diff --git a/final_project/README.md b/final_project/README.md
new file mode 100644
index 0000000..cee3ee3
--- /dev/null
+++ b/final_project/README.md
@@ -0,0 +1,103 @@
+# 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.
+
+
+
+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.
+
+
diff --git a/final_project/beam.png b/final_project/beam.png
new file mode 100644
index 0000000..271d614
Binary files /dev/null and b/final_project/beam.png differ
diff --git a/final_project/beam.svg b/final_project/beam.svg
new file mode 100644
index 0000000..b02b62d
--- /dev/null
+++ b/final_project/beam.svg
@@ -0,0 +1,615 @@
+
+
+
+
diff --git a/final_project/rubric.md b/final_project/rubric.md
new file mode 100644
index 0000000..08248bd
--- /dev/null
+++ b/final_project/rubric.md
@@ -0,0 +1,27 @@
+# Grading Rubric
+
+| Component | Grade | Description |
+| --------- |------ |--------------------------------- |
+|Solutions | 50 % | Problems are solved correctly and solution makes sense |
+|Documentation| 30 % | All files have comments and all functions have help files |
+|Problem Statements | 10% | Descriptions of each problem and the approach |
+|Team Work | 10% | Each group member made commits to repository|
+|Github Bonus | 5% | If a group member opens an [issue](https://guides.github.com/features/issues/) and you commit code to close it|
+
+The Documentation and Problem Statements will be assessed based upon the included m-files
+and the main README.md documentation. This is a final report, so make sure the README.md
+is easy to understand with headers (`#`, `##`, ...) to start each problem and sub-problem.
+
+The Solutions will be graded based upon correctness and the approach. The plots will be
+graded based upon the axis labels, titles, legend, correctness, and design.
+If you are asked to solve for a value, you can choose to put it in a table or in a
+paragraph, but it must be easy to read/understand.
+
+The Team Work section is meant to demonstrate working knowledge of Github (now as a
+collaborative project). If everyone makes commits to the project, then you will receive
+credit.
+
+The Github Bonus requires extra reading. Create an issue and tag your group member. That
+group member can then respond and *fix* the issue with a commit. This is a bonus 5pts (on
+an individual basis, for everyone to get credit, everyone in the group would need to close
+an issue)
diff --git a/lecture_20/lecture_20.ipynb b/lecture_20/lecture_20.ipynb
index 50f858a..da8c4ac 100644
--- a/lecture_20/lecture_20.ipynb
+++ b/lecture_20/lecture_20.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 152,
"metadata": {
"collapsed": true
},
@@ -13,7 +13,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 153,
"metadata": {
"collapsed": true
},
@@ -97,17 +97,19 @@
"source": [
"# Integrals in practice\n",
"\n",
- "### Example: Compare toughness of two steels\n",
+ "### Example: Compare toughness of Stainless steel to Structural steel\n",
"\n",
"\n",
"\n",
+ "### Step 1 - G3Data to get points \n",
+ "\n",
"Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n",
"\n"
]
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 154,
"metadata": {
"collapsed": false
},
@@ -162,7 +164,7 @@
},
{
"cell_type": "code",
- "execution_count": 147,
+ "execution_count": 155,
"metadata": {
"collapsed": false
},
@@ -347,7 +349,7 @@
},
{
"cell_type": "code",
- "execution_count": 148,
+ "execution_count": 156,
"metadata": {
"collapsed": false
},
@@ -429,7 +431,7 @@
},
{
"cell_type": "code",
- "execution_count": 15,
+ "execution_count": 157,
"metadata": {
"collapsed": true
},
@@ -448,7 +450,7 @@
},
{
"cell_type": "code",
- "execution_count": 13,
+ "execution_count": 159,
"metadata": {
"collapsed": false
},
@@ -457,12 +459,12 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "ans = 3.5000\r\n"
+ "ans = 6.6250\r\n"
