calculating shaft critical speed by hand
calculating shaft critical speed by hand
(OP)
I am trying to calculate(estimate) the critical shaft speed of a rotating shaft.
When i did this back in school we solved for the eiganvalues of the 4th order diff equation:
d^4(y)/dx^4 - B^4*y=0 where B^4=rho*A/(E*I)*omega
however this assumes A (shaft area) to be constant. The shaft i an interested in steps from 1" to 2" and back down to 1" between the bearings. Does anyone have some tips or references to solve eigenvalue problems for a shaft with non uniform diameter. thanks
When i did this back in school we solved for the eiganvalues of the 4th order diff equation:
d^4(y)/dx^4 - B^4*y=0 where B^4=rho*A/(E*I)*omega
however this assumes A (shaft area) to be constant. The shaft i an interested in steps from 1" to 2" and back down to 1" between the bearings. Does anyone have some tips or references to solve eigenvalue problems for a shaft with non uniform diameter. thanks





RE: calculating shaft critical speed by hand
RE: calculating shaft critical speed by hand
Blevins has a formula for the frequency of a massless beam with a concentrated mass on it (critical frequency of a shaft is the same as its bending frequency) which may help
2/pi*sqrt(3EI/(M*L^3))
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
Mark's handbook 8th edition page 5-70:
Beam of length l and mass mb simply supported at both ends with mass m in the middle:
wn = sqrt(48*E*I /[(m+0.5mb)*l^3])
A quick comment on iterative methods. I have never solved any yet but I am getting ready to. "Handheld Calculator Programs For Rotating Equipment Design" by Fielding is available used for $10 including shipping. In spite of the title it was geared for the days when programmable calculators were widely available and computers weren't... so the algorithms are suitable for computers. Chapter 1 has a finite element technique called Prohl Myklestad. It looks fairly straightforward. You have to iterate first to find the critical frequencies (the frequencies which satisfy the boundary conditions and make some residual = 0). Then you can easily compute the mode shapes once critical frequencies are known. Some advantages over Raleigh are that you don't need to guess the mode shape, you can calculate higher order critical speeds without knowing the mode shape, and you can calculate the mode shape.
If you could wait about two months I hope to have it programmed in excel and vba.
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RE: calculating shaft critical speed by hand
Beam of length L and mass mb simply supported at both ends with mass m in the middle:
wn = sqrt(48*E*I /[(m+0.5mb)*L^3])
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RE: calculating shaft critical speed by hand
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RE: calculating shaft critical speed by hand
Transfer matrix method - now this almost sounds like the theory of receptances, which is a hand powered method for predicting the dynamic response you get when you join two dynamic systems.
Cheers
Greg Locock
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RE: calculating shaft critical speed by hand
Yeah, i was hoping for a nice closed solution but with the discontinuities of the diameter along the lenght it looks unlikely. My only thought was that you could make the Area a function of length using a step or dirac delta function and solve. But that is pushing the limits of my math skills. The next best bet is to start brushing up on my numerical methods. Thanks
RE: calculating shaft critical speed by hand
Greg's comment makes me think that transfer matrix is a more appropriate term to describe the algorithm I mentioned (Leslie's book) than finite element. You break the shaft into sections and relate the conditions at one end of shaft to other end of shaft via a matrix transformation of the 4 variables ... something like y, displacement, angle, shear, moment.
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RE: calculating shaft critical speed by hand
Cheers
Greg Locock
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RE: calculating shaft critical speed by hand
greg, im coming clean- the dim i gave were just hypothetical because i was too lazy to look at the drawings.
The actual specs are:
4" dia x 9.5" section centered between 50mm (give or take due to bearing stops on shaft) sections
im using two pairs of 210 dulex angular bearings (40mm wide for the two rows) spaced 11.5" apart.
My plan was to model the shaft fixed-fixed.
RE: calculating shaft critical speed by hand
So I think the problem collapses down to the simplest possible - a pinned uniform shaft.
Are you going to get any test work done?
Cheers
Greg Locock
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RE: calculating shaft critical speed by hand
Also, what sort of test work were you referring to?
