An Infinite Dimensional Approach to the Third Fundamental Theorem of Lie

We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Alerting or Somnogenic Light: Pick Your Color.

In mammals, light exerts pervasive effects on physiology and behavior in two ways: indirectly through clock synchronization and the phase adjustment of circadian rhythms, and directly through the promotion of alertness and sleep, respectively, in diurnal and nocturnal species. A recent report by Pilorz and colleagues describes an even more complex role for the acute effects of light. In mice, blue light acutely causes behavioral arousal, whereas green wavelengths promote sleep. These opposing effects are mediated by melanopsin-based phototransduction through different neural pathways. These findings reconcile nocturnal and diurnal species through a common alerting response to blue light. One can hypothesize that the opposite responses to natural polychromatic light in night- or day-active animals may reflect higher sensitivity of nocturnal species to green, and diurnals to blue wavelengths, resulting in hypnogenic and alerting effects, respectively. Additional questions remain to be clarified. How do different light wavelengths affect other behaviors such as mood and cognition? How do those results apply to humans? How does light pose either a risk or benefit, depending on whether one needs to be asleep or alert? Indeed, in addition to timing, luminance levels, and light exposure duration, these findings stress the need to understand how best to adapt the color spectrum of light to our needs and to take this into account for the design of daily lighting concepts-a key challenge for today's society, especially with the emergence of LED light technology

Air entrainment in web handling: To be avoided or mastered?

The presence of ambient air is of prime importance in various industrial processes involving web handling. This paper is an attempt to answer the following questions: how does air influence the quality of the final product? How is it possible to cope with such a situation?After a brief description of a few basic problems of fluid mechanics, namely: (i) the development of boundary layers on moving webs and (ii) the flow structure and pressure generation in wedges (i.e. corner flows ), several illustrative examples are presented.(1) When a flexible web passes over a spindle, a thin air layer is formed between the two surfaces. This is typically a foil bearing configuration, which is important to master in order to reduce wear reduction or to avoid any misfunction at the head-tape interface. A brief survey of the historical works on this topic will be given.(2) In wound roll models, the stress field generated in the roll depends on the winding conditions (i.e. geometry and processing parameters) and on the flexible media bulk properties (elasticity or viscoelasticity) and surface properties (topography). It is well known that there is a strong link between the roughness of a surface (resulting from microparticles added to the resin) and its behavior in terms of air entrainment and evacuation. A first attempt to study the complex mechanisms governing this link is proposed.(3j In high velocity coating flows which are present in numerous processes (magnetic tape manufacturing, paper industry, ... ) some air can be entrained between the solid substrate and the liquid layer being coated on it. After a qualitative description of the complex phenomena occurring in the vicinity of the three-phase junction, the amount of air likely to be entrained is evaluated on the basis of a theoretical model.As a conclusion, a few recommendations for practical applications will be tentatively drawn

Winding plastic films: Experimental study of squeeze film flow between one smooth surface and one "rough" surface

The present paper is concerned with experiments which consist in squeezing an air layer between a rigid, smooth surface and a flexible, rough one.The experimental rig is composed of a smooth glass plate, with a circular slit allowing air aspiration to be done around it. A thin (few microns thick) plastic film is laid on the glass plate and air separating the glass and the film surfaces is removed by means of a vacuum pump. A circular front appears on the film surface, and moves towards the centre, as the film is pressed onto the glass plate.A monochromatic lamp is used to insulate the surfaces from above and Newton rings can be observed as the front moves. The duration of this operation is measured by a chronometer.Typically, the measured time depends on the plate diameter, the sub-ambient pressure exerted, the film flexural rigidity (or its thickness) and its surface roughness.A set of experiments have been carried out for several values of the sub-ambient pressure and of the slit diameter.The results are well reproducible: for a given sample, the characteristic time is proportional to the squared value of the diameter. The dependence on the sub-ambient pressure is more complicated. A simple model using a semi-empirical formulation is suggested on the basis of the experimental data

