In this paper, the phenomena of hysteretic behaviour of friction force observed during experiments are discussed. On the basis of experimental and theoretical analyses, we argue that such behaviour can be considered as a representation of the system dynamics. According to this approach, a classification of friction models, with respect to their sensitivity on the system motion characteristic, is introduced. General friction modelling of the phenomena accompanying dry friction and a simple yet effective approach to capture the hysteretic effect are proposed. Finally, the experimental results are compared with the numerical simulations for the proposed friction model.

In this paper, we re-examine the dynamics of double pendulum in numerical simulations and experimental observations. Typical types of behaviors of the parametrically excited double pendula are presented, including chaos, rotations and periodic oscillations, and the bifurcation analysis is performed, exhibiting complex transitions from one type of motion into another. The character of the observed dynamics is analyzed using Lyapunov exponents, which confirms the hyperchaotic nature of the system. Particular attention is paid to the transient behaviors, showing that the length of the irregular motion can be extremely sensitive to both parameters and initial conditions. Apart from the single double pendulum, we consider also the case of two coupled double pendula, connected by a typical linear scheme. Our results show that depending on the network’s parameters, one can observe the phenomenon of a transient chaotic synchronization, during which the units spontaneously synchronize and desynchronize. The loss of coherence is strictly related to the motion of the pendula around the unstable equilibrium of the system, which has been confirmed in the scenario of pure chaotic oscillations. We determine the regions of the occurrence of transient synchronization in the coupling parameters’ plane, as well as study the statistical properties of the observed patterns. We show that the problem of determining the final dynamical attractor of the system is not straightforward.

The paper explains iterative and non-iterative approaches to control optimization with use of the Fourier series-based method. Both variants of the presented algorithm are used to numerically approximate optimal control of a discontinuous pendulum capsule drive. Firstly, the general algorithm and its two realizations (iterative and non-iterative) are presented. It is shown that the iterative variant assures non-decreasing quality of solutions in subsequent repetitions of the procedure and the background of such guarantees is explained. A numerical example follows: control of a self-propelled capsule drive is optimized using both approaches. Results are compared and discussed. It is expected that the presented methods can be useful in optimal control estimation for complex systems, particularly discontinuous ones.

In this paper, we experimentally verify the phenomenon of chaotic synchronization in coupled forced oscillators. The study is focused on the model of three double pendula locally connected via springs. Each of the individual oscillators can behave both periodically and chaotically, which depends on the parameters of the external excitation (the shaker). We investigate the relation between the strength of coupling between the upper pendulum bobs and the precision of their synchronization, showing that the system can achieve practical synchronization, within which the nodes preserve their chaotic character. We determine the influence of the pendula parameters and the strength of coupling on the synchronization precision, measuring the differences between the nodes’ motion. The results obtained experimentally are confirmed by numerical simulations. We indicate a possible mechanism causing the desynchronization of the system’s smaller elements (lower pendula bobs), which involves their motion around the unstable stationary position and possible transient dynamics. The results presented in this paper may be generalized into typical models of pendula and pendula-like coupled systems, exhibiting chaotic dynamics.