Template:Equations for the eigenfunctions of a free beam: Difference between revisions

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We can find a the eigenfunction which satisfy
We can find eigenfunctions which satisfy
<center>
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n
<math>\partial_x^4 X_n = \lambda_n^4 X_n
\,\,\, -L \leq x \leq L
\,\,\, -L \leq x \leq L
</math>
</math>
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plus the edge conditions of zero bending moment and shear stress
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 X_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{ at } z = 0 \;\;\; x = \pm L.
\partial_x^2 X_n = 0 \;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L.
\end{matrix}</math></center>
\end{matrix}</math></center>

Latest revision as of 10:40, 8 April 2009

We can find eigenfunctions which satisfy

[math]\displaystyle{ \partial_x^4 X_n = \lambda_n^4 X_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 X_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L, \end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix} \partial_x^2 X_n = 0 \;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L. \end{matrix} }[/math]
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