Damping of in-process measuring system through variable stiffness tunable vibration absorber

International audienceIn external cylindrical grinding machines, measuring systems are occasionally mounted on a gantry type frame. The modes of this structure are potentially harmful within the operation range of the grinding wheel in a standard machine configuration, since the rotation of the wheel can enter into resonance, thus avoiding a correct determination of the on-line measurement. The resoncance problem can be succesfully dealt with the use of a variable stiffness vibration absorber which autonomously adapts its stiffness to tune according to the mode to be damped, increasing the dynamic stiffness along the whole operation range of the wheel. In this work, a variable stiffness self-tunable vibration absorber prototype has been built and a new tuning function has been derived in order to minimize the response at the frequency coincident with the rotating speed of the wheel. Finally, validation tests have been performed in a scale supporting structure and the vibration reduction improvement comparison with respect to a standard fixed tuning strategy has been evaluated

Bifurcation analysis of delay-induced resonances of the El-Nino Southern Oscillation

Models of global climate phenomena of low to intermediate complexity are very useful for providing an understanding at a conceptual level. An important aspect of such models is the presence of a number of feedback loops that feature considerable delay times, usually due to the time it takes to transport energy (for example, in the form of hot/cold air or water) around the globe. In this paper we demonstrate how one can perform a bifurcation analysis of the behaviour of a periodically-forced system with delay in dependence on key parameters. As an example we consider the El-Nino Southern Oscillation (ENSO), which is a sea surface temperature oscillation on a multi-year scale in the basin of the Pacific Ocean. One can think of ENSO as being generated by an interplay between two feedback effects, one positive and one negative, which act only after some delay that is determined by the speed of transport of sea-surface temperature anomalies across the Pacific. We perform here a case study of a simple delayed-feedback oscillator model for ENSO (introduced by Tziperman et al, J. Climate 11 (1998)), which is parametrically forced by annual variation. More specifically, we use numerical bifurcation analysis tools to explore directly regions of delay-induced resonances and other stability boundaries in this delay-differential equation model for ENSO.Comment: as accepted for Proc Roy Soc A, 20 pages, 7 figure

Analytical estimations of limit cycle amplitude for delay-differential equations

The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing

Effect of Potential Energy Variation on the Natural Frequency of an Euler–Bernoulli Cantilever Beam Under Lateral Force and Compression

A cantilever beam is subjected to both lateral force and compression under gravity. By taking into account the potential energy variation of the system, we develop a theoretical result that greatly simplifies the bending vibration frequency calculation in agreement with the experiments.</jats:p

Quantization improves stabilization of dynamical systems with delayed feedback

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possible when the fixed point in the absence of feedback is an unstable node, and in the presence of feedback, it is an unstable focus (spiral). The results are illustrated with numerical simulation of the unstable Hayes equation. The solutions of the quantized Hayes equation take the form of oscillations in which the amplitude is a function of the size of the quantization step. If the quantization step is sufficiently small, the amplitude of the oscillations can be small enough to practically approximate the dynamics around a stable fixed point

Effects of bearing clearance on the chatter stability of milling process

In the present study, the influences of the bearing clearance, which is a common fault for machines, to the chatter stability of milling process are examined by using numerical simulation method. The results reveal that the presence of bearing clearance could make the milling process easier to enter the status of chatter instability and can shift the chatter frequency. In addition, the spectra analysis to vibration signals obtained under the instable milling processes show that the presence of bearing clearance could introduce more frequency components to the vibration responses but, however, under both the stable and instable milling processes, the generated frequency components will not violate the ideal spectra structures of the vibration responses of the milling process, which are usually characterized by the tooth passing frequency and its associated higher harmonics for the stable milling process and by the complex coupling of the tooth passing frequency and the chatter frequency for the instable milling process. This implies that, even under the case with bearing clearance fault, the stability of the milling process can still be determined by viewing the frequency spectra of the vibration responses. Moreover, the phenomena of the chatter frequency shift and the generation of more components provide potential ways to detect the bearing clearance in machines. (C) 2010 Elsevier Ltd. All rights reserved

On the stability of periodic orbits in delay equations with large delay

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.Comment: postprint versio