Balancing with Reflex Delay

In sport, unstable equilibria of mechanical systems often have to be stabilized by the human operators, i.e. by the sportsmen. A typical basic example for this is the selfbalancing of the human body. The analysis of the problem of balancing an inverted pendulum proves that the human operator has to apply a quite complicated control strategy if he wants to achieve his goal in the presence of the time delay of his/her reflexes. It is a rule of thumb, that increasing time delay tends to destabilize any dynamical system. To avoid the instability naturally occurring in the mechanical system and also caused by the time delay, the human operator has to choose the control parameters from a narrow region which can be found only after more or less practicing. Above a critical value of the reflex delay, the balancing is impossible. We consider the simplest possible model describing the "man-machine" system when somebody places the end of a stick on his fingertip, and tries to move the lowest point of the stick in a way that its upper position should be stable. The stability analysis of the. mechanical model provides surprisingly simple and clear results in spite of the fact that modelling the time delay makes the corresponding differential equation infinite dimensional in mathematical sense. Conclusions can be derived in the following respects: - The stability chart in the space of the control parameters explains the complicated process the operator should carry out when balancing the stick. - The calculated critical delay of reflexes depends on the mechanical parameters and shows a good agreement with the experimental results. - By means of the above results we also get some in-view into the work of the organ called "labyrinthus" in the inner ear which helps in self balancing of the human body even when our eyes are closed. These conclusions may provide a good basis for developing new tests to check the sportsmen's reflex delays and balancing abilities

On the stability of bodies suspended asymmetrically with an inelastic rope

The stability of a body suspended asymmetrically by means of an inelastic rope is investigated. The rope is attached to the body at two points and passed over a frictionless hook or a nail in the vertical plane. The equilibrium points of the system and their stability are described as a function of the rope length, the distance of the attachment points and the position of the center of mass. Depending on the choice of the parameters, one, two or three equilibrium positions exist: their structural change manifests itself in the form of cusp bifurcations of co-dimension two, which is determined in exact analytical form

Cumulative Surface Location Error for Milling Processes Based on Tool-tip Frequency Response Function

AbstractIn milling processes, the desired machined surface cannot be perfectly achieved even in case of chatter-free machining due to the thermally induced errors, the trajectory following errors and the most significant one: the cutting force induced vibration errors. In case of vibration, the error is represented by the so-called Surface Location Error (SLE), which is the distance between the machined and the required surface position. In case of roughing operations, these errors can have a significant impact on the surface position due to the interaction between the subsequent SLEs. The machined surface depends on the previously resulted SLE through the variation of the radial immersion. In this paper, the series of the consecutive SLEs are investigated in a multi-degree-of-freedom model. The dynamical behaviour of the milling tool is described by frequency response functions. The variation of the SLE values is governed by a discrete map, which may lead to an unpredictable final surface position. The parameter range where this unpredictable final SLE occurs is presented together with the traditional stability chart representing the chatter-free domains of cutting parameters. With the proposed methods, the traditional stability chart can be improved, from which chatter-free and CSLE-stable technological parameters can be selected

Isolated large amplitude periodic motions of towed rigid wheels

This study investigates a low degree-of-freedom (DoF) mechanical model of shimmying wheels. The model is studied using bifurcation theory and numerical continuation. Self-excited vibrations, that is, stable and unstable periodic motions of the wheel, are detected with the help of Hopf bifurcation calculations. These oscillations are then followed over a large parameter range for different damping values by means of the software package AUTO97. For certain parameter regions, the branches representing large amplitude stable and unstable periodic motions become isolated following an isola birth. These regions are extremely dangerous from an engineering view-point if they are not identified and avoided at the design stage.Comment: Appeared online in Nonlinear Dynamics Thursday, April 26, 200

Effects of radial immersion and cutting direction on chatter instability

ABSTRACT Low radial immersion end-milling involves intermittent cutting. If the tool is flexible, its motion in both the x-and ydirections affects the chip load and cutting forces, leading to chatter instability under certain conditions. Interrupted cutting complicates stability analysis by imposing sharp periodic variations in the dynamic model. Stability predictions for the 2-DOF model differ significantly from prior 1-DOF models of interrupted cutting. In this paper stability boundaries of the 2-DOF milling process are determined by three techniques and compared: (1) a frequency-domain technique developed b