Effects of Camber on Vibration

[AlexREI]

Does anyone have a good anwser for how cambering a beam effects it vibration characteristics (in composite steel). The AISC floor vibration guide uses the equation fn=.18*sqrt(gravity acceleration/deflection)) to get the natural frequency of a particular beam or girder.

When a beam is cambered, your deflection will be much less and therefore the natural frequency will be higher. I'm not sure if this is the appropriate value to use though.

If the total uncambered deflection is used, the natural frequency will be lower and the beam would have to be larger to be greater than the forcing frequency.

Does anyone have an opinion on how the natural frequency should be calculated w/ camber? Thanks.

 

[civilperson]

Deflection is identical for cambered and uncambered beams. It is measured from the unloaded position.

 

[SteveGregory]

Ditto

 

[AlexREI]

Maybe I should clarify.
If the beam is cambered, should the value of the camber, say 3/4", be added to the total deflection to find the natural frequency.

So if your total calculated deflection @ vibration level (LL= 10 - 30 psf depending on the activity) is 1.25", would you add the .75" for the camber and get a net deflection of .5".
Thus the natural frequency is higher because of the camber.

 

[jike]

Vibration is related to the amount of movement it goes thru, not its starting point.

 

[271828]

"When a beam is cambered, your deflection will be much less and therefore the natural frequency will be higher."

!!!!! Whoa there buddy, LOL.

This isn't correct at all, and is about the 10th example I've seen of people being WAY misled by the form of that equation.

The natural frequency of a simply supported beam with uniform line mass, m, is

fn = pi/2*sqrt(E*I/(m*L^4)) (1)

The problem is that this equation is hard to remember. However, the equation for deflection of a uniform load is easy to remember, so it's tempting to come up with the equation that you used.

If you manipulate Eq. 1, above and plug in Delta = 5*w*L^4/384EI and recognize that w = m*g, then you get

fn = 0.18*sqrt(g/Delta) (2)

The point is that m is the uniform line mass along the beam. There is no "load" going into any natural frequency equation for a linear system. Natural freq is a function of mass and stiffness and nothing else for linear systems with very light damping. (Talking simple linear systems here before one of our ME friends comes by and smacks me upside the head with the idea of complex modes, LOL). Therefore, camber makes no difference in this calculation. m is from whatever load is attached to the beam, regardless of camber, when the load was put on there (within reason).

Now, it's a separate question, but camber does affect the natural frequency simply because it allows a beam with a smaller Ix to be used.

 

[vandede427]

The live load deflection of a beam is independent of the camber.

A composite beam that was cambered 1" will deflect the exact same amount under Live Load than a beam that wasn't cambered.

 

[vandede427]

To clarify what I meant, that is its measured change in elevation relative to its starting point.

0" down to -1" is the same as -1" down to -2"

 

[AlexREI]

Thanks for the input. That helps a lot.