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First natural frequency of a cantilevered pole
2

First natural frequency of a cantilevered pole

First natural frequency of a cantilevered pole

(OP)
Hi

How do you calculate the first fundamental (natural) frequency of a cantilevered mast or pole (in my case a 13m high crucifix)? I can only find frequency formulas for members where the loading and member stiffness is parallel with the gravity loading (ie horizontal beams subject vertical loads – I have vertical beam subject to horizontal loads).

Also what would be an appropriate serviceability deflection at the tip? I was intending H/125 but I'm struggling....

Thanks!

 
 

RE: First natural frequency of a cantilevered pole

Assuming the axial load is not so high that the column is close to buckling under self weight, the natural frequency and mode shapes are exactly the same for a vertical cantilever as for a horizontal cantilever. That is, natural frequency is a function of geometry, mass and stiffness, but is independent of applied loads.

Hope this helps!  

RE: First natural frequency of a cantilevered pole

You will have to lump the extra mass at the intersection.  

It will not be a stsndard, simple solution due to the extra concentrated mass.  It will be a different equation, or series of them.

However, do you realize that the Greek word, as translated for the cross, is "Staros", transleted "stake" or a single pole, without the crossed member?  

So...the solution to the problem would be simpler without the cross member.

Mike McCann
MMC Engineering

RE: First natural frequency of a cantilevered pole

Sorry, but it is also dependent where the mass is in relation to the base.  Agsain, this is NOT the standard solution here.

Mike McCann
MMC Engineering

RE: First natural frequency of a cantilevered pole

(OP)
Thanks.

I suppose a Greek game of noughts and crosses would look like binary....!

So.. the horizontal frequency is a function of the vertically orientated mass and the horizontal stiffness, it doesn't have anything to do with the mass accelerating the motion or anything? So... a members frequency is independent of it's orientation, loadings and gravity... Just feels like there shouldn't be an 'm' term in the formula....

A bit more background... I am just trying to do a first tier analysis of the crucifix to prove that it has a frequency >1Hz and therefore avoid a heap of painful calculations imposed by my (Australian) standard.

Do you think i can go lower than the H/125?

Thanks again!
 

RE: First natural frequency of a cantilevered pole

Isn't the exact distributed mass solution a partial differential equation??  Those are brutal to solve, but I bet someone has solved it before.

RE: First natural frequency of a cantilevered pole

Check publications on powerline structures with cross arms. Also, as Mike noted, the analysis of the cross structure is a bit more difficult as you may have competing and complementary aeolian vibration occuring at the intersection.  That's a huge crucifix.

RE: First natural frequency of a cantilevered pole

This is very, very easy.  

First off, Julian is right.  Assuming it's not almost unstable (that's the only way "loads" make a difference), it will have the same natural frequency upside down, on its side, up in space, whatever.

If you have a copy of the AISC Design Guide 11, you can use the equations at the start of Chapter 3 to estimate the natural frequency.  *Pretend* that the crucifix is a cantilever beam turned on its side and subject to its self weight.  Compute the deflection at the tip and call this Delta.  The natural frequency in Hz is approximately fn=0.18*sqrt(386/Delta).

Also, any modern structural analysis package will compute this natural frequency also, and be more exact.  RISA, SAP, RAM Advanse...pretty much all of them.

RE: First natural frequency of a cantilevered pole

Oh yeah, the Delta in my post must be in inches.

RE: First natural frequency of a cantilevered pole

"Isn't the exact distributed mass solution a partial differential equation??"
ALL beam, string, shaft, or frame problems end up with PDEs.  The lumped mass just makes it more difficult to solve, so one ends up using some kind of approximate method.  The textbooks spend a lot of time on Rayleigh-Ritz, Galerkin, etc., but we structural guys generally just use the finite element method.


"Those are brutal to solve, but I bet someone has solved it before"
Sure.  We did that in class.  I'd never even think of it without FEA, though.

RE: First natural frequency of a cantilevered pole

A professor of mine once said: "if you can't solve a problem, reduce the problem to something that you can solve", or some such thing...

While this isn't a standard case, it is pretty close to a cantilever with a concentrated load at the top and a uniform mass along it's length.  I would solve that case and see where I stand in relation to the 1 Hz limit.

I've attached the solution to the standard case.  Try yours with "L" equal to the height of the crossing member and "W" including all the material above that point.  If the result is greater than 1 Hz, I'd say you're OK.

