diff --git a/HW5/README.html b/HW5/README.html new file mode 100644 index 0000000..96595a9 --- /dev/null +++ b/HW5/README.html @@ -0,0 +1,53 @@ + + + + + + + + + + + +

Homework #5

+

due 3/28/17 by 11:59pm

+

Include all work as either an m-file script, m-file function, or example code included with ``` and document your code in the README.md file

+
    +

  1. Create a new github repository called ‘linear_algebra’.

    +
      +

    1. Add rcc02007 and pez16103 as collaborators.

      2. +

    2. Clone the repository to your computer.

      3. +
    2. +

  2. Create an LU-decomposition function called lu_tridiag.m that takes 3 vectors as inputs and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower and Upper matrices.

    +

    [ud,uo,lo]=lu_tridiag(e,f,g);

    3. +

  3. Use the output from lu_tridiag.m to create a forward substitution and back-substitution function called solve_tridiag.m that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). Note: do not use the backslash solver \, create an algebraic solution

    +

    x=solve_tridiag(ud,uo,lo,b);

    4. +

  4. Test your function on the matrices A3, A4, …, A10 generated with test_arrays.m Solving for b=ones(N,1); where N is the size of A. In your README.md file, compare the norm of the error between your result and the result of AN.

    5. +
+
| size of A | norm(error) |
+|-----------|-------------|
+|  3 | ... |
+| 4 | ... |
+
+Spring-mass system for analysis +

Spring-mass system for analysis

+
+
    +

  1. In the system shown above, determine the three differential equations for the position of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine the eigenvalues and natural frequencies.

    2. +

  2. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled by

    3. +
+

\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)

+

where \(p^{2} = \frac{P}{EI}\)

+

where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.

+

This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give \(\frac{y_{i+1} − 2y_{i} + y_{i−1} }{\Delta x^{2}}+ p^{2} y_{i}\)

+

where i = a node located at a position along the rod’s interior, and \(\Delta x\) = the spacing between nodes. This equation can be expressed as \(y_{i−1} − (2 − \Delta x^{2} p^{2} )y_{i} + y_{i+1} = 0\) Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equations. (See 13.10 for 4 interior-node example)

+

Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a pole for pole-vaulting ( http://people.bath.ac.uk/taf21/sports_whole.htm) E=76E9 Pa, I=4E-8 m^4, and L= 5m.

+

Include a table in the README.md that shows the following results: What are the largest and smallest eigenvalues for the beam? How many eigenvalues are there?

+
| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |
+

If the segment length approaches 0, how many eigenvalues would there be?

