natural frequency of a radially compressed ring
natural frequency of a radially compressed ring
(OP)
Hi all,
I am looking at the eigen-value analysis of a ring compressed radially using nodal forces. The natural frequencies increased when I used load stiffening option in Algor when compared to a ring with no loads on it. The constraints were the same for both. Is this correct? I am not sure if this is right.
I think for a beam in tension, the natural frequencies would increase and decrease when in compression (when the load is less than the buckling load), but here, in the case of the ring, the frequencies are increasing in compression. Could someone explain this to me?
Thanks,
Kumar
I am looking at the eigen-value analysis of a ring compressed radially using nodal forces. The natural frequencies increased when I used load stiffening option in Algor when compared to a ring with no loads on it. The constraints were the same for both. Is this correct? I am not sure if this is right.
I think for a beam in tension, the natural frequencies would increase and decrease when in compression (when the load is less than the buckling load), but here, in the case of the ring, the frequencies are increasing in compression. Could someone explain this to me?
Thanks,
Kumar





RE: natural frequency of a radially compressed ring
It sounds like a glitch to me. Why don't you try putting the ring in radial tension and see what result you get. That will give an indication of what's going on.
Good luck.
RE: natural frequency of a radially compressed ring
corus
RE: natural frequency of a radially compressed ring
Steve Braune
Tank Industry Consultants
www.tankindustry.com
RE: natural frequency of a radially compressed ring
The static loads superimposed on the structure should have no effect.
As an example, consider the surge mode of a coil spring, with varying amounts of preload. So long as the geometry is unchanged the preload is not part of the equation.
The reason ( I think) is that the resonant frequency is an energy balance, and as much work is done against the preload in one half of the cycle as is regained from it in the other, so it has no net effect.
Cheers
Greg Locock
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
Believe elastic deflections are being dealt with; the equations are non-linear only where the deflections are large.
cspk: You need to specify which modes are involved: extensional, transverse, torsional?
The amount of force required to produce a unit of deflection is greater under radial compression, so your critical frequencies should increase. The analogy with the axial loading of a beam is flawed. It would be more correct to compare your case with a beam with lateral supports.
RE: natural frequency of a radially compressed ring
Explain why you think this should be true for the ring, and if it is true, why it does not imply non-linearity.
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
believe the radial load stiffens motion in the the transverse(radial) direction. but before we get to far along we need a more precise statement from cspk as to what modes he has modeled.
agreed axial loads(compressive) do not increase the criticals. however, this is not the case with transverse stiffening.
linearity in the sense of hooks law.
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
"the so called Babuska's paradox may be mentioned. Whereas the solution of a bending problem for the thin elastic circular plate is unique, it differs from the case of a plate with polygonal boundary even if the number of sides of the polygon tends to infinity. The solution becomes non-unique in this limit."
I assume this is for distributed loading - do you know how big the difference is, typically? What does it depend on?
Here's Roark's formula for an infinite sided polygon, fixed edges, nu=.3
max y= -.171*q*a^4/E/t^3
where a is the inscribed radius
and here's the formula for a disk (table 24 case 10b)
y=-.01563*qa^4/D
and D=Et^3/12/(1-nu^2)
which is an error of 0.2%
Good. I always knew there was something wrong with FEA!
Cheers
Greg Locock
RE: natural frequency of a radially compressed ring
Cheers
Greg Locock
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
As I said I'm not too old to learn ! (I'm serious about this) What sort of structures stiffen under compression ?
By the way Kumar have you lost interest ??
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
I'll watch this space!
RE: natural frequency of a radially compressed ring
Thanks all. I was out of my office and didn't get a chance to access this forum. I am very excited to see so many replies.
I ran the eigen value analysis with different loads and these are the results.
Nodal force: -1.75 lbs (radial compression)
1st Frequency: 3.76 Hz (In-plane bending mode)
Nodal force: -3.5342 lbs (radial compression)
1st Frequency: 5.04 Hz (In-plane bending mode)
Nodal force: -4.36 lbs (radial compression)
1st Frequency: 3.22 Hz (In-plane bending mode)
Nodal force: 4.36 lbs (radial tension)
1st Frequency: 16.1 Hz (In-plane bending mode)
Btw, the carbon-epoxy composite ring is 3 meters in diameter. Cross-section of the ring: O.D: 0.726", thickness: 0.03"
I tried the elastic buckling analysis too. I designed the ring for a total radial compressive load of 558 lbs using the elastic stability equation given by Timoshenko in "Theory of Elastic stability" (page 291). The finite element model had a load of 279 lbs. I was expecting a buckling load multiplier of about 2, but the analysis gave a multiplier of 1.1. I think I am missing something. Thanks for the help.
Regards,
Kumar
RE: natural frequency of a radially compressed ring
What sort of FEA analysis are you doing?
What is the predicted linear first mode, by analysis?
Cheers
Greg Locock
RE: natural frequency of a radially compressed ring
Steve Braune
Tank Industry Consultants
www.tankindustry.com
RE: natural frequency of a radially compressed ring
Questions:
What does the mode shape look like ? I assume it's some sort of elliptical deformation, not the pure radial deformation that most of my comments referred to.
What's the frequency with no load ?
RE: natural frequency of a radially compressed ring
Yes you certainly started an interesting thread ! It looks even more like a glitch to me. The proportions of the structure are a bit unusual, but that shouldn't make any difference in the sort of eigenvalue analyses you're doing. I would suspect some kind of numerical problem in Algor. Why not contact the supplier to see what they say, unless you have access to another analysis package to do a comparison with ? Or you could try running the analysis again using different units, that might shed some light on a numerical problem, if there is one.
