Finding Shaft Critical Speed 1

[sloth4z]

I am trying to compare COSMOS to hand calculations to find max displacement, resonant frequency (no loads) and shaft critical speed (includes loads). I have been able to replicate the first two in COSMOS without a problem but can't seem to figure out the third. Am I misunderstanding the results of COSMOS, or is there a certain way to set the problem up?

Here are the properties..
Shaft OD - 4 7/16"
Shaft Length - 27 3/8"
Bearing located at ends
1400# force located in center of span.
Material - 1045 Steel

I set up one static study and two frequency studies. The static study resulted in a max displacement of 0.0095", which matches calculations using beam theory. The first frequency study was for resonant frequency, and was done without any loads using FFEPlus solver selected, and resulted in a resonant frequency of 116.23 Hz (this also matches hand calculations). The third analysis used Direct Sparse and "In Plane Effect" selected, but the reulting frequency was 115.76 Hz. According to hand calculations it should be around 32 Hz. I used two calculations one using max deflection (32.3 Hz), and one specific to a simply supported beam (32.8 Hz). I also used a program which calculated a critical speed of 1950 rpm which corresponds to 32.51 Hz. Using "In Plane Effect" to include the loads, does not change the frequency as much as theory says it should.

 

[daveboy]

I haven't tried this with COSMOS yet but I used to do this a lot with Ansys designspace (not very successsfully ) and with Ansys mechanical.

Looking at the results from from the resonant frequency and the shaft critical speed they appears to be very similiar and I would suspect although I am not entirly sure that it has something to do with applying loads rather than adding mass.

From my experience trying this with designspace you have to add a lump of material to simulate the mass rather than applying a load.

 

[cbrn]

Hi,
just a little general background: be careful not to confuse external loads (which can cause a "prestress effect" on the shaft, thus varying the stiffness if the calculation of this kind of effect is activated (not even sure it can in COSMOS) with "fictitious" loads which are introduced to simulate the presence of added masses: in this case, you add nothing to the [M] matrix, neither as mass terms nor as inertia terms. As regards bending eigenforms, the inertias are of course the "bending" ones, not the polar ones of the sections. I seem to remember that in COSMOS you have the "mass" element: use it and make sure you not only specify a mass value but also an inertia value (and the same "I" value for the two orthogonal planes containing the shaft axis).
Moreover, in any case for the critical speed search you have to include gyroscopic effects: so you will have to run several eigenmodes extractions with Coriolis effects activated (i.e. calculation of the gyroscopic matrix) and increasing values of the rotation speed: you should find, for each natural frequency, two equipollent eigenmodes (one "bacward whirl" with decresing frequency and the other "forward whirl" with increasing frequency). In rotodynamics calculations, generally only the f.w. are considered. Then, with Excel, find a best-fit curve equation for the calculated values, and search the rotation speed value "omega-k" for which it results omega-k = f(omega-k), where "f" is the equation of the best-fit curve (i.e. you search the value of the rotation speed which is equal to the value of the natural frequency for that rotation speed: this is the definition of "critical speed"). Please note that you will create a loop in Excel, so you will have to activate "iterations" in the "calculation" tab of Excel "options".

Regards, and good luck !

 

[sloth4z]

Daveboy: thank you for your response. That got me closer to the frequency I was looking for. I added a fat disk in the middle, and got 36Hz for the second mode (the first one was 30, but it was the disk's frequency). I will play around with this some more.

CBRN: I appreciate your help, but it's wasted on an idiot like me. To be honest I only understood about half of what you said. I am working with some empirical formulas that are approximations. I don't know anything about an M matrix. I took one course in college about vibration, and it minimal at best. Here are the formulas I am using so you know where I am coming from.

----

Ncr=187.7/sqrt

Ncr = approximate critical speed in rpm
y = the maximum deflection in inches. The book states some correction factors for y. for example 0.789 for the deflection from the wheight of the beam, and 1 for an added weight in the center.


------

fp=f1/sqrt(1+?*P/W)

f1=(?/2/?)*sqrt(g*E*I/?/l^4)

fp = natural freqquency with weight, Hz
f1 = 1st natural frequency without weight, Hz
? = frequency factor (9.87 for first natural frequency)
? = weight of beam per unit length, lbs/in
l = lenght span, inches
E = modulus of elasticity, lb/in^2
I = moment of inertia, in^4
g = gravity (386.09 in/sec^2)
? = weight correction factor (2 for center load)
P = added weight
W = weight of beam

 

[sloth4z]

that's wierd. the greek symbols showed up in the editor. that's what I get for not using preview. Let's try again...

fp=f1/sqrt(1+a*P/W)

f1=(ga/2/pi)*sqrt(g*E*I/w/l^4)

fp = natural freqquency with weight, Hz
f1 = 1st natural frequency without weight, Hz
ga = frequency factor (9.87 for first natural frequency)
w = weight of beam per unit length, lbs/in
l = lenght span, inches
E = modulus of elasticity, lb/in^2
I = moment of inertia, in^4
g = gravity (386.09 in/sec^2)
a = weight correction factor (2 for center load)
P = added weight
W = weight of beam

 

[cbrn]

Hi,
excuse me for having been a little pedant perhaps... Well, rotodynamics is rather a complicated subject and I myself can't be considered an expert.
I'd say that, as long as you can schematize your shaftline in "simple" way, the use of analytical (though approximate) methods are preferable over FEM, where you have the BIG problem of being sure you are modeling a consistent system !
And, anyway, if the geometrical complexity of your system grows up, as far as I know only extremely specialized softwares can handle all rotodynamics implications correctly (for example, MADYN, but it doesn't have any "full-3D" element, so once again you're limited in the geometry you can create...).

Bye, regards !