View source for Template:Solution for a uniform beam in eigenfunctions

If the beam is uniform the equations can be written
<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
<center><math>  \zeta(x,t)=\sum_{n=0}^{\infty} A_n X_n(x) \cos(\lambda_n t) +  
\sum_{n=2}^{\infty}B_n X_n(x) \frac{\sin(\lambda_n t)}{\lambda_n}  </math></center>
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:
<center>
:<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>
:<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 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.