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Research Papers

Accuracy and Reliability of Piecewise-Constant Method in Studying the Responses of Nonlinear Dynamic Systems

[+] Author and Article Information
Liming Dai

Mem. ASME
Fellow ASME
Sino-Canada Research Center for
Noise and Vibration Control,
Xiamen University of Technology
& University of Regina,
University of Regina,
Regina, SK S4S 0A2, Canada
Industrial Systems Engineering,
University of Regina,
Regina, SK S4S 0A2, Canada
e-mail: liming.dai@uregina.ca

Xiaojie Wang

Industrial Systems Engineering,
University of Regina,
Regina, SK S4S 0A2, Canada
e-mail: xiaojie.wang115@gmail.com

Changping Chen

Sino-Canada Research Center for
Noise and Vibration Control,
Xiamen University of Technology
& University of Regina,
Xiamen University of Technology,
Xiamen 361024, China
e-mail: cpchen@163.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear dynamics. Manuscript received October 24, 2013; final manuscript received February 17, 2014; published online January 12, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(2), 021009 (Mar 01, 2015) (10 pages) Paper No: CND-13-1258; doi: 10.1115/1.4026895 History: Received October 24, 2013; Revised February 17, 2014; Online January 12, 2015

Accuracy and reliability of the numerical simulations for nonlinear dynamical systems are investigated with fourth-order Runge–Kutta method and a newly developed piecewise-constant (P-T) method. Nonlinear dynamic systems with external excitations are studied and compared with the two numerical approaches. Semianalytical solutions for the dynamic systems are developed by the P-T approach. With employment of a periodicity-ratio (PR) method, the regions of regular and irregular motions are determined and graphically presented corresponding to the system parameters, for the comparison of accuracy and reliability of the numerical methods considered. Central processing unit (CPU) time executed in the numerical calculations with the two numerical methods are quantitatively investigated and compared under the same computational conditions. Due to its inherent drawbacks, as found in the research, Runge–Kutta method may cause information missing and lead to incorrect conclusions in comparing with the P-T method.

Copyright © 2015 by ASME
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References

Figures

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Fig. 1

Conical pendulum model for rotary sloshing

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Fig. 2

Comparison of the numerical results (line: P-T method; dot: RK4 method), time step 0.01, α0 = 0.01,α·0 = 0.0, g = 8.0, ω = 2.0, F = 0.01, L = 0.5

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Fig. 3

Comparison of numerical results generated by P-T method and Runge–Kutta method with different time steps, α0 = 0.01,α·0 = 0.0, g = 8.0, ω = 2.0, F = 0.01, L = 0.5

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Fig. 4

Regular-irregular region diagram of P-T method (time step 0.001, L = 0.5, α0 = 0.01,α·0 = 0.0)

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Fig. 5

Wave curve of a regular motion at α0 = 0.01,α·0 = 0.0, g = 8.0, ω = 2.0, L = 0.5, time step: 0.001 s

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Fig. 6

Phase diagram of a regular motion at α0 = 0.01,α·0 = 0.0, g = 8.0, ω = 2.0, L = 0.5, time step: 0.001 s

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Fig. 7

Wave curve of an irregular motion at α0 = 0.01,α·0 = 0.0, g = 7.8,ω = 4.0, L = 0.5, time step: 0.001 s

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Fig. 8

Phase diagram of an irregular motion at α0 = 0.01,α·0 = 0.0, g = 7.8,ω = 4.0, L = 0.5, time step: 0.001 s

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Fig. 9

Regular-irregular region diagram of P-T method. Time step 0.1 s, L = 0.5, α0 = 0.01,α·0 = 0.0.

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Fig. 10

Regular-irregular region diagram of Runge–Kutta method. Time step 0.1, L = 0.5, α0 = 0.01,α·0 = 0.0.

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Fig. 11

Regular-irregular region diagram of the fourth order P-T method. Time step 0.00314 s, k = 1.0, ω = 1.0, x(0) = -2.0,x·(0) = 0.0.

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Fig. 12

Regular-irregular region diagram of the fourth order Runge–Kutta method. Time step 0.00314 s, k = 1.0, ω = 1.0,x(0) = -2.0, x·(0) = 0.0.

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Fig. 13

Phase diagram for a chaotic case at k = 1.0,ω = 1.0,A = 8.1,C = 0.15, by P-T method with time step 0.00314 s and initial conditions x(0) = -2.0,x·(0) = 0.0

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Fig. 14

Poincaré map for a chaotic case at k = 1.0,ω = 1.0,A = 8.1,C = 0.15 by P-T method with time step 0.00314 s and initial conditions x(0) = -2.0,x·(0) = 0.0

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Fig. 15

Wave curve for a chaotic case at k = 1.0,ω = 1.0,A = 8.1,C = 0.15 by P-T method, at steady state, with time step 0.00314 s and initial conditions x(0) = -2.0,x·(0) = 0.0

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Fig. 16

Wave curve for a chaotic case at k = 1.0,ω = 1.0,A = 8.1,C = 0.15 by Runge–Kutta method, at steady state, with time step 0.00314 s and initial conditions x(0) = -2.0,x·(0) = 0.0

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Fig. 17

Wave curve for a chaotic case at k = 1.0,ω = 1.0,A = 8.1,C = 0.15 by Runge–Kutta method, at steady state, with initial conditions x(0) = -2.0,x·(0) = 0.0

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