When designing an application, the design engineer must consider proper bearing lubrication. The lubrication method affects the proper operation of the equipment and the cost of maintaining it. Once the lubrication method is determined, selection of the proper lubricant oil viscosity is vital for prolonging the life of the equipment’s rolling element bearings. Rolling element bearings work best when they operate in a lubrication condition called elastohydrodynamic lubrication (EHL).

Bearing life is extended when a rolling element bearing operates with complete separation of the rolling elements from the raceways. Formation of a fully developed EHL layer between the rolling elements and the raceway depends on the speed of rotation, the bearing size and type, and the operating viscosity of the lubricant. Sufficient available oil volume is also necessary for the development of the EHL layer. The film parameter (Λ) evaluates and optimizes this separation given a particular set of application conditions. This article discusses a relatively simple method for calculating the film parameter, with an understanding of the factors that affect it and how to apply it to a design project.

## Oil Film Parameter

The EHL oil film parameter, Λ, represents the ratio between the oil film thickness (h) and the surface roughness (σ) (see Equation 1). It is widely used in the study and application of EHL. The raceway surfaces and rolling surfaces of a bearing are very smooth when viewed with the naked eye. However, when viewed through a microscope, the surface profile is not flat. Each surface has a measurable surface roughness that affects the lubrication performance of the bearing.

Since the scale of the EHL oil film thickness is of the same order of magnitude as the surface roughness, lubricating conditions cannot be determined without considering surface roughness. Given an average oil film thickness, a spectrum of conditions may occur depending on the surface roughness—from complete separation of the two surfaces by the oil film (see Figure 1a) to metal contact between the surface asperities (see Figure 1b) resulting in lubricant degradation and surface damage.

Λ = h / σ Equation 1

Where:

h = EHL oil film thickness

σ = Combined roughness ( √σ_{1}^{2} + σ_{2}^{2} )

σ_{1}, σ_{2} = Root mean square (rms) roughness of each contacting surface

The oil film parameter is correlated to the formation of the oil film as shown in Figure 2, and the degree of lubrication can be divided into three zones (see Figure 2). When Λ is large (around 3.0), bearing life is dominated by sub-surface fatigue. As Λ decreases, surface-originated flaking becomes dominant, reducing the life of the bearing.

## Oil Film Parameter Calculation

The Dowson-Higginson minimum oil film thickness equation (see Equation 2) is the thickness at which the rolling element load is highest and the film is thinnest.

Where:

H_{min} = Non-dimensionalized minimum film thickness

G = Non-dimensionalized factor combining material properties and viscosity

U = Non-dimensionalized entrainment velocity

W = Non-dimensionalized rolling element load

Equation 2 can be rewritten by grouping into terms (R) for speed, (A) for viscosity, (F) for load and (J) for bearing technical specifications, and t is a constant.

Λ = t • R • A • F • J Equation 3

Several factors from Equation 3 can be combined. For example, a rolling element load (P) between 98 Newtons (N) (22 pound force—lbf) and 98,000 N (22,000 lbf), F only increases by a factor of 2.54, given that F α P^{-0.13}. Since the P is determined roughly from the bearing size and type, the actual increase in load (F) is more realistically limited to a range of 20 to 30 percent. Since this variation is relatively small in this case, F is grouped together with J (for example, F ≈ F ∙ J), yielding Equation 4.

Λ = T • R • A • D Equation 4

Where:

T = Factor determined by the bearing type

R = Factor related to the rotation speed

A = Factor related to viscosity (viscosity grade α)

D = Factor related to bearing dimensions

The equation is in terms of factors that the design engineer knows: oil viscosity η_{0} in millipascal-second (mPa∙s) or centipoise (cp), speed n (rpm), bearing bore diameter d in millimeters (mm) and type. The following procedure can be used to calculate Λ:

Determine T from the bearing type with Table 1.

Determine R from n (rpm) with Figure 3.

Determine A from the absolute viscosity (mPa∙s, cp) and oil type in Figure 4. Generally, the kinematic viscosity ν_{0} (square millimeters per second, centistokes) is used and the conversion is shown in Equation 5.

η_{0} = ρ • ν_{0}Equation 5

Where:

ρ = the density (grams per cubic centimeter) and is approximated below:

Mineral oil: ρ = 0.85

Silicone oil: ρ = 1.0

Diester oil: ρ = 0.9