×

On the Reynolds equation and the load problem in lubrication: literature review and mathematical modelling. (English) Zbl 1430.74039

Smith, Frank T. (ed.) et al., Mathematics applied to engineering, modelling, and social issues. Cham: Springer. Stud. Syst. Decis. Control 200, 1-43 (2019).
Summary: In this chapter, we provide a literature review concerning the theory of hydrodynamic lubrication, especially applied to journal bearings. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent contact. In particular, we derive the fluid film thickness model for journal bearings, considering both the parallel and the misaligned case. The hydrodynamic Reynolds equation with cavitation phenomenon, through both Reynolds and Elrod-Adams models are fully derived in this chapter. Subsequently, we pose two suitable variational formulations for the hydrodynamic problem considering both cavitation models. In addition, we present the admissible range of misalignment angle projections for prescribed values of the shaft eccentricity and angular coordinate. Finally, we properly state the problem of a loaded misaligned journal bearing for stationary regime, considering the balance of force and torque components involved.
For the entire collection see [Zbl 1418.76007].

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D08 Lubrication theory
74S99 Numerical and other methods in solid mechanics

Software:

MACS-VRPTW
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abass, B.A., Sahib, M.M.: Effect of bearing compliance on thermo-hydrodynamic lubrication of high speed misaligned journal bearing lubricated with bubbly oil. Ind. Eng. Lett 3, 48-60 (2013)
[2] Álvarez, S.J.: Problemas de frontera libre en teoría de lubrificación. Ph.D. thesis. Universidad Complutense de Madrid (1986)
[3] Asanabe, S., Akakoski, M., Asai, R.: Theoretical and experimental investigation on misaligned journal bearing performance. Ind. Lubr. Tribol. 23(6), 208 (1971)
[4] Bayada, G., Chambat, M.: Sur quelques modélizations de la zone de cavitation en lubrification hydrodynamique. J. Theor. Appl. Mech. 5(5), 703-729 (1986) · Zbl 0621.76030
[5] Bayada, G., Chambat, M.: The transition between the Stokes equation and the Reynolds equation: a mathematical proof. Appl. Math. Opt. 14(1), 73-93 (1986) · Zbl 0701.76039
[6] Bayada, G., Chambat, M., Vázquez, C.: Characteristics method for the formulation and computation of a free boundary cavitation problem. J. Comput. Appl. Math. 98(2), 191-212 (1998) · Zbl 0948.76064
[7] Bayada, G., Martin, S., Vázquez, C.: An average flow model of the Reynolds roughness including a mass-flow preserving cavitation model. J. Tribol-T. ASME. 127(4), 793-802 (2005)
[8] Bayada, G., Martin, S., Vázquez, C.: Homogenization of a nonlocal elastohydrodynamic lubrication problem: a new free boundary model. Math. Mod. Meth. Appl. S. 15(12), 1923-1956 (2005) · Zbl 1080.74041
[9] Bayada, G., Martin, S., Vázquez, C.: Homogneisation du modèle d’Elrod-Adams hydrodynamique. Asymptotic. Anal. 44, 75-110 (2005) · Zbl 1108.35007
[10] Bayada, G., Martin, S., Vázquez, C.: Two-scale homogenization of a hydrodynamic Elrod-Adams model. Asymptotic. Anal. (2005) · Zbl 1108.35007
[11] Bayada, G., Martin, S., Vázquez, C.: Micro-roughness effects in (elasto) hydrodynamic lubrication including a mass-flow preserving cavitation model. Tribol. Int. 39(12), 1707-1718 (2006)
[12] Bayada, G., Vázquez, C.: A survey on mathematical aspects of lubrication problems. Boletín SeMA 39, 37-74 (2007) · Zbl 1242.76040
[13] Bermúdez, A., Durany, J.: Numerical solution of cavitation problems in lubrication. Comput. Methods Appl. Mech. Eng. 75, 455-466 (1989) · Zbl 0687.76030
