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

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We can find eigenfunctions 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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