Rotating Shaft Critical Speed
Rotating Shaft Critical Speed
(OP)
I have a rotating hollow shaft and I'm assuming a fixed end condition. I'm looking for the critical(whirling) speed. Right now I'm using the following equation:
Nc = (3.57*sqrt(EI/m))/L^2
where, E = Young's modulus (lb/in^2)
I = second moment of area (in^4)
m = mass per unit length (lb/in)
L = length (in)
Nc = critical speed (rpm) or (Hz)?
I have attached a PDF with the derivation of the calculation and the equation I'm using. This equation is for a shaft between long bearings and can be found on the last page of the document.
I'm having trouble understanding the units that result from this critical speed calculation. I get the output to be in units of in^(-1/2) and I'm pretty sure it should be either rpm or Hz.
Also, I followed the derivation noted in the PDF for critical speed and agree that it should be:
w = sqrt(KEI/mL^3)
where, K = is a constant depending on the mass and the end fixing conditions.
But I don't know how to find or what units K might be in for my problem.
I greatly appreciate anyone's help in figuring out what I'm doing wrong, or a suggestion on another equation to use.
Nc = (3.57*sqrt(EI/m))/L^2
where, E = Young's modulus (lb/in^2)
I = second moment of area (in^4)
m = mass per unit length (lb/in)
L = length (in)
Nc = critical speed (rpm) or (Hz)?
I have attached a PDF with the derivation of the calculation and the equation I'm using. This equation is for a shaft between long bearings and can be found on the last page of the document.
I'm having trouble understanding the units that result from this critical speed calculation. I get the output to be in units of in^(-1/2) and I'm pretty sure it should be either rpm or Hz.
Also, I followed the derivation noted in the PDF for critical speed and agree that it should be:
w = sqrt(KEI/mL^3)
where, K = is a constant depending on the mass and the end fixing conditions.
But I don't know how to find or what units K might be in for my problem.
I greatly appreciate anyone's help in figuring out what I'm doing wrong, or a suggestion on another equation to use.





RE: Rotating Shaft Critical Speed
Tobalcane
"If you avoid failure, you also avoid success."
RE: Rotating Shaft Critical Speed
E in N/m^2
I in m^4
m in kg/m
L in m
The term under the square root then comes out (converting Newtons to kg-m/s^2) to units of m^4/s^2, taking the root gives units of m^2/s. Dividing by the L^2 term leaves you with units of 1/second, or radians/second. Divide by 2*pi to get rev./second, multiply by 60 to get rpm.
RE: Rotating Shaft Critical Speed
K is unitless
You should be able to solve it in any properly applied unit system although I agree SI is easier.
You can do it with English units also
w = sqrt(KEI/mL^3)
Inside the bracket
K is unitless
E is lbf/inch^2
I is inch^4
m is lbm
L^3 is inch^3
So far we have inside the bracket
lbf/inch^2 * in^4 / [lbm * in^3] = lbf/[lbm*inch}
multiply by [32.2*12 lbm*inch]/[sec^2*lbf] and you will have units of sec^-2 inside bracket which will give the required sec^-1 when you take the sqrt.
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RE: Rotating Shaft Critical Speed
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Eng-tips forums: The best place on the web for engineering discussions.
RE: Rotating Shaft Critical Speed
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Eng-tips forums: The best place on the web for engineering discussions.
RE: Rotating Shaft Critical Speed
Why don't you treat your shaft as an Euler-Bernoulli beam instead? The first harmonic then simply: w1=(4.73/L)^2*sqrt(E*I/(rho*A)). (w1 is in rads/s)
The derivation is in many texts.
Maybe you could use this as a methodology check.
Fe
RE: Rotating Shaft Critical Speed
The formulas you've presented should not be used for calculating the natural frequencies of rotors running on oil film bearings because bearing stiffness has a large effect on the location of the mode. Bearing stiffness is a complex function of detailed bearing geometry.
You say you have a rotating shaft with fixed end conditions. I'm having difficulty envisioning a real world rotor-bearing system that satisfies this assumption. I am concerned that you might be misapplying the simple beam formulas for calculating natural frequencies. The Myklestad-Prohl transfer matrix method is a better approach for modelling most machines of any significance.
Can you tell us more about the machine type, size, speed, power and application?
I do a fair amount of rotordynamic analysis for my clients, but use commercially-available computer codes for this, eg. XLRotor and TLTPAD, which are based on the Myklestad-Prohl method as extended by Lund to include the effects of damping.
Best regards,
Tom McGuinness, PE
Turbosystems Engineering
www.turbosynthesis.com
RE: Rotating Shaft Critical Speed
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Eng-tips forums: The best place on the web for engineering discussions.
RE: Rotating Shaft Critical Speed
Fe
RE: Rotating Shaft Critical Speed
From an academic standpoint it's good to work a few problems using the simple beam equations. But if the OP is actually trying to design a piece of commercial equipment, he really needs to use a better tool....and find somebody who knows how to use that tool. Design of rotating machinery not a field that's very friendly to the trial-and-error approach.
Best regards,
Tom McGuinness, PE
Turbosystems Engineering
www.turbosynthesis.com
RE: Rotating Shaft Critical Speed
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RE: Rotating Shaft Critical Speed
John S. Turner PE
Vibration Consultant
RE: Rotating Shaft Critical Speed
For simple hollow lightly loaded shaft with rolling element bearings (high stiffness, low damping) simple equation should be close enough to real rotor system (if you neglect machine frame and foundation stiffness and damping). This is not some multimass rotor on journal bearings (with oil film stiffness and damping)like steam turbine generator set.
If it is only rotating hollow shaft (paper mill?), and if you have vibration analyzer, you can easy identify the real critical speeds.
Galiano