Problem $2$: Vibrations in a Loaded Elastic Beam

Consider a loaded elastic beam of length $L$ subject to a distributed load $w(x)=w_{0}cos(\frac{x}{L}).$ The deflection $y(x)[m]$ of the beam is governed by the Euler$-$Bernoulli beam equation:

$\frac{d^{4}y}{d{x}^4}=\frac{w(x)}{EI}$

where E is the Young's modulus and I is the moment of inertia of the beam's cross$-$section.

Boundary Conditions:

The beam is clamped at both ends, so the deflection $y$ and its slope $y^{\text{'}}$ are zero at both $x=0$ and $x=L($unit of $x$ is $m)$:

$y(0)=0,y^{\text{'}}(0)=0,y(L)=0,y^{\text{'}}(L)=0$

Using the following methods, solve for the deflection profile $y(x)$ along the beam:

Finite Difference Method

Shooting Method

MATLAB's bvp$4$c Solver

For each method, plot the deflection profile $y(x)$ along the beam.

Parameters:

Set $w_0=200N{m}^{-1},E=2e5Pa,I=1e-4m^4,$ and $L=2m.$