Irreversible reduction of foil tension due to aerodynamical effects

Computation of the residual stress generated during the winding of a plastic film has been coped with by many authors. These studies are based on the assumption that the residual stresses mainly depend on two winding factors: (i) nominal foil tension and (ii) foil mechanical properties. Recently, several authors have introduced the effects of a third winding factor: nip force. But, in all the existing studies, the influence of the entrapped air layer and more generally, the aerodynamical effects are neglected.Such an assumption is reasonable in the case of thick plastic film (thickness about 100 μm) or thin plastic films (thickness about 10 μm) wound under low velocities (about 1 m.s-1). However, in the case of industrial winding conditions in which film thickness is typically about 10 μm and velocity about 5 m.s-1, the effects of aerodynamical phenomena are experimentally known to be as important as the effects of foil tension itself.From a more general point of view, the industrial winding conditions suffer from a lack of theoretical analysis since they are mainly based on empirism, which is not quite satisfactory.We recently proposed a model in order to predict the residual stresses generated under realistic industrial conditions, including the effect of nip roll. This model is based on a new global approach in which the Winding process is seen as a mechanism of air entrainment and air exhaust To set this model in order, we were faced with several problems: (i) computation of the thickness of the entrained air layer, (ii) analysis of the air exhaust phenomena, (iii) analysis of effects due to film roughness ... All the main parameters which govern the winding process (velocity, foil tension, nip force, foil bulk and surface properties ... ) are taken into account.In the present paper, we propose to recall the basis of this new global approach and to focus our attention on one of its most important consequences: the irreversible reduction of foil tension during and after winding. Experimental check is presented for a large set of winding conditions. Comparison is based on the average air layer thickness and on the foil residual tension value

New parameter for dynamic characterization of PET film surface topography

It is well-known that handling and winding flexible media involve aerodynamic phenomena which are crucial for the process. Among those parameters which govern the final thickness value of the air layers separating the film layers in a roll of film (for example PET), surface roughness plays an important role. In order to characterize the surface topography of such materials, in a dynamic way, an original experimental set-up was built. It has been described elsewhere, and only its basic features are recalled here. It consists in a polished glass disc with a circular slit connected to a vacuum pump. Having displayed a sample of PET film onto the glass plate, sub-ambient pressure is applied. The air layer which initially separates the film from the plate is partially reduced due to air aspiration: a circular front starts from the slit and propagates towards the center. For prescribed values of the film thickness, the total propagation time depends on sub-ambient pressure and slit diameter (i.e. squeezing surface) through relationships which involve a single parameter characteristic of film roughness.Here the same experimental set up is used to carry out further investigations dealing with the kinetics of both air layer thinning and front propagation. Using a monochromatic light to insulate the film from above, Newton rings are generated allowing the air gap thickness variation to be measured by means of a CCD camera associated with image processing. The main experimental result is that the air layer at the center decreases linearly versus time, the slope being characteristic of the film surface roughness. A simple theoretical model based on the concept of " equivalent smooth surfaces " is developed in order to predict the circular front propagation. Excellent agreement is observed with the experimental data, namely the front propagation kinetics. These results are extrapolated to the configuration of winding, leading to significant improvement of the existing model for lateral evacuation of the air layers confined between the film layers in a roll of film

Macroscopic effects of surface roughness in confined air-flow

One challenge when processing flexible media such as plastic films is to obtain rolls without any aspect defect : if one considers that a "defect" (i.e. wrinkling or buckling) is due to the fact that the stress generated within the roll is greater than some "plasticity yield", then it is crucial to predict the internal stress state.Several process parameters must be carefully mastered (winding tension, velocity, etc.) as well as the material pertinent properties. One key issue is to optimize the surface topography of the flexible medium so that to improve the quality of the wound roll.We propose here new parameters which describe the surface roughness of plastic films fairly well. The measurements were carried out by using a 3D roughness measurement device.A mathematical model based on homogenization techniques is proposed, where the heights of the roughness peaks, their diameter and their spatial distribution are the governing parameters.Sampling at different levels is carried out by expressing the percentage of peaks which exceed some given threshold value.For each tested film, the threshold value will be the only adjustable parameter.Introducing these parameters into the mathematical model which predicts the evolution of the squeezed air layer and comparing to the experimental data, the following results are obtained :- It is possible to adjust one single parameter so that to obtain a very good agreement between the experimental data and the theoretical results.- The smoother the film, the more important the highest peaks are in terms of air leakage

