On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors

Abstract

This study analyzes a nonlinear system with a crankshaft -slider mechanism linked to a pendulum. The crank is powered by a DC motor that moves horizontally the pendulum pivot. The speed of the motor is influenced by the pendulum dynamics, reason for calling the system as nonideal. In literature, the vibration of the pendulum support by a crank-shaftslider is generally considered as harmonic what neglects the real complexity of the mechanism. It is also common, for the crank speed or the excitation frequency to be considered constant, so a unique degree-of-freedom can take place. The novelty in this research is the analysis of the feedback effect of the pendulum over the crank speed and the complexity of crank-slider mechanism without approaching to a harmonic motion. Different types of motion occur as oscillations, rotations, and chaos. Results with different initial conditions are worked out and basins of attractions plotted. Chaos was obtained when small variations of parameters are made. The methods of analysis include phase portraits, time histories, bifurcations diagrams, and basins of attraction. The verification of chaotic phenomena is performed through the 0-1 test. By the end, the feedback control is applied using the method of Tereshko by altering the energy of the chaotic system, leading the pendulum to a limit cycle.

Comments on the paper “On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors” published in this journal

“…This paper discusses a hypothesis found in recent articles and theses as [1][2][3] that claim the derivative of the current of an electromechanical system can be considered equal to zero into the IVP that gives the dynamics of the system. The hypothesis is based only on some parameter values and, also, does not depend on the electromechanical system being analysed.…”

“…This paper discusses a hypothesis found in recent references of the literature as [1][2][3] and reference number 13 of [2] that affirms that it is possible to consider the derivative of the current equal to zero in the IVP that gives the dynamics of an electromechanical system without changing the system dynamics. The hypothesis is based only on some parameter values and, also, does not depend on the type of electromechanical system analysed.…”

“…There are some references in the literature, as the recent article published by Avanço et al (see [2]) that lead us to write the present paper, two doctoral theses [1,3], and the reference number 13 of [2] among others, that claim that is possible to consider the derivative of the current equal to zero in the IVP that gives the dynamics of an electromechanical system due to the fact that the electrical time constant of the motor l / r is usually much smaller than the mechanical time constant rj m ∕k e…”

Pitfalls in the dynamics of coupled electromechanical systems

“…Observe that the dynamics of the coupled system is given by an initial value problem comprising a set of coupled differential equations. There are some references in the literature, as the recent article published by Avanço R.H. (see [2]) and [1,3,4], that claim that the inductance of the armature can be neglected due to the fact that the electrical time constant of the motor l/r is usually much smaller than the mechanical time constant rj m /k e 2 . Please remark, that this is a hypothesis based only on parameters values, it does not depend on the system being studied.…”

Twofold quantization in digital control: deadzone crisis and switching line collision

“…Note that the explicit control of chaos itself is not our goal with the PD-control. However, an elegant feedback control approach was introduced in [17] and applied to a simple system in [1].…”