Lubricated Impact Dynamics and Pressure Spike Generation: Expanding on Contributions of Dr. R. Gohar

Abstract

The current paper focuses on the research of Dr. R. Gohar and in particular on his impacting ball and pressure spike measurement work. Its scope then expands to discuss contributions from other researchers concerning these two fields. The authors combined the two themes in a numerical study of an impacting contact. This study shows the detailed position of the pressure spike as a function of time. Then, the pressure spike position velocity is derived, and it is demonstrated that this velocity varies with time. As such, the paper concludes that the pressure spike shape itself must vary with time.

1. Introduction

The research interests of Dr. R. Gohar at Imperial College London were centered on ElastoHydrodynamic Lubrication (EHL). He has made significant contributions to film thickness measurements [1,2], as well as to pressure measurements, including the pressure spike [3,4,5]. In addition to the understanding of single contact EHL, he also analysed the effects of the multiple ball-raceway contacts on bearing vibration [6]. His work is largely summarised in [7,8].

In retrospect, Dr. Gohar’s work was certainly pioneering. It covered both extreme regimes of single EHL contact behaviour from the point of view of contact dynamics, the first being the steady state rolling/sliding contact where the contact dynamics of the mutual approach of the contacting elements is assumed to be quasi-static (force balance). The second is the pure impact problem where there is no rolling velocity, and the contact dynamics are determined by Newton’s second law. Both problems were extensively studied by other researchers in later years. The close similarity between these two problems is shown in [9]. Both problems can be described with exactly the same version of the Reynolds equation using the characteristic time. In the rolling contact case, this characteristic time is defined as the ratio of the Hertzian contact radius and the entrainment speed. For the impact case, it is the ratio of the mutual approach at the point of maximum impact (minimum approach) and by the impact velocity. The work on rolling bearing vibration, with the contact response modeled by a nonlinear spring and a viscous damper, with the spring and damping parameters taken from approximate EHL relations [6], preceded the work by Wijnant et al. [10]. Here, using a bearing Finite Element Model, the contacts were modeled using stiffness and damping constants derived from full numerical solutions of the time response of an EHL contact [11,12]. These latter studies benefited a lot from advanced numerical methods, which, aided by increasingly fast digital computers, allowed very detailed solutions of the EHL point problem. This paper focuses on Dr. Gohar’s pressure spike measurements and links it to his impacting ball work.

The pressure spike was predicted by the theoretical work of Petrusevitch [13] and also showed the numerical work of Dowson and Higginson [14,15]. The existence and the nature of the pressure spike has been the topic of many papers in later years. Aspects studied were the spike height and location as well as the parameter regime in which a pressure spike occurred. For example, with increasing piezoviscous effects, the pressure spike gradually develops from a small local pressure maximum into the full scale very narrow spike, which may exceed the Hertz pressure, especially when assuming an incompressible lubricant. Parameter studies have clearly shown that the pressure spike is strongly linked to the piezoviscous behaviour, and it does not occur for isoviscous lubricants. Regarding the understanding of its singular character, papers by Greenwood and Hall substantially added to its understanding [16,17,18].

Thin film transducers were successfully introduced for pressure measurement in EHL contacts by e.g., Kannel [19] and Hamilton et al. [20]. Safa and Gohar [3,4] used thin film transducers to measure the pressure in an impact EHL contact and demonstrated the occurrence of the pressure spike in this contact. Even though this spike is slightly different from the one previously described in rolling-sliding conditions, as will be discussed later. The measurement of a clear local pressure maximum on the outlet side and thereby another indication of the pressure spike not being an artifact, but physically realistic was very significant. The relevance of the ability to measure the pressure profile (and the pressure spike) lies in the fact that it can provide direct information about the rheological behaviour of the lubricant in terms of its shear stress–shear strain behaviour. Unfortunately, the approach with the thin film transducers is very delicate, due to possible damage to the thin film transducer, the averaging over the transducer surface and the fact that transducers are sensitive to both pressure and temperature. Amongst the other techniques applied to directly measure the pressure spike is the one using Raman spectroscopy [21,22,23]. Advantages of this technique include its non-intrusiveness and the “direct” measurement. Disadvantages include the expensive equipment, the long measurement times and the need to use specific lubricants. A final direct technique involves locally measuring the deformation of an elastic layer [24]. Its advantages are non intrusive and fast, its main disadvantage being its relatively low resolution.

As an alternative to these direct measurement techniques, several hybrid methods were introduced, where the film thickness profiles are measured. Then, “calculating backwards”, the pressure distribution (including the pressure spike) is effectively solved from the Fredholm integral equation of the first kind, formed by the elastic deformation integral with the measured film thickness distribution as input [25,26,27,28,29]. The advantages are the quick and “cheap” measurements; however, this inverse calculation of the pressure profile requires the film thickness measurement to be very precise and contain very little noise. This last requirement regarding the data-noise was lessened to some extent by [29]. In spite of the various difficulties, Refs. [25,27] clearly show that pressure information can be obtained for lubricants with different rheological behaviours that can not be obtained from the Reynolds equation, such as greases.

