We assume that the homogeneous Euler-Bernoulli cantilever with constant cross-section length is 1, the material constant EI is 1 and ρS is 1.

[![enter image description here](https://i.stack.imgur.com/6GT1F.png)](https://i.stack.imgur.com/6GT1F.png)

According to the constraints, we can establish the following differential equations:

```
DSolve[{D[D[ω[x, t], {x, 2}], {x, 2}] + 
    D[ω[x, t], {t, 2}] == 0, ω[0, 0] == 
   0, ω[x, 0] == 0, ω[0, t] == 0, 
  D[ω[0, t], {t, 1}] == 
   0(*Constraints on simply supported ends*), 
  D[ω[1, t], {t, 2}] == 0, 
  D[ω[1, t], {t, 3}] == 
   0(*Constraints on fixed end*)}, ω[x, t], {x, t}]

```

But I still can't find the only analytical solution, while the textbook can determine the natural frequencies of each stage of the cantilever according to these conditions (In addition, I find that $\rho$ and $S$ cannot be replaced effectively in the following code).

```
Table[(((2. i - 1) π)/(
    2 l))^2 Sqrt[(\[DoubleStruckCapitalE]*\[DoubleStruckCapitalI])/(\
ρ*S)], {i, 1, 10}] /. {l -> 
   1., \[DoubleStruckCapitalE]*\[DoubleStruckCapitalI] -> 
   1., ρ -> 1., S -> 1}

```

How can I solve the previous differential equation to get similar frequency results?

When I use the relevant code of [this post](https://mathematica.stackexchange.com/questions/190713/euler-bernoulli-beam-equation/190738?noredirect=1#comment557482_190738), I can't even get the numerical solution of the cantilever vibration:

```
ClearAll["Global`*"]
tau = 10;
L = 1;
Elastic = 1;
Imoment = 1;
ρ = 1;
S = 1;
sol = NDSolveValue[{D[Elastic*Imoment*D[w[x, t], {x, 2}], {x, 2}] + 
     S*ρ*D[w[x, t], {t, 2}] == 0, w[0, t] == 0, 
   w[x, 0] == x^2/6 (3 - x), Derivative[0, 1][w][0, t] == 0, 
   Derivative[0, 2][w][L, t] == Derivative[0, 3][w][L, t] == 0}, 
  w, {x, 0, L}, {t, 0, tau}, 
  Method -> {"MethodOfLines", 
    "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> 100},
     "SpatialDiscretization" -> {"TensorProductGrid", 
      "MaxPoints" -> 100, "MinPoints" -> 100, 
      "DifferenceOrder" -> 2}}, MaxSteps -> 10^6]

```