+ + diff --git a/HW5/README.md b/HW5/README.md new file mode 100644 index 0000000..a8495ea --- /dev/null +++ b/HW5/README.md @@ -0,0 +1,83 @@ +# Homework #5 +## due 3/28/17 by 11:59pm + +*Include all work as either an m-file script, m-file function, or example code included +with \`\`\` and document your code in the README.md file* + +1. Create a new github repository called 'linear_algebra'. + + a. Add rcc02007 and pez16103 as collaborators. + + b. Clone the repository to your computer. + +2. Create an LU-decomposition function called `lu_tridiag.m` that takes 3 vectors as inputs +and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 +vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower +and Upper matrices. + + ```[ud,uo,lo]=lu_tridiag(e,f,g);``` + +3. Use the output from `lu_tridiag.m` to create a forward substitution and +back-substitution function called `solve_tridiag.m` that provides the solution of +Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). *Note: do not use +the backslash solver `\`, create an algebraic solution* + + ```x=solve_tridiag(ud,uo,lo,b);``` + +4. Test your function on the matrices A3, A4, ..., A10 generated with `test_arrays.m` +Solving for `b=ones(N,1);` where N is the size of A. In your `README.md` file, compare +the norm of the error between your result and the result of AN\b. + +``` +| size of A | norm(error) | +|-----------|-------------| +| 3 | ... | +| 4 | ... | +``` + +![Spring-mass system for analysis](spring_mass.png) + +5. In the system shown above, determine the three differential equations for the position +of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 +N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine +the eigenvalues and natural frequencies. + +6. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be +modeled by + +$\frac{d^{2}y}{dx^{2}} + p^{2} y = 0$ + +where $p^{2} = \frac{P}{EI}$ + +where E = the modulus of elasticity, and I = the moment of inertia of the cross section +about its neutral axis. + +This model can be converted into an eigenvalue problem by +substituting a centered finite-difference approximation for the second derivative to give +$\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}$ + +where i = a node located at a position along the rod’s interior, and $\Delta x$ = the +spacing between nodes. This equation can be expressed as $y_{i-1} - (2 - \Delta x^{2} +p^{2} )y_{i} y_{i+1} = 0$ Writing this equation for a series of interior nodes along the +axis of the column yields a homogeneous system of equations. (See 13.10 for 4 +interior-node example) + +Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior +nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a +pole for pole-vaulting ( +[http://people.bath.ac.uk/taf21/sports_whole.htm](http://people.bath.ac.uk/taf21/sports_whole.htm)) +E=76E9 Pa, I=4E-8 m^4, and L= 5m. + +Include a table in the `README.md` that shows the following results: +What are the largest and smallest eigenvalues for the beam? How many eigenvalues are +there? + +``` +| # of segments | largest | smallest | # of eigenvalues | +| --- | --- | --- | --- | +| 5 | ... | ... | ... | +| 6 | ... | ... | ... | +| 10 | ... | ... | ... | +``` + +If the segment length ($\Delta x$) approaches 0, how many eigenvalues would there be? diff --git a/HW5/README.pdf b/HW5/README.pdf new file mode 100644 index 0000000..fdb31a3 Binary files /dev/null and b/HW5/README.pdf differ diff --git a/HW5/lu_tridiag.m b/HW5/lu_tridiag.m new file mode 100644 index 0000000..1339973 --- /dev/null +++ b/HW5/lu_tridiag.m @@ -0,0 +1,12 @@ +function [ud,uo,lo]=lu_tridiag(e,f,g);``` + % lu_tridiag calculates the components for LU-decomposition of a tridiagonal matrix + % given its off-diagonal vectors, e and g + % and diagonal vector f + % the output is + % the diagonal of the Upper matrix, ud + % the off-diagonal of the Upper matrix, uo + % and the off-diagonal of the Lower matrix, lo + % note: the diagonal of the Lower matrix is all ones + +end + diff --git a/HW5/solve_tridiag.m b/HW5/solve_tridiag.m new file mode 100644 index 0000000..0c9e510 --- /dev/null +++ b/HW5/solve_tridiag.m @@ -0,0 +1,11 @@ +function x=solve_tridiag(ud,uo,lo,b);``` + % solve_tridiag solves Ax=b for x + % given + % the diagonal of the Upper matrix, ud + % the off-diagonal of the Upper matrix, uo + % the