Validation of a 1-D Elastohydrodynamic lubrication model

The sealing performance of a hydraulic cylinder depends on the characteristics of three essential elements: the rod, the seal and the fluid. To predict the behavior of a hydraulic seal, including the friction force and leakage rate, a series of theoretical and experimental studies have been carried out. In this article, a one-dimensional elastohydrodynamic model of the U-cup hydraulic rod seal is developed taking into account: the roughness of the shaft and lips. The numerical results are validated by experiments previously published.


Introduction
The hydraulic U-rod seal is the most frequently used machine component to prevent leakages with minimal wear effect. Since the 1960s, great importance has been assigned to the modeling of hydraulic joints. The behavior complexity of this device is due to several physical phenomena that could interfere. Indeed, the thermal effect of friction affects simultaneously the elasticity of the elastomeric seal and the rheological law of the lubricant film.
Previously, Lawrie and O'Donoghue [1] have proved experimentally the presence of a thin film throughout the lubricated contact. They also demonstrated that in the outstroke rod motion, the film is thicker and the shape of the lip has a significant effect on the U-cup hydraulic rod seal performances. Earlier, Vissher and Kanters [2] have investigated the rod roughness effect on the performance of rectangular polyurethane seals. Thus, above a critical arithmetic value of roughness, the seal may leak. However, by using a grooved shaft, the friction force decreases substantially as described by Elgadari et al [3]. Therefore, a judicious pattern could improve the lifetime of such a device. To model the film thickness behavior, several approaches have been used. In this article, only two numerical methods have been reported:  Inverse hydrodynamic lubrication (IHL): Based on the assumption that the hydrodynamic pressure is equal to the static pressure calculated on structural computational software (FEM), the Reynolds equation is solved by taking the film thickness as the unknown parameter, as carefully detailed in Crudu thesis [4]. It was underlined that the numerical results are slightly agreed with the measurements.  Elastohydrodynamic lubrication (EHL): Elgadari et al [3] have recently used this approach by solving the Reynolds equation and taking into account the elasticity of the seal, and the roughness of the lip and the shaft. It was proved that asymmetric grooves of the shaft, can improve significantly the sealing performances.
The objective of this work is to perform a parametric study by considering a one-dimensional elastohydrodynamic model that takes into account the elasticity of the lip and the roughness of the shaft.
After validating the current model, numerical simulations were performed and compared with experimental results.  The lubricant side of the seal is submerged entirely in the lubricant with a sealed pressure ps.  The average film thickness is uniform in the axial direction, according to previous numerical and experimental results Crudu [4].  The dry contact is not considered in this theoretical model.  The circumferential shear deformation is considered.

Governing equations
In order to take into account the cavitation effect, the modified Reynolds equation is adopted: A preliminary structural analysis of commercial software was carried out to determine:  Contact pressure field pc and contact width L due to the mounting of the seal on the rod.  The radial and tangential compliance matrices Cz and Cx respectively, based on the study by Elgadari et al [3]. So, the radial and tangential displacements are given by: Where pj is the nodal film pressure, pcj is the nodal contact static pressure, Nx is the number of nodes, and τxzj is the nodal shear stress calculated with: Figure 2 shows that the thickness of the thin film h is written as follows:

3 Film thickness
With h1 the rod roughness, h2 the lip roughness, h0 the average film thickness, δx and δz the axial and normal lip displacement respectively by equations (2) and (3).
To validate the current model, the work of Crudu [4] and Elgadari [3] is reproduced by simulating 4 cases of sealed pressure: 4.5 MPa, 9.5 MPa, 12.5 MPa, and 19.5 MPa and taking into account surface roughness with simple analytical functions similar to the one previously published. The lip roughness is assumed sinusoidal and given by: And the rod roughness is given by: With λ1 and λ2 are the axial periodicities of the rod and seal roughness respectively.

Validation
In order to compare the friction force Ff with the experimental results achieved by Crudu [4], this parameter is calculated by : With R is the radius of the rod and L is the width of contact. Tab.1: Parameters adopted for the parametric study Figure 3 shows a significant correlation between the current model and the experiment.

Conclusions and perspectives
In the present study, we have developed an elastohydrodynamic model (EHL). A structural analysis was performed to determine the width and pressure of contact. These were used as input parameters for resolving the modified Reynolds equation. A comparison between the friction force obtained by the simulation and the other experimental results confirms the validation of the actual model. This work has introduced new ways to explore, by changing the surface states and characteristics of the lubricant.