Natural Frequency hand calc

[trainguy]

To all hotshots out there,

Can anyone tell me a formula for approximating the natural frequency of a simply supported beam, given the length, moment of inertia, material, mass, and first order deflection?

I know of a neat approach for building design called the Rayleigh approximation, using the masses and first order deflections.

Is there a comparable method for beams?

What about for a beam overhanging 2 supports (like a railcar, for example)?

Thank you in advance.

tg

 

[arto]

From Mark's Hdbk [p.5-67 of 8th ed.]:

fn = [1/(2*pi)]*sqrt[g/dst]

g = gravity
dst = static deflection under it's own weight

 

[GBor]

omega = sqrt(k/m) where k is the spring stiffness and m is mass. k has to be defined for the beam. Generally, I seem to recall it is something like EI/L^3 (derived from Force = k * delta, where delta is the deflection). Since beam deflections generally take the form of PL^3/CEI, where C is a constant, solving for force "P" gives delta*(CEI/L^3). If you get the beam deflection equation for a given boundary condition, you should be able to back out a "k" and, with known beam mass, be able to approximate omega.

...or just look it up in Marks' or Roark's

Garland

 

[Denial]

Strictly speaking, the formula Arto gives is correct only if the entire mass of the beam is concentrated at the point that has the greatest deflection (which would be the centre point in the case of a simply supported beam). To give you some idea of the error that might be involved, in the case of a simply supported beam with uniformly distributed mass the formula will under-estimate the natural frequency by about 11%.

(If you apply the formula to a beam that is built-in at each end, the underestimate in the case of uniform mass distribution is about 12%. For a cantilever it is about 19%.)

Once you have some parts of the structure that move upwards when the main part of the structure moves downwards under static gravity loading, then the formula should not be used. In such cases, there is indeed a method known as Raleigh's Method. It involves assuming a deflected shape, then (based on that assumed shape) calculating and equating the strain energy at full deflection and the kinetic energy at zero deflection. The more accurate your assumed deflected shape the more accurate your estimate of the natural frequency. However you usually get adequate estimates with even a crude assumed shape.