off-diagonal of the Lower matrix, lo + % the vector b + % note: the diagonal of the Lower matrix is all ones + +end + diff --git a/HW5/spring_mass.png b/HW5/spring_mass.png new file mode 100644 index 0000000..274c049 Binary files /dev/null and b/HW5/spring_mass.png differ diff --git a/HW5/spring_mass.svg b/HW5/spring_mass.svg new file mode 100644 index 0000000..b854162 --- /dev/null +++ b/HW5/spring_mass.svg @@ -0,0 +1,299 @@ + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + x1 + x2 + x3 + + + + + m1 + m2 + m3 + k1 + k2 + k3 + k4 + + diff --git a/HW5/test_arrays.m b/HW5/test_arrays.m new file mode 100644 index 0000000..002d79c --- /dev/null +++ b/HW5/test_arrays.m @@ -0,0 +1,8 @@ +rand("seed",1); + +for i = 3:10 + e=sort(randi(6,[i-1,1])); + f=sort(randi(10,[i,1])); + g=sort(randi(9,[i-1,1])); + eval(['A',int2str(i),'=diag(f)+diag(e,-1)+diag(g,1);']); +end diff --git a/README.md b/README.md index 8145faf..134beea 100644 --- a/README.md +++ b/README.md @@ -49,7 +49,7 @@ Jupiter notebook (with matlab or octave kernel) ### Note on Homework and online forms -The Homeworks are graded based upon effort and completeness. The forms are not graded at +The Homeworks are graded based upon effort, correctness, and completeness. The forms are not graded at all, if they are completed you get credit. It is *your* responsibility to make sure your answers are correct. Use the homeworks and forms as a study guide for the exams. In general, I will not post homework solutions. diff --git a/lecture_10/octave-workspace b/lecture_10/octave-workspace index 8a9aba2..8c437bb 100644 Binary files a/lecture_10/octave-workspace and b/lecture_10/octave-workspace differ diff --git a/lecture_12/octave-workspace b/lecture_12/octave-workspace index 41ef164..8c437bb 100644 Binary files a/lecture_12/octave-workspace and b/lecture_12/octave-workspace differ diff --git a/lecture_15/eigenvalues.gif b/lecture_15/eigenvalues.gif new file mode 100644 index 0000000..1286762 Binary files /dev/null and b/lecture_15/eigenvalues.gif differ diff --git a/lecture_15/lecture_15.ipynb b/lecture_15/lecture_15.ipynb index 2541de3..49e4b52 100644 --- a/lecture_15/lecture_15.ipynb +++ b/lecture_15/lecture_15.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 40, + "execution_count": 1, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -22,6 +22,15 @@ "setdefaults" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## UConn Hackathon Mar 24-25\n", + "\n", + "[https://www.hackuconn.org/](https://www.hackuconn.org/)" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -115,7 +124,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -182,8 +191,8 @@ "or\n", "\n", "$|K-\\lambda I| = \\left|\\begin{array}{cc}\n", - "\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right) & -\\frac{k}{m_{1}}X_{2} \\\\\n", - "-\\frac{k}{m_{2}}X_{1} & \\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)\n", + "\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right) & -\\frac{k}{m_{1}} \\\\\n", + "-\\frac{k}{m_{2}} & \\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)\n", "\\end{array}\\right|=0$\n", "\n", "where $\\lambda = \\omega^{2}$\n" @@ -191,7 +200,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -200,38 +209,40 @@ "name": "stdout", "output_type": "stream", "text": [ - "K =\n", + "'eig' is a built-in function from the file libinterp/corefcn/eig.cc\n", "\n", - " 10 -5\n", - " -5 10\n", + " -- Built-in Function: LAMBDA = eig (A)\n", + " -- Built-in Function: LAMBDA = eig (A, B)\n", + " -- Built-in Function: [V, LAMBDA] = eig (A)\n", + " -- Built-in Function: [V, LAMBDA] = eig (A, B)\n", + " Compute the eigenvalues (and optionally the eigenvectors) of a\n", + " matrix or a pair of matrices\n", "\n", - "v =\n", + " The algorithm used depends on whether there are one or two input\n", + " matrices, if they are real or complex, and if they are symmetric\n", + " (Hermitian if complex) or non-symmetric.\n", "\n", - " -0.70711 -0.70711\n", - " -0.70711 0.70711\n", + " The eigenvalues returned by 'eig' are not ordered.\n", "\n", - "e =\n", + " See also: eigs, svd.