I want to calculate shear modulus 'G' through the vibration method. Once I get natural frequency corresponding to torsional mode then I'll use it for the expression in order to find G.
I want to calculate shear modulus 'G' through vibration method. Once I get natural frequency corresponding to torsional mode then I'll use it into the expression in order to find G.
Hi Prakash
The first pulsation for a cylindrical cantilever beam is [pi^2*G/(4*L^2*rho)]^(1/2) with L the length and rho the density (kg/m^3)
Best regards
Stephane
Thank you for your reply. Could you give me the reference so that I can see more details about the derivation?
Thank you in advance.....
You can find the solution in a book of vibrations of continuous systems. This is the solution:
In the case of cylindrical beam under torsion, the classical equilibrium equation is: d² theta/dt² = c² d² theta / dx² with c² = G/rho
The general solution is given by theta(x,t)=F(t)*U(x)
with U(x) = A.sin(wx/c)+B.cos(wx/c) and F(t) = C.sin (wt )+D.cos(wt)
For a beam fixed at x=0 and free at x=L, the boundary conditions are:
theta(0,t)=0 then U(0)=0
d theta(x,t) /dx =0 then A.cos(w.L/c) = 0 then the calculation of natural pulsation given by:
wi=(2i-1).Pi/2L * (G/rho)^(1/2) for i=1-> infinity then the first natural pulsation is w1=Pi/2L * (G/rho)^(1/2)
Best regards
Stephane
Sorry I don't see that you consider a thin plate. I think I have maybe the exact solution but you can find an approximated solution with Rayleigh method.
Best regards
Stephane
You can find a solution of your problem in the following paper:
Reissner and Stein
"Torsion and transverse bending of cantilever plates"
in the technical note N°2369 of the NACA in 1951
Best regards
Stephane
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