Kappa – Lambda Enhanced Viscosity Required by Bearing Type (update)

The following data given below is for the various bearing types listed and is very approximate. It gives the minimum oil viscosities at operating temperature when the Mean Bearing Diameter is multiplied by the Rpm (DmN) and is within the range of ≈10000 DmN to ≈400000 DmN.  Bf refers to the Bearing Factor and is the approximate amount that the viscosity has to be increased for the proper oil film thickness generated by the basic Kappa – Lambda formulas based on bearing type.

The following lists bearing types and their minimum viscosity when using simple lubrication (mineral oil only.) It is possible to go below the listed viscosity, but enhanced additives are required to lubricate properly.  (Otherwise the point contact pressure becomes high enough to "punch through" the oil film.)
              
1.0 - Ball bearings

• Mfg01 (Note 1) = 13.2 cSt (centistokes)
• Mfg02 = 13 cSt
• OR-D (Note 2) says Bf = 1 (Base Line)

2.0 - Cylindrical roller bearings

• Mfg01 = 13.2 cSt to 20 cSt (Bf = 1.52) - depending on the cage of the bearing
• Mfg02 = 20 (Bf = 1.53)
• OR-D says Bf = 1.63 (6.5 / 4) see below

3.0 - Spherical (self-aligning), needle (with a cage) and tapered roller bearings

• Mfg01 = 20 cSt (Bf = 1.52)
• Mfg02 = 20 cSt (Bf = 1.53)
• OR-D says Bf = 1.63 (6.5 /4)

4.0 - Spherical (self-aligning) roller thrust bearings and needle bearings without a cage

• Mfg01 = 33 cSt (Bf = 2.5)
• Mfg02 = 30 cSt (Bf = 2.3)
• OR-D = There were no values for these classifications.

(Note 1) – Mfg01 & Mfg02 = Refers to Bearing Manufacturers Number 01 and 02
(Note 2) – OR-D = Other Research and Development

A further delineation came from OR-D where they gave the minimum C/P ratios for standard oil as follows.

• C/P > 4.0 for Ball Bearings.
• C/P > 6.5 for Roller Bearings (Hence 13/8 = 1.625, see above.)
• C/P > 10.5 for spherical thrust bearings, needle bearings without a cage.
• For very high angle tapered roller bearings use C/P ≈ > 6.5 + 4.0[Tan(roller angle with shaft)].

Svenska Kullagerfabriken AB (SKF) data from the early 1960-1970’s also showed approximately the same differentiation by bearing type (values were not quite the same as listed below). After considerable research, the following was concluded and used internally by Vanair Enterprises for the value of “Bf” in the EHL formulas.  This may or may not be the values used by the bearing industry to determine viscosity by bearing type.  However, Vanair has used these values quite successively for many years with Lambda and only recently with Kappa.

k or Λ (enhanced viscosity) ≈ Bf * k or Λ (base viscosity) 

Where Bf (Bearing factor) is equal to >>>  

• 1.00 for ball bearings.
• 1.63 for roller bearings, tapered roller bearings, and needle bearings with a cage.                     
• 2.64 for spherical thrust bearings and needle bearings without a cage.
• For very high angle tapered roller bearings use (1.64) + Tan (roller angle with shaft).

Total System Life Adjustment Factors for Reliability

The formulas below are used to generate the total life of a multibearing system. Many bearing manufacturers give L10 life guidelines and fail to qualify whether it applies to a single bearing or a multibearing system life.  This analysis is only relevant when bearings interact in a single component.  Also, when complete, components interact with each other to form a machine.

1/L (Total) = 1/L1 + 1/L2 + 1/ L3 + 1/L4 + ... + 1/Ln

(Miner’s - Equation - Gaussian Distribution)

The above equation is Gaussian and slightly different from the theoretically correct Weibull distribution which is normally applied to bearings.  However, the difference is minor and is conservative.  The Weibull Distribution is generally not worth the added mathematics involved.  It only becomes relevant when very large bearing systems are involved.

The totally correct Weibull formula is as follows:

L (Total) = [ (1/L1)^e +(1/L2)^e + (1/L3)^e + - - - + (L/n)^e]^ - 1/e

(Weibull Distribution)

Where >>>

L(Total) = This is the total component or system life.
L1 …+ Ln = These are individual element life’s or system Lxx life.
e = 10/9 for ball bearings
e = 9/8 for roller bearings

In theory, if the mathematical model from above would assume that each bearing has the same L10 life. Then a 2- bearing system, such as a spindle would have one-half (½) of the individual bearing L10 life in theory.  When a 4- bearing system is used, such as a gearbox, the bearing L10 life is approximately one-fourth (¼) of the individual bearing L10 life in theory.  Also this applies only to a Gaussian method or Weibull method.  However, in reality the L10 life in components is almost never equal.  Therefore, the above example is used only to illustrate the effects of interaction of a multibearing system.        

Source – SKF, Fafnir, and MPB bearing Corporations.

When the inverse life of the Multibearing system is desired then the following is used. >>>

Ln Multiplier = (Number of bearings in the system) ^1/e

(Weibull Distribution)

Where >>>

e = 10/9 for Ball Bearings
e = 9/8 for Roller Bearings

Therefore, if there are two (2) ball bearings in the system then 2^0.9 is 1.866.  Then the capacity of each bearing must be multiplied be 1.866 in order to have a final result of a factor of one (1) which is the base value of the multibearing system.