first natural frequency of a cantilevered pole

huangyhg

first natural frequency of a cantilevered pole
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!


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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!  
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   
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
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   
isn't the exact distributed mass solution a partial differential equation??  those are brutal to solve, but i bet someone has solved it before.
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.
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.
oh yeah, the delta in my post must be in inches.
"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.
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