diff --git a/README.md b/README.md new file mode 100644 index 0000000..7f03ba6 --- /dev/null +++ b/README.md @@ -0,0 +1,11 @@ +# ME5180 - Dynamics + +## Examples of Numerical Analysis in Dynamics + +### 1- Double Pendulum Kinematics + +[double_pendulum.m](./double_pendulum.m) has a script to plot the velocity over time and +create an animation of the link positions. [double_pendulum.ipynb](double_pendulum.ipynb) +shows the output of the code. + + diff --git a/double_pendulum.ipynb b/double_pendulum.ipynb new file mode 100644 index 0000000..c77b169 --- /dev/null +++ b/double_pendulum.ipynb @@ -0,0 +1,288 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![double pendulum](./figures/p1-10_double_pendulum.png)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "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-1000\n", + "\t\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\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.12\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.14\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.16\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tvelocity (mm/s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\ttime (s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tVelocity of end point\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tx-component\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tx-component\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\ty-component\n", + "\n", + "\t\n", + "\t\ty-component\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; close all\n", + "t=linspace(0,2*pi/50,100); % create time varying from 0-0.126 s (or one period)\n", + "x=20*sin(50*t);dx=20*50*cos(50*t); % define x and dx/dt in terms of time (note dx=dx/dt)\n", + "t1=0.2*pi*cos(50*t); dt1=-10*pi*sin(50*t); % define theta1 (t1) and dtheta1/dt\n", + "t2=0.2*pi*sin(50*t-pi/3); dt2=10*pi*sin(50*t-pi/3); % define theta2 (t2) and dtheta2/dt;\n", + "L1=1;L2=1.5; % set lengths for L1 and L2 (none were given in problem so 1 and 1.5 mm were\n", + " % chosen arbitrarily\n", + "rcc=[x+L1*sin(t1);-L1*cos(t1)]; % position of connection between links\n", + "rco=[x+L1*sin(t1)+L2*sin(t2);-(L1*cos(t1)+L2*cos(t2))]; % create a row vectors of\n", + " % x-component and y-component of\n", + " % pendulum position C (r_C/O) \n", + "vco=[dx+L1*cos(t1).*dt1+L2*cos(t2).*dt2;(L1*sin(t1).*dt1+L2*sin(t2).*dt2)]; % create row\n", + " % vectors of\n", + " % the x- and\n", + " % y-component\n", + " % velocity of\n", + " % point C\n", + "\n", + "\n", + "figure(1)\n", + "plot(t,vco(1,:),t,vco(2,:))\n", + "xlabel('time (s)')\n", + "ylabel('velocity (mm/s)')\n", + "legend('x-component','y-component')\n", + "title('Velocity of end point')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Link positions over time (animation)\n", + "\n", + "![links](./pendulum.gif)\n" + ] + }, + { + "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/double_pendulum.m b/double_pendulum.m new file mode 100644 index 0000000..a74df16 --- /dev/null +++ b/double_pendulum.m @@ -0,0 +1,43 @@ +clear all; close all +t=linspace(0,2*pi/50,100); % create time varying from 0-0.126 s (or one period) +x=20*sin(50*t);dx=20*50*cos(50*t); % define x and dx/dt in terms of time (note dx=dx/dt) +t1=0.2*pi*cos(50*t); dt1=-10*pi*sin(50*t); % define theta1 (t1) and dtheta1/dt +t2=0.2*pi*sin(50*t-pi/3); dt2=10*pi*sin(50*t-pi/3); % define theta2 (t2) and dtheta2/dt; +L1=1;L2=1.5; % set lengths for L1 and L2 (none were given in problem so 1 and 1.5 mm were + % chosen arbitrarily +rcc=[x+L1*sin(t1);-L1*cos(t1)]; % position of connection between links +rco=[x+L1*sin(t1)+L2*sin(t2);-(L1*cos(t1)+L2*cos(t2))]; % create a row vectors of + % x-component and y-component of + % pendulum position C (r_C/O) +vco=[dx+L1*cos(t1).*dt1+L2*cos(t2).*dt2;(L1*sin(t1).*dt1+L2*sin(t2).*dt2)]; % create row + % vectors of + % the x- and + % y-component + % velocity of + % point C + + +figure(1) +plot(t,vco(1,:),t,vco(2,:)) +xlabel('time (s)') +ylabel('velocity (mm/s)') +legend('x-component','y-component') +title('Velocity of end point') + +% Now plot the position each time step +for i =1:length(t) + hold on + plot([x(i),rcc(1,i),rco(1,i)],[0,rcc(2,i),rco(2,i)]) % plot lines for 2 links + plot(rco(1,:),rco(2,:),'--','LineWidth',2); % this plots all of the positions in the x-y-plane + % over time + plot(rco(1,i),rco(2,i),'o','MarkerSize',10); % this plots the position as a circle at + % timestep i + axis([-30 30 -5 0]) % this sets the axis bounds + filename=sprintf('pendulum_output/%05d.png',i); + print(filename); % another option for saving a plot to a png is the 'print' command + hold off + clf % this clears the figure for the next time step +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 pendulum_output/*png pendulum.gif") diff --git a/figures/p1-10_double_pendulum.png b/figures/p1-10_double_pendulum.png new file mode 100644 index 0000000..d20b81d Binary files /dev/null and b/figures/p1-10_double_pendulum.png differ diff --git a/octave-workspace b/octave-workspace new file mode 100644 index 0000000..696cc40 Binary files 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