]
}
],
"source": [
- "gauss_1pt(f1,2,3)"
+ "gauss_1pt(f2,2,3)"
]
},
{
@@ -496,7 +498,7 @@
},
{
"cell_type": "code",
- "execution_count": 149,
+ "execution_count": 164,
"metadata": {
"collapsed": false
},
@@ -507,7 +509,13 @@
"text": [
"ans = 6.6667\n",
"ans = 6.6667\n",
- "ans = 6.6667\n"
+ "ans = 6.6667\n",
+ "f_c =\n",
+ "\n",
+ "@(x) cosh (x / 30) + exp (-10 * x) .* erf (x)\n",
+ "\n",
+ "ans = 2.0048\n",
+ "ans = 2.0048\n"
]
}
],
@@ -515,7 +523,11 @@
"% integral of quadratic\n",
"quad(f2,2,3)\n",
"quadl(f2,2,3)\n",
- "f3(3)-f3(2)"
+ "f3(3)-f3(2)\n",
+ "f_c=@(x) cosh(x/30)+exp(-10*x).*erf(x)\n",
+ "\n",
+ "quad(f_c,1,3)\n",
+ "quadl(f_c,1,3)"
]
},
{
@@ -531,7 +543,7 @@
},
{
"cell_type": "code",
- "execution_count": 82,
+ "execution_count": 165,
"metadata": {
"collapsed": false
},
@@ -682,7 +694,7 @@
"\t\n",
"\tgnuplot_plot_2a\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",
"\n",
@@ -713,7 +725,7 @@
},
{
"cell_type": "code",
- "execution_count": 98,
+ "execution_count": 166,
"metadata": {
"collapsed": false
},
@@ -767,31 +779,36 @@
"\n",
"\n",
"\t\t\n",
+ "\t\t-2\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
"\t\t-1.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-1\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t1\n",
"\t\n",
"\n",
@@ -860,11 +877,11 @@
"\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tgnuplot_plot_2a\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",
"\n",
@@ -919,7 +936,7 @@
},
{
"cell_type": "code",
- "execution_count": 99,
+ "execution_count": 168,
"metadata": {
"collapsed": false
},
@@ -973,31 +990,36 @@
"\n",
"\n",
"\t\t\n",
+ "\t\t-2\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
"\t\t-1.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-1\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t1\n",
"\t\n",
"\n",
@@ -1070,7 +1092,7 @@
"\t\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tno change\n",
"\n",
@@ -1079,7 +1101,7 @@
"\t\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",
"\n",
@@ -1110,7 +1132,7 @@
},
{
"cell_type": "code",
- "execution_count": 100,
+ "execution_count": 169,
"metadata": {
"collapsed": false
},
@@ -1119,7 +1141,7 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "a = 0.99838\r\n"
+ "a = 1.0007\r\n"
]
},
{
@@ -1259,11 +1281,11 @@
"\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tgnuplot_plot_2a\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",
"\n",
@@ -1554,39 +1576,7 @@
},
{
"cell_type": "code",
- "execution_count": 120,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "a =\n",
- "\n",
- " 1 2 3 4 5\n",
- "\n",
- "pa =\n",
- "\n",
- " 2 1 1 2 3 4 5 5 4\n",
- "\n",
- "ans =\n",
- "\n",
- " 1 9\n",
- "\n"
- ]
- }
- ],
- "source": [
- "a=[1,2,3,4,5]\n",
- "pa=[a(4/2:-1:1) a a(end:-1:end-4/2+1)]\n",
- "size(pa)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 137,
+ "execution_count": 170,
"metadata": {
"collapsed": false
},
@@ -1738,111 +1728,111 @@
"\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tgnuplot_plot_2a\n",
"\n",
"\t\n",
- "\t\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",
@@ -1879,7 +1869,7 @@
},
{
"cell_type": "code",
- "execution_count": 138,
+ "execution_count": 171,
"metadata": {
"collapsed": false
},
@@ -1933,31 +1923,36 @@
"\n",
"\n",
"\t\t\n",
+ "\t\t-2\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
"\t\t-1.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-1\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t-0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t0.5\n",
"\t\n",
"\n",
"\n",
- "\t\t\n",
+ "\t\t\n",
"\t\t1\n",
"\t\n",
"\n",
@@ -2026,113 +2021,113 @@
"\n",
"\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tgnuplot_plot_2a\n",
"\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\tgnuplot_plot_3a\n",
"\n",
"\t\n",
- "\t\n",
- "\t\n",
+ "\t\n",
"\t\n",
"\n",