Instability dynamics of a horizontally shaken pendulum

Abstract

We consider the effect of horizontally shaking the pivot point of a damped pendulum. While similar systems have been considered in engineering in specific circumstances, for instance in the design of isolation systems in an earthquake zone, a careful analysis of the shaken damped pendulum behaviour has not been carried out. We identify the periodic solutions in the system, examine their bifurcation behaviour, and explore the development of chaos. We find that the horizontal shaking supports bistability in a critical frequency range and that the development of chaos depends sensitively on the driving frequency. Our work complements recent experiments on coupled pendulums with horizontal shaking, providing a foundation for further study of the pendulum dynamics. References

• Z. Masoud, A. Nayfeh and D. Mook, Cargo pendulation reduction of ship-mounted cranes. Nonlinear Dynamics, 35, 2004, 299--311. doi:10.1023/B:NODY.0000027917.37103.bc
• I. M. Sauer, J. Frank and E. S. Bucherl, Proposal of a new electromechanical total artificial heart: the TAH Serpentina. Artifical Organs, 23, 1999, 290--291. doi:10.1046/j.1525-1594.1999.06226.x
• A. Mokha, M. C. Constantinon, A. M. Reinhorn, et al. Experimental study of friction-pendulum isolation system. Journal of Structural Engineering, 117(4), 1991, 1201--1217. doi:10.1061/(ASCE)0733-9445(1991)117:4(1201)
• R. W. Leven, B. P. Koch, Chaotic behaviour of paramentrically excited damped pendulum. Physics Letters A, 86(2), 1981, 71--74. doi:10.1016/0375-9601(81)90167-5
• H. Heng, W. Martienssen, Analysing the chaotic motion of a driven pendulum. Chaos, Solitons and Fractals, 2(3), 1992, 323--334. doi:10.1016/0960-0779(92)90039-P
• G. L. Baker, Control of the chaotic driven pendulum. American Journal of Physics, 63(9), 1995, 832--838. doi:10.1103/PhysRevLett.74.1974
• R. B. Thakur, L. Q. English and A. J. Sievers, Driven intrinsic localized modes in a coupled pendulum array. Journal of Physics: Applied Physics, 41, 2008, 015503. doi:10.1088/0022-3727/41/1/015503
• Y. Nakamura, M. Saruta, A. Wada, T. Takeuchi, S. Hikone and T. Takahashi, Development of the core-suspended isolation system, Earthquake Engineering and Structural Dynamics 40, 2011, 429--447. doi:10.1002/eqe.1036
• J. M. Schmitt, P. V. Bayly, Bifurcations in the mean angle of a horizontally shaken pendulum: analysis and experiment. Nonlinear Dynamics, 15, 1998, 1--14. doi:10.1023/A:1008279910762