RE: First natural frequency of a cantilevered pole

miecz,

Great advice, and a great help in posting a simple solution to a problem that others are making too complex....a couple questions about your solution...

1) 2000 Kips ?? Is that right?
2) I can't get 4.4 Hz from your formula for angular frequency....?? When I work through your units I'm getting nowhere...which makes me think you made a mistake or I'm missing something.

For the top I get psi x in^4 x ft/sec^2 not considering applicable conversions to inches. On the bottom I get lbs x in^3+ lb/in x in^4...????

Thanks

RE: First natural frequency of a cantilevered pole

miecz,

I cranked through your equation and it seems reasonable to me.

Using my approximate method, I get 0.84 Hz compared to your 0.70 Hz.  If I take away the point mass, I get 21.5 Hz with your equation and 20.4 Hz with my approximate method.

RE: First natural frequency of a cantilevered pole

Hmm. Well I would agree as to the above solutions, I suggest solving the beam PDE is a more informative choice. I do this quite often and the solution to this question could be done by following a simple textbook beam PDE equation with the added lumped mass as a boundary condition to the PDE.
I by no means discount anyone's work. The above posted results are very simple and easy to follow. However, they must include the assumptions that were made in order to be a complete solution.
Try solving: d^2w(x,t)/dt^2+c^2*d^4w(x,t)/dx^4=0, where c=sqrt(EI/rho*A) for a fixed-fixed boundary condition first, then if you are comfortable try setting the mass*accel you gave at a particular point equal to the deflection force.
This should give you a complete solution.

peace
 

Fe

RE: First natural frequency of a cantilevered pole

bigmig-

W=2000kips.  I just picked a number out of the air to show the formulas and give everyone an example to compare to. I get;

3EIg = 38876975 kip-ft^3/sec^2
WL^3 = 2000000 kip-ft^3
wL^4 = 9000 kip-ft^3

Do we agree to there?

RE: First natural frequency of a cantilevered pole

Have you considered an open SS structure to avoid Karman vortices in a strong wind? A tapered column form would be interesting and structurally attractive.

RE: First natural frequency of a cantilevered pole

plasgears brings up something interesting.  I wonder if the fundamental mode could be torsional.  If it's an open section, I could imagine this happening.  Does the first torsional mode also have to have fn > 1 Hz?

RE: First natural frequency of a cantilevered pole

Good point.
Although, usually in strong wind the vortex shedding frequency is significantly higher then the first mode freq. of such a structure. It should be checked.
What is mentioned about the torsional mode is only dangerous if the torsional modal frequency coincides with that of the translational frequency. This is what we call flutter. (I have never seen a crucifix flutter however)
The uncoupled fundamental mode is not likely to be torsional because torsional stiffness is likely to be higher then that of the bending mode.
Usually one could first find out what the maximum and average wind speeds are for the area then correlate this to the vortex shedding frequency. If this simple relation comes out close or relatively close to any of your calculated natural frequencies then a more rigorous analysis should be done. This is likely to not be the case though.





 

Fe

RE: First natural frequency of a cantilevered pole

(OP)
Thanks for all your inputs.

The crucifix is an open section, a 250UC90 - pretty close to a W10x49 (I think) which results in a service deflection at the tip of about h/140 (8m cantilever). The major axis frequencies are both greater than 1, the torsional frequency is very small but has low associated deflections.
The way my code handles member design with frequencies less than 1Hz is to amplify the static load. Given that the section is obviously sized for deflection (and I was willing to go down to about 1/100) I believe there is sufficient redundancy to account for 'unanticipated' effects in both the strength and service states.

Thanks again for all the assistance it was very informative!
 

RE: First natural frequency of a cantilevered pole

OzEng80

How did you finally determine your natural frequencies?

RE: First natural frequency of a cantilevered pole

OzEng80,

You have not provided sufficient information to enable a full natural frequency analysis (e.g. height and span of cross arm, orientation of web in cross arm and mast, etc).

However, I would have some concerns about the torsional mode using an open UC section for the mast. A quick check suggests the torsional natural frequency could be as low as 0.25 Hz. While a simple wind load assessment would suggest the wind load should be balanced on the cross-arm (i.e. equal mean load on each side), so there is no steady-state torsion due to wind load, it is pretty apparent that there will actually be dynamic fluctuations in the wind loads on the two arms, and this could generate some response to the torsional vibration mode. I am not sure how to calculate this; just flagging a potential issue.

