We investigated dynamic responses of a parametric pendulum obtained experimentally. Using the recurrence plot technique designed to analyze experimental time series we have distinguished different types of motion. This method, supplemented by recurrence quantification analysis (RQA), has been used to identify oscillations, rotations, and transient chaotic vibrations for relatively short time series composed of only few cycles.

Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability.

The aim of this study is to provide a simple, yet effective and generally applicable technique for determining damping for parametric pendula. The proposed model is more representative of system dynamics because the numerical results describe the qualitative features of experimentally exhibited transient tumbling chaotic motions well. The assumption made is that the system is accurately modelled by a combination of viscous and Coulomb dampings; a parameter identification procedure is developed from this basis. The results of numerical and experimental time histories of free oscillations are compared with the model produced from the parameters identified by the classic logarithmic decrement technique. The merits of the present method are discussed before the model is verified against experimental results. Finally, emphasis is placed on a close corroboration between the experimental and theoretical transient tumbling chaotic trajectories.

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.

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