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.

Critical Frequency Design Formulas (20091117) R1

01.0 – Critical Speed –

ω_n = (π/l)2 * k * (g * E/γ)^0.5

Where >>>

ω_n = Frequency in Cycles per second

k = Radius of Gyration

g = Acceleration due to Gravity

E = Young’s Modulus of Elasticity

γ = Weight Density

l = Length of shaft in inches

f_n = Cycles per minute or revolutions per minute

Rpm = Revolutions per minute

02.0 - ωn = 2πf_n

03.0 - f_n (cycles per minute or Rpm) = 30π (gE/ γ) ^0.5 * k/l² when reduced using at least 4 decimal places and U S Bureau of Standards values for all of the constants it becomes >>>

19,074,672.75 * k/l²

04.0 - By substituting the k (k = d / 4 for a solid shaft) for a solid shaft the formula becomes >>>

N1 (Revolutions per minute) = 4768668.188 d/l² (As appears in the Machinery’s Handbook for Steel))

However, any “k” will give the resonant Rpm frequency of the so called “shaft”. Whether it is a tube, triangle, or square all that is required is the proper “k”.

Once the resonant Rpm is known then the maximum operating Rpm is 2^-½ or 0.707 of the resonant Rpm. This is the resonant curve half power point – sometimes referred to as the “3db down point” on resonant curves. Some screw manufacturer’s use 0.8 instead of the 0.707 number. Both seem to work, but Vanair has found that on some rare occasions the 0.8 can lead to a slight wobble of the shaft. This is especially true if it is a “fixed-free” shaft mount.

A PDF version of this article can be found at "http://vanairent.com/Critical-Frequency-of-any-Shaft-20091117.pdf".

Note – Shaft end connections are addressed in another Technical Design Manual (TDM)

The Kappa System Viscosity for Synthetic Oils - Issue 20110714

Vanair Enterprises again wishes to thank Markus Raabe of MESYS in Switzerland (www.mesys.ch) which writes bearing design programs and other machine design programs. He has supplied the missing formulas that allow a user of the Kappa (k) oil film system* to transpose the mineral oil system values to synthetic or other oil film values. 

In order to do this, the following equations were used to generate the end formula >>> 

From ISO 281 – 2007, the following approximate formula was used (the Dawson & Higginson equation for line contact) >>> 

k ≈ Λ^1.3 (1)

Where >>>

k = is the scalar value in the kappa oil film value system.
Λ = is the scalar value in the Lambda oil film value system. 

The following formula establishes the kappa relationship between synthetic and mineral oils >>>

k_syn = k_mineral [α _syn / α _mineral ]^(x)(y) (2)

Where >>>

k_syn = is the synthetic oil scalar value for in the kappa oil film value system.
k_mineral = is the mineral oil scalar value for in the kappa oil film value system.
x = is the exponent for G in the oil film thickness equation and is equal to 0.54.
y = is the exponent of Lambda in k – Λ (1) equation and is equal to 1.3.
α = is the pressure viscosity coefficient for each type of oil.

Therefore, the approximate formula has the final form of >>> 

k_syn = k_mineral [α_syn / α_mineral ]^(0.7) (3) 

(*) For the original natural or mineral oil formulas see Vanair Enterprises’ original TDM on this topic on its website.

A PDF version of this article is available at http://vanairent.com/Kappa-Viscosity-for-Synthetic-Oils-20110706.pdf.

Kappa Oil Film Formulas in accordance with ISO 281 — 20110117

In the mid 2000’s Vanair became fully aware of the Kappa system, which negated all of the complex parameters of the “classic” Lambda system.  Again, all of the methods of evaluation used graphs and charts.  There were no apparent mathematical formulas.  There was no way to make a computer program.  Vanair used Lambda method until the spring of 2010.  At this time a concerted search was undertaken to find the Kappa formulas if they existed.  The major bearing manufactures were contacted and again there were only the graphs.  By chance, Vanair made contact with a Markus Raabe of MESYS in Switzerland (info@mesys.ch) which writes bearing design programs and other machine design programs.  He supplied the  missing  SKF documentation on the Kappa System from 1980 – for which I am most grateful and wish to thank him.  From the documentation, Vanair obtained the following formulas.

	(1)
	(2)
	(3)
	(4)

Where >>>

v₁ = Viscosity in cSt or ISO Viscosity Index of the oil at operating temperature.
N = Revolutions per Minute (rpm)
D_m = Diameter Mean of the bearing where: in mm.

The above formulas generate the “classic” graph, which appears in the ISO 281-2007(1) and in many bearing companies’ catalogs.   Again, from MESYS in Switzerland, Vanair also learned that the above “classic” Kappa formulas were for natural oil.

Source via MESYS in Switzerland  – TRIBOLIGIA e LUBRIFICAZIONE  – Vol XV, Ehd lubrication in rolling bearings, by R. Heenskerk, from SKF Engineering and Research Center B. V. Netherlands, December 1980

(1) – This standard has not been adopted by the ABMA/ANSI Committee as of early 2011. Also see STLE July and August 2010 issues of TLT (stle.org) or >>>

(http://www.stle.org/assets/document/tlt_July_cover_story_article.pdf)

A PDF version of this article is available at http://vanairent.com/Kappa-Viscosity-Formulas-20110117.pdf.