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Research Papers

Nonlinear Dynamics of a Rotating Flexible Link

[+] Author and Article Information
B. Sandeep Reddy

Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: bsandeep@mecheng.iisc.ernet.in

Ashitava Ghosal

Professor
Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: asitava@mecheng.iisc.ernet.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 10, 2014; final manuscript received October 24, 2014; published online April 9, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(6), 061014 (Nov 01, 2015) (8 pages) Paper No: CND-14-1151; doi: 10.1115/1.4028929 History: Received June 10, 2014; Revised October 24, 2014; Online April 09, 2015

This paper deals with the study of the nonlinear dynamics of a rotating flexible link modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial differential equation of motion is discretized using a finite element approach to yield four nonlinear, nonautonomous and coupled ordinary differential equations (ODEs). The equations are nondimensionalized using two characteristic velocities—the speed of sound in the material and a velocity associated with the transverse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator.

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References

Hilborn, R. C., 2000, Chaos and Nonlinear Dynamics: An Introduction to Scientists and Engineers, Oxford University Press, New York.
Nikolai, A. M., and Sidorov, S. V., 2006, New Methods for Chaotic Dynamics (World Scientific Series on Nonlinear Science: Series A), World Scientific Publishing Company, Singapore.
Nayfeh, A. H., and Balachandran, B., 2004, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim, Germany.
Strogatz, S. H., 2007, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Westview Press, Boulder, CO.
Sun, J. Q., and Luo, A. C. J., 2006, Bifurcations and Chaos in Complex Systems (Edited Series on Advances in Nonlinear Science and Complexity), Vol. 1, Elsevier, NY.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences), Vol. 42, Springer-Verlag, NY.
Thompson, J. M. T., and Stewart, H. B., 2002, Nonlinear Dynamics and Chaos, 2nd ed., John Wiley & Sons, Chichester, UK.
Kovacic, I., and Brennan, M. J., 2011, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, 1st ed., John Wiley & Sons, Ltd., New York.
Burov, A. A., 1986, “On the Non-Existence of a Supplementary Integral in the Problem of a Heavy Two-Link Plane Pendulum,” J. Appl. Math. Mech., 50(1), pp. 123–125. [CrossRef]
Lankalapalli, S., and Ghosal, A., 1996, “Possible Chaotic Motion in a Feedback Controlled 2R Robot,” Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN, Apr. 24–26, N. Caplan and T. J. Tarn, eds., IEEE Press, NY, pp. 1241–1246.
Nakamura, Y., Suzuki, T., and Koinuma, M., 1997, “Nonlinear Behavior and Control of a Nonholonomic Free-Joint Manipulator,” IEEE Trans. Rob. Autom., 13(6), pp. 853–862. [CrossRef]
Mahout, V., Lopez, P., Carcasss, J. P., and Mira, C., 1993, “Complex Behaviours of a Two-Revolute Joints Robot: Harmonic, Subharmonic, Higher Harmonic, Fractional Harmonic, Chaotic Responses,” Proceedings of the IEEE Systems, Man & Cybernetics'93 Conference, Le Touquet, France, Oct. 17–20, pp. 201–205.
Verduzco, F., and Alvarez, J., 1999, “Bifurcation Analysis of a 2-DOF Robot Manipulator Driven by Constant Torques,” Int. J. Bifurcation Chaos, 9(4), pp. 617–627. [CrossRef]
Li, K. F., Li, L., and Chen, Y., 2002, “Chaotic Motion of a Planar 2-DOF Robot,” J. Sichuan Univ. Sci. Technol., 21(1), pp. 6–9.
Yin, Z., and Ge, X., 2011, “Chaotic Self-Motion of a Spatial Redundant Robotic Manipulator,” Res. J. Appl. Sci., Eng. Technol., 3(9), pp. 993–999.
Yang, X. D., and Chen, L. Q., 2005, “Bifurcation and Chaos of an Axially Accelerating Viscoelastic Beam,” Chaos, Solitons Fractals, 23(1), pp. 249–258. [CrossRef]
Yu, P., and Bi, Q., 1988, “Analysis of Nonlinear Dynamics and Bifurcations of a Double Pendulum,” J. Sound Vib., 217(4), pp. 691–736. [CrossRef]
Balachandran, B., and Nayfeh, A. H., 1992, “Cyclic Motions Near a Hopf Bifurcation of a Four-Dimensional System,” Nonlinear Dyn., 3(1), pp. 19–39. [CrossRef]
Nayfeh, A. H., 1993, Introduction to Perturbation Techniques, John Wiley and Sons Inc., New York.
Wahi, P., and Kumawat, V., 2011, “Nonlinear Stability Analysis of a Reduced Order Model of Nuclear Reactors: A Parametric Study Relevant to the Advanced Heavy Water Reactor,” Nucl. Eng. Des., 241(1), pp. 134–143. [CrossRef]
Chandra Shaker, M., and Ghosal, A., 2006, “Nonlinear Modeling of Flexible Link Manipulators Using Non-Dimensional Variables,” Trans. ASME J. Comput. Nonlinear Dyn., 1(2), pp. 123–134. [CrossRef]
Endurance Wind Power Ltd., “Endurancewindpower,” Last Accessed June 6, 2014, http://www.endurancewindpower.com/e3120.html
Nandakumar, K., 2009, “A study of Four Problems in Nonlinear Vibrations Via the Method of Multiple Scales,” Doctor of Philosophy thesis, IISc, Bangalore.
El-Bassiouny, A. F., 1999, “Response of a Three-Degree-of-Freedom System With Cubic Nonlinearities to Harmonic Excitation,” Appl. Math. Comput., 104(1), pp. 65–84. [CrossRef]
Cao, D. X., and Zhang, W., 2006, “Global Bifurcations and Chaotic Dynamics for a String-Beam Coupled System,” Chaos, Solitons Fractals, 37(3), pp. 858–875. [CrossRef]
Jinchen, J., and Yushu, C., 1999, “Bifurcation in a Parametrically Excited Two Degree of Freedom Nonlinear Oscillating System With 1:2 Internal Resonance,” Appl. Math. Mech., 20(4), pp. 350–359. [CrossRef]
Parker, T. S., and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, Springer Verlag Inc., NY.
Sandri, M., 1996, “Numerical Calculation of Lyapunov Exponents,” Math. J., 6(3), pp. 78–84.
Tsumoto, K., Ueta, T., Yoshinaga, T., and Kawakami, H., 2012, “Bifurcation Analyses of Nonlinear Dynamical Systems: From Theory to Numerical Computations,” Nonlinear Theory Appl. IEICE, 3(4), pp. 458–476. [CrossRef]
MATLAB, Version 8 (R2012b), 2012, The MathWorks, Inc., Natick, MA.
Monagan, M. B., Geddes, K. O., Heal, K. M., Labahn, G., Vorkoetter, S. M., McCarron, J., and DeMarco, P., 2012, Maple 14 Programming Guide, Maplesoft, Waterloo, ON, Canada.

Figures

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Fig. 1

Schematic of a rotating flexible beam and an ith element: (a) flexible rotating beam and (b) planar beam element

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Fig. 2

Poincaré maps for various Ug—undamped case: (a) Ug = 250, (b) Ug = 150, (c) Ug = 100, and (d) Ug = 50

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Fig. 3

Plot of Lyapunov exponents at Ug = 50—undamped case

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Fig. 4

Phase plots for various Ug—damped case: (a) Ug = 155, (b) Ug = 105, and (c) Ug = 50

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Fig. 5

Spectra of Lyapunov exponents at Ug = 105—damped case

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