Template:Solution for a uniform beam in eigenfunctions
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Jump to navigationJump to searchIf the beam is uniform the equations can be written
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{ \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.