Experimental Determination of Oil Rheology Parameters to be Implemented in Power Loss Predictions of Gears and Rolling Element Bearings

Introduction

    Rheology models express the way tribological conditions translate to shear stress of the lubricant and friction force on the interacting surfaces. Due to the complexity of the lubricant rheology, the friction coefficient is usually obtained experimentally either under the same operating conditions or by curve fitting in a properly chosen friction map. The current study aims at determining the rheological parameters of a lubricant based on friction measurements carried out on a commercial, readily available ball-on-disc machine. They can then be implemented in power loss prediction methods that utilize state of the art thermo-elastohydrodynamic numerical models considering the non-Newtonian lubricant behaviour and the dependency on pressure and temperature of the lubricant properties. Lubricants are commonly macro-molecular chains that behave like polymers in elastohydrodynamic lubrication. These chains follow a Newtonian linear law during steady state or shear rates close to zero. However, they deviate greatly under conditions of high pressure and high shear rate such as those in a typical gear mesh or a rolling element bearing. Under those conditions, the maximum friction typically peaks and reaches a plateau at around 0.08, far less compared to that a linear law would predict. Evans and Johnson (Ref.1) using the extended rheology equation by Johnson and Tevaarwerk (Ref.2) classified the behavior of the lubricant into four distinct regions, which indicate if it stems from the non-linear viscous or linear-elastic regime. The classification is based on the Deborah number defined as the ratio of the relaxation time of the lubricant to the time needed to pass through the contact. When it becomes greater than unity, which is typical for EHD contacts, the traction curve (friction coefficient over slide-to-roll ratio) is linear-elastic at first and then non-linear with a potential peak. The extended rheology equation uses a hyperbolic sinus function which is attributed to the studies of Eyring (Ref.3) on polymers. The sinh() function is used to model the thermal activation theory of a molecule which defines the amount of work a molecule must perform to jump from an energy well to the next. This aspect is known to be affected by both temperature and pressure and hence it is reasonable to expect a similar dependence in lubricants as well. Indeed, there are many different models proposed, such as those by Johnson and Tevaarwerk (Ref.2), Houpert et al (Ref.4), and Mihailidis and Panagiotidis (Ref.5). Some contain both parameters while others omit temperature in favour of a simpler formulation.

    Friction would still rise with the increase of shear rate even if the thermal influence on viscosity were negligible, due to the term. Limiting shear stress introduces a threshold to the maximum shear stress that a material could sustain before actually deforming as a “plastic” one. The theory was first proposed in 1960 by Smith (Ref.6), although hinted in a previous work of Petrusevich in 1951 (Ref. 7). The flow mechanism of thermally activated zones and viscous flow has been shown in polymers to give its place to a different one in shear stress above G/30, where G is the shear modulus. The new mechanism is the formation of a shear band. A straightforward separation of the thermal effects due to shearing is almost impossible (Johnson and Greenwood (Ref.8)). Experiments by Bair and Winer (Ref.9) in low shear rates but very high pressures in an isothermal disc machine showed a clear and distinct maximum of the friction coefficient indicating a limiting shear stress. Further calculations and later microflow images of shear bands have been presented by Bair (Ref.10). On the other hand, there have been additional phenomena observed, such as wall slip—especially in dissimilar, interfacing materials (Guo et al (Ref.11)) that may also contribute to the reduction of the friction. Despite that, in steel on steel friction these phenomena have been only observed under extreme sliding, thus the limiting shear stress can arguably be considered the most probable explanation.

    Various models for implementing in simulation the theory of the limiting shear stress have been proposed. Initially, Bair (Ref.12) attempted an analytic approach in order to develop a parameter that would contain physical properties such as bond strength of the molecular structure. The calculation of such parameter is quite difficult and in practice it was experimentally obtained. In fact, this is an issue, since the temperature effect on the oil viscosity under high pressure is very difficult to isolate and calculate outside of an EHL contact. The second issue relates to the fact that an EHL contact is not at constant pressure overall, which in turn means that the total friction force is a sum over the contact area that includes both thermally activated zones and shear bands. Temperature and pressure have a strong impact on the shear band formation by affecting the shear modulus. Houpert (Ref. 4) presented in his dissertation an exponential model concerning the effect of temperature on the limiting shear stress. Wang and Zhang in 1987 (Ref.13) also utilized an exponential law, which was later modified by Hsiao and Hamrock in 1992 (Ref.14). Roshetov and Gryazon, as mentioned by Wikstro?m and Ho?glund (Ref.15), presented an equation that includes first degree factors concerning pressure and temperature, which are multiplied. Kleemola and Lehtovaara in 2008 (Ref.16) presented two models, a simplified and a more complex multi-parametric one including both second order and exponential laws.

     The present study incorporates a simple relationship to model the influence of temperature and pressure without exponential function that could generate instability in a solver. The present study describes an experimental-analytical procedure that has been developed, in order determine the Eyring stress and the limiting shear stress of a given oil as functions of temperature and pressure. Special care has been taken to use commercially available equipment and standard experimental procedures. These data could then be used as input to any EHL model that would require them in order to calculate the friction coefficient accurately. As an example, the rheological parameters of the FVA (Forschungsvereinigung Antriebstechnik) reference oil Nr. 4 are determined.

Conclusions

    The method, outlined in the present study, achieved to extract the rheological parameters needed to describe the oil behavior in elastohydrodynamic contacts. The Eyring and the limiting shear stress, as well as the factors considering the temperature and pressure influence, are obtained by evaluating the friction coefficient measurements conducted under nearly isothermal, fullyflooded EHL conditions.

    Based on the friction coefficient over slide-toroll ratio measurements, obtained from 18 test runs following the proposed procedure, the rheology parameters for the FVA4 reference oil were extracted. They can be used in advanced EHL models to calculate the friction coefficient.

    As a preliminary validation of the method, these parameters were then fed in a thermo-EHL solver and the friction coefficient was calculated. The results showed very good agreement with measurements carried out under conditions outside the range of those used to extract the rheology parameters. A final validation incorporating experiments on a two-disk machine is on the way.

    Summing up, the proposed method allows the use of a commercial, readily available test rig with automated process and minimum oil requirements to extract rheology parameters needed for advanced thermos-EHL simulation models and for conditions commonly observed in rolling element bearing and gears.