Question: Problem with initial/boundary conditions solving two PDE's

December 05 2012 maschoemaker 0
Maple 16

0

Hello,

I'm trying to solve the effects (deflictions, tensions, etc) of a load on a timoshenko beam. It uses two partial diffential equations, wherein q is the load:

> PDE1 := kappa*G*A*(diff(y(x, t), x, x)-(diff(theta(x, t), x)))-rho*A*(diff(y(x, t), t, t))-q;

> PDE2 := E*J*(diff(theta(x, t), x, x))+kappa*G*A*(diff(y(x, t), x)-theta(x, t))-rho*J*(diff(theta(y, x), t, t));

The boundary conditions are that in the corners the moments (derative of theta) en the deflections (y) are zero, therefore:

> BCS := {y(0, t) = 0, y(L, t) = 0, (D[1](theta))(0, t) = 0, (D[1](theta))(L, t) = 0};
The initial conditions are that there is no movement in the beginning, so y has to be zero everywhere, the derative of y (the velocity). Also the rotation of the normal starts of as zero, so theta at t=0 and the derative of theta also  has to be zero.
> ICS := {theta(x, 0) = 0, y(x, 0) = 0, (D[2](theta))(x, 0) = 0, (D[2](y))(x, 0) = 0};

Solving this PDE system numerically for y and theta, ie:

> sol := pdsolve({PDE1, PDE2}, {BCS, ICS}, {theta(x, t), y(x, t)}, numeric, time = t, range = 0 .. L);

Gives me this error:

Error, (in pdsolve/numeric/process_IBCs) invalid initial/boundary condition: {theta(x, 0) = 0, y(x, 0) = 0, (D[2](theta))(x, 0) = 0, (D[2](y))(x, 0) = 0}

And whatever I change, I cannot get it right. I've tried numorous things, but I keep getting this error... If I loose some boundary or initial conditions (I believe you need 8), doesn't make a difference, same error...

 

What am I doing wrong?

Thank you!

 

PS: every variable which should be a number is already defined in a previous part in the maple sheet.

shear_beam_pin_pin_t.mw

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