0
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
Your Session has timed out. Please sign back in to continue.

References

Nakamura, S., 1991, Applied Numerical Methods With Software, Prentice Hall, New Jersey.
Zingg, D. W., and Chisholm, T. T., 1999, “Runge–Kutta Methods for Linear Ordinary Differential Equations,” Appl. Numer. Math., 31, pp. 227–238. [CrossRef]
Abukhaled, M. I., and Allen, E. J., 1998, “A Class of Second-Order Runge–Kutta Methods for Numerical Solution of Stochastic Differential Equations,” Stochastic Anal. App., 16, pp. 977–992. [CrossRef]
Zhang, S., and Li, J., 2011, “Explicit Numerical Methods for Solving Stiff Dynamical Systems,” ASME J. Comput. Nonlinear Dyn., 6(4), p. 041008. [CrossRef]
Wang, J., Rodriguez, J., and Keribar, R., 2010, “Integration of Flexible Multibody Systems Using Radau IIA Algorithms,” ASME J. Comput. Nonlinear Dyn., 5(4), p. 041008. [CrossRef]
Kuinian, L., and Antony, P., 2009, “A High Precision Direct Integration Scheme for Nonlinear Dynamic Systems,” ASME J. Comput. Nonlinear Dyn., 4(4), p. 041008. [CrossRef]
Shampine, L. F., and Watts, H. A., 1971, “Comparing Error Estimators for Runge–Kutta Methods,” Math. Comput., 25(115), pp. 445–455. [CrossRef]
Hull, T. E., Enright, B. M., Fellen, B. M., and Sedgwick, A. E., 1972, “Comparing Numerical Methods for Ordinary Differential Equations,” SIAM J. Numer. Anal., 9(4), pp. 603–637. [CrossRef]
Enright, W. H., and Hayes, W. B., 2007, “Robust and Reliable Defect Control for Runge–Kutta Methods,” ACM Trans. Math. Software, 33(1), pp. 1–19. [CrossRef]
Dai, L., and Singh, M. C., 1997, “An Analytical and Numerical Method for Solving Linear and Nonlinear Vibration Problems,” Int. J. Solid Struct., 34, pp. 2709–2731. [CrossRef]
Dai, L., and Singh, M. C., 2003, “A New Approach With Piecewise-Constant Arguments to Approximate and Numerical Solutions of Oscillatory Problems,” J. Sound Vib., 263(3), pp. 535–548. [CrossRef]
Abramson, H. N., and Silverman, S., 1966, “The Dynamic Behavior of Liquids in Moving Containers,” Paper No. NASA SP-106.
Dodge, F. T., 2000, “ The New Dynamic Behavior of Liquids in Moving Containers,” Southwest Research Institute, San Antonio, TX.
Berlot, R. R., 1959, “Production of Rotation in a Confined Liquid Through Translational Motion of the Boundaries,” ASME J. Appl. Mech., 26, pp. 513–516. [CrossRef]
Dai, L., and Singh, M. C., 1997b, “Diagnosis of Periodic and Chaotic Responses in Vibratory Systems,” J. Acoust. Soc. Am., 102(6), pp. 3361–3371. [CrossRef]
Dai, L., Xu, L., and Han, Q., 2006, “Semi-Analytical and Numerical Solutions of Multi-Degree-of-Freedom Nonlinear Oscillation Systems With Linear Coupling,” Commun. Nonlinear Sci. Numer. Sci., 11, pp. 831–844. [CrossRef]
Dai, L., and Wang, G., 2007, “Implementation of Periodicity-Ratio in Analyzing Nonlinear Dynamic Systems—A Comparison With Lyapunov-Exponent,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011006. [CrossRef]
Dai, L., and Wang, X., 2013, “Diagnosis of Nonlinear Oscillatory Behavior of a Fluttering Plate With a Periodicity Ratio Approach,” Nonlinear Eng., 1(3–4), pp. 67–76. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
Brennan, M. J., Kovacic, I., Carrella, A., and Waters, T. P., 2008, “On the Jump-up and Jump-Down Frequencies of the Duffing Oscillator,” J. Sound Vib., 318, pp. 1250–1261. [CrossRef]
Junyi, C., Chengbin, M., Hang, X., and Zhuangde, J., “Nonlinear Dynamics of Duffing System With Fractional Order Damping,” ASME: J. Comput. Nonlinear Dyn., 5(4), p. 041012. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Conical pendulum model for rotary sloshing

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 10

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In