using Newton’s method at each point on the curve. This process poses a great challenge in terms of efficiency and robustness, because rough predictive values for Newton’s iterations (especially at turning points) lead to poor convergence when large step sizes are adopted to reduce computational cost. Therefore, efficient explicit methods tailored for accurately acquiring frequency-response curves are highly attractive. This work develops a new and robust method that allows for efficient explicit calculation of frequency-response curves. The key is to transform the previous process, where nonlinear algebraic equations are implicitly solved at each point on the response curve, into solving ordinary differential equations (ODEs) that are well-suited for explicit integrators. The explicit Runge-Kutta-Chebyshev integrator with an adaptive-step-size strategy is adopted to solve the ODEs. Its stability domain can be adaptively extended during the simulation, reaching a good tradeoff between stability and efficiency while simultaneously keeping local errors bounded. This capability ensures efficient explicit calculation of response curves with accessible large step sizes. Several numerical examples demonstrate the advantages and feasibility of the proposed method. The proposed method contributes to the efficient analysis of the frequency response of nonlinear systems, which is crucial in potential applications such as the agile design of structure parameters.