\n", "\n", - "Diagonal Matrix\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", "\n", - " 5 0\n", - " 0 15\n", - "\n" + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" ] } ], "source": [ - "m=40; % mass in kg\n", - "k=200; % spring constant in N/m\n", - "\n", - "K=[2*k/m,-k/m;-k/m,2*k/m]\n", - "\n", - "[v,e]=eig(K)" + "help eig" ] }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -240,59 +251,38 @@ "name": "stdout", "output_type": "stream", "text": [ - "ans =\n", - "\n", - " -10.607\n", - " 10.607\n", + "K =\n", "\n", - "ans =\n", + " 10 -5\n", + " -5 10\n", "\n", - " -10.607\n", - " 10.607\n", - "\n" - ] - } - ], - "source": [ - "K*v(:,2)\n", - "e(2,2)*v(:,2)" - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ "v =\n", "\n", - " 0.78868 -0.57735 0.21132\n", - " -0.57735 -0.57735 0.57735\n", - " 0.21132 0.57735 0.78868\n", + " -0.70711 -0.70711\n", + " -0.70711 0.70711\n", "\n", "e =\n", "\n", "Diagonal Matrix\n", "\n", - " 1.2679 0 0\n", - " 0 3.0000 0\n", - " 0 0 4.7321\n", + " 5 0\n", + " 0 15\n", "\n" ] } ], "source": [ - "[v,e]=eig(A)" + "m=40; % mass in kg\n", + "k=200; % spring constant in N/m\n", + "\n", + "K=[2*k/m,-k/m;-k/m,2*k/m]\n", + "\n", + "[v,e]=eig(K)" ] }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -303,22 +293,20 @@ "text": [ "ans =\n", "\n", - " 1.00000\n", - " -0.73205\n", - " 0.26795\n", + " -10.607\n", + " 10.607\n", "\n", "ans =\n", "\n", - " 1.00000\n", - " -0.73205\n", - " 0.26795\n", + " -10.607\n", + " 10.607\n", "\n" ] } ], "source": [ - "A*v(:,1)\n", - "e(1,1)*v(:,1)" + "K*v(:,2)\n", + "e(2,2)*v(:,2)" ] }, { @@ -364,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -411,7 +399,7 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -563,6 +551,39 @@ "![solution to principle stresses](stress_soln.svg)" ] }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "e =\n", + "\n", + " -10\n", + " 0\n", + " 10\n", + "\n" + ] + } + ], + "source": [ + "e=eig(s)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, diff --git a/lecture_15/lecture_15.pdf b/lecture_15/lecture_15.pdf new file mode 100644 index 0000000..94d5f8c Binary files /dev/null and b/lecture_15/lecture_15.pdf differ diff --git a/lecture_15/octave-workspace b/lecture_15/octave-workspace index d68e895..8c437bb 100644 Binary files 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"cells": [ + { + "cell_type": "code", + "execution_count": 171, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 172, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![question 1](q1.png)\n", + "\n", + "![question 2](q2.png)\n", + "\n", + "### Project Ideas so far\n", + "\n", + "- Nothing yet...probably something heat transfer related\n", + "\n", + "- Modeling Propulsion or Propagation of Sound Waves\n", + "\n", + "- Low Thrust Orbital Transfer\n", + "\n", + "- Tracking motion of a satellite entering orbit until impact\n", + "\n", + "- What ever you think is best.\n", + "\n", + "- You had heat transfer project as an option; that sounded cool\n", + "\n", + "- Heat transfer through a pipe\n", + "\n", + "- I would prefer to do something with beam/plate mechanics or vibrations than a heat transfer or thermo problem\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Questions from you:\n", + "\n", + "- Is attempting to divide by zero an acceptable project idea?\n", + "\n", + "- Would it be alright if we worked in a group of 4?\n", + "\n", + "- What are acceptable project topics?\n", + "\n", + "- How do the exams look? \n", + "\n", + "- Is there no pdf for the lecture today?\n", + "\n", + "- Thank you for making the formatted lectures available!\n", + "\n", + "- did you do anything cool over spring break?\n", + "\n", + "- Could we have a group of 4? We don't want to have to choose which one of us is on their own.\n", + "\n", + "- In HW 5 should there be 4 vectors as an input?\n", + "\n", + "- Would it be possible for me to join a group of 3? I seem to be the odd man out in two 3 member groups that my friends are in." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Statistics and Curve-Fitting\n", + "## Linear Regression\n", + "\n", + "Often, we have a model with empirical parameters. (e.g. Young's modulus, Poisson's ratio, drag coefficient, coefficient of restitution, spring constant)\n", + "\n", + "Experimental measurements are prone to a number of stochastic (random) factors:\n", + "\n", + "- Environmental noise\n", + " \n", + "- Measurement Uncertainty\n", + " \n", + "- Factors not accounted in model (e.g. 2D effects of 1D approximation)\n", + "\n", + "These can lead to **noise** (lack of precision) and **bias** (lack of accuracy)\n", + "\n", + "Consider a piece of glass being stretched. \n", + "\n", + "![movie of stretching glass in microtensile machine](sgs_strain.gif)\n", + "\n", + "It is clear that a straight line is a \"good\" fit, but how good and what line?