A modified incremental harmonic balance method based on the fast Fourier transform and Broyden’s method
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A modified incremental harmonic balance (IHB) method is introduced, where Fourier coefficients of the residual of nonlinear algebraic equations are approximated by the fast Fourier transform, and the Jacobian of the nonlinear algebraic equations is approximated by Broyden’s method. The modified IHB method is first illustrated by solving Duffing’s equation, whose solutions from the modified IHB method are in excellent agreement with that from Runge–Kutta method. The calculation time for the modified IHB method is almost two orders of magnitude less than that for the original IHB method. By showing that the Jacobian of the path function in Broyden’s method is invariant, the arc-length method with the path-following technique is used to calculate an amplitude–frequency response curve of Duffing’s equation. Bifurcations of Mathieu–Duffing equation are also studied using the modified IHB method.
KeywordsHarmonic balance Nonlinear system Fast Fourier transform Broyden’s method Jacobian Arc-length method Bifurcation
The authors would like to thank the support from the National Science Foundation through Grant No. CMMI-1000830. They would also like to thank Jianliang Huang for some valuable discussion on bifurcations of Mathieu–Duffing equation.
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