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.