I want to calculate the natural frequency of the cantilever beam according to the theoretical method.

I find an article on how to solve the free vibration frequency of a cantilever beam with the finite element method.

```
(*https://mathematica.stackexchange.com/questions/99724/finite-
element-boundary-breaking*)
ps = {Inactive(
Div)({{0, -((Y*ν)/(1 - ν^2))}, {-(Y*(1 - ν))/(2*(1
- ν^2)), 0}}.Inactive(Grad)(v(x, y), {x, y}), {x, y}) +
Inactive(
Div)({{-(Y/(1 - ν^2)),
0}, {0, -(Y*(1 - ν))/(2*(1 - ν^2))}}.Inactive(Grad)(
u(x, y), {x, y}), {x, y}),
Inactive(
Div)({{0, -(Y*(1 - ν))/(2*(1 - ν^2))}, {-((Y*ν)/(1
- ν^2)), 0}}.Inactive(Grad)(u(x, y), {x, y}), {x, y}) +
Inactive(
Div)({{-(Y*(1 - ν))/(2*(1 - ν^2)),
0}, {0, -(Y/(1 - ν^2))}}.Inactive(Grad)(
v(x, y), {x, y}), {x, y})} /. {Y -> 10^3, ν -> 33/100}
{vals, funs} =
NDEigensystem({ps}, {u, v}, {x, y} ∈
Rectangle({0, 0}, {5, 0.25}), 8)
theory = {0, 0, 0, 22/L^2 Sqrt((Y d^2)/(12 1)),
61.7/L^2 Sqrt((Y d^2)/(12 1)), 121/L^2 Sqrt((Y d^2)/(12 1)),
200/L^2 Sqrt((Y d^2)/(12 1)), π/L Sqrt(Y/1.)} /. {Y -> 10^3,
d -> 0.25, L -> 5}
TableForm(Transpose({Sqrt(Abs(vals)), theory}),
TableHeadings -> {Automatic, {"Calculated", "Theory"}})
bcs = DirichletCondition({u(x, y) == 0, v(x, y) == 0}, x == 0);
{vals, funs} =
NDEigensystem({ps, bcs}, {u, v}, {x, y} ∈
Rectangle({0, 0}, {5, 0.25}), 5);
theory = {3.52 Sqrt((Y d^2)/(12 L^4)), 22 Sqrt((Y d^2)/(12 L^4)),
61.7 Sqrt((Y d^2)/(12 L^4)), π/2 Sqrt(Y/L^2),
121 Sqrt((Y d^2)/(12 L^4))} /. {Y -> 10^3, d -> 0.25, L -> 5.};
TableForm(Transpose({Sqrt(Abs(vals)), theory}),
TableHeadings -> {Automatic, {"Calculated", "Theory"}})
Needs("NDSolve`FEM`")
mesh = funs((1, 1))("ElementMesh");
Column(Table(uif = funs((n, 1));
vif = funs((n, 2));
dmesh =
ElementMeshDeformation(mesh, {uif, vif}, "ScalingFactor" -> 0.1);
Show({mesh("Wireframe"),
dmesh("Wireframe"(
"ElementMeshDirective" ->
Directive(EdgeForm(Red), FaceForm())))}), {n, 5}))
```

But I would like to know how to solve the natural frequency of the model in the post above according to the method of solving the partial differential equation.

**Other related links:**

How do you find the eigenvalues of a PDE (Dynamic Euler-Bernoulli beam)?

Inhomogeneous dynamic Euler-Bernoulli beam equation with discontinuous parameters

Analytical solution of the dynamic equation of the Euler-Bernoulli beam with compatibility condition