Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T11:21:42.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Model of an electro-rheological shock absorber and coupled problem for partial and ordinary differential equations with variable unknown domain

Published online by Cambridge University Press:  01 August 2007

W. G. LITVINOV ,
T. RAHMAN  and
R. H. W. HOPPE
W. G. LITVINOV
Affiliation:
Institute of Mathematics, University of Augsburg, Universitätsstr. 14, D-86159 Augsburg, Germany email: [email protected]
T. RAHMAN
Affiliation:
Department of Mathematics, University of Bergen, Allégaten 41, N-5007 Bergen, Norway email: [email protected]
R. H. W. HOPPE
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, U.S.A., and Institute of Mathematics, University of Augsburg, Universitä tsstr. 14, D-86159 Augsburg, Germany email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Amortization of a shock in an electro-rheological shock absorber is carried out in the motion of a piston in an electrorheological fluid. The drag force acting on the piston is regulated by varying the voltage applied to electrodes. A model of an electrorheological shock absorber is constructed. A problem on shock absorber reduces to the solution of a coupled problem for motion equation of the piston and non-linear equations of fluid flow in an unknown domain that varies with the time. A method of semi-discretization for approximate solution of the coupled problem is considered. Results on the existence and on the uniqueness of the solution of the coupled problem are obtained. Convergence of approximate solutions to the exact solution is proved. Numerical simulation of the operation of the shock absorber is performed.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 4 , August 2007 , pp. 513 - 536
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baiocchi, C. & Capelo, A. (1984) Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, John Wiley and Sons.Google Scholar
[2]Bayer, . Provisional Product Information of Rheobay TP AI 3565 and Rheobay TP AI 3566. Bayer Silicones Rheobay, Bayer.Google Scholar
[3]Belonosov, M. S. & Litvinov, W. G. (1996) Finite element method for nonlinearly viscous fluids. Zeitschrift für Angewandte Mathematik und Mechanik 76, 307–320.CrossRefGoogle Scholar
[4]Block, H. & Kelly, J. P. (1988) Electro-rheology. J. Phys. D. Appl. Phys. 21, 1661–1677.CrossRefGoogle Scholar
[5]Bossis, G. (editor) (2002) Electro-rheological fluids and magnetorheological suspensions. In: Proceedings of the Eight International Conference, Nice, France, 9–13 July 2001, World Scientific Singapore.Google Scholar
[6]Ellam, D. J., Atkin, R. J. & Bullough, W. A. (2005) Analysis of a smart clutch with cooling flow using two-dimensional Bingham plastic analysis and computational fluid dynamics. Proc. Inst. Mech. Eng. Pt. A J. Power Energ 219, 639–652, 2005.CrossRefGoogle Scholar
[7]Ellam, D. J., Bullough, W. A. & Atkin, R. J. (2005) Modelling the flow of an electrostructured fluid in transient operation. Proc. Inst. Mech. Eng. Pt. A J. Power Energy 219, 61–76.CrossRefGoogle Scholar
[8]Filisko, F. (1995) Overview of ER technology. In: Havelka, K. (ed), Progress in ER Technology, Plenum Press, New York.Google Scholar
[9]Fučik, S., Krtochvil, A. & Nečas, J. (1973) Kachanov–Galerkin method. Commentationes Mathematicae Universitatis Carolinae 14, 651–659.Google Scholar
[10]Gajewski, H., Gröger, K. & Zacharias, (1974) Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin.Google Scholar
[11]Girault, V. & Raviart, P. (1986) Finite Element Approximation of the Navier–Stokes Equations, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[12]Güntner, M. & Prokert, G. (1997) Existence for the quasistationary motion of a free capillary liquid drop. Zeitschrift für Analysis und ihre Anwendungen 16, 311–348.CrossRefGoogle Scholar
