ENG  Vol.2 No.12 , December 2010
Classical and Fractional-Order Analysis of the Free and Forced Double Pendulum
ABSTRACT
This paper presents the fractional-order dynamics of the double pendulum by means of fractional-order modeling. Equations of motion have been derived for cases with and without external forcing. Generalized force terms have been obtained for five different cases of forcing. Both integer and fractional-order analysis have been carried out. Phase diagrams have been plotted to visualize the effect of fractional order approach. The originality of this work arises from the fact that the double pendulum has been modeled with the fractional dynamics approach. The governing equations of motion of the system have been obtained through fractional variational principles.
Cite this paper
nullE. Anli and I. Ozkol, "Classical and Fractional-Order Analysis of the Free and Forced Double Pendulum," Engineering, Vol. 2 No. 12, 2010, pp. 935-949. doi: 10.4236/eng.2010.212118.
References
[1]   A. Ohlhoff and P. H. Richter, “Forces in the Double Pendulum, Zeitschrift für Angewandte Mathematik und Mechanik,” Journal of Applied Mathematics and Mechanics, Vol. 80, No. 8, 2000, pp. 517-534.

[2]   A. A. Martynyuk and N. V. Nikitina, “The Theory of Motion of a Double Mathematical Pendulum,” International Applied Mechanics, Vol. 36, No. 9, 2000, pp. 1252-1258.

[3]   U. Mackenroth, “Robust Stabilization of a Double Pendulum with an Elastic Joint and Uncertain High-Frequency Dynamics,” 16th Mediterrenean Conference on Control and Automation, France, 2008, pp. 1490-1495.

[4]   D. Sado and K. Gajos, “Note on Chaos in Three Degree of Freedom Dynamical System with Double Pendulum,” Meccanica, Vol. 38, No. 6, 2003, pp. 719-729.

[5]   M. P. Hanias, “Chaotic Behavior of an Electrical Analog to the Mechanical Double Pendulum,” Journal of Engi-neering Science and Technology Review, Vol. 1, No. 1, 2008, pp. 33-37.

[6]   T. Stachowiak and T. Okada, “A Numerical Analysis of Chaos in the Double Pendulum,” Chaos, Solitons and Fractals, Vol. 29, No. 2, 2006, pp. 417-422.

[7]   L. Sheu, H. Chen, J. Chen and L. Tam, “Chaotic Dynamics of the Fractionally Damped Duffing Equation,” Chaos, Solitons and Fractals, Vol. 32, No. 4, 2007, pp. 1459-1468.

[8]   X. Gao, J. Yu, “Chaos in the Fractional Order Periodically Forced Complex Duffing’s Oscillators,” Chaos, Solitons and Fractals, Vol. 24, No. 4, 2005, pp. 1097-1104.

[9]   R. S. Barbosa, J. A. T. Machado, I. M. Ferreiraa and J. K. Tar, “Dynamics of the Fractional-Order Van der Pol Oscillator,” Second IEEE International Conference on Computational Cybernetics, 2004, pp. 373-378.

[10]   M. Seredynska, “Nonlinear Differential Equations with Fractional Damping with Applications to the 1dof and 2dof Pendulum,” Acta Mechanica, Vol. 176, No. 3-4, 2005, pp. 169-183.

[11]   S. I. Muslih and D. Baleanu, “Fractional Euler-Lagrange Equations of Motion in Fractional Space,” Journal of Vibration and Control, Vol. 13, 2007, No. 9-10, pp. 1209-1216.

[12]   I. Podlubny, “Fractional Differential Equati Equations,” Academic Press, San Diego, 1999.

[13]   A. A. Kilbas, H. M. Srivastava and J. J Trujillo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.

[14]   L. Munteanu, T. Badea and V. Chiroiu, “Linear Equivalence Method for the Analysis of the Double Pendulum’s Motion,” Complexity International, Vol. 9, 2002, pp. 1-17.

 
 
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