[FlexPDE] Not sure if displacement equations are right

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Hi guys, I'm recently using FlexPDE to find the resonant frequencies of a torsional oscillator, but it does not match the analytic solution. It uses a Finite Element Method to find solutions.

Is there something wrong with the equations I'm using?

simple_oscillator2.png




C11 = G*(1-nu)
C12 = G*nu
C13 = G*nu
C22 = G*(1-nu)
C23 = G*nu
C33 = G*(1-nu)
C44 = G*(1-2*nu)/2

Strains

ex = dx(U)
ey = dy(V)
ez = dz(W)
gxy = dy(U) + dx(V)
gyz = dz(V) + dy(W)
gzx = dx(W) + dz(U)

VARIABLES

U { X displacement }
V { Y displacement }
W { Z displacement }



Stresses

Sx = C11*ex + C12*ey + C13*ez
Sy = C12*ex + C22*ey + C23*ez
Sz = C13*ex + C23*ey + C33*ez
Txy = C44*gxy
Tyz = C44*gyz
Tzx = C44*gzx

EQUATIONS

U: dx(Sx) + dy(Txy) + dz(Tzx) + lambda*rho*U = 0 { the U-displacement equation }
V: dx(Txy) + dy(Sy) + dz(Tyz) + lambda*rho*V = 0 { the V-displacement equation }
W: dx(Tzx) + dy(Tyz) + dz(Sz) + lambda*rho*W = 0 { the W-displacement equation }
 

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