[13]Hoppe, R. H. W. & Litvinov, W. G. (2004) Problems on electrorheological fluid flows. Commun. Pure Appl. Analy. 3 (4)809–848.CrossRefGoogle Scholar
[14]Hoppe, R. H. W., Litvinov, W. G. & Rahman, T. (2005) Problems of stationary flow of electrorheological fluids in the cylindrical coordinate system. SIAM J. Appl. Math. 65 (5)1633–1656.CrossRefGoogle Scholar
[15]Ladyzhenskaya, O. A. & Solonnikov, V. A. (1976) Some problems of vector analysis and generalized formulation of boundary value problems for the Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel Math. Inst. Steklov (LOMI) 59, 81–116 (in Russian).Google Scholar
[16]Landau, L. D. & Lifshitz, E. M. (1984) Electrodynamics of Continuous Media, Pergamon, Oxford.Google Scholar
[17]Lions, J.-L. (1969) Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris.Google Scholar
[18]Litvinov, W. G. (1982) Motion of Nonlinearly Viscous Fluid, Nauka, Moscow (in Russian).Google Scholar
[19]Litvinov, W. G. (1996) A problems on nonsteady flow of a nonlinear viscous fluid in a deformable pipe. Methods Funct. Analy. Topol. 2 (3–4), 85–113.Google Scholar
[20]Litvinov, W. G. (2000) Optimization in Elliptic Problems With Applications to Mechanics of Deformable Bodies and Fluid Mechanics, Birkhäuser.CrossRefGoogle Scholar
[21]Litvinov, W. G. (2003) A model and a general problem on plastic flow under large deformations. Zeitschrift für Angewandte Mathematik und Mechanik 83, 291–310.CrossRefGoogle Scholar
[22]Parthasarathy, M. & Kleingenberg, D. J. (1996) Electrorheology: Mechanisms, and models. Mater. Sci. Eng. R17, 57–103.CrossRefGoogle Scholar
[23]Pukhnachov, V. V. (1989) Motion of Viscous Fluid With Free Boundaries, Novosibirsk University Publication, Novosibirsk (in Russian).Google Scholar
[24]Pukhnachov, V. V. (1998) The quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid. J. Appl. Math. Mech. 62, 927–937.CrossRefGoogle Scholar
[25]Pukhnachov, V. V. (1999) On a problem of viscous strip deformation with a free boundary. C. R. Acad. Sci. Paris 328 (Série 1), 357–362.CrossRefGoogle Scholar
[26]Schwartz, L. (1967) Cours d'Analyse Analyse Mathématique. Hermann Paris.Google Scholar
[27]Shulman, Z. P. & Kordonskii, V. I. (1982) Magneto-rheological effect. Nauka i Technika 146. p. 72. Minsk (in Russian).Google Scholar
[28]Shulman, Z. P. & Nosov, B. M. (1996) Rotation of the axisymmetric dielectric bodies in electrorheological suspensions. Int. J. Mod. Phys. B 10, 2903–2915.CrossRefGoogle Scholar
[29]Solonnikov, V. A. (1977) The solvability of a problem on motion of viscous incompressible fluid bounded by a free surface. Izv. Akad. Nauk. SSSR, Ser. Mat. 41, 1388–1424 (in Russian).Google Scholar
[30]Solonnikov, V. A. (1984) The solvability of the problem on the evolution of an isolated volume of a viscous incompressible capillary liquid. Zap. Nauchn. Seminarov Leningr, Optiko–Mekh. Inst. 140, 179–186 (in Russian).Google Scholar
[31]Vainberg, M. M. (1964) Variational Methods for the Study of Nonlinear Operators, Holden Day, San Francisco.Google Scholar
[32] >Whittle, M., Atkin, R. J. & Bullough, W. A. (1995) Fluid dynamic limitations on the performance of an electrorheological clutch. J Non-Newtonian Fluid Mech. 57, 61–81.CrossRefGoogle Scholar
[33]Whittle, M., Atkin, R. J. & Bullough, W. A. (1996) Dynamics of an electrorheological valve. Int. J. Mod. Phy. B 10, 2933–2950.CrossRefGoogle Scholar
[34]Whittle, M., Atkin, R. J. & Bullough, W. A. (1999) Dynamics of a radial electrorheological clutch. Int. J. Mod. Phys. B 13, 2119–2126.CrossRefGoogle Scholar
[35]Whittle, M., Firoozian, R., Peel, D. J. & Bullough, W. A. (1995) Electro-rheological relaxation times derived from pressure response experiments in the flow mode. J. Non-Newtonian Fluid Mech. 57, 1–25.CrossRefGoogle Scholar