Difference between revisions of "Template:Solution for a uniform beam in eigenfunctions"

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<math> D \frac{\partial^{4}\zeta}{\partial x^{4}}+\rho_i h \frac{\partial^{2}\zeta}{\partial t^{2}}=0</math> </center>
 
<math> D \frac{\partial^{4}\zeta}{\partial x^{4}}+\rho_i h \frac{\partial^{2}\zeta}{\partial t^{2}}=0</math> </center>
 
We can express the deflection as the series
 
We can express the deflection as the series
−
<center><math>  \zeta(x,t)=\sum_{n=0}^{\infty} \left( A_n w_n(x) \sin(\lambda_n t) +   
+
<center><math>  \zeta(x,t)=\sum_{n=0}^{\infty} \left( A_n w_n(x) \cos(\lambda_n t) +   
−
A_n w_n(x) \frac{\cos(\lambda_n t)}{\lambda_n} \right) </math></center>
+
\sum_{n=0}^{\infty}B_n w_n(x) \frac{\sin(\lambda_n t)}{\lambda_n} \right) </math></center>
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where <math>w_n</math> are the [[Eigenfunctions for a Uniform Free Beam]] and <math>\lambda_n</math>
+
where <math>X_n</math> are the [[Eigenfunctions for a Uniform Free Beam]] and <math>\lambda_n</math>
  
 
Then <math>  A_n \,\!</math> and <math> B_n \,\!</math> can be found using orthogonality properties:
 
Then <math>  A_n \,\!</math> and <math> B_n \,\!</math> can be found using orthogonality properties:
 
<center>
 
<center>
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:<math>  A_n=\frac{\int_{-L}^{L}f(x)w_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} \,\! </math>  
+
:<math>  A_n=\frac{\int_{-L}^{L}f(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} \,\! </math>  
 
</center>
 
</center>
 
<center>
 
<center>
−
:<math>  B_n=\frac{\int_{-L}^{L}g(x)w_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x}  </math></center>
+
:<math>  B_n=\frac{\int_{-L}^{L}g(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x}  </math></center>
−
Note that <br />
+
Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum
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- these modes drop off very quickly (ie <math> v_4 \,\!</math> oscillates about zero with negligible amplitude), so the higher order vibration modes can be ignored.<br />
+
starts at <math>n=2</math> for time derivative.
−
- As time progresses (<math> t \rightarrow \infty \,\!</math>), each mode will vibrate around the zero displacement line with frequency <math> \overline{\omega}_{n}\,\!</math>.<br />
 
−
- Having obtained eigenvalues <math> k_n \,\!</math>, the natural frequencies can easily be obtained <math>\overline{\omega}_{n}=k_{n}^{2}\sqrt{\frac{D}{m}}\,\!</math>.
 
−
<br />
 

Revision as of 08:29, 7 April 2009

If the beam is uniform the equations can be written

[math]\displaystyle{ D \frac{\partial^{4}\zeta}{\partial x^{4}}+\rho_i h \frac{\partial^{2}\zeta}{\partial t^{2}}=0 }[/math]

We can express the deflection as the series

[math]\displaystyle{ \zeta(x,t)=\sum_{n=0}^{\infty} \left( A_n w_n(x) \cos(\lambda_n t) + \sum_{n=0}^{\infty}B_n w_n(x) \frac{\sin(\lambda_n t)}{\lambda_n} \right) }[/math]

where [math]\displaystyle{ X_n }[/math] are the Eigenfunctions for a Uniform Free Beam and [math]\displaystyle{ \lambda_n }[/math]

Then [math]\displaystyle{ A_n \,\! }[/math] and [math]\displaystyle{ B_n \,\! }[/math] can be found using orthogonality properties:

[math]\displaystyle{ A_n=\frac{\int_{-L}^{L}f(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} \,\! }[/math]
[math]\displaystyle{ B_n=\frac{\int_{-L}^{L}g(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} }[/math]

Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum starts at [math]\displaystyle{ n=2 }[/math] for time derivative.

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