First natural frequency of a cantilevered pole 2

[OzEng80]

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!

 

[jhardy1]

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.

 

[OzEng80]

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

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

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






Fe

 

[OzEng80]

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

 

[GregLocock]

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

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[FeX32]

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,







Fe