On-line control of tension in web winding systems based on wound roll internal stress computation

One of the key challenges in the processing of flexible media such as plastic films is to obtain rolls without any aspect defect: if one considers that a "defect" (i.e. wrinkling or buckling) is due to the fact that the stress generated within the roll is greater than some "plasticity threshold", then it is crucial to predict the internal stress. Several process parameters must be carefully mastered, among which the winding tension is very important. Offline optimization of the tension can a priori guarantee the production of perfect rolls, with respect to the internal stress. Nevertheless, the industrial control systems never generate perfect follow-up of the tension reference value, because the tension which is actually imposed (i.e. measured) exhibits oscillations due to the imperfections of the winding system, including geometrical irregularities of the rolls. The fluctuations about the tension nominal value induce variations in the stress within the roll as compared to the value which would result from an ideal control. As a consequence, it is judicious to change the tension reference value during the winding process, according to some criterion defined from the stress computed within the roll, and then to apply this new "up-dated" reference to the forthcoming web layers. This new way of online tension control requires new concepts such as "robust multivariable control", because distributed control may not work as well.The first step consists in computing the internal stress generated within a roll of a wound web (for instance plastic film). For that purpose, a modified non-linear model is developed in the spirit of Hakiel's. The web's winding process can be considered as a continuous accretion process, in the sense that the stress components at a given point are continuously modified by the upper superimposed layers. In addition, the residual air films which separate the web layers are taken into account in an indirect way through the radial Young's modulus of the roll which is a non-linear (polynomial) function of the compressive stress component. Several illustrative examples are presented and commented. Then, having prescribed an optimization criterion for the winding tension, an optimization algorithm based on the simplex principle is described. Finally, a new concept of online tension control, based on prediction-correction is proposed. Dividing the roll radius into several segments, the tension reference is computed and corrected for each range of roll radius values, by using the predictive model for the stress within the roll. The adjusted tension is reactualized step by step, following the optimization principle as described above and it will be considered as the new tension reference value for the coming layers. A comparison between offline and online tension controls clearly shows the improvement given by the new optimization technique (online)

Control and online tension reference optimization in winding systems: Application to an identified three-motors simulator

It is well known that the tension reference value, which a priori guarantees a good quality roll, is based on the stress generated within the roll. However, due to the imperfections of the winding systems and to the limited performances of the disturbances rejection controllers, a control with fixed reference never generates perfect follow-up of the tension. A solution would consist in adjusting the tension reference online, according to real measurements.In a previous paper, the criterion for tension adjustment was the tangential stress. A method for online control based on prediction-correction using the simplex algorithm was presented. This method was tested numerically.In the present paper, we propose to generalize the criterion of tension reference optimization by considering both the tangential and the radial stress within the roll during winding. The same optimization algorithm is used, taking into account the dynamic tension model. Moreover, a dynamic gauge is now introduced, so that it can vary during the winding process. It generally represents the limits of elastic deformations of the web.The new optimization algorithm for the on-line reference tension calculation has been validated on a dynamic non-linear winding model. This complete model used for simulations was validated on a three-motor setup using brushless motors. The setup is with PI controllers, where the web velocity is imposed by master traction motor and the tension is controlled by unwinding and winding motors.In this approach, a new tension-prediction algorithm using a linear parameter varying (LPV) model is used. The influence of the tension prediction algorithm is also analyzed.Several illustrative examples will be presented and the improvement as compared to an offline control will be commented

Gauge optimization of the reference tension in winding systems using wound internal stresses calculation

In winding process, the quality of the roll is directly connected to its stress state. The winding tension is the most significant parameter which plays an important role in the stresses generated within a roll, during winding. If the stresses exceed a critical value, defects can appear in the roll and make the web non usable.This work concerns the estimation and optimization of the maximal dispersion of the reference tension, so that the tangential and radial stresses values remain in a gauge. It aims to find automatically the maximum and minimum limits for the reference tension, so that all curves ranging between these two limits or thresholds, generate radial and tangential stresses, theirs selves included in a gauge fixed in advance. The results lead to a practical gauge optimization of the reference tension for industrial applications