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**1. Homework Statement**

The problem concerns the free undamped oscialtions of an elactic beam, clamped at one end.

The system is governed by a partial differential equation (one spatial dimension + time) which are to be solved by separation of the variables.

Constants:

E Young's Modulus, Real, and positive

I: Moment of inertia, real and positive

L: Lenght of the beam, real and positive

**2. Homework Equations**

after some preparations, one ends with the following eigenvalue equation for the spatial component:

[tex] EI\frac{d^4}{dz^4}Z(z)=\lambda Z(z)[/tex]

Where lambda is the seperation constant and eigenvalue of the problem

With the boundary conditions:

- [tex]Z(0)=0[/tex]
- [tex]\left.\frac{d}{dz}Z(z)\right|_{z=0}=0[/tex]
- [tex]\left.EI\frac{d^2}{dz^2}Z(z)\right|_{z=L}=0[/tex]
- [tex]\left.EI\frac{d^3}{dz^3}Z(z)\right|_{z=L}=0[/tex]

**3. The Attempt at a Solution**

I have now tried every trick 4 years of physics education have taught me, and I still cannot find any nontrivial solution to this equation. I only seem to get an insane misture of exponential functions, real and complex, with no obvious way to fit the BC's. Any thought or suggestions on how to tackle this one will be most helpfull.