Abstract
This paper examines the oscillations of a spherical pendulum with horizontal Lissajous excitation. The pendulum has two degrees of freedom: a rotational angle defined in the horizontal plane and an inclination angle defined by the pendulum with respect to the vertical z axis. The results of numerical simulations are illustrated with the mathematical model in the form of multi-colored maps of the largest Lyapunov exponent. The graphical images of geometrical structures of the attractors placed on Poincaré cross sections are shown against the maps of the resolution density of the trajectory points passing through a control plane. Drawn for a steady-state, the graphical images of the trajectory of a tip mass are shown in a three-dimensional space. The obtained trajectories of the moving tip mass are referred to a constructed bifurcation diagram.
Highlights
Some dynamical systems are very sensitive to small changes of initial conditions leading the system to different responses
Bifurcation diagrams and Lyapunov exponents are most often used to distinguish a chaotic response from a periodic response
In our modeling studies, bifurcation diagrams were generated using an algorithm based on Poincarecross-sectional points
Summary
Some dynamical systems are very sensitive to small changes of initial conditions leading the system to different responses. Even subtle changes of parameters may cause huge deviations in the response of such a system making the long-term predictions of the system response impossible Such systems are called chaotic, and their mathematical model, i.e., their governing differential equations of motion, has no analytical solutions. The spherical pendulum excited vertically was studied experimentally and theoretically by Naprstek and Fischer [11, 12], Pospısil et al [13], and Fischer et al [14] They provided a large number of solutions with some approximate considerations. The phenomenological model of the spherical pendulum, which is affected by kinematic excitations along the x and y axes, has been formulated Such model can simulate overhead-traveling cranes, which are characterized by negligible small deflections, and the impact of rail unevenness is neglected. Such model can simulate overhead-traveling cranes, which are characterized by negligible small deflections, and the impact of rail unevenness is neglected. (Kinematic vertical excitations are absent.)
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