Investigation of the source and improvement of non-physical solutions in high-order harmonic balance

The time derivatives in unsteady equations are eliminated by high-order harmonic balance HOHB method by expanding solutions into Fourier series containing several harmonics, which can reduce computational consumes of periodic unsteady problems significantly. In this paper, the source of non-physical solutions in HOHB method is investigated by Duffing oscillator. It is illustrated that the left and right terms of equations are not strictly equal because of the processing of nonlinear terms in the derivation process, which induces non-physical solutions. According to the characteristics of nonlinear term, sub-time solutions are extended. Besides, higher order harmonics of nonlinear term are also truncated. Thus, the left and right sides of HOHB equations are enforced strictly to be equal. It is manifested that not only non-physical solutions are eliminated, but also the numbers of required harmonics are reduced through the numerical simulation of Duffing oscillator equation. Comparing with results in references, the accuracy and simulation ability of improved method and classical harmonic balance method with same number of harmonics are almost equivalent, which proves the feasibility of the improved method. Lastly the improved method is applied in nonlinear aeroelastic system with cubic nonlinearity, which validates its engineering applicability. However, when there are excessive number of nonlinear terms in dynamic system, the computational consume of improved method will increase.