View source for Eigenfunctions for a Uniform Free Beam

{{complete pages}}

== Introduction ==

We show here how to find the eigenfunction for a beam with free edge conditions. 

== Equations ==

{{equations for a eigenfunction of a free beam}}

== Solution ==

General solution of the differential equation for <math>\lambda \neq 0</math> is
<center>
<math>X_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>
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.

== Modes for <math>\lambda = 0</math> ==

There are two modes for <math>\lambda = 0</math>  which are the two rigid body motions; they are given by 
{{rigid modes for an elastic plate}}

== 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> :
<center>
<math>
\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>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_0 L = 0, \lambda_2 L = 2.365, \lambda_4 L = 5.497\,</math>
</center>


Symmetric natural modes can be written in normalized form as :
<center>
<math>X_{2n}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\cos(\lambda_{2n} x)}{\cos(\lambda_{2n} L)}+\frac{\cosh(\lambda_{2n} x)}{\cosh(\lambda_{2n} L)} \right )
\,\,\,n\geq 1
</math>
</center>
where the
The symmetric modes have been normalised so that their inner products equal the Kronecker delta.

== Anti-symmetric modes ==

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

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n L)&\sinh(\lambda_n L)\\
-\cos(\lambda_n L)&\cosh(\lambda_n L)\\ 
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\ 
\end{bmatrix}
 = 
\begin{bmatrix}
0\\
0\\ 
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_1 L = 0, \lambda_3 L = 3.925, \lambda_5 L = 7.068\,</math>
</center>

Anti-symmetric natural modes can be written in normalized form as :
<center>
<math>X_{2n+1}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\sin(\lambda_{2n+1} x)}{\sin(\lambda_{2n+1} L)}+\frac{\sinh(\lambda_{2n+1} x)}{\sinh(\lambda_{2n+1} L)} \right )
\,\,\,n\geq 1
</math>
</center>
where the eigenfunctions have been chosen so that their inner products equal the Kronecker delta.

== Equations for a beam ==
{{equations for a beam}}

== Solution for a uniform beam in [[Eigenfunctions for a Uniform Free Beam|eigenfunctions]] ==

{{solution for a uniform beam in eigenfunctions}}



== Matlab Code ==

A program to calculate the eigenvalues can be found here
{{eigenvalues beam}}

A program to calculate the eigenvectors can be found here
{{eigenvectors beam}}

[[Category:Complete Pages]]
[[Category:Simple Linear Waves]]