RE: calculating shaft critical speed by hand
"it seem that bending wise that that 4" section would be very stiff and the majority of the bending would occur in the 50mm sections." exactly, so the ends would be like pin joints, even if the bearings are rigid, which in the context of a 4 inch shaft is unlikely...
as to test work, run it through the speed range to find where the critical speeds are, or hit it with a modal hammer to measure the bending frequencies, or both.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
TITLE: Handheld Calculator Programs for Rotating Equipment Design
by Leslie Fielding
ISBN: 00702-0695-3
Publisher: McGraw-Hill
Publish Date: February, 1983
Binding: Hardcover
List Price: USD 52.95
If this links works correctly you will see it begins at $10 for used:
http://www
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RE: calculating shaft critical speed by hand
With a shaft that is almost all 4" wide, how likely is it that you'll spin it up to anything near its critical frequency?
Two quick checks: If it were pinned at the ends, and had a uniform 50mm section, would you get close to the critical speed? What if you had a uniform 100mm cross section? If the answer to the first check is "no," then why worry about the real shaft? If the answer to the second check is "yes," then you know you'll have a problem with the stepped shaft, right?
RE: calculating shaft critical speed by hand
RE: calculating shaft critical speed by hand
greg, no physical tests yet as i am still in the design stages of all this.
((what i am trying to go is find a maximum operating speed for my rig so i can open up the radial clearances of my fluid bearing-so when i measure displacements runout and other background noises will comparatively smaller to increase SN ratio)).
ivymike, i like te idea of the quick check- i think something along these lines will be good enough for my purposes. However is that sound logic that if a 50mm uniform shaft is OK that my 50,4",50 shaft will be OK too, i would guess that adding mass in the middle would lower the critical speed, but maybe not since the 4" section runs almost the majority of the length.
RE: calculating shaft critical speed by hand
Above is output of a critical speed/mode shape program that uses the algorithm I mentioned above.
Of course the output is only as good as the input.
Check the first figure to see if the geometry is right.
Mode shapes are shown in the 2nd figure. 1st critical at 68000cpm and 2nd crit above 100kcpm.
Below that is program output which describes the options I used in running the program. See if you agree.
I think I got carried away cranking up bearing stiffness. At lower stiffnesses I was getting a very flat looking first critical mode shape, so I cranked up the stiffness to get a pretty bowed shape. But now that I think about it, this rotor without disks may not deform much at first critical if most of deflection is occuring in the bearings.
If you have some changes to suggest, I will be glad to rerun the program.
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RE: calculating shaft critical speed by hand
I'm not an expert on this subject. Oil whirl of course occurs at a frequency slightly below half running speed. Wouldn't oil whip then occur when running speed slightly above twice critical speed so that the whirl frequency is near the rotor critical frequency?
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RE: calculating shaft critical speed by hand
Your correct-- whirl is loss of load capacity when an external load oscillates at .47 times the rotating frequency. Whip is fluid film bearing instabilty at starting ~ 2x crital speed. Thanks
Also that shaft mode shape program is great! Did you make that or is it a commerically avail program?
I plan on calculating a pinned uniform shaft both 50mm and 4" and see where the critical speeds of those fall just out of curiousity.
RE: calculating shaft critical speed by hand
API 684 Tutorial on rotodynamics doesn't say much on the subject. They have a table 1-3 of "Typical Stiffness and Damping Properties of Common Bearings (Comparison Only)" which lists 3,000,000 lbf / inch for the broad category of antifriction bearings.
I don't have any better references for selecting bearing stiffness. Does anyone know of good references on this subject?
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RE: calculating shaft critical speed by hand
good point - you could throw in some extra mass in your calculation if you're worried about it, but then you might be "fudging" a bit too much for comfort.
RE: calculating shaft critical speed by hand
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RE: calculating shaft critical speed by hand
You can also tell that by looking at the mode shape... there is a lot of displacement at the bearing compared to the bending of the shaft. If the bearings were stiff compared to the rotor, the 1st critical mode shape trace would go through shaft centerline near the bearings... but in the attachment the whole mode shape trace is offset from the centerline indicating substantial flexing of the bearings.
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RE: calculating shaft critical speed by hand
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RE: calculating shaft critical speed by hand
Unfortunately, to the extent that the rotor flexes, this formulation also overestimates the critical speed and does not provide a lower bound.
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RE: calculating shaft critical speed by hand
RE: calculating shaft critical speed by hand
I didn't write that program.