[14] Bermúdez, A., Moreno, C.: Duality methods for solving variational inequalities. Comp. Math. Appl. 7, 43-58 (1981) · Zbl 0456.65036
[15] Blum, C., Vallès, M.Y., Blesa, M.J.: An ant colony optimization algorithm for DNA sequencing by hybridization. Comput. Oper. Res. 35(11), 3620-3635 (2008) · Zbl 1167.92011
[16] Boedo, S., Booker, J.F.: Classical bearing misalignment and edge loading: a numerical study of limiting cases. J. Tribol-T. ASME. 126(3), 535-541 (2004)
[17] Boucherit, H., Lahmar, M., Bou-Said, B.: Misalignment effects on steady-state and dynamic behaviour of compliant journal bearings lubricated with couple stress fluids. Lubr. Sci. 20(3), 241-268 (2008)
[18] Bouyer, J., Fillon, M.: An experimental analysis of misalignment effects on hydrodynamic plain journal bearing performances. J. Tribol. 124(2), 313-319 (2002)
[19] Bouyer, J., Fillon, M.: Improvement of the THD performance of a misaligned plain journal bearing. J. Tribol. 125(2), 334-342 (2003)
[20] Brito, F.P.: Thermohydrodynamic performance of twin groove journal bearings considering realistic lubricant supply conditions: a theoretical and experimental study. Ph.D. thesis. Universidade do Minho (2009)
[21] Calvo, N., Durany, J., Vázquez, C.: Comparación de algoritmos numéricos en problemas de lubricación hidrodinámica con cavitación en dimensión uno. Rev. Int. Metod. Numer. 13(2), 185-209 (1997)
[22] Capriz, G., Cimatti, G.: Free boundary problems in the theory of hydrodynamic lubrication: a survey. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, vol. 2. Research Notes in Mathematics 79. Pitman, pp. 613-635 (1983) · Zbl 0557.76038
[23] Chelouah, R., Siarry, P.: Enhanced continuous tabu search: an algorithm for optimizing multiminima functions. In: Meta-Heuristics. Advances and Trends in Local Search Paradigms for Optimization (Chap. 4), pp. 49-61. Springer, Berlin (1999) · Zbl 1073.90578
[24] Chelouah, Rachid, Siarry, Patrick: A continuous genetic algorithm designed for the global optimization of multimodal functions. J. Heuristics 6(2), 191-213 (2000) · Zbl 0969.68641
[25] Christopherson, D.G.: A new mathematical method for the solution of film lubrication problems. Inst. Mech. Engrs. J. Proc. 146, 126-135 (1941) · Zbl 0063.00888
[26] Cimatti, G.: On certain nonlinear problems arising in the theory of lubrication. Appl. Math. Opt. 11(1), 227-245 (1984) · Zbl 0597.76066
[27] Cimatti, G.: Existence and uniqueness for nonlinear Reynolds equations. Int. J. Eng. Sci. 24(5), 827-834 (1986) · Zbl 0624.76090
[28] Ciuperca, I.S., Hafidi, I., Jai, M.: Singular perturbation problem for the incompressible Reynolds equation. Electr. J. Differ. Equ. 2006(83), 1-19 (2006) · Zbl 1109.76018
[29] Ciuperca, I.S., Jai, M., Tello, J.I.: On the existence of solutions of equilibria in lubricated journal bearings. SIAM J. Math. Anal. 40(6), 2316-2327 (2009) · Zbl 1180.35423
[30] Ciuperca, I.S., Jai, M., Tello, J.I.: Equilibrium analysis for a mass-conserving model in presence of cavitation. Nonlinear Anal.: R. World Appl. 35, 250-264 (2017) · Zbl 1367.35119
[31] Ciuperca, I.S., Tello, J.I.: Lack of contact in a lubricated system. Q. Appl. Math. 69(2), 357-378 (2011a) · Zbl 1219.35065
[32] Ciuperca, I.S., Tello, J.I.: On a variational inequality on elasto-hydrodynamic lubrication. J. Math. Anal. Appl. 383(2), 597-607 (2011b) · Zbl 1229.35099
[33] Cryer, C.W.: The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences. Math. Comput. 25(115), 435-443 (1971) · Zbl 0223.65044
[34] Cuvelier, C.: A free boundary problem in hydrodynamic lubrication including surface tension. In: Cabannes, H., Holt, M., Rusanov, V. (eds.) Sixth International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, vol. 90, pp. 143-148. Springer, Berlin (1979)
