Research Papers

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

[+] Author and Article Information
R. Ju

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China

W. Fan, W. D. Zhu

Division of Dynamics and Control,
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

J. L. Huang

Department of Applied Mechanics
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 12(5), 051007 (Apr 18, 2017) (12 pages) Paper No: CND-16-1578; doi: 10.1115/1.4036118 History: Received November 23, 2016; Revised February 09, 2017

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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Fig. 1

Truncation of the 2D Fourier series to p=5

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Fig. 2

(a) Partial mapping strategy used for the cubic stiffness term and (b) the undersampling strategy in generating the 1D time-domain sequence

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Fig. 3

Schematic of the incremental arc-length method for one timescale

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Fig. 4

Periodic response of Duffing equation with f2=0: (a) time history and (b) phase diagram

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Fig. 5

Quasi-periodic response of Duffing equation with f2=0.004: (a) time history, (b) phase diagram, and (c) Poincare diagram

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Fig. 6

Convergence comparison between the original IHB method and modified two-timescale IHB method

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Fig. 7

Fourier spectrum of the quasi-periodic response of Duffing equation with ω1=(2/10 ) and ω2=1

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Fig. 8

Amplitude–frequency response curves of Duffing equation corresponding to ω1 (solid line) and ω2 (dashed line) with ω2=1

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Fig. 10

Quasi-periodic response of van der Pol equation with f2=0.3, ω1=1.2, and ω2=0.2: (a) time history and (b) phase diagram

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Fig. 11

Poincare diagrams with (a) p=3, (b) p=5, (c) p=7, and (d) p=9

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Fig. 9

Periodic response of van der Pol equation with ω1=1.0 and f2=0: (a) time history and (b) phase diagram

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Fig. 12

Amplitude–frequency response surfaces corresponding to six main frequency components of van der Pol equation: (a) ω1, (b) 2ω1, (c) ω1−ω2, (d) ω2, (e) ω1+ω2, and (f) 2ω2



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