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Statistics\n", + "\n", + "How do we describe *precision* and *accuracy*?\n", + "\n", + "- mean\n", + "\n", + "- standard deviation\n", + "\n", + "Take our class dart problem\n" + ] + }, + { + "cell_type": "code", + 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} + ], + "source": [ + "darts=dlmread('compiled_data.csv',',');\n", + "x_darts=darts(:,1).*cosd(darts(:,2));\n", + "y_darts=darts(:,1).*sind(darts(:,2));\n", + "\n", + "colormap(colorcube(length(darts(:,3))))\n", + "\n", + "scatter(x_darts, y_darts, [], darts(:,3))\n", + "xlabel('x position (cm)')\n", + "ylabel('y position (cm)')" + ] + }, + { + "cell_type": "code", + "execution_count": 174, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "mu_x = 0.90447\r\n" + ] + } + ], + "source": [ + "mu_x=mean(x_darts)" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "s_x = 4.2747\r\n" + ] + } + ], + "source": [ + "s_x=std(x_darts)" + ] + }, + { + "cell_type": "code", + "execution_count": 175, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": 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"source": [ + "hist(darts(:,2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Monte Carlo Simulations\n", + "\n", + "Monte Carlo models use random numbers to either understand statistics or generate a solution. \n", + "\n", + "### Example 1:\n", + "#### Calculate $\\pi$ with random numbers. \n", + "\n", + "Assuming we can actually generate random numbers (a topic of philosophical and heated debates) we can populate a unit square with random points and determine the ratio of points inside and outside of a circle.\n", + "\n", + "![Unit circle and unit square](MonteCarloPi.gif)\n", + "\n", + "![1/4 Unit circle and 1/4 unit square](MonteCarloPi_rand.gif)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The ratio of the area of the circle to the square is:\n", + "\n", + "$\\frac{\\pi r^{2}}{4r^{2}}=\\frac{\\pi}{4}$\n", + "\n", + "So if we know the fraction of random points that are within the unit circle, then we can calculate $\\pi$\n", + "\n", + "(number of points in circle)/(total number of points)=$\\pi/4$" + ] + }, + { + "cell_type": "code", + "execution_count": 191, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "function our_pi=montecarlopi(N)\n", + " % Create random x-y-coordinates\n", + "\n", + " x=rand(N,1);\n", + " y=rand(N,1);\n", + " R=sqrt(x.^2+y.^2); % compute radius\n", + " num_in_circle=sum(R<1);\n", + " total_num_pts =length(R);\n", + " our_pi = 4*num_in_circle/total_num_pts;\n", + "end\n" + ] + }, + { + "cell_type": "code", + "execution_count": 194, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "mean value for pi = 3.138880\n", + "standard deviation is 0.009956\n" + ] + } + ], + "source": [ + "test_pi=zeros(10,1);\n", + "for i=1:10\n", + " test_pi(i)=montecarlopi(10000);\n", + "end\n", + "fprintf('mean value for pi = %f\\n',mean(test_pi))\n", + "fprintf('standard deviation is %f',std(test_pi))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example 2\n", + "#### Determine uncertainty in failure stress based on geometry\n", + "\n", + "In this example, we know that a steel bar will break under 940 MPa tensile stress. The bar is 1 mm by 2 mm with a tolerance of 1 %. What is the range of tensile loads that can be safely applied to the beam?\n", + "\n", + "$\\sigma_{UTS}=\\frac{F_{fail}}{wh}$\n", + "\n", + "$F_{fail}=\\sigma_{UTS}wh$" + ] + }, + { + "cell_type": "code", + "execution_count": 196, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.01\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.03\n", + "\t\n", + "\n", + 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"\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_37a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_38a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_39a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_40a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_41a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "N=1000;\n", + "r=rand(N,1);\n", + "wmean=1; % in mm\n", + "wmin=wmean-wmean*0.01;\n", + "wmax=wmean+wmean*0.01;\n", + "hmean=2; % in mm\n", + "hmin=hmean-hmean*0.01;\n", + "hmax=hmean+hmean*0.01;\n", + "\n", + "wrand=wmin+(wmax-wmin)*r;\n", + "hrand=hmin+(hmax-hmin)*r;\n", + "\n", + "uts=940; % in N/mm^2=MPa\n", + "\n", + "Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n", + "hist(Ffail,20,1)\n", + "xlabel('failure load (kN)')\n", + "ylabel('relative counts')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Normally, the tolerance is not a maximum/minimum specification, but instead a normal distribution that describes the standard deviation, or the 68 % confidence interval.