Dynamic analysis of rotating pendulum by Hamiltonian approach. (English) Zbl 1391.70012

Summary: A conservative system always admits Hamiltonian invariant, which is kept unchanged during oscillation. This property is used to obtain the approximate frequency-amplitude relationship of the governing equation with sinusoidal nonlinearity. Here, we applied Hamiltonian approach to obtain natural frequency of the nonlinear rotating pendulum. The problem has been solved without series approximation and other restrictive assumptions. Numerical simulations are then conducted to prove the efficiency of the suggested technique.


70E17 Motion of a rigid body with a fixed point
70H99 Hamiltonian and Lagrangian mechanics
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[1] A. M. M. Abdel-Rahman, “The simple pendulum in a rotating frame,” American Journal of Physics, vol. 51, no. 8, pp. 721-724, 1983.
[2] S. J. Liao and A. T. Chwang, “Application of homotopy Analysis Method in nonlinear oscillations,” Journal of Applied Mechanics, Transactions ASME, vol. 65, no. 4, pp. 914-922, 1998.
[3] S. T. Wu, “Active pendulum vibration absorbers with a spinning support,” Journal of Sound and Vibration, vol. 323, no. 1-2, pp. 1-16, 2009.
[4] A. S. Alsuwaiyan and S. W. Shaw, “Performance and dynamics stability of general-path centrifugal pendulum vibration absorbers,” Journal of Sound and Vibration, vol. 252, no. 5, pp. 791-815, 2002.
[5] O. Fischer, “Wind-excited vibrations-solution by passive dynamic vibration absorbers of different types,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 95, no. 9-11, pp. 1028-1039, 2007.
[6] S. K. Lai, C. W. Lim, Z. Lin, and W. Zhang, “Analytical analysis for large-amplitude oscillation of a rotational pendulum system,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6115-6124, 2011. · Zbl 1215.34044
[7] C. W. Lim and B. S. Wu, “A modified mickens procedure for certain non-linear oscillators,” Journal of Sound and Vibration, vol. 257, no. 1, pp. 202-206, 2002. · Zbl 1237.70109
[8] S.-Q. Wang and J.-H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,” Chaos, Solitons & Fractals, vol. 35, no. 4, pp. 688-691, 2008. · Zbl 1210.70023
[9] J. Fan, “Application of He’s frequency-amplitude formulation to the duffing-harmonic oscillator,” Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 389-391, 2008. · Zbl 1146.01302
[10] J. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207-209, 2008. · Zbl 06942339
[11] A. Chatterjee, “Harmonic balance based averaging: approximate realizations of an asymptotic technique,” Nonlinear Dynamics, vol. 32, no. 4, pp. 323-343, 2003. · Zbl 1042.70017
[12] J. H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430-1439, 2007. · Zbl 1152.34327
[13] J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052
[14] N. A. Khan, M. Jamil, S. A. Ali, and A. Nadeem Khan, “Solutions of the force free Duffing van der pol oscillator equation,” International Journal of Differential Equations, vol. 2011, Article ID 852919, 9 pages, 2011. · Zbl 1239.34015
[15] N. A. Khan, A. Ara, S. A. Ali, and M. Jamil, “Orthognal flow impinging on a wall with suction or blowing,” International Journal of Chemical Reactor Engineering, vol. 9, no. 1, 2011.
[16] H. M. Liu, “Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 577-579, 2005. · Zbl 1078.34509
[17] J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312-2314, 2010. · Zbl 1237.70036
[18] N. A. Khan, M. Jamil, and A. Ara, “Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity,” International Journal of Differential Equations, vol. 2011, Article ID 649748, 7 pages, 2011. · Zbl 1239.34033
[19] A. Yildirim, Z. Saddatania, H. Askri, Y. Khan, and M. K. Yazdi, “Higher order approximate periodic solution for nonlinear oscillators with Hamiltonian approach,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2042-2051, 2011. · Zbl 1272.70110
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