The numerical study of the EHL impact problem was pioneered by Dowson et al. [30] and by Larsson et al. [31]. Meanwhile, the experimental side of the problem still remained significantly interesting. For example, Workel et al. [32] describe the development of an experimental setup using ball impact for the direct measurement of lubricant friction under high pressures and shear rates. Recent theoretical work on film prediction under impact loading was published in [9]. As a tribute to the pioneering work of Dr. Gohar, on the impact problem and on measuring the pressure spike, the current paper builds on earlier work [9] by the authors and extends it by focusing on the study of the pressure spike and its position during the rebound phase of the impact problem.

2. Theory

Scaling the equations using dimensionless variables based on the Hertzian contact parameters as mutual approach, contact radius, and maximum contact pressure at the point of maximum deformation during impact, see [9], and the dimensionless equations describing the problem are the Reynolds equation:

$$\frac{\partial }{\partial X}\left(\frac{\overline{\rho }{H}^3}{\overline{\eta }\overline{\lambda }}\frac{\partial P}{\partial X}\right)+\frac{\partial }{\partial Y}\left(\frac{\overline{\rho }{H}^3}{\overline{\eta }\overline{\lambda }}\frac{\partial P}{\partial Y}\right)-\frac{\partial \left(\overline{\rho }H\right)}{\partial T}=0$$

(1)

with

$$\overline{\lambda }=12\frac{{\eta }_0{R}^2}{{a^{*}}^2\tilde{t}{p}_h^{*}}$$

(2)

in which the parameters $a^{*}$ and $p_h^{*}$ refer to the Hertzian maximum contact radius and the maximum pressure (in the impact cycle), respectively. The boundary condition in space is $P=0$ on the edges of the domain where these should be taken sufficiently far from the center. The second equation is the gap height equation:

$$H=-\Delta +\frac{X^2}{2}+\frac{Y^2}{2}+\frac{2}{{\pi }^2}\int {\int }_S\frac{P(X^{\prime },Y^{\prime },T)d{X}^{\prime }d{Y}^{\prime }}{\sqrt{{(X-X^{\prime })}^2+{(Y-Y^{\prime })}^2}},$$

(3)

assuming a parabolic undeformed gap shape, and approximating the elastic deformation by the surface deformation integral of a semi-infinite halfspace. $\Delta$ is the mutual approach of two remote points in the solids which is determined by the contact dynamics equation:

$$\frac{d^2\Delta }{d{T}^2}+\frac{3}{2\pi }\int {\int }_{S}P(X,Y)dXdY=0.$$

(4)

The dimensionless initial conditions are :

$$\frac{d\Delta }{dT}=V_0$$

(5)

and $\Delta (T=0)={\Delta }_0$.

Note that Equation (1) is in fact the same as the steady state Reynolds equation for the rolling contact but with T replacing X and defining the characteristic time $\overline{t}$ based on the ratio of contact size to entrainment velocity rather than the ratio of maximum mutual approach to impact velocity in the present problem. When the value of $\lambda$ is the same for both problems, the results of central film thickness versus time in impact and film thickness versus X in rolling/sliding should be very similar as is shown in [9]. The difference arises from the fact that the actual surface speed in the impact problem varies during impact, whereas it is constant in the rolling/sliding problem. For dry contact impact, the integral over the pressure in (4) can be replaced by the Hertzian nonlinear spring, i.e., the impact is fully determined by elastic stiffness. For the lubricated problem, this term represents the combined stiffness and damping of the lubricated contact. For high impact velocities, the stiffness will be close to the Hertzian stiffness. The damping is due to the viscous effects which are for the nearly elastic case mainly localized in the region around the momentary contact radius; see [11].

3. Calculation

The Reynolds equation was discretized with finite differences using a second order approximation in space on a uniform grid with equal mesh size in both spatial directions. Rotational symmetry as in [31] was not used, as the program was developed to include the effect of surface waviness with different orientations on the impact. The discretization in time was a second order backward with a constant timestep. The elastic deformation integral was discretized using a second order approximation, assuming a piecewise constant pressure at a region $\Delta X$, $\Delta Y$ centered around each grid point. The contact dynamics equation was discretized using a Newmark scheme [11,33]. Starting from the initial condition at $\Delta =1.0$, $V_0=1.0$, the resulting equations were solved marching in time (timestepping) at each timestep using a Multigrid coarse grid correction cycle algorithm based on a mixed Gauss–Seidel/distributive line relaxation scheme. This allows for fast solution of the equations per timestep so that dense grids can be used for accuracy. For the fast evaluation of the multisummations resulting from the discretization of the elastic deformation integrals, Multilevel Multi-Integration was used. For details regarding these methods, see [9,11,34].

For the calculations presented in this work, a domain $(X,Y)\in [-1.5:1.5][-1.5:1.5]$ was used with ${1025}^2$ grid points. This results in a mesh size $\Delta X=\Delta Y=0.00293$. The dimensionless time step was chosen to be equal to the mesh size: $\Delta T=\Delta X$. As from the dry impact problem, the total (dimensionless) impact time is known to be about $T_n=3$, and some 900 time steps are required to cover the period from the start of the impact ($\Delta =1.0$) to the end of the rebound. The exact number of time steps varies with the operating conditions as the code stops calculating once no points of positive pressure remain in the calculational domain. The computing time is of the order of 10 hours on a standard computer.

The similarity between the rolling and the impact problem allows the use of the Moes [35] parameters M and L to describe this impact problem similar to the rolling/sliding problem. Furthermore, Ref. [9] shows that the minimum film thickness during impact can be described by a graph similar to the rolling/sliding case, and that, for high M values (small $\lambda$ values), the total deformation approaches that of the dry contact case.

However, during the impact and rebound, the contact zone does not expand/retract at a constant velocity. Something similar happens in the lubricated case. This means that, during rebound, the relative flow velocity in the pressure spike region changes considerably. As such, the pressure spike behaviour (position and height) will vary during the rebound. Accurately monitoring the location of a very local phenomenon such as the pressure spike in a numerical solution is a challenge, particularly of only data on a single line were available. which would be the case when rotational symmetry would have been assumed and the problem solved as an effectively 1D problem. Now, in order to study the spike variation behavior in detail, it was decided to use the information of the entire calculational domain $P(X,Y)$, instead of e.g., just $P(X,Y=0)$. Using $P\left(R\right)$ with $R=\sqrt{X^2+Y^2}$, for $X\ge 0$ and $Y\ge 0$, all the information of the domain was used. This means that the $P\left(R\right)$ contains roughly 50 times as much data as the $P\left(X\right)$ file (the exact number varies with time and the size of the pressurised domain).

Obviously, high frequency noise arises from this operation as locally the discretisation error varies because of the different directions. After sorting the $P\left(R\right)$ signal, it was averaged using a moving 20 point average and finally a low pass filter was applied.

Figure 1 shows that $P\left(X\right)$ and $P\left(R\right)$ are very close. Figure 2 shows a detail from the pressure spike zone. Please note that the authors do not claim that the spike height is accurately approximated; however, they strongly believe that the spike position is accurate, or at least the evolution of its position.

4. Results

The evolution of the pressure distribution $\left(P\right(R\left)\right)$ as a function of time is shown in Figure 3.

Figure 3 shows the pressure distribution $P\left(R\right)$ as a function of time T. $T=3.223$ represents the pressure distribution with the right-most spike. For increasing time, the spike moves towards $X=0$. Please note that the central pressure is decreasing with time, as the ball rebounds and the elastic deformation diminishes and thus the central pressure diminishes as well. The time increments are of 20 times steps.

Figure 4 focusses on the last pressure distributions, close to the complete lift-off point. The time increment here is only two time steps.

Both figures show that, using the $P\left(R\right)$ representation, the pressure spike is very detailed. The two figures show that the distance the spike moves increases with time. Once again, the authors would like to stress that they do not claim that the spike height is correctly represented! That would require a much finer grid; however, they think that the spike position is correctly represented, see Figure 5. This figure shows the spike position $X_s$ as a function of time. It is of interest to note that, if the authors would have limited their analysis to $P\left(X\right)$, this position would be in multiples of $\Delta X$. The curves from bottom to top represent $M=10,20,50,100$. For increasing M values, the Hertz deformation, caused by the impact, increases. Hence, the curves are higher and broader. In the case that $M\to \infty$, the pressure spike would occur on the limit of the contact area $A\left(T\right)$, which can be approximated by

$$A\left(t\right)=sin\left(\frac{\pi T}{2T_c}\right)$$

(6)

where $Tc=3.218$ for the dry contact problem according to K.L. Johnson [36]. For the mutual approach, Wijnant predicts a Weierstrass function [11]. The dry contact curve has been shifted in time to accommodate the fact that the impact does not start at $T=0$ as the ball starts at a height $\Delta =1$.

As the spike position as a function of time is relatively smooth, it is possible to compute the time derivative of the spike position to obtain the spike velocity $V_s$ as a function of time. Note that no filtering was applied in the last two steps.

Figure 6 shows that, close to the maximum deformation point $T=3$, the spike velocity $V_s$ is low and not very well defined. As time progresses, this speed increases and becomes more continuous. Just before lift-off, the spike velocity is some ten times larger than at maximum deformation. It should be noted that, when using the data from $P\left(X\right)$ only, the velocity $V_s$ would be discrete with values $[0,1,2,\dots ]\times \Delta X/\Delta T$.

5. Conclusions

The current paper focuses on the research contribution of Dr. R. Gohar concerning film thickness and pressure measurements in EHL point contacts. It shows that pressure spikes occur during the rebound phase of a ball impact. However, as the velocity of the receding contact edge varies, the spike shape itself varies with time. As such, it can not be directly linked to the spike shape of an equivalent rolling-sliding contact with the same M and L values. Predicting the spike evolution as a function of the impact time seems a formidable challenge but would provide rheological lubricant information over a certain speed range. Furthermore, the impacting point contact allows for full angular averaging, contrary to the rolling sliding point contact. As such, the impact problem is better suited for rheological analysis than the rolling sliding problem. The authors would have loved to discuss the fine points of these measurements with Dr. R. Gohar and hear his suggestions.