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physics - Dynamic Euler–Bernoulli beam equation


I am trying to solve for the vibration of a Euler–Bernoulli beam. The equation is


$\frac{\partial ^2u(t,x)}{\partial t^2}+\frac{\partial ^4u(t,x)}{\partial x^4}=0$


For the boundary conditions I would like the displacement to be zero at the ends and with zero second derivative. This corresponds to pinned-pinned conditions. For time I will start with a displacement and no velocity.


In the future I would like to solve for a beam that is not uniform in thickness along the x-axis and for general initial conditions.


There is a similar problem in the NDEigensystem documentation here but this is for the standard wave equation which is only second order in space. However, I follow that example. First I define an initial displacement and try to solve the pde.



ClearAll[f];
f[x_] := x (1 - x)

tu = NDSolveValue[{
    D[u[t, x], {t, 2}] + D[u[t, x], {x, 4}] == 0,
    u[0, x] == f[x],
    Derivative[1, 0][u][0, x] == 0,
    DirichletCondition[u[t, x] == 0, True],
    DirichletCondition[D[u[t, x], {x, 2}] == 0, True]
    }, u, {t, 0, 1}, {x, 0, 1}, 

   Method -> {"PDEDiscretization" -> "MethodOfLines"}];

This gives me the error


NDSolveValue::femcmsd: The spatial derivative order of the PDE may not exceed two.

Thus I proceed to supply two coupled differential equations one for displacement one for the second derivative (which is the bending moment). Thus I try to solve


tu = NDSolveValue[{
    D[u[t, x], {t, 2}] + D[m[t, x], {x, 2}] == 0,
    D[u[t, x], {x, 2}] == m[t, x],
    u[0, x] == f[x],

    Derivative[1, 0][u][0, x] == 0,
    DirichletCondition[u[t, x] == 0, True],
    DirichletCondition[m[t, x] == 0, True]
    }, {u, m}, {t, 0, 1}, {x, 0, 1}, 
   Method -> {"PDEDiscretization" -> "MethodOfLines"}];

However this also gives an error


NDSolveValue::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable.

I don't understand this error because I think I have done as asked... Can you help? Thanks





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