3d_plate

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{ 3D_PLATE.PDE

 

 This problem considers the oscillation modes of a glass plate in space

 ( no mountings to constrain motion ).

           -- Submitted by John Trenholme, Lawrence Livermore Nat'l Lab.

}

 

TITLE 'Oscillation of a Glass Plate'

 

COORDINATES

 cartesian3

 

SELECT

   modes = 5

   ngrid=10

   errlim = 0.01 { 1 percent is good enough }

 

VARIABLES

   U           { X displacement }

   V           { Y displacement }

   W           { Z displacement }

 

DEFINITIONS

   cm = 0.01       { converts centimeters to meters }

 

   long = 20*cm   { length of plate along Y axis }

   wide = 10*cm   { width of plate along X axis }

   thick = 1.2*cm { thickness of plate along Z axis }

 

   E = 50e9       { Youngs modulus in Pascals }

   nu = 0.256     { Poisson's ratio  }

   rho = 2500     { density in kg/m^3 = 1000*[g/cc] }

 

  { constitutive relations - isotropic material }

   G = E/((1+nu)*(1-2*nu))

   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)

 

  { 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

 

  { find mean Y and Z translation and X rotation }

   Vol = Integral(1)

 

  { scaling factor for displacement plots }

   Mt =0.1*globalmax(magnitude(x,y,z))/globalmax(magnitude(U,V,W))

 

INITIAL VALUES

   U = 1.0e-5    V = 1.0e-5    W = 1.0e-5

 

EQUATIONS

  { we assume sinusoidal oscillation at angular frequency omega =sqrt(lambda) }

   U:  dx(Sx) + dy(Txy) + dz(Tzx) + lambda*rho*U = 0   { X-displacement equation }

   V:  dx(Txy) + dy(Sy) + dz(Tyz) + lambda*rho*V = 0   { Y-displacement equation }

   W:  dx(Tzx) + dy(Tyz) + dz(Sz) + lambda*rho*W = 0   { Z-displacement equation }

 

CONSTRAINTS

   integral(U)=0               { eliminate translations }

   integral(V)=0

   integral(W)=0

   integral(dx(V)-dy(U)) = 0   { eliminate rotations }

   integral(dy(W) - dz(V)) = 0

   integral(dz(U) - dx(W))  = 0

 

EXTRUSION

  surface "bottom" z = -thick / 2

  layer "plate"

  surface "top" z = thick / 2

 

BOUNDARIES

  region 1 { all sides, and top and bottom, are free }

      start( -wide/2, -long/2 )

      line to ( wide/2, -long/2 )

      line to ( wide/2, long/2 )

      line to ( -wide/2, long/2 )

      line to close

 

MONITORS

  grid(x+Mt*U,y+Mt*V,z+Mt*W) as "Shape"

      report sqrt(lambda)/(2*pi) as "Frequency in Hz"

 

PLOTS

  contour( W ) on z = 0 as "Mid-plane Displacement"

      report sqrt(lambda)/(2*pi) as "Frequency in Hz"

  grid(x+Mt*U,y+Mt*V,z+Mt*W) as "Shape"

      report sqrt(lambda)/(2*pi) as "Frequency in Hz"

 

  summary

      report lambda

      report sqrt(lambda)/(2*pi) as "Frequency in Hz"

 

END