On the equivalence of modes of convergence for log-concave measures

An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note, we explore the extent to which different notions of distance between probability measures are comparable for log-concave distributions. Our results imply that weak convergence of isotropic log-concave distributions is equivalent to convergence in total variation, and is further equivalent to convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note

A hypoid gear pair tribo-dynamic model taking into account the rheological behaviour of fully formulated gear lubricants

A fully coupled tribo-dynamic model, capable of predicting the inefficiency and dynamic response of automotive differential hypoid gear pairs, is presented in this study. A gear dynamics solver is coupled with an analytical friction solver, which calculates the viscous shear, as well as the boundary conjunctional friction force. The time varying geometry and contact characteristics of the hypoid gear pair are taken into account by using realistic data available in the literature. The rheological models employed cover a range of two different behaviors: Newtonian and non-Newtonian Eyring (shear thinning). The Chittenden-Dowson equation is used to calculate the central film thickness of the elasto-hydrodynamic teeth conjunctions. The boundary friction force is calculated using the Greenwood & Tripp model. Finally, the actual surface topography of a run-in hypoid gear is obtained using a stylus profilometer. The results indicate an overestimation of the viscous friction by the Newtonian model, as opposed with the non-Newtonian model, mainly due to shear thinning effects. Comparative studies are performed for different operating conditions, namely near or away from resonance, as well as for conditions corresponding to a non-linear sub-harmonic resonance. The frictional damping effect on the dynamic transmission error, which is an indication of the NVH response of the gear pair, is also examin

Isothermal Elastohydrodynamic Lubrication Analysis of Heavily Loaded Hypoid Gear Pairs

A numerical model able to predict the pressure distribution and the film thickness in heavily loaded elliptical EHL contacts is developed and presented in this study. The operating conditions, such as the contact load and the velocities of the mating surfaces, are representative of the corresponding conditions present in automotive differential hypoid gear pair units. The EHL solver presented is able to predict the minimum and central film thickness of the lubricating oil as well as the pressure distribution assuming isothermal and Newtonian conditions. Results are presented for a full quasi-static meshing cycle. A comparison between the numerically calculated values of the central and the minimum film thickness is performed against the corresponding values produced using the Chittenden-Dowson formula. A very good agreement is observed between the values of the central film thickness. However, it is shown that the minimum film thickness values using the Chittenden-Dowson formula can deviate up to 40% compared with the corresponding values which are calculated numerically

Film Thickness Investigation in Heavily Loaded Hypoid Gear Pair Elastohydrodynamic Conjunctions

Introduction: Hypoid gear pairs are some of the most highly loaded components of the differential unit in modern automobiles. Prediction of wear rate and generated friction require determination of lubricant film thickness. However, only very few investigations have addressed the issue of thin elastohydrodynamic films in hypoid gear pairs. The main reason for dearth of analysis in this regard has been the need for accurate determination of transient contact geometry and kinematics of interacting surfaces throughout a typical meshing cycle. Furthermore, combined gear dynamics and lubrication analysis of any pairs of simultaneous meshing teeth pairs is required. Simon [1] was among the first to deal with these issues. He used Tooth Contact Analysis (TCA) in order to calculate the instantaneous contact geometry and load for any teeth pair during their meshing cycle. However, in his study, the load carried by the hypoid pair was quite low, making the application of the results limited and not entirely suitable for real life operating conditions of typical hypoid gear pairs of vehicular differentials, which is of interest in the current paper. Xu and Kahraman [2] performed numerical prediction of power losses and consequently the film thickness for highly loaded hypoid gear pairs. However, in their study only the one-dimensional Reynolds equation was employed. Consequently, the effect of lubricant side leakage in the passage through the contact was ignored. A more recent study by Mohammadpour et al. [3] employed realistic gear geometry data (through the use of TCA) for calculation of film thickness time history through mesh. The two-dimensional Reynolds equation, accounting for the side leakage of the lubricant, was solved numerically. It was shown that the side leakage component of the entraining velocity can significantly influence the film thickness. With regard to hypoid gear dynamics, several studies should be mentioned. Wang and Lim [4] studied the dynamic response of hypoid gear pairs under the influence of time varying meshing stiffness. Yang and Lim [5] created a model able to predict the dynamic response of a hypoid gear pair by taking into account the lateral translations of their shafts due to the compliance of the supporting bearings. Karagiannis et al. [6-7] studied the dynamics of automotive differential hypoid gear pairs by taking into account the velocity dependent resistive torque at the gear caused by aerodynamic drag and tyre-road rolling resistance. The study integrated the gear dynamics with the generated viscous and boundary conjunctional friction

Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below. As our main application, we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of various conservative spin system models, consisting of non-interacting and weakly-interacting particles, constrained to conserve the mean-spin. When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg and Yau. When the self-interaction is only assumed to be (non-strongly) convex, as in the case of the two-sided exponential measure, we obtain sharp estimates on the system's spectral-gap as a function of the mean-spin, independently of the size of the system.Comment: 57 page

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

Heat generation and transfer in automotive dry clutch engagement

Dynamic behaviour of automotive dry clutches depends on the frictional characteristics of the contact between the friction lining material, the flywheel, and the pressure plate during the clutch engagement process. During engagement due to high interfacial slip and relatively high contact pressures, generated friction gives rise to contact heat, which affects the material behaviour and the associated frictional characteristics. In practice excess interfacial slipping and generated heat during torque transmission can result in wear of the lining, thermal distortion of the friction disc, and reduced useful life of the clutch. This paper provides measurement of friction lining characteristics for dry clutches for new and worn state under representative operating conditions pertaining to interfacial slipping during clutch engagement, applied contact pressures, and generated temperatures. An analytical thermal partitioning network model of the clutch assembly, incorporating the flywheel, friction lining, and the pressure plate is presented, based upon the principle of conservation of energy. The results of the analysis show a higher coefficient of friction for the new lining material which reduces the extent of interfacial slipping during clutch engagement, thus reducing the frictional power loss and generated interfacial heating. The generated heat is removed less efficiently from worn lining. This might be affected by different factors observed such as the reduced lining thickness and the reduction of density of the material but mainly because of poorer thermal conductivity due to the depletion of copper particles in its microstructure as the result of wear. The study integrates frictional characteristics, microstructural composition, mechanisms of heat generation, effect of lining wear, and heat transfer in a fundamental manner, an approach not hitherto reported in literature

Multiscale friction in lubricant-surface systems for high performance transmissions under mild wear

The lubricant-surface system is complex in nature and can significantly affect the frictional performance of high-performance transmission systems. The complexity stems from the coupled mechanical and chemical phenomena that occur at the interfacial tooth conjunctions. A combined analytical and precision experimental approach is presented to analyse the salient parameters of the lubricant-surface system. A multiscale procedure comprising topographical measurement, pin-on-disc tribometry, atomic force microscopy in lateral force mode, X-ray photo-electron spectroscopy and continuum contact mechanics analysis under mixed non-Newtonian thermo-elastohydrodynamics is used to describe the formation of a tribo-film, as well as wear and frictional characteristics of the lubricant-surface system. The contribution of chemisorbed and physisorbed bonded tribo-film on the boundary coefficient of friction is ascertained at different physical scales. Therefore, the paper presents a novel multiscale analysis, promoting improved understanding of the complex interactions between mechanisms of friction, wear and surface chemistry

Parameter distributions for the drag-based modeling of CME propagation

In recent years, ensemble modeling has been widely employed in space weather to estimate uncertainties in forecasts. We here focus on the ensemble modeling of Coronal Mass Ejections (CME) arrival times and arrival velocities using a drag-based model, which is well-suited for this purpose due to its simplicity and low computational cost. Although ensemble techniques have previously been applied to the drag-based model, it is still not clear how to best determine distributions for its input parameters, namely the drag parameter and the solar wind speed. The aim of this work is to evaluate statistical distributions for these model parameters starting from a list of past CME-ICME events. We employ LASCO coronagraph observations to measure initial CME position and speed, and in situ data to associate them with an arrival date and arrival speed. For each event we ran a statistical procedure to invert the model equations, producing parameters distributions as output. Our results indicate that the distributions employed in previous works were appropriately selected, even though they were based on restricted samples and heuristic considerations. On the other hand, possible refinements to the current method are also identified, such as the dependence of the drag parameter distribution on the CME being accelerated or decelerated by the solar wind, which deserve further investigation

An analytical approach for prediction of elastohydrodynamic friction with inlet shear heating and starvation

An analytical friction model is presented, predicting the coefficient of friction in elastohydrodynamic (EHD) contacts. Three fully formulated SAE 75W-90 axle lubricants are examined. The effect of inlet shear heating (ISH) and starvation is accounted for in the developed friction model. The film thickness and the predicted friction are compared with experimental measurements obtained through optical interferometry and use of a mini traction machine. The results indicate the significant contribution of ISH and starvation on both the film thickness and coefficient of friction. A strong interaction between those two phenomena is also demonstrated, along with their individual and combined contribution on the EHD friction