[35] Das, S., Guha, S.K., Chattopadhyay, A.K.: On the steady-state performance of misaligned hydrodynamic journal bearings lubricated with micropolar fluids. Tribol. Int. 35(4), 201-210 (2002)
[36] Díaz, J.I., Tello, J.I.: A note on some inverse problems arising in lubrication theory. Differ. Integral. Equ. 17, 583-592 (2004) · Zbl 1224.35335
[37] Díaz, M., Lombera, H., et al.: An approach for assembly sequence planning based on MAX-MIN Ant System. IEEE Latin Am. Trans. 13(4), 907-912 (2015)
[38] Dorigo, M.: Optimization, learning and natural algorithms. Italian. Ph.D. thesis. Dipartimento di Elettronica, Politecnico di Milano, Milan, Italy (1992)
[39] Dorigo, M., Maniezzo, V., Colorni, A.: Ant system: optimization by a colony of cooperating agents. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 26(1), 29-41 (1996)
[40] Dowson, D.: A generalized Reynolds equation for fluid film lubrication. Int. J. Mech. Sci. 4, 159-170 (1962)
[41] Dubois, G.B., Mabie, H.H., Ocvirk, F.W.: Experimental investigation of oil film pressure distribution for misaligned plain bearings. Technical Report. National Advisory Committee for Aeronautics, KittyHawk, NC, USA (1951)
[42] Durany, J., García, G., Vázquez, C.: Numerical simulation of a lubricated hertzian contact problem under imposed load. Finite. Elem. Anal. Des. 38(7), 645-658 (2002) · Zbl 1073.74048
[43] Durany, J., Pereira, J., Varas, F.: Numerical solution to steady and transient problems in thermohydrodynamic lubrication using a combination of finite element, finite volume and boundary element methods. Finite. Elem. Anal. Des. 44(11), 686-695 (2008)
[44] Durany, J., Pereira, J., Varas, F.: Dynamical stability of journal-bearing devices through numerical simulation of thermohydrodynamic models. Tribol. Int. 43, 1703-1718 (2010)
[45] Durany, J., Vázquez, C.: Numerical approach of lubrication problems in journal bearing devices with axial supply. Numer. Methods Eng. 92, 839-844 (1992)
[46] El Alaoui Talibi, M., Bayada, G.: Une méthode du type caractéristique pour la résolution d’un problème de lubrification hydrodynamique en régime transitoire. ESAIM-Math. Model. Num. 25(4), 395-423 (1991) · Zbl 0748.76040
[47] Elrod, H.G., Adams, M.L.: A computer program for cavitation and starvation problems. Cavitation and related phenomena in lubrication: proceedings of the 1st Leeds-Lyon Symposium on Tribology. Mech. Eng. Publ. pp. 37-42 (1975)
[48] Frêne, J., Nicolas, D., et al.: Hydrodynamic Lubrication: Bearings and Thrust Bearings, vol. 33. Elsevier, Amsterdam (1997) · Zbl 0904.76002
[49] Gambardella, L.M., Taillard, É., Agazzi, G.: MACS-VRPTW: a multiple ant colony system for vehicle routing problems with time windows. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization (Chap. 5), pp. 63-76. McGraw-Hill, London (1999)
[50] Garcia-Najera, A., Bullinaria, J.A.: Extending ACOR to solve multi-objective problems. In: Proceedings of the UK Workshop on Computational Intelligence, London (2007)
[51] Gómez-Mancilla, J., Nosov, V.: Short journal bearing with misaligned axes. In: Proceedings 1st International Symposium on Control of Rotating Machinery (2001)
[52] Gómez-Mancilla, J., Nosov, V.: Perturbed pressure field solution for misaligned short journal bearings. In: Proceedings 9th International Symposium on Transport Phenomena and Rotating Machinery. ISROMAC-9, Hawaii, EU (2002)
[53] Guha, S.K.: Analysis of steady-state characteristics of misaligned hydrodynamic journal bearings with isotropic roughness effect. Tribol. Int. 33(1), 1-12 (2000)
[54] He, Z., Zhang, J., et al.: Misalignment analysis of journal bearing influenced by asymmetric deflection, based on a simple stepped shaft model. J. Zhejiang Univ. Sci. A 13(9), 647-664 (2012)
