diff --git a/HW4/README.md b/HW4/README.md index 7f04a9a..4a06ed0 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -1,4 +1,4 @@ -# Homework #3 +# Homework #4 ## due 3/1/17 by 11:59pm diff --git a/lecture_13/octave-workspace b/lecture_13/octave-workspace index be23e10..8c437bb 100644 Binary files a/lecture_13/octave-workspace and b/lecture_13/octave-workspace differ diff --git a/lecture_15/.ipynb_checkpoints/lecture_15-checkpoint.ipynb b/lecture_15/.ipynb_checkpoints/lecture_15-checkpoint.ipynb new file mode 100644 index 0000000..2fd6442 --- /dev/null +++ b/lecture_15/.ipynb_checkpoints/lecture_15-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_15/animate_eig.m b/lecture_15/animate_eig.m new file mode 100644 index 0000000..a4721a5 --- /dev/null +++ b/lecture_15/animate_eig.m @@ -0,0 +1,56 @@ +function [v,e]=animate_eig(k,m) + % create a series of png's for use in an animation based on + % spring constants [k1,k2,k3]=k + % and + % masses + % [m1,m2]=m + + % check inputs for m and k + if length(m)==1 + m1=m; + m2=m; + else + m1=m(1); + m2=m(2); + end + if length(k)==1 + k1=k; k2=k; k3=k; + else + k1=k(1); + k2=k(2); + k3=k(3); + end + setdefaults + K=[k1/m1+k2/m1,-k2/m1;-k3/m2,k2/m2+k3/m2]; + [v,e]=eig(K); + w1=e(1,1); w2=e(2,2); + scale = 0.5; % the magnitude of oscillations is not important for vibrational modes + % just set scale to 1/2 for nice plots + % the eigenvector magnitude is independent of the solution + X11=v(1,1)*scale; X12=v(2,1)*scale; + X21=v(1,2)*scale; X22=v(2,2)*scale; + f11=@(t) X11*sin(w1*t); f12=@(t) X12*sin(w1*t); + f21=@(t) X21*sin(w2*t); f22=@(t) X22*sin(w2*t); + f=figure(); + time=linspace(-1,4); + % create a loop to plot the position over time where mass 1 and 2 start at x=1 and x=2 m + % then for the next vibrational mode, mass 1 and 2 start at x=3 and x=4 m + + for i=1:length(time) + t=time(i); + plot(f11(t)+1,0,'rs',f11(time+t)+1,-time,... + f12(t)+2,0,'rs',f12(time+t)+2,-time,... + f21(t)+3,0,'bo',f21(time+t)+3,-time,... + f22(t)+4,0,'bo',f22(time+t)+4,-time) + axis([0 6 -4 1]) + title('Vibration Modes') + xlabel('position (m)') + ylabel('time (s)') + filename=sprintf('output/%05d.png',i); + % another option for saving a plot to a png is the 'print' command + print(filename) + end + % this is a system command to use ImageMagick's cli 'convert' to create an animated gif + % if you don't have ImageMagick installed, comment this next line + system ("convert -delay 10 -loop 0 output/*png eigenvalues.gif") +end diff --git a/lecture_15/eig_200_100_40_20.gif b/lecture_15/eig_200_100_40_20.gif new file mode 100644 index 0000000..1b583a7 Binary files /dev/null and b/lecture_15/eig_200_100_40_20.gif differ diff --git a/lecture_15/eig_200_100_40_50.gif b/lecture_15/eig_200_100_40_50.gif new file mode 100644 index 0000000..ba05f59 Binary files /dev/null and b/lecture_15/eig_200_100_40_50.gif differ diff --git a/lecture_15/eig_200_40.gif b/lecture_15/eig_200_40.gif new file mode 100644 index 0000000..f29f27e Binary files /dev/null and b/lecture_15/eig_200_40.gif differ diff --git a/lecture_15/lecture_15.ipynb b/lecture_15/lecture_15.ipynb new file mode 100644 index 0000000..2541de3 --- /dev/null +++ b/lecture_15/lecture_15.ipynb @@ -0,0 +1,597 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Final Project\n", + "\n", + "1. Will be a team project (select team of 2-3 students)\n", + "\n", + "2. You will create a repository and each of you will contribute code and documentation\n", + "\n", + "3. If you have an idea feel free to suggest it, otherwise I will come up with a project, possible topics include:\n", + "\n", + " a. Conduction of heat through simple geometry\n", + " \n", + " b. Plate or beam mechanics (1-D and 2-D geometries)\n", + " \n", + " c. \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Eigenvalues\n", + "\n", + "Eigenvalues and eigen vectors are the solution to the set of equations where\n", + "\n", + "$Ax=\\lambda x$\n", + "\n", + "or \n", + "\n", + "$A-I \\lambda=0$\n", + "\n", + "Where A is the description of the system and I is the identity matrix with the same dimensions as A and $\\lambda$ is an eigenvalue of A. \n", + "\n", + "These problems are seen in a number of engineering practices:\n", + "\n", + "1. Determining vibrational modes in structural devices\n", + "\n", + "2. Material science - vibrational modes of crystal lattices (phonons)\n", + "\n", + "3. [Google searches - http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf](http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf)\n", + "\n", + "4. Quantum mechanics - many solutions are eigenvalue problems\n", + "\n", + "5. Solid mechanics, principle stresses and principle stress directions are eigenvalues and eigenvectors\n", + "\n", + "One way of determining the eigenvalues is taking the determinant:\n", + "\n", + "$|A-\\lambda I|=0$\n", + "\n", + "This will result in an $n^{th}$-order polynomial where A is $n \\times n$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Take, A\n", + "\n", + "$A=\\left[\\begin{array}{ccc}\n", + "2 & 1 & 0 \\\\\n", + "1 & 3 & 1 \\\\\n", + "0 & 1 & 4 \\end{array}\\right]$\n", + "\n", + "$|A-\\lambda I| = \\left|\\begin{array}{ccc}\n", + "2-\\lambda & 1 & 0 \\\\\n", + "1 & 3-\\lambda & 1 \\\\\n", + "0 & 1 & 4-\\lambda \\end{array}\\right|=0$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$(2-\\lambda)(3-\\lambda)(4-\\lambda)-(4-\\lambda)-(2-\\lambda)=0$\n", + "\n", + "$-\\lambda^{3}+9\\lambda^{2}-24\\lambda+18 =0$\n", + "\n", + "$\\lambda = 3,~\\sqrt{3}+3,-\\sqrt{3}+3$\n", + "\n", + "in Matlab/Octave:\n", + "\n", + "```matlab\n", + "A=[2,1,0;1,3,1;0,1,4];\n", + "pA=poly(A);\n", + "lambda = roots(pA)\n", + "```\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "lambda =\n", + "\n", + " 4.7321\n", + " 3.0000\n", + " 1.2679\n", + "\n" + ] + } + ], + "source": [ + "A=[2,1,0;1,3,1;0,1,4];\n", + "pA=poly(A); % characteristic polynomial of A, e.g. l^3-9*l^2+24*l-18=0\n", + "lambda = roots(pA)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Applications of Eigenvalue analysis\n", + "\n", + "Consider the 2-mass, 3-spring system shown below\n", + "\n", + "![masses and springs in series](springs_masses.png)\n", + "\n", + "It might not be immediately obvious, but there are two resonant frequencies for these masses connected in series. \n", + "\n", + "Take the two FBD solutions:\n", + "\n", + "$m_{1}\\frac{d^{2}x_{1}}{dt^{2}}=-kx_{1}+k(x_{2}-x_{1})$\n", + "\n", + "$m_{2}\\frac{d^{2}x_{2}}{dt^{2}}=-k(x_{2}-x_{1})-kx_{2}$\n", + "\n", + "we know that $x_{i}(t)\\propto sin(\\omega t)$ so we can substitute\n", + "\n", + "$x_{i}=X_{i}sin(\\omega t)$\n", + "\n", + "$-m_{1}X_{1}\\omega^{2}sin(\\omega t) = -kX_{1}sin(\\omega t) +k(X_{2}-X_{1})sin(\\omega t)$\n", + "\n", + "$-m_{2}X_{2}\\omega^{2}sin(\\omega t) = -kX_{2}sin(\\omega t) - k(X_{2}-X_{1})sin(\\omega t)$\n", + "\n", + "where $X_{1}$ and $X_{2}$ are the amplitude of oscillations and $\\omega$ is the frequency of oscillations. