Simply supported beam—Fourier series solution of the ODE

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re finally here, at the end of all things. In this post, we’ll use a Fourier series to get the formula for the center deflection of a simply supported beam with a uniformly distributed load. We’ll see some of the same math that we saw in the previous Fourier series solution, but the fundamental approach will be different. Let’s start with the fourth-order ordinary differential equation for a beam with a general loading function, q(x): EIyiv=q The simple supports at both ends give us these boundary conditions: y(0)=y(L)=y″(0)=y″(L)=0 Let’s work out the solution for a loading condition we haven’t considered before, a distributed load in the form of a sine wave: q(x)=qmsin⁡mπxL where m is a positive integer and qm is some constant amplitude. Given the form of the governing ODE and the boundary conditions, it seems likely that the solution will…