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 (2D) 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 nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and 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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September 2017
Research-Article
A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems
R. Ju,
R. Ju
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China
Search for other works by this author on:
W. Fan,
W. Fan
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
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W. D. Zhu,
W. D. Zhu
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
Search for other works by this author on:
J. L. Huang
J. L. Huang
Department of Applied Mechanics
and Engineering,
Sun Yat-Sen University,
Guangzhou 510275, China
and Engineering,
Sun Yat-Sen University,
Guangzhou 510275, China
Search for other works by this author on:
R. Ju
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China
W. Fan
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
W. D. Zhu
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
J. L. Huang
Department of Applied Mechanics
and Engineering,
Sun Yat-Sen University,
Guangzhou 510275, China
and Engineering,
Sun Yat-Sen University,
Guangzhou 510275, China
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 23, 2016; final manuscript received February 9, 2017; published online April 18, 2017. Assoc. Editor: Hiroshi Yabuno.
J. Comput. Nonlinear Dynam. Sep 2017, 12(5): 051007 (12 pages)
Published Online: April 18, 2017
Article history
Received:
November 23, 2016
Revised:
February 9, 2017
Citation
Ju, R., Fan, W., Zhu, W. D., and Huang, J. L. (April 18, 2017). "A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems." ASME. J. Comput. Nonlinear Dynam. September 2017; 12(5): 051007. https://doi.org/10.1115/1.4036118
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