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

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If the beam is uniform the equations can be written
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If the beam is uniform the equations can be written as
 
<center>
 
<center>
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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>
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<math> \beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \frac{\partial^{2}\zeta}{\partial t^{2}}=0
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</math> </center>
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We can express the deflection as the series
 
We can express the deflection as the series
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<center><math>  \zeta(x,t)=\sum_{n=0}^{\infty} \left( A_n w_n(x) \sin(\lambda_n t) +   
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<center><math>  \zeta(x,t)=\sum_{n=0}^{\infty} A_n X_n(x) \cos(k_n t) +   
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A_n w_n(x) \frac{\cos(\lambda_n t)}{\lambda_n} \right) </math></center>
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\sum_{n=2}^{\infty}B_n X_n(x) \frac{\sin(k_n t)}{k_n} </math></center>
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where <math>w_n</math> are the [[Eigenfunctions for a Uniform Free Beam]] and <math>\lambda_n</math>
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where <math>X_n</math> are the [[Eigenfunctions for a Uniform Free Beam]] and <math>k_m = \lambda^2_n \sqrt{\beta/\gamma}</math>
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where <math>\lambda_n</math> are the eigenfunctions.
  
 
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>  
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:<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>
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:<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>
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:<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>
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Note that <br />
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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 />
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starts at <math>n=2</math> for time derivative.
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- 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 />
 
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- 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>.
 
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<br />
 

Latest revision as of 23:41, 2 July 2009

If the beam is uniform the equations can be written as

[math]\displaystyle{ \beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \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} A_n X_n(x) \cos(k_n t) + \sum_{n=2}^{\infty}B_n X_n(x) \frac{\sin(k_n t)}{k_n} }[/math]

where [math]\displaystyle{ X_n }[/math] are the Eigenfunctions for a Uniform Free Beam and [math]\displaystyle{ k_m = \lambda^2_n \sqrt{\beta/\gamma} }[/math] where [math]\displaystyle{ \lambda_n }[/math] are the eigenfunctions.

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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