A harmonic balance method combined with dimension reduction and FFT for nonlinear dynamic simulation

This paper proposes a harmonic balance method combined with dimension reduction procedure and fast Fourier transform (FFT) technique (DRF-HB) to efficiently analyze and rapidly apprehend the periodic responses of nonlinear systems. The dimension reduction procedure is employed to reduce the dimensionality of the Jacobian matrix required during the Newton-Raphson iterative procedure, while the FFT technique is utilized to expeditiously calculate the Jacobian matrix, facilitating the swift determination of the periodic responses of nonlinear systems. Computational results from two typical nonlinear systems underscore the superior performance of DRF-HB method in attaining the periodic responses for nonlinear systems. Specifically, in the case of the two-degree-of-freedom nonlinear oscillator, the DRF-HB method demonstrates an average computational efficiency 68-fold higher than that of the earlier HB-AFT (harmonic balance-alternating frequency/time domain) method and 182-fold higher than that of the RK4 (4th order Runge-Kutta) method. Similarly, in the scenario of the high-dimensional nonlinear Bernoulli beam, the DRF-HB method achieves significant computational efficiency advantage over both the HB-AFT and RK4 methods, and the superiority becoming increasingly pronounced with the augmentation of the system's degrees-of-freedom. The superior computational performance underscores the potential of the DRF-HB method presented here for application in the dynamic characteristic analysis of high-dimensional systems with complex nonlinearities in practical engineering context.