, Volume 53, Issue 11–12, pp 2833–2860 | Cite as

Modeling of radial piston machines considering elastohydrodynamic effects in both cam–piston and piston–cylinder lubricating interfaces

  • Gautham Ramchandran
  • Pulkit Agarwal
  • Andrea Vacca
  • Kwang Sun Kim
  • TaeGul Kim


This study illustrates a novel numerical approach for investigating the complex physics involved in radial piston pump (rotating cam type) operation. This approach is based on a fully-coupled model that combines the evaluation of the main flow through the unit, realized by the pistons’ displacement, with the simulation of the internal lubricating interfaces, given by the piston–cylinder interface and the cam–piston interface. These interfaces represent the main source of power dissipation due to leakages and shear. The comprehensive multi-domain simulation tool presented in this paper incorporates a robust fluid–structure interaction based numerical model for the piston–cylinder lubricating interface as well as a model for the lubricant flow in the cam–piston interface. Since the approach used for piston–cylinder interface was previously presented by the authors, in this work particular emphasis is given to the description of the elastohydrodynamic lubrication model for the cam–piston interface. The overall coupling between these models enables an accurate estimation of the piston micro-motion that is critical to analyzing fluid flow in the lubricating gaps during pump operation. Results are shown with reference to a pump with four pistons designed to reach pressures up to 2500 bar. For this unit, the outer race of the cam—which is in contact with multiple pistons during pump operation—is unconstrained in its rotation, and it undergoes a complex motion due to which experimental measurements of its angular velocity were performed. The results from the study confirm the utility of the numerical model as an effective tool for modeling radial piston machines.


Radial piston pump Lubricating gap Piston micro-motions Cam piston interface Fluid–structure interaction (FSI) Elastohydrodynamic lubrication (EHL) 

List of symbols


Pressure (Pa)


Viscosity of hydraulic oil (Pa s)


Entrainment velocity of the lubricant (m/s)


Density (kg/m3)


Effective modulus of elasticity (Pa), \(\frac{1}{{E^{'} }} = \frac{1}{2}\left( {\frac{{1 - \nu_{a}^{2} }}{{E_{a} }} + \frac{{1 - \nu_{b}^{2} }}{{E_{b} }}} \right)\)


Load per unit width (N/m)


Pressure coefficient for viscosity


Pressure coefficient for density


Radius of cylindrical surface in x direction (m)


Dimensionless load, \(W' = \frac{w'}{{E^{'} R_{x} }}\)


Maximum Hertzian pressure (Pa), \(p_{H} = E^{'} \sqrt {\frac{{W^{'} }}{2\pi }}\)


Half-width Hertzian contact region (m), \(b = R_{x} \sqrt {\frac{{8W^{'} }}{\pi }}\)


Dimensionless pressure, \(\bar{P} = \frac{p}{{p_{H} }}\)


Length along the width of Hertzian contact region (m)


Start (Inlet) of Hertzian contact region (m)


End (Outlet) of Hertzian contact region (m)


Dimensionless coordinate, \(X = \frac{x}{{R_{x} }}\)

\(\bar{\rho }\)

Dimensionless density, \(\bar{\rho } = \frac{\rho }{{\rho_{0} }}\)

\(\bar{\eta }\)

Dimensionless viscosity, \(\bar{\eta } = \frac{\eta }{{\eta_{0} }}\)


Dimensionless film thickness, \(H = \frac{{hR_{x} }}{{b^{2} }}\)


Dimensionless speed parameter, \(U_{e} = \frac{{\eta_{0} u_{e} }}{{E^{'} R_{x} }}\)


Dimensionless material parameter, \(G = \alpha_{p} E^{'}\)


Integration constant in dimensionless film thickness equation

\({\bar{\uptau }}\)

Dimensionless shear stress, \({\bar{\uptau }} = \frac{\tau }{{E^{'} }}\)

\(\overline{{{\uptau }_{\text{L}} }}\)

Dimensionless limiting shear stress, \(\overline{{\tau_{\text{L}} }} = \overline{{\tau_{0} }} + \gamma \bar{P}\)


Reaction force exerted by the piston on the outer race (N)


Friction force due to the rolling elements (N)


Friction force between the outer race and each piston (N)


Force exerted by the spring (N)


Displacement chamber pressure (Pa)


Force exerted by the displacement chamber pressure (N)


Normal contact force from the cam acting on the piston (N)


Inertial force acting on the piston (N)


Viscous friction from the fluid in the piston–cylinder interface (N)


Contact friction force of the cam acting on the piston (N)


Normal force exerted by the fluid film in the piston–cylinder interface (N)


Angle made by the center of the eccentric cam with the shaft center (degrees)


Instantaneous velocity of the upper surface of the line contact from a stationary frame of reference (m/s)


Instantaneous velocity of the lower surface of the line contact from a stationary frame of reference (m/s)

\((u_{1} )_{C}\)

Instantaneous velocity of the upper surface of the line contact as seen with respect to the contact point (m/s)

\((u_{2} )_{C}\)

