×
INTELLIGENT WORK FORUMS
FOR ENGINEERING PROFESSIONALS

Log In

Come Join Us!

Are you an
Engineering professional?
Join Eng-Tips Forums!
  • Talk With Other Members
  • Be Notified Of Responses
    To Your Posts
  • Keyword Search
  • One-Click Access To Your
    Favorite Forums
  • Automated Signatures
    On Your Posts
  • Best Of All, It's Free!
  • Students Click Here

*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.

Posting Guidelines

Promoting, selling, recruiting, coursework and thesis posting is forbidden.

Students Click Here

Jobs

Natural Frequency hand calc

Natural Frequency hand calc

Natural Frequency hand calc

(OP)
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

RE: Natural Frequency hand calc

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

RE: Natural Frequency hand calc

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

RE: Natural Frequency hand calc

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.

Red Flag This Post

Please let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.

Red Flag Submitted

Thank you for helping keep Eng-Tips Forums free from inappropriate posts.
The Eng-Tips staff will check this out and take appropriate action.

Reply To This Thread

Posting in the Eng-Tips forums is a member-only feature.

Click Here to join Eng-Tips and talk with other members!


Resources