\n", + "\n", + "So instead, we should generate normally distributed dimensions." + ] + }, + { + "cell_type": "code", + "execution_count": 197, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + 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"\t\t\n", + "\t\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_36a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_37a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_38a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_39a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_40a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_41a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "N=1000;\n", + "wmean=1; % in mm\n", + "wstd=wmean*0.01; % standard deviation in mm\n", + "hmean=2; % in mm\n", + "hstd=hmean*0.01; % standard deviation in mm\n", + "\n", + "\n", + "wrand=normrnd(wmean,wstd,[N,1]);\n", + "hrand=normrnd(hmean,hstd,[N,1]);\n", + "uts=940; % in N/mm^2=MPa\n", + "\n", + "Ffail=uts.*wrand.*hrand*1e-3; % force in kN\n", + "hist(Ffail,20,1)\n", + "xlabel('failure load (kN)')\n", + "ylabel('relative counts')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear Least Squares Regression\n", + "\n", + "When approximating a set of data as a polynomial, there will always be error introduced (except in 2 cases). \n", + "\n", + "For a straight line, the actual data points, $y_{i}$ as a function of the independent variable, $x_{i}$, is:\n", + "\n", + "$y_{i}=a_{0}+a_{1}x_{i}+e_{i}$\n", + "\n", + "where $a_{0}$ and $a_{1}$ are the intercept and slope of the line and $e_{i}$ is the error between the approximate function and the recorded data point. \n", + "\n", + "We make the following assumptions in this analysis:\n", + "\n", + "1. Each x has a fixed value; it is not random and is known without error.\n", + "\n", + "2. The y values are independent random variables and all have the same variance.\n", + "\n", + "3. The y values for a given x must be normally distributed.\n", + "\n", + "The total error is \n", + "\n", + "$\\sum_{i=1}^{n}e_{i}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})$\n", + "\n", + "we don't care about the sign though. One approach has been demonstrated to provide a unique solution is minimizing the sum of squares error or\n", + "\n", + "$S_{r}=\\sum_{i=1}^{n}e_{i}^{2}=\\sum_{i=1}^{n}(y_{i}-a_{0}-a_{1}x_{i})^{2}$\n", + "\n", + "Where, $S_{r}$ is the sum of squares error (SSE). \n", + "\n", + "$\\frac{\\partial S_{r}}{\\partial a_{0}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})$\n", + "\n", + "$\\frac{\\partial S_{r}}{\\partial a_{1}}=-2\\sum(y_{i}-a_{0}-a_{1}x_{i})x_{i}$\n", + "\n", + "The minimum $S_{r}$ occurrs when the partial derivatives are 0. \n", + "\n", + "$\\sum y_{i}= \\sum a_{0}+\\sum a_{1}x_{i}$\n", + "\n", + "$\\sum x_{i}y_{i}= \\sum a_{0}x_{i}+\\sum a_{1}x_{i}^{2}$\n", + "\n", + "$\\left[\\begin{array}{c}\n", + "\\sum y_{i}\\\\\n", + "\\sum x_{i}y_{i}\\end{array}\\right]=\n", + "\\left[\\begin{array}{cc}\n", + "n & \\sum x_{i}\\\\\n", + "\\sum x_{i} & \\sum x_{i}^{2}\\end{array}\\right]\n", + "\\left[\\begin{array}{c}\n", + "a_{0}\\\\\n", + "a_{1}\\end{array}\\right]$\n", + "\n", + "\n", + "$b=Ax$\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example \n", + "\n", + "Find drag coefficient with best-fit line to experimental data\n", + "\n", + "|i | v (m/s) | F (N) |\n", + "|---|---|---|\n", + "|1 | 10 | 25 |\n", + "|2 | 20 | 70 |\n", + "|3 | 30 | 380|\n", + "|4 | 40 | 550|\n", + "|5 | 50 | 610|\n", + "|6 | 60 | 1220|\n", + "|7 | 70 | 830 |\n", + "|8 |80 | 1450|" + ] + }, + { + "cell_type": "code", + "execution_count": 123, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a =\n", + "\n", + " -234.286\n", + " 19.470\n", + "\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t30\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t70\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tvelocity (m/s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tForce (N)\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "drag_data=[...