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

From WikiWaves
Jump to navigationJump to search
Line 28: Line 28:
 
By imposing boundary conditions at <math>x = L</math> :
 
By imposing boundary conditions at <math>x = L</math> :
  
 +
<math>
 
\[ \left( \begin{array}{ccc}
 
\[ \left( \begin{array}{ccc}
 
a & b & c \\
 
a & b & c \\
 
d & e & f \\
 
d & e & f \\
 
g & h & i \end{array} \right)\]  
 
g & h & i \end{array} \right)\]  
is given by the formula
+
</math>
\[ \chi(\lambda) = \left| \begin{array}{ccc}
 
\lambda - a & -b & -c \\
 
-d & \lambda - e & -f \\
 
-g & -h & \lambda - i \end{array} \right|.\]
 
  
 
<center><math>
 
<center><math>

Revision as of 22:00, 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) + C_2 cos(\lambda_n x) + C_3 sinh(\lambda_n x) + C_4 cosh(\lambda_n x) }[/math]

Symmetric modes

[math]\displaystyle{ C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 cos(\lambda_n x) + C_4 cosh(\lambda_n x) }[/math]

By imposing boundary conditions at [math]\displaystyle{ x = L }[/math] :

[math]\displaystyle{ \[ \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right)\] }[/math]

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