[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Vibration Testing and System Dynamics
ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

A Period-1Motion to Chaos in a Periodically Forced, Damped, Double-Pendulum

Journal of Vcibration Testing and System Dynamics 3(3) (2019) 259--280 | DOI:10.5890/JVTSD.2019.09.002

Albert C.J. Luo, Chuan Guo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

Download Full Text PDF

 

Abstract

In this paper, periodic motions in a periodically forced, damped double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is developed from discretization of the corresponding differential equation. From the mapping structures, period-1 to period-4 motions are obtained and the corresponding stability and bifurcation analysis of the periodic motions are completed through eigenvalue analysis. Using the finite Fourier series, nonlinear frequency-amplitude characteristics of period-1 to period-4 motions are presented. Numerical simulations of period motions in the double pendulum is completed, and the initial conditions are obtained from the analytical predictions. The harmonic amplitude spectrums are also presented for showing harmonic term effects on periodic motions.

References

  1. [1]  Ziegler, H. (1952), Die Stabiltatskriterien der Elastomechanik, Ing. Arch. 20(1), 49-56.
  2. [2]  Herrmann, G. and Jong, I.C. (1966), On non-conservative stability problems of elastic systems with slight damping, Journal of Applied Mechanics, 33, 125-133.
  3. [3]  Roorda, J. and Nemat-Nasser, S. (1967), An energy method for stability analysis of nonlinear, non- conservative systems, American Institute of Aeronautics and Astronautics, 5, 1262-1268.
  4. [4]  Scheidl, R., Troger, H. and Zeman, K. (1983), Coupled flutter and divergence bifurcation of a double pendu- lum, International Journal of Non-Linear Mechanics, 19, 163-176.
  5. [5]  Jin, J.-D. and Matsuzaki, Y. (1988), Bifurcations in a two-degree-of-freedom elastic system with follower forces, Journal of Sound and Vibration, 126, 265-277.
  6. [6]  Jin, J.-D. and Matsuzaki, Y. (1992), Bifurcation analysis of double pendulum with a follower force, Journal of Sound and Vibration, 154, 191-204.
  7. [7]  Skeldon, A.C. and Mullin, T. (1992), Mode interaction in a double pendulum, Physics Letter A, 166, 224-229.
  8. [8]  Skeldon, A.C. (1994), Dynamics of a parametrically excited double pendulum, Physica D, 75, 541-558.
  9. [9]  Zhou Z. and Whiteman, C. (1996), Motions of a double pendulum, Nonlinear Analysis, Theory, Methods, & Applications, 26(7), 1177-1191.
  10. [10]  Shinbrot, A., Grebogi, C., Wisdon, J., Yorke, and J.A. (1992), Chaos in a double pendulum, American Journal of Physics, 60(6), 491-499.
  11. [11]  Levien, R.B. and Tan, S.M. (1993), Double pendulum: An experiment in Chaos, American Journal of Physics, 61(11), 491-499.[12] Stochowiak, T. and Okada, T. (2006), A numerical analysis of chaos in the double pendulum, Chaos, Solitons and Fractals, 29, 417-422.
  12. [12]  Stachowiak, T. and Szuminski, W. (2015), Non-integrability of restricted double pendula, Physics Letters A, 379, 3017-3024.
  13. [13]  Bhattacharjee, J.K. and Roy, J. (2013), Oscillators: Old and new perspectives, Nonlinear Dynamics, 73, 993-1004.
  14. [14]  Awrejcewicz, J., Kudra, G., and Lamarque, C.-H, (2003), Dynamics investigation of three coupled rods with a horizontal barrier, Meccanica, 38(6), 687-698.
  15. [15]  Awrejcewicz, J., Kudra, G., and Lamarque, C.-H. (2004), Investigation of triple pendulum with impacts using fundamental solution matrices, International Journal of Bifurcation and Chaos, 14(12), 4191-4213.
  16. [16]  De Paula, A.S., Savi, M.A., and Pereira-Pinto, F.H.I. (2006), Chaos and transient chaos in an experimental nonlinear pendulum, Journal of Sound and Vibration, 294, 585-595.
  17. [17]  J. Awrejcewicz, D. Sendkowski and M. Kazmierczak, 2006, Geometrical approach to the swinging pendulum dynamics, Computers & Structures, 84(24-25), 1577-1583.
  18. [18]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon.
  19. [19]  Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25(3), Article No: 1550044 (62 pages).
  20. [20]  Luo, A.C.J. (2015), Discretization and Implicit Mapping Dynamics, Beijing/Dordrecht: HEP/Springer.
  21. [21]  Luo, A C. J. and Guo, Y, (2016), Periodic motions to chaos in pendulum, International Journal of Bifurcation and Chaos, 26(9), Article No: 1650159 (64 pages).
  22. [22]  Guo, Y. and Luo, A.C.J., (2017), Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3). 551-569.
  23. [23]  Guo, Y. and Luo, A.C.J. (2017), Complete bifurcation trees of a parametrically driven pendulum, Journal of Vibration Testing and System Dynamics, 1(2), 93-134.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates