Starvation

Tribology Wikipedia > Starvation

It is said that a contact operates under “starvation conditions” when an increase of oil supply at the contact would result in an increase of the film thickness at the contact. In other words, when the availability of the lubricant is limited and hence determines the lubriaction regime at the contact (1).

Starvation Model of Kingsbury

In a “starved” bearing, the oil on the track is about as thick outside the contact as in it. Stroboscopic observation of interference fringes confirms that such thin films flow very slowly in the centrifugal fields due to the bearing rotation. The working oil is thus stationary with respect to the track outside the contact, and remains unloaded most of the time. The oil within the contact may flow in the direction of rolling and also across of it. Its viscosity might singificantly increase due to the high pressure. The flow in the rolling direction cannot vary the average film thickness since it does not remove any oil from the track. Only the transverse oil flow can change the overall film thickness. The long-term decay in film thickness observed with oil jags in starved EHD contact, thus imply that transverse oil flowing out of the Hertzian contact area must be considered.

Starvation model of Kingsbury
Schematic representation of a starved EHL contact. The dashed area corresponds to the Hertzian contact area.

From the Reynolds equation (see equation 5) the speed of the flow perpendicular to the rolling direction is known:

(1)   \begin{eqnarray*} \omega = \frac{1}{2\mu}\frac{\partial p}{\partial z}(y^2-yh) \\ \end{eqnarray*}

To calculate the flow out from the contact, it is necessary to integrate the speed of flow in the direction of rolling/sliding and across the film thickness:

(2)   \begin{eqnarray*} Q_{out} = \int_{-a_h}^{a_h} \int_{0}^{h} \frac{1}{2\mu}\frac{\partial p}{\partial z}(y^2-yh) dy dx =  \frac{a_h h^3}{6\mu}\frac{\partial p}{\partial z} \\ \end{eqnarray*}

The term \frac{\partial p}{\partial z} is approximated as \frac{P_m^h}{b_h} and P_m^h is the mean Hertz pressure. Thus, the flow rate out of the contact in the direction perpendicular to rolling/sliding can be written as follows:

(3)   \begin{eqnarray*} Q_{out} = \frac{a_h h^3}{6\mu}\frac{P_m^h}{b_h} \\ \end{eqnarray*}

This equation gives the amount of liquid flow out from the Hertzian contact for the given film thickness h.

The flow in the rolling/sliding direction is neglected in the model, since in the starved EHL contacts the film thickness outside the contact is the same as inside and therefore the average flow in this direction must be cancelled.

Any change in the volume of the lubricant (film thickness) in time, i.e., flowrate, has to be balanced by Q_{out}:

(4)   \begin{eqnarray*} dV = 2a_hb_hdh=-Q_{out}dt=-\frac{a_h h^3}{6\mu}\frac{P_m^h}{b_h} \\ \end{eqnarray*}

Hence,

(5)   \begin{eqnarray*} \frac{ 12\mu b_h^2}{P_m^h} \frac{dh}{h^3} = -dt \\ \end{eqnarray*}

Integration of both sides of this equation gives:

(6)   \begin{eqnarray*} -\frac{6\mu b_h^2}{P_m^h} \frac{1}{h^2} = -t + C \\ \end{eqnarray*}

where C is a constant. This constant can be obtained from the initial condition: t=0, h = h_0. Finally, the equation gets the following form:

(7)   \begin{eqnarray*} \frac{h_0^2}{h^2} =  \frac{P_m^h h_0^2 t}{ 6\mu b_h^2} + 1\\ \end{eqnarray*}

Rearranging this equation gives:

(8)   \begin{eqnarray*} \frac{h}{h_0} =  {(\frac{P_m^h h_0^2 t}{ 6\mu b_h^2} + 1)}^{-1/2}\\ \end{eqnarray*}

Using this equation one can find the amount of oil that flows out of the contact for the given time t. This equation can be applied to the case of bearings or multiple passages of a ball or a sphere on the same spot of the disk. To do so, we can take the time t as t=nT, where n is the number of passages and T is the period of time between two passages. Then our equation becomes:

(9)   \begin{eqnarray*} \frac{h}{h_0} =  {(\frac{P_m^h h_0^2 n}{ 6\mu b_h^2} T + 1)}^{-1/2}\\ \end{eqnarray*}

In this last form the equation allows one to calculate readily the loss of lubricant from the contact after several passages of a ball or sphere through the contact zone. It should be mentioned that this model assumes no reflow of oil back to the contact, which is quite unrealistic from most of the practical problems. In reality, once the oil has left the contact, it is forced to flow back int it by surface tension driven flows.

Chevalier/Damiens Starvation Model

The approach of Kingsbury was further generalized by Chevalier and Damiens (2,3). By performing a large number of numerical and experimental work, they found that the folowing model can be used to describe the outflow of oil from the contact:

(10)   \begin{eqnarray*} Q_{out} = \alpha h^{\gamma + 1} \\ \end{eqnarray*}

Following the same approach as in Kingsbury model, one can get the following expression for the film thickness:

(11)   \begin{eqnarray*} \frac{1}{\gamma h^\gamma}  = \alpha' t + C \\ \end{eqnarray*}

where C is the integration constant. If now t=nT and the initial oil level at n=0 is h_{oil}, then C=\frac{1}{\gamma h_{oil}^{\gamma}}  and hence:

(12)   \begin{eqnarray*} \frac{1}{\gamma h^\gamma}  = \alpha' T + \frac{1}{\gamma h_{oil}^{\gamma}} \\ \end{eqnarray*}

References

  1. Kingsbury, E. P., “Experimental Observations on Instrument Ball Bearings,” Bearing Conjerence Proceedings, Dartmouth College, Hanover, N.H., 1968.

  2. Damiens, Venner, Lubrecht, Cann, Starved Lubrication of Elliptical EHD Contacts, DOI: 10.1115/1.1631020, 2004.

  3. Chevalier, F., Lubrecht, A. A., Cann, P. M. E., Colin, F., and Dalmaz, G., 1998, ‘‘Film Thickness in Starved EHL Point Contacts,’’ ASME J. Tribol., 120, pp. 126–133.

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