You might want to consider using a closed section for the mast at least, to significantly increase the torsional stiffness and frequency. Just boxing up your UC mast probably won't change the two main sway frequencies much, but it will probably raise the frequency of the torsional mode substantially. Just a thought.

I agree that the major axis frequencies are likely to be around the 1 Hz mark.

RE: First natural frequency of a cantilevered pole

(OP)
miecz - I'm ashamed to admit that I got it out of a black box... My attempts to do this by hand generally supported the output, and the propagation of a headache.

Julian - Unfortunately I am stuck with the open section. I have already informed the architect that I could do it for about half the weight if he went with a hollow section... waste of steel... Regarding possible torsional amplifications I have satisfied myself that due to the mast being significantly (and the connections will be too) over designed for strength that there will be sufficient redundancy for 'uncertainties'. My code accounts for vibration effects by increasing the static load - I account for this (in strength) up to a factor of about 2.5. I would welcome comments to this approach?

Cheers

 

RE: First natural frequency of a cantilevered pole

Problem with torsionals is that they tend to run very high Q values, and you have an interesting possible self-excitation method with a vortex street off the two arms as the thing twists. You also have a geometry that means that it will be able to tune into a greater range of windspeeds.

Calculating the excitation frequency produced by a vortex street is possible for some geometries, related to the Strouhal number. Sorry I don't have any direct references, but we see it on whip aerials on cars and the frequency prediction is pretty good.

http://en.wikipedia.org/wiki/Vortex_shedding




 

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: First natural frequency of a cantilevered pole

Vortex shedding frequency is related to the Strouhal number by the simple relation: fs=S*U/D, where S is the Strouhal number, U is the mean flow vel., D is the nominal cross-sectional thickness/diameter.
Greg points out something interesting. The fact that because of the span of the horizontal member you could see a 'lock-in' regime at a range of frequencies.
Something to look into.
What I mentioned above about the torsional frequency may not be the case. I went through a quick program I wrote and it seems that the torsional rigidity of many structural members may not be as high as I thought.

P.S. to save you time, the universal Strouhal number is 0.2

peace


 

Fe

RE: First natural frequency of a cantilevered pole

(OP)
Fex32

I don't understand how the vortex frequency corresponds to a design state. If i set the frequency as 1Hz this gives me a corresponding velocity of 1.25m/s (0.25/0.2), which is obviously very low. Any increase in the velocity results in a higher frequency (good?). Does this mean (assuming 1Hz) is the magical number, that the mast (this still ignores the crossbar) get excited at wind velocities less than 1.25m/s and stabilizes as the wind increases? Assuming this only amplifies loadings for this wind speed (very small load)and doesn't result in some sort of cumulative harmonic amplification then this should be acceptable - is this your interpretation?

I am considering closing the W section with some continuous plates between the flanges (set back so that it still looks like a W section). This would give me the 'box' torsional stiffness without compromising (in my opinion) the architecture. Hopefully this will raise the torsional stiffness to greater than 1Hz.

Cheers
 

RE: First natural frequency of a cantilevered pole

As the structure vibrates torsionally the airspeed will vary along the arms. At some radius the critical St number will occuur and so vortex shedding will start. Even if the windspeed varies, or the amplitude of the torsion varies, the shedding region will just move along the arms and will still excite torsion.

Now, it could be that it is nowhere near your resonant frequency, but given the absence of details, it could be.

Incidenatally what does the 'mast' consist of? How did you calculate the torsional constant?
 

 

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

RE: First natural frequency of a cantilevered pole

Exactly,

Also I would keep in mind that the 'alternating force' created by the vortex shedding should increase in magnitude with an increase in frequency. So a low fs is sometimes a good thing.
To answer your question, from my perspective, an increase in velocity is usually good as long as you also check coincidence with the other natural frequencies.
However, if your structure has low damping I would look into 'galloping' which occurs when the vortex shedding frequency is much higher then the first nat. freq.
Well, now that I think about it make sure you have the fundamentals down packed first then you can worry about the special cases such as galloping.

And I don't know what you mean by: cumulative harmonic amplification.

ttyl,

peace



 

Fe

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