Event-driven Gauss quadrature and stability analysis for fast alternating frequency–time harmonic balance of non-smooth systems

Abstract

The alternating frequency–time harmonic balance method (AFT-HB) has been widely recognized as a practical tool for periodic solution of nonlinear systems due to its ability to deal with general smooth and non-smooth nonlinear problems through discrete Fourier transform (DFT). However, when the nonlinear terms are of some low smoothness (e.g., discontinuous), a large number of sampling points are required to achieve accurate Fourier transform and this significantly increases the computation cost. In addition, though Hill’s method is commonly used in conjunction with the HB to predict the solution stability, wrong stability results may be obtained especially for non-smooth systems. Aiming to circumvent the above limitations of the AFT-HB for non-smooth systems, an event-driven Gauss quadrature scheme is firstly proposed instead of the DFT to further accelerate the Fourier integral computation of non-smooth terms. The idea behind is that non-smoothness generally occurs at the transition points (or events) and by dividing the whole time period into sub-intervals with these transition points, the Gauss quadrature on every sub-interval is well applicable. It is proved that for an H-order harmonic solution of non-smooth systems, the event-driven Gauss quadrature requires only linear-scale (O(H)) sampling points to achieve exponential-order integration accuracy of non-smooth terms, while much more (O(Hs) with generally s ≥2) sampling points are demanded for the DFT to get the desired convergence rate. Next, with the transition points, the monodromy matrix can also be easily calculated through simple integration over every sub-interval and possible jumps at transition points, with which the stability is directly predicted by the Floquet theory. Numerical examples are studied to demonstrate the remarkable efficiency, accuracy and stability prediction of the proposed event-driven Gauss quadrature and stability analysis scheme for harmonic balance solution of non-smooth systems.