Based on the inputs and outputs produced by the program I have deduced that it uses the same algorithm shown in Chapter 1 of Fielding's book.
I am planning on programming it within excel in the near future but I have some other pressing projects in my spare computer time first.
I am very happy with Fielding's book. My review:
TOC:
Lateral Critical Speed Calc
Torsional Crit Speed Calc
Rotor Bearing Stability Program
Blad Vibration (Flexural, Torsional)
Disk Weight and Inertia Properities Progra
Disk Stresses
Blade Stresses
Stationary Shrink Fit calcs
Rotating Shrink Fit calcs
Fluid dynamics stuff that I haven't looked at.
Each of the chapters has a mathematical development of the algorithm and then some examples of output expected for a given input so you can double-check your program.
The exact programs are written in TI-59 programming language which is meaningless to me. There is enough info to figure it out from the math descritpion without looking at the code but the discussion is a little terse (brief) and I have had to work at it. It's a good book to use along with other texts and the price is definitely right.
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RE: calculating shaft critical speed by hand
For the case in the word file above k = 3E6 lbf/ft and
M = 37.4*lbm
With extra factor of 60 to convert to rpm it gives:
f = (60/2*Pi)*sqrt(2*3E6*lbf/inch/(37.4*lbm)*(32.2*12*inch/sec^2*lbm/lbf))
= 74,000 rpm.
As expected, higher than the transfer matrix model which included shaft flexing. But only about 10%. Again not good for finding a lower bound but I just wanted to mention it.
If you want to see what is the lowest that the critical speed can be, you need to estimate how low can bearing stiffness be (and put it into transfer matrix)
A side note - this algorithm models bearing as simple spring stiffness with spring perpendicular to shaft. There was some discussion about resistance of bearing to angular bending perpendicular to original shaft. I would think this assumption would be pretty good for simple conrad deep groove bearing or for face-to-face duplex angle contact, and certainly for self-aligning bearings like spherical roller bearing. But for typical duplex angle contact bearings mounted back to back, that configuration is very intolerant to misalignment (produces reaction moment) and this assumption of the model is least accurate in this bearing configuraiton.
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RE: calculating shaft critical speed by hand
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RE: calculating shaft critical speed by hand
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RE: calculating shaft critical speed by hand
In particular, the stiffnss of the bearing will be bounded by the stiffness of its mounting system. I rarely see a structural part with a stiffness in excess of 100 000 N/mm, and half that would be more typical.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
Non-linearity of bearing does not compute (literally... no room for that in the model).
100 000 N / mm * (1lbf/4.482N) * (25.4mm/inch) = 570,000 lbf/inch.
If I plug 500,000 lbf/inch into the program (all other inputs same as the word file), the first critical comes down to
As I mentioned 5E6 lbf/inch came out of the API document. Maybe this value was intended to be combined with another value for support stiffness.
If you run the program again with K = 500,000 lbf/inch, then first critical speed becomes 30,131 rpm. Mode shape looks even more like rigid rotor.
The rigid rotor approximation which had 10% error at K=3E6 gets better and better as we reduce K: For K=5E5
f = evalf(60/2/Pi)*sqrt(2*5E5*lbf/inch/(37.4*lbm)*(32.2*12*inch/sec^2*lbm/lbf)) = 30,694 rpm.
Now only a 2% error compared to the program.
If you want to consider K an unknown and plot the values for various of first critical speed for values of K, use the rigid body approximation above. We know the accuracy (compared to program) is 10% for K=3E6, 2% for K=5E5 and even better as we reduce K further.
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RE: calculating shaft critical speed by hand
Thanks Greg. I think you are right. I have seen examples worked with a range of stiffness far lower than what I used. I thought my value was high until I saw the API number. Maybe that number is supposed to be combined with a support stiffness.
Non-linearity of bearing does not compute (literally... no room for that in the model).
100 000 N / mm * (1lbf/4.482N) * (25.4mm/inch) = 570,000 lbf/inch.
If you run the program again with K = 500,000 lbf/inch, then first critical speed becomes 30,131 rpm. Mode shape looks even more like rigid rotor.
