I used Buckinghum Pi theorem. I derived pi groups for the natural frequency phenomena.
The first pi group was function of (natural frequency, E,Rho,Cantilever length)
The second pi group was function of (moment of inertia and L^4)
I made a scaled down beam (length model/length full scale = S) and did modal experiments to get the natural frequency. After that I used this natural frequency to predict the natural frequency of the large prototype.
Due to machine limitations, The small model was with moment of inertia not scaled S^4
But using the first pi group , I predicted natural frequency of large beam and it was close to the FE solution of this full scale beam.
Is that incorrect procedure?
what should I do?
Hi there,
I wouldn't say that your procedure is incorrect... it is simply incomplete. When you have scaled Pi groups that differ from large scale, you should consider the "distorted approach".
Basically, if you're not respecting PI2, your result should consider this fact. For instance, if Moment of Inertia is kept equal in small and large scale, you should derive equations direct from the natural frequency formula... (link attached)
f=alpha*sqrt(E I /mL4).
If you write scaled F / Large scale F for same E, rho and moment of inertia as stated, you'll obtain
scaling factor for frequency beta_f=sqrt(rho_p*A_p*L_pˆ4/(rho_m*A_m*L_mˆ4))=sqrt(A_p*L_pˆ4/(A_m*L_mˆ4))=sqrt(1/(betaˆ2*betaˆ4))=1/betaˆ3.
Summarising, not modifying moment of inertia will make your model betaˆ2 stiffer.
If you also scale moment of inertia, beta_f=sqrt(A_p*L_pˆ4*I_m/(A_m*L_mˆ4*I_p))=sqrt(betaˆ4/(betaˆ2*betaˆ4))=1/beta.
This last result is the traditional based on simply Mass-Length-Time basis.
Note that deriving relations from Natural frequency equation allow us to use different Elastic Modulus as well as density/mass. You can also change the beam section a bit and just consider the variation of inertia on your model.
Hope that helps...
http://iitg.vlab.co.in/?sub=62&brch=175&sim=1080&cnt=1
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