Difference between revisions of "Eigenfunctions for a Uniform Free Beam"

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Line 17: Line 17:
  
 
<center><math>
 
<center><math>
w_n(x)=C_1*
+
w_n(x)=C_1 sin(\lambda_n x)
 
</math></center>
 
</math></center>
  

Revision as of 21:46, 6 November 2008

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) }[/math]

Symmetric modes


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