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.