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resonance (natural) frequency of a cantilever beam

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?  

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Ji Wang's picture

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.

Thanks and regards
Somashekara Bhat

ramdas chennamsetti's picture

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.


Thanks and regards
Somashekara Bhat

Can any one suggest me how to model a functionally graded material (FGM) using ANSYS software.

thank u

could u suggest any formula for calculating frequency at mode  2,3,4,.......etc?

Dear sir, 

Different modes shapes do mean that the body which we are considering is
having more than one degree of freedom. That means cantilever beam can’t be
considered as a 1 DoF structure. Either you can analyze it using FEM (in which
the DoF = No of nodes* DoF per node - constraints) or by using analytical
methods. You may have to refer the vibration of continuous systems for
analytical methods.

Asst. Professor
Mechanical Engineering Department
National Institute of Technology

I read comments on this page and the people who commented seemed to have worked on field of aeroelasticity/fsi/flutter, I am a BE student and doing my final semester project on fsi of multi cell box beam using ansys-cfx, can you please me on that whether to use ansys-fluent and what benifits I will get if I use ansys-flent over ansys cfx.

waiting for reply


Muhammad amir.

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