project

final_project/README.md

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+# 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. 
+
+3. 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)
+
+4. 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,...)
+
+5. 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
+
+6. (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. 
+
+