Life Expectancy Calculations of Transient Chaotic Behaviour in the Lorenz Model

In engineering practice, chaotic oscillations are often observed which disappear suddenly. This phenomenon is often referred to as transient chaos. The life expectancy of these oscillations varies stochastically. In this work, a method is presented for the simple estimation of the expected length of the chaotic behaviour. As an example, the Lorenz system is considered at some specific parameter values

Life expectancy calculation of transient chaos in the 2D micro-chaos map

We analyse a simple model of a digitally controlled mechanical system, which may perform chaotic vibrations. As a consequence of the digital effects, i.e., the sampling and the round-off error, the behaviour of this system can be described by the so-called micro-chaos map. If dry friction is present in the system, it can stop the motion. In such cases the resulting behaviour is referred to as transient chaos, the duration of which can be closely related to the control time. We developed a method for the exact calculation of the mean lifetime Nm of transient chaos in case of the 1D micro-chaos map, and showed that in certain cases Nm characterizes the duration of chaotic transients better than the so-called escape rate. In the present paper, we try to extend these results to a 2D version of the micro-chaos map

INSTABILITY CAUSED BY DELAY IN ROBOT SYSTEMS

It is well-known that delayed feedback in the control of robots may cause stability problems. This paper presents an analytical investigation of this effect by means of stability charts on the plane of the parameters of simple but typical robot systems

FRACTAL DIMENSION AS MEASURE OF CONTROL TIME

The nonlinearity caused by the application of digital control may lead to chaotic behaviour. There are several cases, when these chaotic oscillations disappear suddenly. This phenomenon is referred to as transient chaos. In the present paper, we analyse a simple model of a digitally controlled mechanical system, which may perform transient chaotic vibrations, and propose a new procedure for the estimation of the duration of these transients. The relation between the mean lifetime and the so-called escape rate is also examined. As a result, a new formula is introduced, whose reliability is validated with the help of the new lifetime estimation method

A MODEL OF BALANCING

Place the end of a rod on your fingertip and move this lowest point of the rod to a degree that the upper vertical position of it should be stable. It is obvious, that it is not possible to equilibrate it if one's reflexes are slow. The paper shows the determination of the critical delay of the reflexes where this balancing is still possible

STABILITY TRANSITION BETWEEN 1 AND 2 DEGREE-OF-FREEDOM MODELS OF MILLING

Chatter prediction for 2 degree of freedom (DOF) milling model is presented. The workpiece is assumed to be flexible and the tool to be stiff. Non-linear cutting force model is used, and the linearized equation of motion is derived. Stability charts are constructed for different stiffness values in the directions x and y. The charts for the 1 DOF models associated with the x and the y directions are also given. It is shown that the 2-DOF case can not be given via the pure overlaying of the charts of the two single DOF cases

Stability of towed wheels with elastic steering mechanism and shimmy damper

This paper investigates a low degree-of-freedom wheel model, which describes the lateral vibration of towed wheels, called the shimmy. The model takes into account the lateral deformation of the tyre and also the torsional elasticity and the damping of the steering mechanism. The lateral deformation of the wheel is modeled by the stretched string-like tyre model, which considers the relaxation of the tyre during rolling. The linear stability analysis of this shimmy model is presented and the stability properties are examined in different parameter ranges of the model