Template:Solution for a uniform beam in eigenfunctions

If the beam is uniform the equations can be written as

$\beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \frac{\partial^{2}\zeta}{\partial t^{2}}=0$

We can express the deflection as the series

$\zeta(x,t)=\sum_{n=0}^{\infty} A_n X_n(x) \cos(k_n t) + \sum_{n=2}^{\infty}B_n X_n(x) \frac{\sin(k_n t)}{k_n}$

where $X n$ are the Eigenfunctions for a Uniform Free Beam and $k_m = \lambda^2_n \sqrt{\beta/\gamma}$ where $λ n$ are the eigenfunctions.

Then $A_n \,\!$ and $B_n \,\!$ can be found using orthogonality properties:

$A_n=\frac{\int_{-L}^{L}f(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} \,\!$

$B_n=\frac{\int_{-L}^{L}g(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x}$

Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum starts at $n = 2$ for time derivative.