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research-article

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

[+] Author and Article Information
Ren Ju

Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
767819680@qq.com

Wei Fan

Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
fanwei@umbc.edu

Weidong D. Zhu

Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
wzhu@umbc.edu

Jianliang L. Huang

Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, China
huangjl@mail.sysu.edu.cn

1Corresponding author.

ASME doi:10.1115/1.4036118 History: Received November 23, 2016; Revised February 09, 2017

Abstract

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure and complex two-dimensional integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude-frequency response surfaces of quasi-periodic responses and avoid non-convergent points at peaks, an incremental arc-length method for one time-scale is extended for quasi-periodic responses with two timescales. Two examples: a Duffing equation and a van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge-Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

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