Template:Equations for a 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} \partial_x^3 w_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \partial_x^2 w_n = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l. \end{matrix} }[/math]

Note that other boundary conditions can be applied at the ends of the beam.