{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Solving a variant of problem 6.3, Gray, Costanzo, and Plesha, where kinetics are added (Intertias 1, 2, 3). (ME 3210 Quiz 2 2016)\n", "The problem statment is \"A Moment $M$ is applied to the right most gear (gear 3). Find the acceleration of the gear as a resuling moment presuming intertias $I_1, I_2, I_3$.\n", "\n", "\n", "\n", "I am using Python 3.5 in iPython (now called JuPyter). The notebook is available by emailing me. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# import symbolic capability to Python\n", "from sympy import *" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# print things all pretty\n", "from sympy.abc import *\n", "init_printing()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Need to define variables as symbolic for sympy to use them. \n", "r_A, r_B, r_C, r_D, F_AB, F_DC, M= symbols(\"r_A r_B r_C r_D F_AB F_DC M\", real = True)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "I_1, I_2, I_3 = symbols(\"I_1 I_2 I_3\", real = True)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "theta_1, theta_2, theta_3 = symbols(\"theta_1 theta_2 theta_3\", real = True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Defining equations. In SymPy, and equation is an expression that is equal to zero. All terms must be brought to the same side in defining it. " ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$F_{DC} r_{D} - I_{3} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )}$$" ], "text/plain": [ " 2 \n", " d \n", "F_DC⋅r_D - I₃⋅───(θ₃(t))\n", " 2 \n", " dt " ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EA = F_DC *r_D - I_3 * diff(theta_3(t),t,t)\n", "EA" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$F_{AB} r_{B} - F_{DC} r_{C} - I_{2} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )}$$" ], "text/plain": [ " 2 \n", " d \n", "F_AB⋅r_B - F_DC⋅r_C - I₂⋅───(θ₂(t))\n", " 2 \n", " dt " ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EB = r_B* F_AB - r_C* F_DC - I_2 * diff(theta_2(t),t,t)\n", "EB" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- F_{AB} r_{A} - I_{1} \\frac{d^{2}}{d t^{2}} \\theta_{1}{\\left (t \\right )} + M$$" ], "text/plain": [ " 2 \n", " d \n", "-F_AB⋅r_A - I₁⋅───(θ₁(t)) + M\n", " 2 \n", " dt " ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EC = r_A *(-F_AB) + M - I_1 * diff(theta_1(t),t,t)\n", "EC" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- r_{C} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )} + r_{D} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )}$$" ], "text/plain": [ " 2 2 \n", " d d \n", "- r_C⋅───(θ₂(t)) + r_D⋅───(θ₃(t))\n", " 2 2 \n", " dt dt " ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E1 = diff(theta_3(t),t,t)* r_D - diff(theta_2(t),t,t) *r_C\n", "E1" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- r_{A} \\frac{d^{2}}{d t^{2}} \\theta_{1}{\\left (t \\right )} + r_{B} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )}$$" ], "text/plain": [ " 2 2 \n", " d d \n", "- r_A⋅───(θ₁(t)) + r_B⋅───(θ₂(t))\n", " 2 2 \n", " dt dt " ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E2 = r_B * diff(theta_2(t),t,t) - diff(theta_1(t),t,t) *r_A\n", "E2" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAyBAMAAADSNPrMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2u6vNZkQi\n3ZlVwJ/bAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACKUlEQVRIDdWWwU/TUBzHvzzWja2treJNE4YH\no4lk4y9gCnqlxmhMPNBw0oNGPXkwgQthJGRZQsKRjZNHuXFcj3pi8eDBmGBIiOhhIegEYqT+ft2s\na/dgjyMv6dv39+3n/fp7v27tMOTvofcQvm9jcOJ2bxLaxLiNiwGYnNkvzg+5JyxK/ENhlYFUVQmt\nZYHMmhKaywP9BSW04sDrd5TQPzCqUVC/9uiK+98Kt6UfYcCDPlPaeU1nM6+2HngitZS2sX7QmPd4\nQYiKw+J5ileBmx6MZWgvMG7O6gVkbCQuR9B0Ew/JKAIDs9igTryBM8UNoZDdjqxWHfco/ggk6/hA\n6g4wxoTpAk9ZhAVwWwHtJyDsZJPkfeAbW1MFYJhFiOb4YtB36Ypl8zlr4AJPNToWWIToJkcQZaAv\na31hDeM3z9fpCNa0UW3BL3nkpbLAKEyajSr0Ojl4STfRZhFm5QCw8tAu4RwVIFoG8AO4u8ZBDJ10\n8D4PfIJ2o40mmjCWAh1FxWZjZ5t8UbrqtNGRo0appaNo+7T84wyilt9j7Hb1Vb73lnsGOxDdzmN/\nxYs6HMm3ddgNHoPqe8qoCH4pcVxaQJruTfeQovyo7R5StOZCPJuLw1I0V8B3fFVCK8AW3iqh+0wp\nZdV/Eam1XjwdqWW1CnrG4YnbQQVShppBW1UKeFc5yFKayXwsrSwrIVTpRlUNxWesOororenFGHnM\n9zVOBTHVeoqXvPpfh7+to5gHQfW1OQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\frac{F_{DC} r_{D}}{I_{3}}\\right ]$$" ], "text/plain": [ "⎡F_DC⋅r_D⎤\n", "⎢────────⎥\n", "⎣ I₃ ⎦" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(EA,diff(theta_3(t),t,t))" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAA1BAMAAACNXnumAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzOoSMZ4Uhv8BZEyDgna6mQkAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\frac{1}{r_{B}} \\left(F_{DC} r_{C} + I_{2} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )}\\right)$$" ], "text/plain": [ " 2 \n", " d \n", "F_DC⋅r_C + I₂⋅───(θ₂(t))\n", " 2 \n", " dt \n", "────────────────────────\n", " r_B " ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(EB,F_AB)[0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I'm going to do substitutions in stages, eliminating one variable at a time for clarity." ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- I_{1} \\frac{d^{2}}{d t^{2}} \\theta_{1}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(F_{DC} r_{C} + I_{2} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )}\\right)$$" ], "text/plain": [ " ⎛ 2 ⎞\n", " ⎜ d ⎟\n", " r_A⋅⎜F_DC⋅r_C + I₂⋅───(θ₂(t))⎟\n", " 2 ⎜ 2 ⎟\n", " d ⎝ dt ⎠\n", "- I₁⋅───(θ₁(t)) + M - ──────────────────────────────\n", " 2 r_B \n", " dt " ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "step1 = EC.subs(F_AB, solve(EB,F_AB)[0])\n", "step1" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- I_{1} \\frac{d^{2}}{d t^{2}} \\theta_{1}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(I_{2} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )}\\right)$$" ], "text/plain": [ " ⎛ 2 ⎞\n", " ⎜ d ⎟\n", " ⎜ I₃⋅r_C⋅───(θ₃(t))⎟\n", " ⎜ 2 2 ⎟\n", " ⎜ d dt ⎟\n", " r_A⋅⎜I₂⋅───(θ₂(t)) + ─────────────────⎟\n", " 2 ⎜ 2 r_D ⎟\n", " d ⎝ dt ⎠\n", "- I₁⋅───(θ₁(t)) + M - ───────────────────────────────────────\n", " 2 r_B \n", " dt " ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "step2 = step1.subs(F_DC,solve(EA,F_DC)[0])\n", "step2" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- \\frac{I_{1} r_{B}}{r_{A}} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(I_{2} \\frac{d^{2}}{d t^{2}} \\theta_{2}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )}\\right)$$" ], "text/plain": [ " ⎛ 2 ⎞\n", " ⎜ d ⎟\n", " ⎜ I₃⋅r_C⋅───(θ₃(t))⎟\n", " 2 ⎜ 2 2 ⎟\n", " d ⎜ d dt ⎟\n", " I₁⋅r_B⋅───(θ₂(t)) r_A⋅⎜I₂⋅───(θ₂(t)) + ─────────────────⎟\n", " 2 ⎜ 2 r_D ⎟\n", " dt ⎝ dt ⎠\n", "- ───────────────── + M - ───────────────────────────────────────\n", " r_A r_B " ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "step3 = step2.subs(diff(theta_1(t),t,t),solve(E2,diff(theta_1(t),t,t))[0])\n", "step3" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- \\frac{I_{1} r_{B} r_{D}}{r_{A} r_{C}} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(\\frac{I_{2} r_{D}}{r_{C}} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}} \\theta_{3}{\\left (t \\right )}\\right)$$" ], "text/plain": [ " ⎛ 2 2 ⎞\n", " ⎜ d d ⎟\n", " 2 ⎜I₂⋅r_D⋅───(θ₃(t)) I₃⋅r_C⋅───(θ₃(t))⎟\n", " d ⎜ 2 2 ⎟\n", " I₁⋅r_B⋅r_D⋅───(θ₃(t)) ⎜ dt dt ⎟\n", " 2 r_A⋅⎜───────────────── + ─────────────────⎟\n", " dt ⎝ r_C r_D ⎠\n", "- ───────────────────── + M - ───────────────────────────────────────────\n", " r_A⋅r_C r_B " ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "step4 = step3.subs(diff(theta_2(t),t,t),solve(E1,diff(theta_2(t),t,t))[0])\n", "step4" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, the acceleration of the right most gear as a function of the input moment" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left [ \\frac{M r_{A} r_{B} r_{C} r_{D}}{I_{1} r_{B}^{2} r_{D}^{2} + I_{2} r_{A}^{2} r_{D}^{2} + I_{3} r_{A}^{2} r_{C}^{2}}\\right ]$$" ], "text/plain": [ "⎡ M⋅r_A⋅r_B⋅r_C⋅r_D ⎤\n", "⎢──────────────────────────────────────────⎥\n", "⎢ 2 2 2 2 2 2⎥\n", "⎣I₁⋅r_B ⋅r_D + I₂⋅r_A ⋅r_D + I₃⋅r_A ⋅r_C ⎦" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(step4,diff(theta_3(t),t,t))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" }, "latex_envs": { "bibliofile": "biblio.bib", "cite_by": "apalike", 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