Rotating Shaft Critical Speed

[mrjonadam]

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.

 

[Twoballcane]

Hmmm... I am not alowed to open files and I don't reconized this equiaition, however, the critical speed of a shaft is the same as the natural frequency (Fn)of the shaft. So if you can calculate the Fn (cycle per sec)of the shaft, multiply it by 60s/1min to get the rpm.

Tobalcane
"If you avoid failure, you also avoid success."

 

[btrueblood]

Convert your units to SI, and the first equation works.

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.

 

[electricpete]

But I don't know how to find or what units K might be in for my problem.

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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[electricpete]

Nc = critical speed (rpm) or (Hz)?

Depends on how you plug your units. You are solving for radian frequency w = 2*pi*f. If you follow either metric or english solution as above and get w in sec^-1, then you need to divide by 2*pi to get f in hz (note btrueblood had already mentioned the 2*pi but I wanted to higlight it also)


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[electricpete]

Actually it looks like your document uses "N" for rotations per time and w for radians per time. In a formula for w, you need to divide the result by 2*pi to get rotations per time. In a formul for N, it should already be taken care of for you.

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[FeX32]

Surly there is another method to calculate the 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

 

[tgmcg]

The formulas presented ignore the effects of bearing stiffness, damping and gyroscopic effects of the mounted components. Most process turbomachinery running on sleeve-type or tilt-pad oil-film bearings operate between the 1st "rigid rotor" bearing mode and the 1st flexible rotor bending mode. They often run close to the 2nd "rigid rotor" bearing mode, but due to damping from bearings and seals, the vibrational amplitudes are low and are considered non-critical.

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

http://www.turbosynthesis.com

 

[electricpete]

You have to learn to walk before you can run.

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[FeX32]

Good point electricpete. Everything in the real world is more complex.


Fe

 

[tgmcg]

I'm sorry for budding in like that.

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

http://www.turbosynthesis.com

 

[electricpete]

Those are all good points and there was certainly nothing wrong with mentioning them. I was just feeling a little punchy.


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[VibesGuy]

I have a program I wrote that can calculate critical speeds of hollow shafts. Send me an example and I'll run the numbers for you so you can compare the results with the general cases on the PDF you attached. The general case formulas tend to overestimate the critical speed and as you can read you have to have a special fudge factor depending on the loading case.

John S. Turner PE
Vibration Consultant

 

[galiano]

mrjonadam,

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