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

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## Abstract

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.

## Keywords

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

- \(p\)
Pressure (Pa)

- \(\eta\)
Viscosity of hydraulic oil (Pa s)

- \(u_{e}\)
Entrainment velocity of the lubricant (m/s)

- \(\rho\)
Density (kg/m

^{3})- \(E'\)
Effective modulus of elasticity (Pa), \(\frac{1}{{E^{'} }} = \frac{1}{2}\left( {\frac{{1 - \nu_{a}^{2} }}{{E_{a} }} + \frac{{1 - \nu_{b}^{2} }}{{E_{b} }}} \right)\)

- \(w'\)
Load per unit width (N/m)

- \(\alpha_{p}\)
Pressure coefficient for viscosity

- \(\beta_{p}\)
Pressure coefficient for density

- \(R_{x}\)
Radius of cylindrical surface in x direction (m)

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

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

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

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

- \(x\)
Length along the width of Hertzian contact region (m)

- \(x_{min}\)
Start (Inlet) of Hertzian contact region (m)

- \(x_{max}\)
End (Outlet) of Hertzian contact region (m)

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

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

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

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

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

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

- \(H_{o}\)
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, \(\overline{{\tau_{\text{L}} }} = \overline{{\tau_{0} }} + \gamma \bar{P}\)

- \(F_{PN}\)
Reaction force exerted by the piston on the outer race (N)

- \(F_{Bf}\)
Friction force due to the rolling elements (N)

- \(F_{Of}\)
Friction force between the outer race and each piston (N)

- \(F_{spring}\)
Force exerted by the spring (N)

- \(P_{DC}\)
Displacement chamber pressure (Pa)

- \(F_{DK}\)
Force exerted by the displacement chamber pressure (N)

- \(F_{CN}\)
Normal contact force from the cam acting on the piston (N)

- \(F_{aK}\)
Inertial force acting on the piston (N)

- \(F_{Tf}\)
Viscous friction from the fluid in the piston–cylinder interface (N)

- \(F_{Cf}\)
Contact friction force of the cam acting on the piston (N)

- \(F_{TN}\)
Normal force exerted by the fluid film in the piston–cylinder interface (N)

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

- \(u_{1}\)
Instantaneous velocity of the upper surface of the line contact from a stationary frame of reference (m/s)

- \(u_{2}\)
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)

## Notes

### Acknowledgement

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.

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