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 | + | If the beam is uniform the equations can be written as |
<center> | <center> | ||
− | <math> | + | <math> \beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \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} | + | <center><math> \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></center> | |
− | where <math> | + | where <math>X_n</math> are the [[Eigenfunctions for a Uniform Free Beam]] and <math>k_m = \lambda^2_n \sqrt{\beta/\gamma}</math> |
+ | 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> | ||
− | :<math> A_n=\frac{\int_{-L}^{L}f(x) | + | :<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) | + | :<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 | + | Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum |
− | + | starts at <math>n=2</math> for time derivative. | |
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Latest revision as of 23:41, 2 July 2009
If the beam is uniform the equations can be written as
We can express the deflection as the series
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.