\n", + "1 , 10 , 25 \n", + "2 , 20 , 70 \n", + "3 , 30 , 380\n", + "4 , 40 , 550\n", + "5 , 50 , 610\n", + "6 , 60 , 1220\n", + "7 , 70 , 830 \n", + "8 ,80 , 1450];\n", + "x=drag_data(:,2);\n", + "y=drag_data(:,3);\n", + "\n", + "b=[sum(y);sum(x.*y)];\n", + "A=[length(x),sum(x);\n", + " sum(x), sum(x.^2)];\n", + " \n", + "a=A\\b\n", + "\n", + "plot(x,y,'o',x,a(1)+a(2)*x)\n", + "xlabel('Force (N)')\n", + "ylabel('velocity (m/s)')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "How do we know its a \"good\" fit? \n", + "\n", + "Can compare the sum of squares error to the total sum of squares of the dependent variable (here F). \n", + "\n", + "$S_{r}=\\sum(y_{i}-a_{0}-a_{1}x_{i})^{2}$\n", + "\n", + "$S_{t}=\\sum(y_{i}-\\bar{y})^{2}$\n", + "\n", + "Then, we can calculate the *coefficient of determination*, $r^{2}$ or *correlation coefficient*, r. \n", + "\n", + "$r^{2}=\\frac{S_{t}-S_{r}}{S_{t}}$\n", + "\n", + "This represents the relative improvement of assuming that y is a function of x (if the x-values are not random and the y-values are random)\n", + "\n", + "For further information regarding statistical work on regression, look at \n", + "[NIST Statistics Handbook](http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd44.htm)" + ] + }, + { + "cell_type": "code", + "execution_count": 128, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "St = 1.8083e+06\n", + "St = 1.8083e+06\n" + ] + } + ], + "source": [ + "Sr=sum((y-a(1)-a(2)*x).^2);\n", + "St=std(y)^2*(length(y)-1)\n", + "St=sum((y-mean(y)).^2)" + ] + }, + { + "cell_type": "code", + "execution_count": 130, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "r2 = 0.88049\n", + "r = 0.93834\n" + ] + } + ], + "source": [ + "r2=(St-Sr)/St\n", + "r=sqrt(r2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Limiting cases \n", + "\n", + "#### $r^{2}=0$ $S_{r}=S{t}$\n", + "\n", + "#### $r^{2}=1$ $S_{r}=0$" + ] + }, + { + "cell_type": "code", + "execution_count": 152, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tCase 1\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tCase 1\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tCase 2\n", + "\n", + "\t\n", + "\t\tCase 2\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "gen_examples\n", + "plot(x1(1:50:end),y1(1:50:end),'s',x2,y2,'o')\n", + "legend('Case 1','Case 2','Location','NorthWest')" + ] + }, + { + "cell_type": "code", + "execution_count": 157, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a1 =\n", + "\n", + " 0.0497269\n", + " 0.0016818\n", + "\n", + "a2 =\n", + "\n", + " 0\n", + " 1\n", + "\n", + "Sr1 = 0.82607\n", + "St1 = 0.82631\n", + "coefficient of determination in Case 1 is 0.000286\n", + "Sr2 = 0\n", + "St2 = 1.0185\n", + "coefficient of determination in Case 2 is 1.000000\n" + ] + } + ], + "source": [ + "b1=[sum(y1);sum(x1.*y1)];\n", + "A1=[length(x1),sum(x1);\n", + " sum(x1), sum(x1.^2)];\n", + " \n", + "a1=A1\\b1\n", + "\n", + "b2=[sum(y2);sum(x2.*y2)];\n", + "A2=[length(x2),sum(x2);\n", + " sum(x2), sum(x2.^2)];\n", + " \n", + "a2=A2\\b2\n", + "\n", + "Sr1=sum((y1-a1(1)-a1(2)*x1).^2)\n", + "St1=sum((y1-mean(y1)).^2)\n", + "fprintf('coefficient of determination in Case 1 is %f\\n',1-Sr1/St1)\n", + "\n", + "Sr2=sum((y2-a2(1)-a2(2)*x2).^2)\n", + "St2=sum((y2-mean(y2)).^2)\n", + "\n", + "fprintf('coefficient of determination in Case 2 is %f\\n',1-Sr2/St2)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_16/octave-workspace b/lecture_16/octave-workspace new file mode 100644 index 0000000..8c437bb Binary files /dev/null and b/lecture_16/octave-workspace differ diff --git a/lecture_16/q1.png b/lecture_16/q1.png new file mode 100644 index 0000000..f086fc8 Binary files /dev/null and b/lecture_16/q1.png differ diff --git a/lecture_16/q2.png b/lecture_16/q2.png new file mode 100644 index 0000000..6f91d19 Binary files /dev/null and b/lecture_16/q2.png differ diff --git a/lecture_16/sgs_strain.avi b/lecture_16/sgs_strain.avi new file mode 100755 index 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