Approximate rotational solutions of pendulum under combined vertical and horizontal excitation

Ekaterina Pavlovskaia, Bryan Horton, Marian Wiercigroch, Stefano Lenci, Giuseppe Rega

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29 Citations (Scopus)


A pendulum excited by the combination of vertical and horizontal forcing at the pivot point was considered and the period-1 rotational motion was studied. Analytical approximations of period-1 rotations and their stability boundary on the excitation parameters (ω, p)-plane are derived using asymptotic analysis for the pendulum excited elliptically and along a tilted axis. It was assumed that the damping is small and the frequency of the base excitation is relatively high. The accuracy of the approximations was examined for different values of the parameters e and κ controlling the shape of excitation, and it was found that using the second and third order approximations ensures a good correspondence between analytical and numerical results in the majority of cases. Basins of attractions of the coexisting solutions were constructed numerically to evaluate the robustness of the obtained rotational solutions. It was found that the horizontal component of excitation has a larger effect on the shift in position of the saddle node bifurcations for the elliptically excited case than for the pendulum excited along a tilted axis. For the elliptically excited pendulum with pivot rotating in the same direction as the pendulum the stability boundary is shifted downwards providing a larger region of the solution existence. When the pendulum and the pivot rotate in opposite directions, the boundary is shifted upwards significantly limiting the region of the solution existence. In contrast, for the pendulum excited along the tilted axis, the direction of the rotation has a minor effect for low frequency values and the addition of the horizontal component always results in a larger region of the solution existence.
Original languageEnglish
Article number1250100
Number of pages13
JournalInternational Journal of Bifurcation and Chaos
Issue number5
Publication statusPublished - May 2012


  • pendulum
  • rotational solutions
  • perturbation methods
  • saddle node bifurcation


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