Grzegorz Kudra

Application of the generalized Prony spectrum for extraction of information hidden in chaotic trajectories of triple pendulum

In this paper we apply a new method of analysis of random behavior of chaotic systems based on the Prony decomposition. The generalized Prony spectrum (GPS) is used for quantitative description of a wide class of random functions when information about their probability distribution function is absent. The scaling properties of the random functions that keep their invariant properties on some range of scales help to fit the compressed function based on the Prony’s decomposition. In paper [1] the first author (RRN) found the physical interpretation of this decomposition that includes the conventional Fourier decomposition as a partial case. It has been proved also that the GPS can be used for detection of quasi-periodic processes that are appeared usually in the repeated or similar measurements. A triple physical pendulum is used as a chaotic system to obtain a chaotic behavior of displacement angles with one, two and three positive Lyapunov’s exponents (LEs). The chaotic behavior of these angles can be expressed in the form of amplitude-frequency response (AFR) that is extracted from the corresponding GPS and can serve as a specific ”fingerprint” characterizing the random behavior of the triple-pendulum system studied. This new quantitative presentation of random data opens additional possibilities in classification of chaotic responses and random behaviors of different complex systems.

On the Dynamics of the Rigid Body Lying on the Vibrating Table with the Use of Special Approximations of the Resulting Friction Forces

This paper is a presentation of the simulation and dynamical behaviour of a rigid body lying on a vibrating table. Shaping and control of the body’s dynamical behaviour, by the use of manipulation of parameters of the table oscillations, are presented. This article includes an implementation of the specially prepared mathematical models of friction between table and the moving body as well. The above-mentioned models are based on the integral model assuming the fully developed sliding on the plane contact area with the foundation of any pressure distribution. In order to simplify the calculations and reduce their computational cost, special approximations of the integral models of friction force and moment are proposed. They are based on Pade approximants and their generalizations.

A Pendulum Driven by a Crank-Shaft-Slider Mechanism and a DC Motor—Mathematical Modeling, Parameter Identification, and Experimental Validation of Bifurcational Dynamics

In this work, we investigate numerically and experimentally the dynamics of a pendulum vertically excited by a crank-shaft-slider mechanism driven by a DC motor. The power supplied to the DC is small enough to observe return influence of the pendulum dynamics on the motor angular velocity. In the performed experiments, the motor is supplied with constant time voltages. A series of experimental periodic solutions allowed to estimate the model parameters and, in the further step, predict bifurcation phenomena observed in the real object.