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Hydrodynamic Lubrication, Volume 33
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You do not currently have access to this content. The flow is regarded as compressible, the viscosity can be shear rate and pressure dependent non-Newtonian and piezo-viscous and elastic deformation of the contacting surfaces can be considered.
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The hydrodynamic pressures that develop may be large enough to deform the surfaces. The most frequently used approach for describing the pressure dependence of the film thickness, is based on standard contact mechanics results for line and point loading of elastic half-spaces. These are detailed in Chapters 2 and 3 in the book by Johnson [ 16 ] and in [ 2 ]. The classical thin film approximation, presented in e.
Remark: an alternative, one-dimensional, thin film approximation was introduced in [ 17 ]. Indeed, let. In this case, the correspondence to Eq. Integrating Eq. With the aim set to derive an Reynolds type of equation for the pressure in the lubricant, the analysis continues by first formulating an expression for the mass flow and thereafter requiring continuity of the mass flow. By using Eq. With this expression for q , the requirement for conservation of mass reads. In order to incorporate the effect of cavitation, we assume that the following holds.
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In the full film zones, the pressure is larger than the cavitation pressure and the density is expressed by Eq. Hence the pressure—density relationship in both the full film and the cavitated zones becomes. In this notation, Eq. Preservation of mass flow is ensured by inserting Eq.
The system in Eq. In this section, we present a numerical solution procedure for the cavitation model Eq. In this notation the system Eq. A spatial finite difference discretization of the problem Eq. Since the finite difference approximations of the partial derivatives w. A central difference scheme is therefore used to approximate the derivatives of the first term in the right hand side of Eq.
In this notation, the first term in the right hand side of Eq. Since h and f depend on p , a numerical solution procedure can be posed as follows:. Solve the now obtained linear complementarity problem corresponding to Eq. To fully discretize the problem at hand several approaches can be applied. For instance, first order forward explicit or backward implicit Euler, the second order implicit Crank-Nicolson method.
In this section, the numerical solution procedure, for the cavitation model in Eq. In all four examples, only the lower surface is moving, i.
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In the first example, results obtained with the present approach are compared to results for the one-dimensional parabolic slider in [ 8 ]. Two different pressure—density relationships are examined. The second example extends the first one in the sense that besides cavitation the bearing surfaces are also assumed to be linear elastic. In particular, this means that the film thickness in addition depends on the pressure.
nttsystem.xsrv.jp/libraries/75/moc-handys-ausspionieren.php The film thickness, in this case, consists of three parts; a parameter related to the minimum film thickness h 0 , the bearing geometry g , and the elastic deformation of the bearing surfaces. The Boussinesq-Cerruti half space solution is used to model the elastic deformation [ 2 ]. Indeed, h is given by.
The third example considers a double parabolic slider, which in addition to the single parabolic slider, exhibits reformation and highlight that mass is conserved. The fourth and last example considers a quadruple parabolic slider bearing. The reason for choosing this configuration is to test the hypothesis that an elastically deformable bearing, in general, does not generate more film than the corresponding rigid one.
In all examples, the initial undeformed bearing geometry consists of 1, 2 or 4 parabolic parts. These bearing geometries, g n , can then be described via the L -periodic auxiliary function G. In this example, a model problem with rigid surfaces and two different Newtonian lubricants is considered in order to compare with previous results presented in [ 8 ]. Indeed, a single parabolic slider bearing of length L with rigid surfaces and film thickness of the form. Two different Newtonian lubricants are studied, one which obeys the constant bulk modulus pressure—density relationship;.
In the case with the Dowson-Higginson compressibility, force-balance was included and the load carrying capacity obtained for the constant bulk modulus case was used as the applied load. It should be noted, that it is typically desired to have a sufficiently thick lubricant film, so that the surface are completely separated to reduce wear. At the same time, it is clear, that the friction hydrodynamic will increase with the increase of both, sliding speed and viscosity. This in turn leads to energy losses.
Therefore, there will be a trade-off between the wear performance and the optimization of energy losses. Currently, continuous efforts are undertaken to reduce the energy losses and to move towards a sustainable society and at the same time increase the reliability of tribological devices. According to the discussion above it is clear that there is a contradiction between the wear performance and the energetic performance.
Therefore, classical lubrication theory has reached its fundamental limit in the energy losses reduction and new theories have to be developed.
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