column natural frequency

[structSU10]

Lets say you have a steel column, of height L, and need to know its vertical natural frequency. using f = sqrt(k/m), k = A*E/L and m = ρ*A*(L/2) [assume half of mass is effective due to variation across height] , then f=sqrt((A*E/L)/(ρ*A*(L/2))) which reduces to 2*E/L^2*ρ - essentially showing that no matter how stocky the member, the natural frequency for the steel columns will be the same. Is this strictly correct?

I have a frame that must meet strict natural frequency criteria, and is tall enough that the column alone cannot meet the criteria. Should I be pushing to get a more refined criteria, or is there a way to improve the frames frequency even with the issues at the column?

 

[JStephen]

If I understand that right, you're calculating the natural frequency of a free-standing pillar, but that's not the same as a column with additional mass attached at the top, which in most cases would be considerably different.

 

[WARose]

I'd question that the vertical stiffness would be "AE/L". More than likely, the vertical (dynamic) stiffness of the foundation involved (or the connections) would be less.

Also, you'd have to consider other modes involved. I'm not sure what your "strict natural frequency criteria" is.....but more than likely, they don't have just the vertical mode(s) in mind.

[red]EDIT[/red]: Messing with a column (with a spring base in the vertical direction) in a FEA program I have, as I expected, I am not getting significant differences in response (vertically) based on the locations of the lumped masses along the column. The total amount matters.....just not it's locations. The mode shape vectors are nearly identical for each node. (Meaning it all moves together in that direction.) This also presupposes a concentric load which if often times difficult to achieve.

 

[Agent666]

I believe if its only the freestanding column (no other lumped mass situated on top), then you are correct because area and mass are directly proportional to one another in terms of the vertical frequency, increase the area and the distributed mass increases at the same rate, so the frequency must be the same?

In reality the mass is distributed vertically along the column rather than being half of the mass lumped at the top and the whole length of column acting as a vertical spring with all these little lumped masses along the entire length.

What does the strict criteria stem from? Might pay to give some background so people can offer alternatives in context of the actual problem.

 

[GregLocock]

A section with a greater I/A (that is generally larger) will have a higher frequency, in the absence of additional masses. The reason is that the elastic stiffness is proportional to I and the mass is proportional to A and obviously f is proportional to root(k/m)

Unfortunately I cannot access my notes or my copy of RD Blevins Natural Frequencies and Modeshapes to get the exact equation.

However equation 4.7 here looks right to me

http://vlab.amrita.edu/?sub=62&brch=175&sim=1080&cnt=1

Cheers

Greg Locock


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[structSU10]

They want a frame with a vertical natural frequency of 200 Hz. They claim the horizontal modes do not bother the testing they perform on this frame. There is very little mass to the remainder of the structure, and as I would imagine extra mass will only reduce the frequency.

I was attempting to simplify the equation just for comparison. I do have a very small additional mass at the top of my column, but the actual weight of the column is more significant. My test models of two columns of the same height, vibrating under own weight only, give me roughly the same frequency.

My real issue is trying to find any way to increase the vertical frequency for the required height.

This also is a system that already exists, and does not match their criteria. They had a vertical frequency test done in the field, at the location of their test, and my current model matches that test within 5%.

 

[canwesteng]

You could yield the column and stretch it until you get some strain hardening, or built some kind of truss tower/bracing system that would decouple the frequency from the mass. Or shorten the columns.

 

[structSU10]

Thanks for some suggestions. Those are along the lines of what I am trying at this point. If I find something that works best I will update things here.

 

[JoshPlumSE]

X bracing. Concrete filled tube columns.

Also, you could put a nice thick concrete slab on top of the frame to add mass. Or, a concrete frame instead. Lots of options to stay away from that particular frequency.

 

[JStephen]

A quick check in the Vibrations textbook shows this is "Longitudinal Vibration of Rods". For free-free end conditions, they derive frequency as N/(2L) x sqrt(E/rho), where N is 1, 2, 3, etc. I assume for your case, your L is half of their L so there would be a factor of 2 difference.

 

[structSU10]

I did find the case in Roarks, for a fixed-free system, with own weight only and with an additional. mass at the top. for a bar with own weight only its (1.57/2*PI)*SQRT(A*E*g/w*L^2) where w = ρ*A , so it becomes (1.57/2*PI)*SQRT(E*g/ρ*L^2). When another mass is added increasing area does make a difference, but it reaches an asymptote based on the equation above, as the extra mass becomes 'negligible' to the own weight.

 

[Agent666]

Tuned mass damper?

EDIT - I don't know if a damper would be any help in this situation, but just throwing it out there.

 

[Leftwow]

Staadpro can calculate the Rayleigh Frequency if you use that.