Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators

Chein-Shan Liu, Chung-Lun Kuo and Chih-Wen Chang ()
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Chein-Shan Liu: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Chung-Lun Kuo: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Chih-Wen Chang: Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan

Mathematics, 2025, vol. 13, issue 1, 1-29

Abstract: To solve the nonlinear vibration problems of second- and third-order nonlinear oscillators, a modified harmonic balance method (HBM) is developed in this paper. In the linearized technique, we decompose the nonlinear terms of the governing equation on two sides via a constant weight factor; then, they are linearized with respect to a fundamental periodic function satisfying the specified initial conditions. The periodicity of nonlinear oscillation is reflected in the Mathieu-type ordinary differential equation (ODE) with periodic forcing terms appeared on the right-hand side. In each iteration of the linearized harmonic balance method (LHBM), we simply solve a small-size linear system to determine the Fourier coefficients and the vibration frequency. Because the algebraic manipulations required for the LHBM are quite saving, it converges fast with a few iterations. For the Duffing oscillator, a frequency–amplitude formula is derived in closed form, which improves the accuracy of frequency by about three orders compared to that obtained by the Hamiltonian-based frequency–amplitude formula. To reduce the computational cost of analytically solving the third-order nonlinear jerk equations, the LHBM invoking a linearization technique results in the Mathieu-type ODE again, of which the harmonic balance equations are easily deduced and solved. The LHBM can achieve quite accurate periodic solutions, whose accuracy is assessed by using the fourth-order Runge–Kutta numerical integration method. The optimal value of weight factor is chosen such that the absolute error of the periodic solution is minimized.

Keywords: strongly nonlinear oscillators; analytic periodic solution; harmonic balance method; jerk equation; Duffing equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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