Template:Solution for a uniform beam in eigenfunctions

If the beam is uniform the equations can be written

[math]\displaystyle{ D \frac{\partial^{4}\zeta}{\partial x^{4}}+m\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} \left( A_n w_n(x) \sin(\lambda_n t) + A_n w_n(x) \frac{\cos(\lambda_n t)}{\lambda_n} \right) }[/math]

where [math]\displaystyle{ w_n }[/math] are the Eigenfunctions for a Uniform Free Beam and [math]\displaystyle{ \lambda_n }[/math]

If we introduce the initial conditions

[math]\displaystyle{ \zeta(x,0)=f(x) \,\! }[/math]

[math]\displaystyle{ \frac{\partial \zeta(x,0)}{\partial t}=g(x) }[/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)w_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)w_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} }[/math]

Note that
- these modes drop off very quickly (ie [math]\displaystyle{ v_4 \,\! }[/math] oscillates about zero with negligible amplitude), so the higher order vibration modes can be ignored.
- As time progresses ([math]\displaystyle{ t \rightarrow \infty \,\! }[/math]), each mode will vibrate around the zero displacement line with frequency [math]\displaystyle{ \overline{\omega}_{n}\,\! }[/math].
- Having obtained eigenvalues [math]\displaystyle{ k_n \,\! }[/math], the natural frequencies can easily be obtained [math]\displaystyle{ \overline{\omega}_{n}=k_{n}^{2}\sqrt{\frac{D}{m}}\,\! }[/math].