natural frequency of a pinned beam 2

[lvkb]

Can any one tell me how to work out the natural frequency of a beam pinned at both ends and subject to axial tension?

Thanks

 

[Drej]

You can model it using FE software, although only in its pre-tensioned state and not with the load applied actually applied (this is down to the "free vibration" theory used to obtain the eigenvalues of the system). As far as hand calcs on a pre-stiffened beam is concerned, that's not straightforward. If the loading is "easy", then you may be able to make some assumptions in your calcs to incorporate the added stiffness of the system due to the load (in FE eigenvalue analysis, the stiffness matrix is "updated" (added) with the extra stiffness of the load). I would seek some guidance from the bible that is Blevins (if you have access to this). Otherwise, modal analysis (eigenvalue) with pre-stress effects turned on using FE software maybe the way to go to get the definitive solution.

Cheers,

-- drej --

 

[lvkb]

Thanks for the help - unfortunatley we don't have a copy of Blevins in the office and we don't have a FE program with the option to take into account pre stressed effects. Any advice on how to make assumptions of the increased stiffness due to tension loads? (structural dynamics are not my forte )

 

[Qshake]

Try to window the solution from Roark's Formulas for Stress and Strain. This is a readily available industry standard. In this reference you'll find a case for a string vibrating laterally under a tenion, T, with both ends fixed. So what you'll need to do is look back at the example of a beam with ends pinned and see how that impacts the formula and then work in the tension as per the case noted above.

Another option is the Vibrations HandBook which is a three volume set.



Regards,
Qshake

Eng-Tips Forums:Real Solutions for Real Problems Really Quick.

 

[rlnorton]

Take a look at Timoshenko's "Vibration Problems in Engineering." In the 4th edition the solution is on page 454.

 

[Denial]

I tried to work this one out for myself many years ago, and concluded that the natural circular frequency (ie the frequency in radians per second) is given by
(p/L)*sqrt[(p^2EI/L^2+T)/m]
where
p = pi
L = length of bar
E & I = the usual
m = mass per unit length
T = axial tension

Unfortunately I do not have ready access to the Timoshenko book suggested by rlnorton above, but this result looks and smells right. (It gives the right result when the axial compression equals the Euler buckling load, and it gives the classic frequency for a taut string when EI is zero.)