Eigenfunctions for a Uniform Free Beam

We can find a the eigenfunction which satisfy

[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n }[/math]

plus the edge conditions.

[math]\displaystyle{ \begin{matrix} \frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L. \end{matrix} }[/math]

This solution is discussed further in Eigenfunctions for a Free Beam.

Expanding

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = i\omega \sum_{n=0}^{\infty} \xi_n w_n }[/math]