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Concept# Mechanical equilibrium

Summary

In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero.
In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent. In terms of momentum, a system is in equilibrium if the momentum of its parts is all constant. In terms of velocity, the system is in equilibrium if velocity is constant. In a rotational mechanical equilibrium the angular momentum of the object is conserved and the net torque is zero. More generally in conservative systems, equilibrium is established at a point in configuration space where the gradient of the potential energy with respect to the generalized coordinates is zero.
If a particle in equilibrium has zero velocity, that pa

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Reducing friction and wear is essential for building efficient systems with low energy consumption and a long lifetime. Surface texturing is one of the methods to reduce friction and wear, especially in oil-lubricated systems. However, there is still a lack of knowledge about the fundamental aspects involved in the improvement of tribological performance, especially for the mechanical parts sliding under starved lubrication conditions. Moreover, failure in such lubricated contacts is poorly understood and there are little agreements regarding the mechanisms leading to scuffing. The present research focuses on three main objectives. The first is to characterize a cast iron-steel tribo-system under starved lubrication conditions, and to improve its tribological performance by laser surface texturing. All tribo-tests are performed using a flat-on-flat set-up with reciprocal movements. This set-up simulates specific industrial operating conditions of a semi-journal bearing used in a cutting machine. A Design of Experiments (DoE) approach is selected to investigate the effect of the different geometrical micro-texture parameters on the coefficient of friction (COF) and the lifetime of the cast iron samples. An optimum micro-texture geometry leading to a low COF and a long lifetime is determined based on a fractional factorial design. The second goal is to investigate the failure mechanisms of the selected tribo-pair. Failure of the tribo-system is characterized for both textured and un-textured surfaces. Scuffing is found to be the failure mechanism for the un-textured cast iron samples. For the textured cast iron samples, in contrast, two different failure mechanisms are observed depending on the distance between the micro-textures in direction of sliding (DMS). The textured surfaces with a DMS > 3.5 mm fail via the scuffing mechanism, whereas the surfaces with a DMS < 3 mm fail by a different mechanism named the oxidation mechanism. In contrary to the scuffing mechanism, the oxidation failure is found to be more a gradual failure. Based on the experimental observations, two hypotheses are suggested for the scuffing mechanism, and one for the oxidation mechanism. Finally, predicting the lifetime of a tribo-system is of the utmost importance to save costs. The DoE model developed for the lifetime was found to be a weak predictive model due to the sudden nature of scuffing. The accuracy of this model could only be improved by performing large statistics, which is not time and cost effective. Consequently, the last objective of this work is to employ an acoustic emission (AE) technique to detect precisely the surface state of our tribo-system. The application of wavelet packet decomposition, as an advanced signal processing method used for AE signals, is found to be promising for early detection of scuffing. The extension of this work using random forest regression shows the possibility of predicting scuffing 5 min before it occurs.

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model. (C) 2016 Elsevier B.V. All rights reserved.

We study the evolution of a system composed of N non-interacting particles of mass m distributed in a cylinder of length L. The cylinder is separated into two parts by an adiabatic piston of a mass M ≫ m. The length of the cylinder is a fix parameter and can be finite or infinite (in this case N is infinite). For the infinite case we carry out a perturbative analysis using Boltzmann's equation based on a development of the velocity distribution of the piston in function of a small dimensionless parameter ε = √(m/M). The non-stationary case is solved up to the order ε ;; our analysis shows that the system tends exponentially fast towards a stationary state where the piston has an average velocity V. The characteristic time scale for this relaxation is proportional to the mass of the piston (τ0 = M/A where A is the cross-section of the piston). We show that for equal pressures the collisions of the particles induce asymmetric fluctuations of the velocity of the piston which leads to a macroscopic movement of the piston in the direction of the higher temperature. In the case of the finite model a perturbative approach based on Liouville's equation (using the parameter α = 2m/(M + m)) shows that the evolution towards thermal equilibrium happens on two well separated time scales. The first relaxation step is a fast, deterministic and adiabatic evolution towards a state of mechanical equilibrium with approximately equal pressures but different temperatures. The movement of the piston is more or less damped. This damping qualitatively depends on whether the ratio R = Mgas/M between the total mass of the gas and the mass of the piston is small (R < 2) or large (R > 4). The second part of the evolution is much slower ; the typical time scales are proportional to the mass of the piston. There is a stochastic evolution including heat transfer leading to thermal equilibrium. A microscopic analysis yields the relation XM(t) = L(1/2 - ξ(at)) where the function ξ is independent of M. Using the hypothesis of homogeneity (i.e. the values of the densities, pressures and temperatures at the surface of the piston can be replaced by their respective average values) introduced in the previous analysis the observed damping does not show up. This can be explained by shock waves propagating between the piston and the walls at the extremities of the cylinder. In order to study the behaviour of the system there is hence a need to adequately describe the non-equilibrium fluids around the piston. We carry out an analysis of the infinite case, based on the perturbative approach introduced earlier. In this case the initial conditions are chosen in such a manner that the piston on average stays at the origin. It is shown that it is possible to describe the evolution of the fluids in such a way that it is coherent with the two laws of thermodynamics and the phenomenological relationships. Finally we study the case of a constant velocity of the piston in a finite cylinder. Such a condition and elastic collisions allow us to derive an explicit expression for the distribution of the fluids and hence for the hydrodynamics fields. This expression reveals the presence of shock waves between the piston and the extremities of the cylinder.

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