[55] Jang, J.Y., Khonsari, M.M.: Design of bearings on the basis of thermohydrodynamic analysis. In: Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, vol. 218, No. 5, pp. 355-363. Sage, Beverley Hills (2004)
[56] Jang, J.Y., Khonsari, M.M.: On the characteristics of misaligned journal bearings. Lubricants 3(1), 27-53 (2015)
[57] Jang, J.Y., Khonsari, M.M., Bair, S.: On the elastohydrodynamic analysis of shear-thinning fluids. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 463, No. 2088, pp. 3271-3290. The Royal Society (2007)
[58] Khonsari, M.M., Booser, E.R.: Applied Tribology: Bearing Design and Lubrication, 2nd edn. Wiley, New York (2008). ISBN: 978-0-470-05711-7
[59] Kumar, P., Khonsari, M.M.: Traction in EHL line contacts using free-volume pressureviscosity relationship with thermal and shear-thinning effects. J. Tribol. 131(1), 011503 (2009)
[60] Leguizamón, G., Coello, C.A.: A study of the scalability of ACOR for continuous optimization problems. Technical report, The Evolutionary Computation Group at CINVESTAV-IPN (2010)
[61] Leguizamón, G., Coello, C.A.: An alternative ACOR algorithm for continuous optimization problems. In: Swarm Intelligence: 7th International Conference, ANTS 2010, vol. 6234, pp. 48-59. Lecture Notes in Computer Science, Brussels, Belgium. Springer, Berlin (2010)
[62] Liao, T.: Improved ant colony optimization algorithms for continuous and mixed discrete continuous optimization problems. MA thesis. Université Libre de Bruxelles, Belgium (2011)
[63] Liao, T., Montes de Oca, M.A., et al.: An incremental ant colony algorithm with local search for continuous optimization. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, pp. 125-132. ACM, New York (2011)
[64] Liao, T., Stützle, T., et al.: A unified ant colony optimization algorithm for continuous optimization. Euro. J. Oper. Res. 234(3), 597-609 (2014) · Zbl 1304.90221
[65] Liñan, A.: Curso de Mecánica de Fluidos. Technical Report, Universidad Politécnica de Madrid (1999)
[66] Liu, G.: Dual variational principles for the free-boundary problem of cavitated bearing lubrication. In: Gao, D.Y. (ed.) Complementarity, Duality and Symmetry in Nonlinear Mechanics: Proceedings of the IUTAM Symposium, vol. 6, pp. 179-189. Springer Science & Business Media (2004) · Zbl 1118.76316
[67] Lombera, H., Tello, J.I.: A numerical approach to solve an inverse problem in lubrication theory. RACSAM. Rev. R. Acad. A. 108(2), 617-631 (2014) · Zbl 1432.65140
[68] Martin, S.: Influence of multiscale roughness patterns in cavitated flows: applications to journal bearings. Math. Probl. Eng. 2008, 1-26 (2008) · Zbl 1151.76430
[69] Maru, M.M., Tanaka, D.K.: Consideration of stribeck diagram parameters in the investigation on wear and friction behavior in lubricated sliding. J. Braz. Soc. Mech. Sci. Eng. 29(1), 55-62 (2007)
[70] McKee, S.A., McKee, T.R.: Pressure distribution in oil films of journal bearings. ASME 54, 149-165 (1932)
[71] Nikolakopoulos, P.G., Papadopoulos, C.A.: Non-linearities in misaligned journal bearings. Tribol. Int. 27(4), 243-257 (1994)
[72] Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Science + Business Media, LLC, Berlin (2006) · Zbl 1104.65059
[73] Ocvirk, F.W., DuBois, G.B.: Analytical derivation and experimental evaluation of shortbearing approximation for full journal bearings. Technical Report 1157. NACA (1953)
[74] Osman, T.A.: Misalignment effect on the static characteristics of magnetized journal bearing lubricated with ferrofluid. Tribol. Lett. 11(3-4), 195-203 (2001)
[75] Pedraza, G., Díaz, M., Lombera, H.: An approach for assembly sequence planning by genetic algorithms. IEEE Latin Am. Trans. 14(5), 2066-2071 (2016)
