Template:Equations for a free beam: Difference between revisions

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We can find a the eigenfunction which satisfy
We can find a the eigenfunction which satisfy
<center>
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
<math>\partial_x^4 w_n = \lambda_n^4 w_n
\,\,\, -L \leq x \leq L
</math>
</center>
</center>
plus the edge conditions.
plus the edge conditions of zero bending moment and shear stress
<center><math>\begin{matrix}
<center><math>\begin{matrix}
\partial_x^3 w_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\partial_x^3 w_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L,
\end{matrix}</math></center>
\end{matrix}</math></center>
<center><math>\begin{matrix}
<center><math>\begin{matrix}
\partial_x^2 w_n = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\partial_x^2 w_n = 0 \;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L.
\end{matrix}</math></center>
\end{matrix}</math></center>
Note that other boundary conditions can be applied at the ends of the beam.

Latest revision as of 08:33, 7 November 2008

We can find a the eigenfunction which satisfy

[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n \,\,\, -L \leq x \leq L }[/math]

plus the edge conditions of zero bending moment and shear stress

[math]\displaystyle{ \begin{matrix} \partial_x^3 w_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L, \end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix} \partial_x^2 w_n = 0 \;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L. \end{matrix} }[/math]
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