The rigid rotor approximation which had 10% error at K=3E6 gets better and better as we reduce K: For K=5E5
f = evalf(60/2/Pi)*sqrt(2*5E5*lbf/inch/(37.4*lbm)*(32.2*12*inch/sec^2*lbm/lbf)) = 30,694 rpm.
Now only a 2% error compared to the program.
If you want to consider K an unknown and plot the values for various of first critical speed for values of K, use the rigid body approximation above. We know the accuracy (compared to program) is 10% for K=3E6, 2% for K=5E5 and even better as we reduce K further.
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RE: calculating shaft critical speed by hand
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
I presume the large shaft diameter is to simulate the journal of a larger machine.
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RE: calculating shaft critical speed by hand
Can the OP decribe how the bearings are mounted? All the way to the ground.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
I will be interested to see how the support stiffness is calculated. The only form I can come up with is for a uniformly loaded rectangular block of steel... the vertical stiffness would be E * A / h. That suggests starting with the numerical value of E English units 3E7, and then multiply it by the ratio A/h where A is horizontal area and h is height. Unfortunately not only is the geometry simplistic, but the lower stiffness in horizontal direction associated with bending of the support will be more relevant and that will quickly get a lot more complicated.
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RE: calculating shaft critical speed by hand
How will it be driven... belt?
If you have two antifriction bearings on the end and a sleeve bearing tested in the middle, how will you control the load on the sleeve bearing?
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RE: calculating shaft critical speed by hand
yes the 4" diameter shaft is to simulate larger diameter fluid bearing applications as well as allow for some space to instrument the bearing and shaft
RE: calculating shaft critical speed by hand
Since the bearing and support stiffness will represent a fairly big uncertainty, a bump test would of course be a great way to check the results if you have the luxury of access to an assembled unit at this point in time. Try to bump the shaft horizontally near the center and measure vibration response on the bearing housing. Also can try vertical but expect that’ll be higher.
I'll be interested to hear what Greg and others estimate for support stiffness. Seems pretty low to the ground and stout compared to most machines.
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RE: calculating shaft critical speed by hand
Question though, in your 'bump test' why would you bump the shaft horizontally- it seems that the direction of interest would be the vertical, longitudinal modes of the supports (yes the transverse/bending modes would of the pillars would be lower-but there really is no excitation in that direction)- maybe i didnt descibe the rig very well. The shaft is mounted horizontally, supported by two vertical bearing pillars. Any static or dynamic loads would be applied perpendicular to the shaft axis
when i get home im going to calc the longitudinal modes for a basic 2"x7"x15" steel bar (thats basically what the pillars are just with a 90mm hole bored at the top)- im guessing this is going to be quite high and what you guys mentioned before-- that its going to be the stiffness of the bearings that will be the weak link. I guess a call to the manufacturer will be in order.
RE: calculating shaft critical speed by hand
Of the pedestal modes, vertical is axial compression of the structure, which is inherently stiff, whereas horizonatl is bending of the structure (typically) which is likely to be less stiff.
For our work at 500 hz we use cast iron bedplates tied to the reinforcement in the concrete floor and grouted in place- anything less and we see spurious effects due to the mounting system.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: calculating shaft critical speed by hand
If it were a flexible rotor and bearing/support very stiff in comparison, there is only one first critical speed.
In this case with the rotor very rigid compared to bearings/support, the bearings and support determine the critical. The support as Greg described typically much lower stiffness in the horizontal direction. So you have a "split" critical, with the lower frequency determined by the horizontal direction stiffness.
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RE: calculating shaft critical speed by hand
"in your 'bump test' why would you bump the shaft horizontally- it seems that the direction of interest would be the vertical, longitudinal modes of the supports (yes the transverse/bending modes would of the pillars would be lower-but there really is no excitation in that direction)"
#1 - Mode shapes and frequencies are characteristic of the system, not the excitation.
#2 - A mode shape is of concern if excitation is present to excite that mode shape at it's natural frequency. In the case of this horizontal rotor, there are numerous potential sources of running speed radial force which can excite resonance in either radial direction H or V. These would include unbalance, eccentricity, bowed shaft, misalignment of drive etc. That's why we never like to operate machines near critical speed or resonance. In the specific case you mentioned, the water is alittle muddier. I believe there was a concern for 1/2 speed whirl to excite critical or resonance (oil whip).
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