deflection and critical speeds of a stepped shaft 1

[paulratledge]

I would really love an equation to figure out the deflection of a rotating shaft.
The shaft is 76" from bearing to bearing.
9" at 2.950" diameter
64" at 9.685" d
9.5" at 2.95" d
the 64"section is a hollow shaft and has about 52" at 3"d missing from middle.
density of material is .28lb/in^3 and we spin it at about 3000rpm.

there is no 'load' so to speak of on the object besides the weight of the object its self which if you calculate the volume and density is about 880lbs.
criticals would also be lovely. thank you

 

[Tmoose]

Got a vibration issue?

what are the end support conditions ? simple supports like u-joints, or single row ball bearings, or ???

Reduced diameter ends don't usually change the critical speed much.
As a first cut I'd try an online calculator like this one with the middle dimensions .

http://www.wallaceracing.com/driveshaftspeed.htm

to immediately followed by a 2nd cut using the formula in the SAE u-joint handbook.

which would be immediately followed with a 3rd cut using a basic FEA simulation plugging in various end support stiffnesses, which the first 2 assume are too stiff.

4th cut would be a bonk with a spectrum analyzer listening to an accelerometer stuck on at mid span

 

[electricpete]

Attached is a spreadsheet where the problem has been solved numerically.

Everything is worked in SI units.

Tab Main shows the “parameters” of the calculation, including E and rho.
I assumed a bearing stiffness of 1E7 N/m, and then swept that value by applying multipliers 0.1, 1, 10,100,1000,10000.

Tab rotorsections is my description of your rotor.

Tab rotor picture is a rough picture of upper half of rotor, generated from the data in rotorsections.

Tab outsheet is a critical speed map. On the left side for very low bearing stiffnesses, we see the lower two natural frequencies are linearly increasing with bearing stiffness... we can guess the shaft is not bending here but just acting like a mass on springs, with both bearings moving same direction for first mode and opposite for 2nd mode.

As we move to the right all critical frequencies level out, indicating the bearings no longer play any role, as if it is a rigid mount.

Plots of modeshapes and frequencies are given in tabls Mult0.1, Mult1, Mult10, Mult10000.

From Mult1000, the values for roughly infinite beairrng stiffness are:
f1=73.23hz f2=180.21hz f3=526.04hz

There is a bearing stiffness where your first resonant freq will cross over your running speed (running at resonance). The would occur for bearing stiffness multiplier somewhere between 1 and 10 (correspondign to bearing stiffness between 1E7 and 1E8 N/m)


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[electricpete]

The bearing stiffness is intended to represent a composite of the bearing and support, assuming there is no significant moving mass present on the bearing housings/supports.

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[electricpete]

Also I put the simple supports representing the bearings at the extreme outer position of the 9" long ~" diameter sections.

Now I am thinking these are the bearing journals. In that case I probably should have put the simple support somewhere in the middle of these 9" sections. Right?

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[electricpete]

the object its self which if you calculate the volume and density is about 880lbs.

I calcualte 543 kg ~ 1200 pounds. One of us made a mistake. Could indicate I modeled your rotor wrong, I don't know. I'll let you figure that out.



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[paulratledge]

The bearings are ballbearing style.

I would put the ends someplace on the 9" sections. The rotor is allowed to move a little laterally with floating bearings. The bearings in the frame sit 76" apart. the rotor is longer than 76 total inches but it floats a few tenths to allow for thermal expansion.

The rotor spins inside of another tube and i want to see how close i can safely get the two before i make contact. that is why i want deflection. i want critical speeds just to know to know.

hope this helps. i can also be emailed at paulratledge@yahoo.com

 

[electricpete]

that is why i want deflection

You could calculate static deflection.

I don't think there is anything to work with to calculate dynamic deflection. There is a standard rotordynamic analysis which will calculate a deflection (under idealized conditions) in response to a specified unbalance at a specified location. You have not specified any unbalance.

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[paulratledge]

I balance the rotors on a case by case basis and as such do not have a standardized unbalance. I have thought about putting a mic close to the rotor and just measuring the deflection per piece also.

What about criticals?

 

[paulratledge]

Sorry.. how about 1mil at 3000rpm

 

[Tmoose]

How was the roll manufactured, especially the creation of the 3 inch bore?

If the wall thickness is uneven it can make the roll "whip" with an onset well below the first bending critical.
I believe that Thumb rules that prohibit operating speeds within 10/20/etc% of resonant frequencies are based on amplification by resonance, but can not take whipping or similar dynamic deflection in to account.

Whipping Below 50% of the FBCs is very possible with very uneven wall thickness, and rolls less chunky than yours.

 

[paulratledge]

The smaller diameter pieces are inserted 6 inches into each side. that is why the 64" piece only have 52" hollow. i cant really discuss the 64" piece in great detail only know that it is machined stainless and 4140. treat everything as 4140 for density.

thanks a million I love this site.