[76] Pierre, I., Bouyer, J., Fillon, M.: Thermohydrodynamic study of misaligned plain journal bearings-comparison between experimental data and theoretical results. Appl. Mech. Eng. 7(3), 949-960 (2002)
[77] Pierre, I., de France, E., et al.: Thermohydrodynamic behavior of misaligned plain journal bearings: theoretical and experimental approaches. Tribol. Trans. 47(4), 594-604 (2004)
[78] Piggott, R.J.S.: Bearings and lubrication. Bearing troubles traceable to design can be avoided by engineering study. Mech. Eng. 64, 259 (1942)
[79] Piniganti, L.: A Survey of Tabu search in combinatorial optimization. MA thesis. University of Nevada, Las Vegas (2014)
[80] Pinkus, O., Bupara, S.S.: Analysis of misaligned grooved journal bearings. J. Tribol. 101(4), 503-509 (1979)
[81] Pinkus, O., Sternlicht, B.: Theory of Hydrodynamic Lubrication. McGraw-Hill, New York (1961) · Zbl 0100.23001
[82] Reynolds, O.: On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. Roy. Soc. A 117, 157-234 (1886) · JFM 18.0946.04
[83] Rodrigues, J.F.: Remarks on the Reynolds problem of elastohydrodynamic lubrication. Eur. J. Appl. Math. 4(01), 83-96 (1993) · Zbl 0768.76016
[84] Rohde, S.M., McAllister, G.T.: A variational formulation for a class of free boundary problems arising in hydrodynamic lubrication. Int. J. Eng. Sci. 13(9-10), 841-850 (1975) · Zbl 0307.76018
[85] Sharma, S.C., Jain, S.C., Nagaraju, T.: Combined influence of journal misalignment and surface roughness on the performance of an orifice compensated non-recessed hybrid journal bearing. Tribol. Trans. 45(4), 457-463 (2002)
[86] Siarry, Patrick, Berthiau, Gérard, et al.: Enhanced simulated annealing for globally minimizing functions of many-continuous variables. ACM Trans. Math. Softw. (TOMS) 23(2), 209-228 (1997) · Zbl 0887.65067
[87] Socha, K.: Ant colony optimization for continuous and mixed-variable domains. PhD thesis. Université Libre de Bruxelles, Belgium (2008) · Zbl 1146.90537
[88] Socha, K., Dorigo, M.: Ant colony optimization for continuous domains. Euro. J. Oper. Res. 185(3), 1155-1173 (2008) · Zbl 1146.90537
[89] Srinivas, M., Patnaik, L.M.: Genetic algorithms: a survey. Computer 27(6), 17-26 (1994)
[90] Stachowiak, G., Batchelor, A.W.: Engineering Tribology. Butterworth-Heinemann, Oxford (2013)
[91] Stieber, W.: Das Schwimmlager. Technical Report Ver. Dtsch. Ing. Berlin (1933) · JFM 59.1472.09
[92] Stützle, T., Hoos, H.: MAX-MIN Ant System. Futur. Gener. Comput. Syst. 16(8), 889-914 (2000)
[93] Sun, J., Deng, M., et al.: Thermohydrodynamic lubrication. Analysis of misaligned plain journal bearing with rough surface. J. Tribol. 132(1), 011704 (2010)
[94] Sun, J., Zhu, X., et al.: Effect of surface roughness, viscosity-pressure relationship and elastic deformation on lubrication performance of misaligned journal bearings. Ind. Lubr. Tribol. 66(3), 337-345 (2014)
[95] Swift, H.W.: The stability of lubricating films in journal bearings. J. Inst. Civ. Eng. 233(Part 1), 267-288 (1932)
[96] Thomsen, K., Klit, P.: Improvement of journal bearing operation at heavy misalignment using bearing flexibility and compliant liners. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 226(8), 651-660 (2012)
[97] Vijayaraghavan, D., Keith, T.G.: Effect of cavitation on the performance of a grooved misaligned journal bearing. Wear 134(2), 377-397 (1989)
[98] Vijayaraghavan, D., Keith, T.G.: Analysis of a finite grooved misaligned journal bearing considering cavitation and starvation effects. J. Tribol-T. ASME. 112(1), 60-67 (1990)
[99] Xu, G., Zhou, J., et al.: Research on the static and dynamic characteristics of misaligned journal bearing considering the turbulent and thermohydrodynamic effects. J. Tribol. 137(2), 024504 (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.