There is another possibility, and I don't have time to do any arithmetic myself on this (it's a long shot too), if the total radial compression you are using in any of the nat freq analyses is anywhere near the buckling load, you could get strange results.
Best of luck
RE: natural frequency of a radially compressed ring
Now that I've read the question (again), my comments do not apply to a ring loaded at the nodes. You might look into "shear locking" if your stiffness is increasing with node count.
RE: natural frequency of a radially compressed ring
The mode shape is elliptical deformation. One thing which I forgot to mention is the constraints in the ring. The ring is in the XY plane. The model (beam elements) was constrained with Txz at the top and bottom nodes and Tyz at the two side nodes. The load's been distributed in 64 nodes on the circumference.
Frequency with no load is about 0.008 Hz (a 3-m ring vibrating at 0.48 cycles every minute, I don't understand). Unfortunately, I don't have access to any other software. I will try to post my findings as I work on this.
Thanks all..
Kumar
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
Have you tried increasing the number of nodes yet ? My guess would be that at least some of the anomalies would disappear. But none of it makes sense to me so far.
RE: natural frequency of a radially compressed ring
f(Hz) = i*(i^2-1)/(2*pi*R^2*sqrt(i^2+1))*sqrt(E*Iz*gc/m)
where i = 1,2,3 etc (in this case i=2, since i=1 is pure translation)
R = radius of ring
E = youngs modulus
Iz = second moment of area of ring section (using your xyz)
m = mass density per unit length
I put the gc in for clarity.
How does that stack up against your .008 Hz ?
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
I get 12.75 Hz for a steel ring - carbon won't be much better.
I think the non linear solver is probably getting confused by the relatively large static forces compared with the stiffness of the structure, small number of nodes, and rather odd constraints.
Anyway, it sounds as though either you have very fundamental modelling problems, or (Heaven forbid) I've made a mistake.
Cheers
Greg Locock
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
RE: natural frequency of a radially compressed ring
I get 5.994 Hz for an unloaded STEEL ring, using a diameter of 3 metres, cross section is circular tube, OD 0.726" (18.44 mm), wall thickness is 0.03" (0.762 mm). (I used steel because it has relatively "standard" material properties. E = 200 GPa, Density = 7,870 kg/m3, Poisson's Ratio = 0.25.) The first mode shape is elliptical in-plane vibration, as expected.
My value for the frequency of the same ring using EnglishMuffin's formula is 5.986 Hz, which agrees to 0.13% with my FEA result.
In my FEA analysis, the frequency drops SLIGHTLY when the ring goes into compression, and rises SLIGHTLY when the ring goes into tension, all as expected. I kept my ring compression load significantly less than the buckling capacity of the ring. You would expect to see a dramatic drop in frequency if the ring compression approaches the ring buckling load. Similarly, a significant increase in frequency could arise if the ring tension becomes “significant”. I kept my axial loads (tension and compression) to less than 15% of the buckling load. My frequencies only changed by about 1.5%. I didn’t see any frequency changes of the order of magnitude of those reported by cspkumar.
I suspect the very low frequency of 0.008 Hz is actually a free-body mode. Check your constraint conditions to make sure your model is actually constrained properly. When I analyse my ring using 2D beam DOFs, and no other restraints at all, I get 3 zero natural frequencies (X translation, Y translation, and Z rotation), before my first “real” frequency of 5.994 Hz. If I constrain my ring with minimum constraints to permit normal linear static analysis (similar to cspkumar’s description), my first reported mode shape is an elliptical mode at 5.994 Hz.
Apart from this, I can't account for the apparent behaviour of frequency increasing under slight compression, and then decreasing again. It sounds like a modelling problem to me. I would check all units carefully, as well as model constraints.
In particular, is there a possible problem with confusing mass, weight and force units? In the metric world, we are lucky that the only mass unit we need to know is the kilogram, and the only force/weight unit we need is the Newton. In the foot-pound-second / inch-pound-second world, you need to be VERY careful to not confuse the pound-mass, and the pound-force – they are NOT equivalent. I believe that the “pound mass” is defined as that mass which has a weight of one “pound force” when accelerated at one inch/s/s. This is approximately equivalent to 386.4 “pounds” (as in “a pound of sugar”), or 175.24 kilograms. An error in density or force of this order of magnitude could result in alls orts of unexpected results!
(My understanding of the “pound mass” could be wrong, or perhaps there is more than one “common” definition of the “pound mass”. My understanding comes from “Building Better Products with Finite Element Analysis” by Adams & Askenazi. As I said, in the metric world, we rarely have to deal with this confusion.)
RE: natural frequency of a radially compressed ring
As far as the definition of pound mass goes, the situation is confused because there are two different systems employing english units, one based on force and the other on mass. In the UK, the lbm used to be based on a physical standard made of platinum. Today, for the f lbm sec system it is simply defined in terms of the Kilogram. In the case of the f lbf sec system, your definition is not even approximately correct. You should have said that a pound mass is defined as that mass to which a pound force imparts an acceleration of 386.088 in/sec^2. However, we seem to be in basic agreement. I suspect that with more nodes, kumars results will look better.
RE: natural frequency of a radially compressed ring
It was a software glitch. Somehow it got corrupt and I had to delete and load it again. Now, the first real mode for an unloaded ring is out-of-plane flexure mode with a frequency of 11.75 Hz. The first in-plane flexure mode frequency is 12.1 Hz, which agrees with Englishmuffin's equation(3% error, I am sure like what englishmuffin said, if I use more nodes, I would converge to the theory). Even the buckling analysis worked. The code came back with 2.013, which is what I expected.
Thanks for all the replies. It was very very informative. Thanks again,
Kumar
RE: natural frequency of a radially compressed ring