Suppose a uniform beam of length is simply supported at and . If the load per unit length of the beam is given by w(x), , then the differential equation modeling the deflection of the beam y(x) is given by Because the beam is simply supported at both ends we know that
We are going to use a Fourier series to solve this equation. To satisfy the initial conditions we should use a
Fourier Sine
Fourier Cosine
Fourier
expansion of :
Then
Plugging the last sum into the differential equation we get that
Using a Fourier Sine series expansion on we know that
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