README.md

# ME 3255 - Final Project
## Due May 1 by 11:59pm

In this project you are going to solve for the shape of a beam under different loading
conditions. The shape of the beam varies along the x-axis and as a function of time.  

Notes: Label the plots with legends, x- and y-axis labels and make sure the plots are easy
to read (you can use the `setdefaults.m` script we have used in class). All functions
should have a help file and your README.md should describe each file in your repository
and provide a description of each problem and each solution (use `#`-headings in your file
to show the start of new problems)

You will be graded both on documentation and implementation of the solutions. 

![Diagram of beam and loading conditions](beam.png)

We will use the Euler-Bernoulli beam equation to describe the shape of the beam, the
differential equation that governs the solution is:

$\frac{\partial^4 w}{\partial x^4}-\frac{P}{EI}\frac{\partial^2 w}{\partial
x^2}+\frac{\rho A}{EI}\frac{\partial^2 w}{\partial t^2}=q(x)$     (1)

Where w(x,t) is the displacement of the beam away from the neutral axis as a function of
position along the beam, x, and time, t, P is the transverse loading of the beam, E is the
Young's modulus, I is the second moment of Inertia of the beam, $\rho$ is the density, A
is the cross-sectional area, and q(x) is the transverse distributed load (for a uniform
pressure, it is the applied pressure times the width of the beam, in units of
force/length).

We can separate the function $w(x,t)=w(x)e^{i\omega t}$, now equation (1) becomes

$\left(\frac{\partial^4 w}{\partial x^4}-\frac{P}{EI}\frac{\partial^2 w}{\partial
x^2}-\frac{\rho A \omega^{2}}{EI}w\right)e^{i\omega t}=\frac{q(x)}{EI}$    (2)

For the simply-supported beam shown in Figure 1, the boundary conditions are:

$w(0)=w(L)=0$

$w''(0)=w''(L)=0$

The material is aluminum, E=70 GPa, $\rho$=2700 kg/m$^3$. The bar is 1-m-long with a base
width, b=0.1 m, and height, h=0.01 m, and the second moment of inertia,
I=$\frac{bh^3}{12}$.

1. Analytically solve for the shape of the beam if q(x)=cst, P=0, and $\omega$=0 and
create a function called `shape_simple_support.m` that returns the displacement w(x) given
q and x

```
w=shape_simple_support(x,q);
```

a. Plot q vs the maximum deflection, $\delta x$, of the beam

b. Use a Monte Carlo model to determine the mean and standard deviation  for the
maximum deflection $\delta x$ if b and h are normally distributed random variables
with 0.1 % standard deviations at q=50 N/m. 

2. Now use the central difference approximation to set up a system of equations for the
beam for q(x)=cst, P=0, and $\omega=0$. Use the boundary conditions with a numerical
differentiation to determine the valuea of the end points

    a. set up the system of equations for 6 segments as a function of q

    b. set up the system of equations for 10 segments as a function of q 

    c. set up the system of equations for 20 segments as a function of q

    d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs $\delta x$
    
    e. Comment on the results from the analytical and numerical approaches (if you used
    functions then provide help files, if you used scripts, then describe the steps used)

3. Now set up the system of equations using a central difference method if P>0 and
$\omega=0$

    a. set up the system of equations for 6 segments as a function of q and P

    b. set up the system of equations for 10 segments as a function of q and P

    c. set up the system of equations for 20 segments as a function of q and P

    d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs $\delta x$ for
    P=0, 100, 200, 300 (4 lines, labeled as P=0,P=100,...)

4. Now set up an eigenvalue problem to solve for the natural frequencies of the simply
supported beam if P=0 and q=0. 

    a. set up the system of equations for 6 segments

    b. set up the system of equations for 10 segments 

    c. set up the system of equations for 20 segments

    d. solve for the natural frequencies ($\omega_{1}$, $\omega_{2}$,...)

    e. Plot the shape of the beam for the first 3 natural frequencies

5. (Bonus 5pt) Create a function to return the system of equations for the eigenvalue
problem as a function of P, if P>0. Then, plot the lowest natural frequency vs the applied
load P.