View source for Eigenfunctions for a Uniform Free Beam

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions. 
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L,
\end{matrix}</math></center>
<center><math>\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></center>
This solution is discussed further in [[Eigenfunctions for a Free Beam]].

Due to symmetry of the problem, dry natural vibrations of a free beam can be split into two different sets of modes, symmetric (even) modes and skew-symmetric (odd) modes.

General solution of the above stated equation is:

<center>
<math>w_n(x) = C_1 sin(\lambda_n x) + C_2 cos(\lambda_n x) + C_3 sinh(\lambda_n x) + C_4 cosh(\lambda_n x)</math>
</center>

Symmetric modes

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 cos(\lambda_n x) + C_4 cosh(\lambda_n x)</math>
</center>

By imposing boundary conditions at <math>x = L</math> :

\[ \left( \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right)\] 
is given by the formula
\[ \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right|.\]

<center><math>
\frac{1}{2}
xx
</math></center>