 

[electricpete]

I would say your best bet is to listen to someone knowledgable and experienced with similar equipment, like Tmoose. Also some vibration measurements certainly would be helpful to begin to understand the system response.

I have attached a somewhat academic excercize to estimate the balance sensitivity using a direct-stiffness approach in Matlab. The results suggest that 1 kg-m unbalance at center of the rotor would cause 0.9mm deflection at the center of the rotor at running speed of 50hz (and that resonant frequency is around 96hz.

I "validated" this calculation to a small extent by comparing the critical speeds indicated in the direct-stiffness based unbalance-response during speed sweep to the critical speeds based on transfer matrix approach. (first and 2nd critical speeds matched very well). This gives me higher confidence that I have not made a silly programming error, like leaving out a minus sign somewhere.

However of course more important are the assumptions. The assumptions are listed - I doubt your rotor will meet all of them. For example permanent bow is not considered at all.

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[electricpete]

The previous attachment had an error in the magnitude of off-center unbalances. (should have been 0.01 kg-m). This is corrected in the new attachment here.

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[electricpete]

Also, the word file says "0.009m".... that should be corrected to say "0.0009m" (0.9mm).

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[electricpete]

Reduced diameter ends don't usually change the critical speed much.

I think that is a reasonable starting point for order-of magnitude.

In this particular case, when we look at the numbers, a fairly small tweak in the length of those thinner-diameter end stubs out to the bearing (from 9/9.5” down to 7.2”) results in significant increase in resonant frequency (73hz to 96hz). Specifically:

[ul]
[li]On the first transfer matrix spreadsheet ( 29 Sep 11 13:15), there was 9” of smaller-diameter shaft on one end and 9.5” on the other end. The resulting first resonant frequency was 73.2 hz. [/li]
[li]On the first transfer matrix spreadsheet (included within attachment 1 Oct 11 20:28), there was 7.2” of smaller-diameter shaft on both ends and the resulting first resonant frequency was 96.4hz[/li]
[/ul]

So did I make an error? . I don’t think so. In this particular case the change in diameter is more than a factor of 3, which means the change in diametrical area moment of inertia is more than a factor of 3^4 = 81. This is evident in the modeshapes where we see significant bending occurs in the thin region very near where it meets the thicker region.

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Paul

There may be some simpler ways to attack the problem than what I posted.

If we believe the shaft is operating a reasonable margin below critical speed as suggested by my analysis, we can get close using a simple strength of materials approach – examining how this system would respond to force F = m*r*w^2 while neglecting mass acceleration effects. Finding response to applied force while neglecting mass/dynamic effects is very similar problem to solving static defelction.

If the stiffness of the bearings is lower than I assumed such that machine is near critical speed, a reasonable model would be to combine the stiffness of the shaft ends and the bearing into one equivalent spring, and then treat the 9” diameter as a rigid mass (as means of looking for simpler solution methods than the one I posted).

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[electricpete]

If we believe the shaft is operating a reasonable margin below critical speed as suggested by my analysis, we can get close using a simple strength of materials approach – examining how this system would respond to force F = m*r*w^2 while neglecting mass acceleration effects. Finding response to applied force while neglecting mass/dynamic effects

Attached I have done this analysis using Euler-Bernoulli beam theory, for the same rotor dimensions as analysed in my attachment of 1 Oct 11 20:22.

This analysis predicts that for a force applied at rotor center, with magnitude equivalent to 1kg-m unbalance at 50hz rotational speed, the resulting displacement at the center is 0.0007m = 0.7mm. This results is only slightly lower than the previous prediction of 0.9mm using dynamic analysis. It is expected that the dynamic analysis gives a slightly higher number than the static analysis, because the dynamic stiffness associated with mass acceleration acts vectoriarlly opposite to dynamic stiffness assocaited with material stiffness... the total dynamic stiffness ends up slightly lower than static stiffness when analysing a scenario far below resonance.

I also double-checked that at even lower frequency (10hz), the agreement between static and dynamic analysis were very good, suggesting again there is no silly programming error. (But many assumptions are shared and still deserve to be scrutinized).

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[paulratledge]

wow, you did a lot of work. thank you a lot. i feel like i should be buying you dinner or something. thanks again. i am going to need to read over this a few times

 

[electricpete]

No problem. I took the opportunity to try to validate the direct-stiffness method (which is something I am trying to learn) against some more familiar methods.

I have in mind to do one more thing: static deflection under weight of the rotor. It should be a relatively straightforward modification of the last file that I posted.


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[electricpete]

Attached is the static gravity deflection calc... very similar to the previous static center-unbalance deflection calc. The result is that the deflection at center of the rotor under it's own weight is 33 microns (assumes infinite bearing stiffness).

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