iMechanica - resonance (natural) frequency of a cantilever beam - Comments

resonance (natural) frequency of a cantilever beam

Somashekara Bhat — Sat, 03 Nov 2007 07:58:03 -0400

Thanks and regards
Somashekara Bhat

resonance (natural) frequency of a cantilever beam

Somashekara Bhat — Sat, 03 Nov 2007 07:56:39 -0400

Thanks and regards
Somashekara Bhat

Natural frequency....

ramdas chennamsetti — Fri, 02 Nov 2007 14:14:51 -0400

R. Chennamsetti, Scientist, R&D Engineers, India

The procedure is as following.

(a) Derive governing equation of a beam for lateral vibration. You need to use Newton's second law. This will be fourth order in space and second order in time.

(b) Assume there is no external loading on the beam. Use separation of variables technique. We get two separate equations, one is in space and the other in temporal domain. Both equal to some constant (this is sqaure of circular frequency).

(c) In spatial domain, the equation is fourth order. Four constants appear in the solution. Use four boundary conditions of beam, two at each end. For a cantilever (fixed-free) beam, in first mode (1.875)^2 = 3.52 (approx) appears.

(d) In time domain, the equation is second order. Two initial conditions are required.

(e) f = {(1.875)^2/2pi}sqrt(EI/(m*L^3)) = {(1.875)^2/2pi}sqrt(EI/(rho*A*L^4))

You may check your term in the denominator, which is in square root.

For more details refer the following.

S. S. Rao, 'Mechanical Vibrations' 4th edition, Pearson Edition, Page - 609 - 613.

How to get the flexural vibration frequency of a beam

Ji Wang — Fri, 02 Nov 2007 02:45:36 -0400

Indeed this is a simple procedure. You can start from the flexural vibration equation of a beam, then applying the boundary conditions (one fixed end and one free end). You can find the complete procedure in books by Timoshenko like the Vibration one.

Also many books on elementary vibration and structural analysis should have the procedure given.

resonance (natural) frequency of a cantilever beam

Somashekara Bhat — Tue, 30 Oct 2007 06:22:49 -0400

resonance (natural) frequency of a cantilever beam is given by

f=[kn/2pi][sqrt(EI/wL^4)] where, kn=3.52 for mode 1, E is Young's modulus, I is moment of Inertia, w is beam width, L is beam length. (this is from Formulas for Stress and Strain, 5th edition by Raymond J. Roark and Warren C. Young).

I would like to derive this formula. Can any one suugest me any book or any link?