Period-One Rotating Solutions of Horizontally Excited Pendulum Based on Iterative Harmonic Balance
Affiliation(s)
Department of Civil and Environmental Engineering, University of Hawai’i at Mānoa, Honolulu, USA.
Department of Civil and Environmental Engineering, University of Hawai’i at Mānoa, Honolulu, USA.
ABSTRACT
In this study, the iterative harmonic balance method was used to develop analytical solutions of period-one rotations of a pendulum driven horizontally by harmonic excitations. The performance of the method was evaluated by two criteria, one based on the system energy error and the other based on the global residual error. As a comparison, analytical solutions based on the multi-scale method were also considered. Numerical solutions obtained from the Dormand-Prince method (ODE45 in MATLAB©) were used as the baseline for evaluation. It was found that under lower-level excitations, the multi-scale method performed better than the iterative method. At higher-level excitations, however, the performance of the iterative method was noticeably more accurate.
In this study, the iterative harmonic balance method was used to develop analytical solutions of period-one rotations of a pendulum driven horizontally by harmonic excitations. The performance of the method was evaluated by two criteria, one based on the system energy error and the other based on the global residual error. As a comparison, analytical solutions based on the multi-scale method were also considered. Numerical solutions obtained from the Dormand-Prince method (ODE45 in MATLAB©) were used as the baseline for evaluation. It was found that under lower-level excitations, the multi-scale method performed better than the iterative method. At higher-level excitations, however, the performance of the iterative method was noticeably more accurate.
Cite this paper
Zhang, H. and Ma, T. (2015) Period-One Rotating Solutions of Horizontally Excited Pendulum Based on Iterative Harmonic Balance. Advances in Pure Mathematics, 5, 413-427. doi: 10.4236/apm.2015.58041.
Zhang, H. and Ma, T. (2015) Period-One Rotating Solutions of Horizontally Excited Pendulum Based on Iterative Harmonic Balance. Advances in Pure Mathematics, 5, 413-427. doi: 10.4236/apm.2015.58041.
References
[1] Kautz, R.L. and Macfarlane, J.C. (1986) Onset of Chaos in the rf-Biased Josephson Junction. Physical Review A, 33, 498-509. http://dx.doi.org/10.1103/PhysRevA.33.498 [2] Thompson, J.M.T. (1989) Chaotic Phenomena Triggering the Escape from a Potential Well. Proceedings of the Royal Society of London A, 421, 195-225.
http://dx.doi.org/10.1098/rspa.1989.0009 [3] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 1: Applied Torque. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 1-22.
http://dx.doi.org/10.1017/S0334270000008183 [4] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 2: Horizontal Forcing. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 23-41.
http://dx.doi.org/10.1017/S0334270000008195 [5] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 3: Vertical Forcing. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 42-60.
http://dx.doi.org/10.1017/S0334270000008201 [6] Koch, B.P. and Leven, R.W. (1985) Subharmonic and Homoclinic Bifurcations in a Parametrically Forced Pendulum. Physica, 16D, 1-13. [7] Leven, R.W., Pompe, B., Wilke, C. and Koch, B.P. (1985) Experiments on Periodic and Chaotic Motions of Aparametrically Forced Pendulum. Physica, 16D, 371-384. [8] Clifford, M.J. and Bishop, S.R. (1995) Rotating Periodic Orbits of the Parametrically Excited Pendulum. Physics Letters A, 201, 191-196.
http://dx.doi.org/10.1016/0375-9601(95)00255-2 [9] Garira, W. and Bishop, S.R. (2003) Rotating Solutions of the Parametrically Excited Pendulum. Journal of Sound and Vibration, 263, 233-239.
http://dx.doi.org/10.1016/S0022-460X(02)01435-9 [10] Ma, T.W. and Zhang, H. (2012) Enhancing Mechanical Energy Harvesting with Dynamics Escaped from Potential Well. Applied Physics Letters, 100, Article ID: 114107.
http://dx.doi.org/10.1063/1.3694272 [11] Xu, X. and Wiercigroch, M. (2007) Approximate Analytical Solutions for Oscillatory and Rotational Motion of Aparametric Pendulum. Nonlinear Dynamics, 47, 311-320.
