Determination of high-order frequency response of nonlinear systems using the arc-length method

Abstract

While linear systems have been extensively studied in the past few decades, systems with strong nonlinearities are not as widely investigated. One of the reasons why these are less applied in engineering is due to the fact that solving nonlinear equations usually demands a steep increase in processing capacity and time when compared to their linear counterparts. As computers become more powerful, significantly more advanced nonlinear systems can be studied and analyzed, providing solutions that are more accurate for real-world applications. When it comes to frequency analysis, it is possible that nonlinear frequency response shows the phenomena of hardening or softening. In this case, the resonance peaks of the frequency response are tilted to the right or left, respectively, in comparison with linear frequency response. The consequences of these phenomena might prove essential to safety assessment of real-life structures as the resonance peak might greatly differ from those obtained in a linear analysis. In this study, the high-order frequency response of the Helmholtz-Duffing oscillator is analyzed in order to evaluate its influence on the system. The oscillator exhibits Duffing nonlinearities represented by cubic springs and Helmholtz nonlinearities represented by quadratic springs. The high-order harmonic balance method was used to determine the dynamic equation in the frequency domain. The nonlinearities were numerically integrated based on the coefficients of the Fourier series. Originally used to find the solution path of nonlinear static structural analyses, the arc-length method was adapted to determine the nonlinear frequency response

Full Paper