A variational formulation for a class of free boundary problems arising in hydrodynamic lubrication

Rohde, S. M.; McAllister, G. T.

doi:10.1016/0020-7225(75)90084-1

Cited by 39 publications

(11 citation statements)

References 4 publications

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“…[15][16][17] The cavitation problem has been widely considered by practitioners interested in free boundary problems for which the mathematical concept of complementarity is a powerful tool. In particular, a number of studies have formulated the cavitation problem in terms of variational inequalities, [18][19][20][21] while a parallel group of studies focused on the use of complementarity. [22][23][24][25] In these studies, the problem of determining the cavitated area was solved using complementarity; however, these methods do not inherently ensure the conservation of mass.…”

Section: Introductionmentioning

confidence: 99%

Transient effects in lubricated textured bearings

Medina

Fowell

Vlădescu

et al. 2015

Proceedings of the Institution of Mechanical Engineers, Part J:

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The aim of this paper is to study transient phenomena in hydrodynamic textured bearings. Both convergent and convergent-divergent reciprocating textured bearings are considered. A massconserving formulation of the Reynolds equation recently proposed by some of the authors and used to capture cavitation in steady-state lubricated contacts has been implemented to study transient effects in lubricated textured bearings. It is shown that the proposed solver is capable of capturing the frictional response of bearings characterised by various geometries and loading conditions in both steady-state and transient configurations.Depending on the boundary conditions governing the problems under investigation, changes in load support or film thickness variations are correctly predicted, demonstrating that the methodology developed in this paper is suitable to provide an efficient tool for the design and optimisation of textured bearings. A qualitative comparison with preliminary experimental data obtained using an apparatus developed to study reciprocating textured surfaces is performed, showing that the characteristic transient behaviour of such surfaces in different loading regimes can be correctly captured using the proposed numerical implementation. Current limitations of the proposed methodology and possible extensions of the technique to study elasto-hydrodynamic and mixed lubrication regimes are also discussed.

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Section: Introductionmentioning

confidence: 99%

Transient effects in lubricated textured bearings

Medina

Fowell

Vlădescu

et al. 2015

Proceedings of the Institution of Mechanical Engineers, Part J:

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“…These necessary and sufficient conditions (15) were proved first by Rohde and Mallister [8] where, however, only one-dimensional problem was discussed. Now we prove (15) in general case.…”

Section: Fundamental Theorem For Nonlinear Forces Of Bearingsmentioning

confidence: 99%

Nonlinear modeling and dynamic analysis of the rotor-bearing system

et al. 2008

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To overcome the shortcomings of extreme time-consuming in solving the Reynolds equation, two efficient calculation methods, based on the free boundary theory and variational principles for the unsteady nonlinear Reynolds equation in the condition of Reynolds boundary, are presented in the paper. By employing the two mentioned methods, the nonlinear dynamic forces as well as their Jacobians of the journal bearing can be calculated saving time but with the same accuracy. Of these two methods, the one is called a Ritz model which manipulates the cavitation region by simply introducing a parameter to match the free boundary condition and, as a result, a very simple approximate formulae of oil-film pressure is being obtained. The other one is a one-dimensional FEM method which reduces the two-dimensional variational inequality to the one-dimensional algebraic complementary equations, and then a direct method is being used to solve these complementary equations, without the need of iterations, and the free boundary condition can be automatically satisfied. Meanwhile, a new order reduction method is contributed to reduce the degrees of freedom of a complex rotor-bearing system. Thus the nonlinear behavior analysis of the rotorbearing system can be studied time-sparingly. The results in the paper show the high efficiency of the two methods as well as the abundant nonlinear phenomenon of the system, compared with the results obtained by the usual numerical solution of the Reynolds equation.

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“…The possibility of studying the free boundary problem A with the techniques described by H. Lewy and G. Stampacchia in [6] has been noticed first in [5] and subsequently investigated in [2], [10].…”

Section: To Find a Function P Defined In Q And An Open Set F~ C Q Sucmentioning

confidence: 99%