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

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== Introduction ==
 +
 +
We show here how to find the eigenfunction for a beam with free edge conditions.
 +
 +
== Equations ==
 +
 
We can find a the eigenfunction which satisfy
 
We can find a the eigenfunction which satisfy
 
<center>
 
<center>
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plus the edge conditions.  
 
plus the edge conditions.  
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
+
\frac{\partial^3}{\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}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
+
\frac{\partial^2}{\partial x^2} w_n = 0\mbox{ for } \;\;\;\; \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.
 +
 +
== Solution ==
  
 
General solution of the differential equation is :
 
General solution of the differential equation is :
 
 
<center>
 
<center>
 
<math>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>
 
<math>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>
 
</center>
 
</center>
 
 
 
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.
 
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.
 
 
  
 
== Symmetric modes ==
 
== Symmetric modes ==
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<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
 
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
 
</center>
 
</center>
 
 
By imposing boundary conditions at <math>x = l</math> :
 
By imposing boundary conditions at <math>x = l</math> :
 
 
<center>
 
<center>
 
<math>
 
<math>
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\sin(\lambda_n l)&\sinh(\lambda_n l)\\  
 
\sin(\lambda_n l)&\sinh(\lambda_n l)\\  
 
\end{bmatrix}
 
\end{bmatrix}
 
 
\begin{bmatrix}
 
\begin{bmatrix}
 
C_2\\
 
C_2\\
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</center>
 
</center>
  
Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
+
Having obtained eigenvalues <math>\lambda_n</math>, natural frequencies can be readily calculated :
 
<center>
 
<center>
 
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
 
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>

Revision as of 08:27, 7 November 2008

Introduction

We show here how to find the eigenfunction for a beam with free edge conditions.

Equations

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} w_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l, \end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix} \frac{\partial^2}{\partial x^2} w_n = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l. \end{matrix} }[/math]

Note that other boundary conditions can be applied at the ends of the beam.

Solution

General solution of the differential 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]

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.

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{ \begin{bmatrix} - \cos(\lambda_n l)&\cosh(\lambda_n l)\\ \sin(\lambda_n l)&\sinh(\lambda_n l)\\ \end{bmatrix} \begin{bmatrix} C_2\\ C_4\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} }[/math]

For a nontrivial solution one gets:

[math]\displaystyle{ \tan(\lambda_n l)+\tanh(\lambda_n l)=0\, }[/math]

The first three roots are :

[math]\displaystyle{ \lambda_0 l = 0, \lambda_2 l = 2.365, \lambda_4 l = 5.497\, }[/math]

Having obtained eigenvalues [math]\displaystyle{ \lambda_n }[/math], natural frequencies can be readily calculated :

[math]\displaystyle{ \omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m} }[/math]

Symmetric natural modes can be written in normalized form as :

[math]\displaystyle{ w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right ) }[/math]

Skew-symmetric modes

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

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

[math]\displaystyle{ \begin{bmatrix} - \sin(\lambda_n l)&\sinh(\lambda_n l)\\ -\cos(\lambda_n l)&\cosh(\lambda_n l)\\ \end{bmatrix} \begin{bmatrix} C_1\\ C_3\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} }[/math]

For a nontrivial solution one gets:

[math]\displaystyle{ -\tan(\lambda_n l)+\tanh(\lambda_n l)=0\, }[/math]

The first three roots are :

[math]\displaystyle{ \lambda_1 l = 0, \lambda_3 l = 3.925, \lambda_5 l = 7.068\, }[/math]

Having obtained eigenvalues [math]\displaystyle{ \lambda_n }[/math], natural requency can be readily calculated :

[math]\displaystyle{ \omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m} }[/math]

Skew-symmetric natural modes can be written in normalized form as :

[math]\displaystyle{ w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right ) }[/math]