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "now, \n", + "\n", + "$\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right)X_{1}-\\frac{k}{m_{1}}X_{2} = 0$\n", + "\n", + "\n", + "$-\\frac{k}{m_{2}}X_{1}+\\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)X_{2} = 0$\n", + "\n", + "or\n", + "\n", + "$|K-\\lambda I| = \\left|\\begin{array}{cc}\n", + "\\left(\\frac{2k}{m_{1}}-\\omega^{2}\\right) & -\\frac{k}{m_{1}}X_{2} \\\\\n", + "-\\frac{k}{m_{2}}X_{1} & \\left(\\frac{2k}{m_{2}}-\\omega^{2}\\right)\n", + "\\end{array}\\right|=0$\n", + "\n", + "where $\\lambda = \\omega^{2}$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "K =\n", + "\n", + " 10 -5\n", + " -5 10\n", + "\n", + "v =\n", + "\n", + " -0.70711 -0.70711\n", + " -0.70711 0.70711\n", + "\n", + "e =\n", + "\n", + "Diagonal Matrix\n", + "\n", + " 5 0\n", + " 0 15\n", + "\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)" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " -10.607\n", + " 10.607\n", + "\n", + "ans =\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", + "\n", + "e =\n", + "\n", + "Diagonal Matrix\n", + "\n", + " 1.2679 0 0\n", + " 0 3.0000 0\n", + " 0 0 4.7321\n", + "\n" + ] + } + ], + "source": [ + "[v,e]=eig(A)" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 1.00000\n", + " -0.73205\n", + " 0.26795\n", + "\n", + "ans =\n", + "\n", + " 1.00000\n", + " -0.73205\n", + " 0.26795\n", + "\n" + ] + } + ], + "source": [ + "A*v(:,1)\n", + "e(1,1)*v(:,1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### $m_{1}=m_{2}=$40 kg and $k_{1}=k_{2}=k_{3}=$200 N/m\n", + "![animation](./eig_200_40.gif)\n", + "\n", + "### $m_{1}=$40 kg, $m_{2}=$50 kg, $k_{1}=$200 N/m, and $k_{2}=k_{3}=$100 N/m\n", + "![animation](./eig_200_100_40_50.gif)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solid Mechanics\n", + "\n", + "Many times you want to know the \"principle\" components of stress or strain. \n", + "\n", + "stress and strain are second order tensors:\n", + "\n", + "$\\sigma_{ij} =\\frac{F_{i}}{A_{j}}$ (engineering stress)\n", + "\n", + "$\\epsilon_{ij}=\\frac{\\Delta l_{i}}{l_{j}}$ (engineering strain)\n", + "\n", + "Imagine you can apply a force in two directions normal to each other, could you create a shear stress? \n", + "\n", + "![Desired stress state and with unknown applied stresses](stress.svg)\n", + "\n", + "Let's try to create the stress state of 10 MPa shear stress with two normal stresses. \n", + "\n", + "that means, $\\sigma_{12}=\\sigma_{21}=$10 MPa, and $\\sigma_{11}=\\sigma_{22}=\\sigma_{33}=\\sigma_{23}=\\sigma_{13}=0$ MPa\n", + "\n", + "in matrix form:\n", + "\n", + "$[\\sigma]=\\left[\\begin{array}{ccc}\n", + "0 & 10 & 0 \\\\\n", + "10 & 0 & 0 \\\\\n", + "0 & 0 & 0 \\end{array} \\right]$ MPa" + ] + }, + { + "cell_type": "code", + "execution_count": 55, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "v =\n", + "\n", + " -0.70711 0.00000 0.70711\n", + " 0.70711 0.00000 0.70711\n", + " 0.00000 1.00000 0.00000\n", + "\n", + "e =\n", + "\n", + "Diagonal Matrix\n", + "\n", + " -10 0 0\n", + " 0 0 0\n", + " 0 0 10\n", + "\n", + "v1 =\n", + "\n", + " 7.07107\n", + " -7.07107\n", + " 0.00000\n", + "\n", + "v2 =\n", + "\n", + " 7.07107\n", + " 7.07107\n", + " 0.00000\n", + "\n" + ] + } + ], + "source": [ + "s=[0,10,0;10,0,0;0,0,0];\n", + "[v,e]=eig(s)\n", + "v1=s*v(:,1)\n", + "v2=s*v(:,3)" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "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\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\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": [ + "v1=s*v(:,1);\n", + "v2=s*v(:,3);\n", + "%quiver([0,0],[0,0],v(1,[1,3]),v(2,[1,3]))\n", + "quiver([0,0],[0,0],[v1(1),v2(1)],[v1(2),v2(2)])\n", + "axis([-10,10,-10,10])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![solution to principle stresses](stress_soln.svg)" + ] + }, + { + "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_15/octave-workspace b/lecture_15/octave-workspace new file mode 100644 index 0000000..d68e895 Binary files /dev/null and b/lecture_15/octave-workspace differ diff --git 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