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.

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:

[math]\displaystyle{ 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]

Symmetric modes

[math]\displaystyle{ C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 cos(\lambda_n x) + C_4 cosh(\lambda_n x) }[/math]

By imposing boundary conditions at [math]\displaystyle{ x = L }[/math] :

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

[math]\displaystyle{ \frac{1}{2} xx }[/math]