Instantaneous velocity of the lower surface of the line contact as seen with respect to the contact point (m/s)



The authors are thankful to Daejin Hydraulic Machinery Inc., South Korea for their valuable support and guidance.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ivantysyn J, Ivantysynova M (2003) Hydrostatic pumps and motors. Tech Books International, New DelhiGoogle Scholar
  2. 2.
    Pelosi M, Ivantysynova M (2012) A geometric multigrid solver for the piston–cylinder interface of axial piston machines. Tribol Trans 55(2):163–174CrossRefGoogle Scholar
  3. 3.
    Bergada JM, Watton J, Haynes JM, Davies DL (2010) The hydrostatic/hydrodynamic behavior of an axial piston pump slipper with multiple lands. Meccanica 45(2):585–602CrossRefzbMATHGoogle Scholar
  4. 4.
    Dhar S, Vacca A (2013) A fluid structure interaction-EHD model of the lubricating gaps in external gear machines: formulation and validation. Tribol Int 62:78–90CrossRefGoogle Scholar
  5. 5.
    Chapple P (1992) Modeling of a radial-piston hydraulic motor. Proc Inst Mech Eng Part I J Syst Control Eng 206(1):171–180CrossRefGoogle Scholar
  6. 6.
    Kleist A (1997) Design of hydrostatic bearing and sealing gaps in hydraulic machines—a new simulation tool. In: Proceedings of the 5th scandinavian international conference on fluid power ICFP, Linköping, SwedenGoogle Scholar
  7. 7.
    Kleist A (1995) Berechung von hydrostatischen Dichtstellen in hydraulischen Maschinen. Olhydraulik und Pneumatik 39:767–771Google Scholar
  8. 8.
    Agarwal P, Vacca A, Kim K, Kim T (2014) A numerical model for the simulation of flow in radial piston machines. In: Proceedings of the international exposition for fluid power, Las Vegas, USAGoogle Scholar
  9. 9.
    Agarwal P, Vacca A, Wang K, Kim K et al (2014) An analysis of lubricating gap flow in radial piston machines. SAE Int J Commer Veh 7(2):524–534CrossRefGoogle Scholar
  10. 10.
    Venner CH, Ten Napel WE (1992) Multilevel solution of the elastohydrodynamically lubricated circular contact problem Part 2: smooth surface results. Wear 152(2):369–381CrossRefGoogle Scholar
  11. 11.
    Jacobson BO, Hamrock BJ (1984) Non-Newtonian fluid model incorporated into elastohydrodynamic lubrication of rectangular contacts. J Tribol 106(2):275–282CrossRefGoogle Scholar
  12. 12.
    Hamrock BJ, Schmid SR, Jacobson BO (2004) Fundamentals of fluid film lubrication, 2nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  13. 13.
    Dowson D, Higginson GR (1966) Elasto-hydrodynamic lubrication: the fundamentals of roller and gear lubrication, 2nd edn. Pergamon Press, OxfordGoogle Scholar
  14. 14.
    Barus C (1893) Isothermals, isopiestics and isometrics relative to viscosity. Am J Sci 266:87–96ADSCrossRefGoogle Scholar
  15. 15.
    Wang J, Venner CH, Lubrecht AA (2013) Influence of surface waviness on the thermal elastohydrodynamic lubrication of an eccentric-tappet pair. J Tribol 135(2):021101-1–021101-12Google Scholar
  16. 16.
    Conry TF, Wang S, Cusano C (1987) A Reynolds-Eyring equation for elastohydrodynamic lubrication in line contacts. J Tribol 109(4):648–654CrossRefGoogle Scholar
  17. 17.
    Peiran Y, Shizhu W (1990) A generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic lubrication. J Tribol 112(4):631–636CrossRefzbMATHGoogle Scholar
  18. 18.
    Salehizadeh H, Saka N (1991) Thermal non-Newtonian elastohydrodynamic lubrication of rolling line contacts. J Tribol 113(3):481–491CrossRefGoogle Scholar
  19. 19.
    Jacobson BO, Hamrock BJ (1984) Non-Newtonian fluid model incorporated into elastohydrodynamic lubrication of rectangular contacts. J Tribol 106(2):275–282CrossRefGoogle Scholar
  20. 20.
    Venner CH, ten Napel WT, Bosma R (1990) Advanced multilevel solution of the EHL line contact problem. J Tribol 112(3):426–431CrossRefGoogle Scholar
  21. 21.
    Venner CH, Lubrecht AA (2000) Multi-level methods in lubrication, 1st edn. Elsevier, AmsterdamGoogle Scholar
  22. 22.
    Gupta PK (1979) Dynamics of rolling-element bearings—Part III: ball bearing analysis. J Tribol 101(3):312–318Google Scholar
  23. 23.
    Gupta PK (1984) Advanced dynamics of rolling elements, vol 39. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Maha Fluid Power Research CenterPurdue UniversityWest LafayetteUSA
  2. 2.Korea University of Technology and EducationCheonanSouth Korea
  3. 3.Daejin Hydraulic Machinery Co. LtdBusanSouth Korea

Personalised recommendations