Dynamic response of the spherical pendulum subjected to horizontal Lissajous excitation
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ORIGINAL PAPER
Dynamic response of the spherical pendulum subjected to horizontal Lissajous excitation Grzegorz Litak . Jerzy Margielewicz Krzysztof Da˛bek
. Damian Ga˛ska
. Daniil Yurchenko
.
Received: 29 May 2020 / Accepted: 12 October 2020 The Author(s) 2020
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 Poincare´ 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.
G. Litak (&) K. Da˛bek Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland e-mail: [email protected] J. Margielewicz D. Ga˛ska Faculty of Transport an Aviation Engineering, Silesian University of Technology, Krasin´skiego 8, 40-019 Katowice, Poland D. Yurchenko Institute of Mechanical, Process and Energy Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK
Keywords Nonlinear oscillations Spherical pendulum Strange attractor Chaos Lyapunov exponents Lissajous curves Amplitude–frequency spectrum
1 Introduction Some dynamical systems are very sensitive to small changes of initial conditions leading the system to different responses. Consequently, 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. Besides, several solutions can coexist for various initial conditions. The influence of initial conditions on coexisting solutions is most often presented in the form of multi-colored maps of basins of attraction. However, the basins of attraction are difficult to implement in higher-dimensional phase space. Bifurcation diagrams and Lyapunov exponents (which shows divergence in close trajectories defined in phase space) are most often used to distinguish a chaotic response from a periodic response. Another popular numerical tool used to analyze nonlinear systems is the Poincare cross section. In the case of periodic responses, there are single points, the number of which corresponds to the
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periodicity. On the other hand, in the case of chaotic solutions, on Poincare cross sections, complex geometric structures
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