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.
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]
[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] :
[math]\displaystyle{ \begin{bmatrix} - \cos(\lambda_n l)&\cosh(\lambda_n l)\\ \sin(\lambda_n l)&\sinh(\lambda_n l)\\ \end{bmatrix} \begin{bmatrix} C_2\\ C_4\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} }[/math]
For a nontrivial solution one gets:
[math]\displaystyle{ \tan(\lambda_n l)+\tanh(\lambda_n l)=0 }[/math]