http://dx.doi.org/10.1007/s11071-006-9074-4 [12] Lenci, S., Pavlovskaia, E., Rega, G. and Wiercigroch, M. (2008) Rotating Solutions and Stability of Parametric Pendulum by Perturbation Method. Journal of Sound and Vibration, 310, 243-259.
http://dx.doi.org/10.1016/j.jsv.2007.07.069 [13] Zhang, H. and Ma, T.W. (2012) Iterative Harmonic Balance for Period-One Rotating Solution of Parametric Pendulum. Nonlinear Dynamics, 70, 2433-2444.
http://dx.doi.org/10.1007/s11071-012-0631-8 [14] Pavlovskaia, E., Horton, B., Wiercigroch, M., Lenci, S. and Rega, G. (2012) Approximate Rotational Solutions of Pendulum under Combined Vertical and Horizontal Excitation. International Journal of Bifurcation and Chaos, 22, 1250100-1250113.
[1] Kautz, R.L. and Macfarlane, J.C. (1986) Onset of Chaos in the rf-Biased Josephson Junction. Physical Review A, 33, 498-509. http://dx.doi.org/10.1103/PhysRevA.33.498 [2] Thompson, J.M.T. (1989) Chaotic Phenomena Triggering the Escape from a Potential Well. Proceedings of the Royal Society of London A, 421, 195-225.
http://dx.doi.org/10.1098/rspa.1989.0009 [3] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 1: Applied Torque. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 1-22.
http://dx.doi.org/10.1017/S0334270000008183 [4] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 2: Horizontal Forcing. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 23-41.
http://dx.doi.org/10.1017/S0334270000008195 [5] Bryant, P.J. and Miles, J.W. (1990) On a Periodically Forced, Weakly Damped Pendulum. Part 3: Vertical Forcing. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 42-60.
http://dx.doi.org/10.1017/S0334270000008201 [6] Koch, B.P. and Leven, R.W. (1985) Subharmonic and Homoclinic Bifurcations in a Parametrically Forced Pendulum. Physica, 16D, 1-13. [7] Leven, R.W., Pompe, B., Wilke, C. and Koch, B.P. (1985) Experiments on Periodic and Chaotic Motions of Aparametrically Forced Pendulum. Physica, 16D, 371-384. [8] Clifford, M.J. and Bishop, S.R. (1995) Rotating Periodic Orbits of the Parametrically Excited Pendulum. Physics Letters A, 201, 191-196.
http://dx.doi.org/10.1016/0375-9601(95)00255-2 [9] Garira, W. and Bishop, S.R. (2003) Rotating Solutions of the Parametrically Excited Pendulum. Journal of Sound and Vibration, 263, 233-239.
http://dx.doi.org/10.1016/S0022-460X(02)01435-9 [10] Ma, T.W. and Zhang, H. (2012) Enhancing Mechanical Energy Harvesting with Dynamics Escaped from Potential Well. Applied Physics Letters, 100, Article ID: 114107.
http://dx.doi.org/10.1063/1.3694272 [11] Xu, X. and Wiercigroch, M. (2007) Approximate Analytical Solutions for Oscillatory and Rotational Motion of Aparametric Pendulum. Nonlinear Dynamics, 47, 311-320.
http://dx.doi.org/10.1007/s11071-006-9074-4 [12] Lenci, S., Pavlovskaia, E., Rega, G. and Wiercigroch, M. (2008) Rotating Solutions and Stability of Parametric Pendulum by Perturbation Method. Journal of Sound and Vibration, 310, 243-259.
http://dx.doi.org/10.1016/j.jsv.2007.07.069 [13] Zhang, H. and Ma, T.W. (2012) Iterative Harmonic Balance for Period-One Rotating Solution of Parametric Pendulum. Nonlinear Dynamics, 70, 2433-2444.
http://dx.doi.org/10.1007/s11071-012-0631-8 [14] Pavlovskaia, E., Horton, B., Wiercigroch, M., Lenci, S. and Rega, G. (2012) Approximate Rotational Solutions of Pendulum under Combined Vertical and Horizontal Excitation. International Journal of Bifurcation and Chaos, 22, 1250100-1250113.