Difference between revisions of "Eigenfunctions for a Uniform Free Beam"
Marko tomic (talk | contribs) |
|||
| Line 1: | Line 1: | ||
| + | == 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> | ||
| Line 5: | Line 11: | ||
plus the edge conditions. | plus the edge conditions. | ||
<center><math>\begin{matrix} | <center><math>\begin{matrix} | ||
| − | \frac{\partial^3}{\partial x^3} | + | \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^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 == | ||
| Line 27: | Line 31: | ||
<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> | ||
| Line 36: | Line 38: | ||
\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\\ | ||
| Line 62: | Line 63: | ||
</center> | </center> | ||
| − | Having obtained eigenvalues <math>\lambda_n</math>, natural | + | 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.
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]
