Harmonic Balance Methods: A Review and Recent Developments

The harmonic balance (HB) method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems. However, it is confined to low-order approximations due to complex symbolic operations. Many variants have been developed to improve the HB method, among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation. So far, there are two problems remaining to be addressed. i) A dozen of different versions of HB-like methods, in frequency domain or time domain or in hybrid, have been developed; unfortunately, misclassification pervades among them due to the unclear borderlines of different methods. ii) The time domain HB-like methods suffer from non-physical solutions, which have been shown to be caused by aliasing (mixture of the high-order into the low-order harmonics). Although a series of dealiasing techniques have been developed over the past two decades, the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community. This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences. In particular, the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.