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diff --git a/Theory_Of_Machines/ch13.ipynb b/Theory_Of_Machines/ch13.ipynb new file mode 100755 index 00000000..d6a66198 --- /dev/null +++ b/Theory_Of_Machines/ch13.ipynb @@ -0,0 +1,1411 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:e7234bd223a2d57f8ee3c0d73106a260e10c7534a13d5c3bf25633720527eb77" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter 13 : Gear Trains" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.1 Page No : 432" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# Variables:\n", + "NA = 975 \t\t\t#rpm\n", + "TA = 20.\n", + "TB = 50.\n", + "TC = 25\n", + "TD = 75\n", + "TE = 26\n", + "TF = 65\n", + "\n", + "#Solution:\n", + "#Calculating the speed of gear F\n", + "NF = NA*(TA*TC*TE)/(TB*TD*TF) \t\t\t#rpm\n", + "\n", + "#Results:\n", + "print \" Speed of gear F, NF = %d rpm.\"%(NF)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of gear F, NF = 52 rpm.\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.2 Page No : 433" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "from numpy import linalg\n", + "\n", + "# Variables:\n", + "x = 600.\n", + "pc = 25. \t\t\t#mm\n", + "N1 = 360.\n", + "N2 = 120. \t\t\t#rpm\n", + "#Solution:\n", + "#Calculating the pitch circle diameters of each gear\n", + "#Speed ratio N1/N2 = d2/d1 or N1*d1-N2*d2 = 0 .....(i)\n", + "#Centre distance between the shafts x = 1/2*(d1+d2) or d1+d2 = 600*2 .....(ii)\n", + "A = [[N1, -N2],[ 1, 1]]\n", + "B = [0, 600*2]\n", + "V = linalg.solve(A,B)\n", + "d1 = V[0] \t\t\t#mm\n", + "d2 = V[1] \t\t\t#mm\n", + "#Calculating the number of teeth on the first gear\n", + "T1 = round(math.pi*d1/pc)\n", + "#Calculating the number of teeth on the second gear\n", + "T2 = int(math.pi*d2/pc+1)\n", + "#Calculating the pitch circle diameter of the first gear\n", + "d1dash = T1*pc/math.pi \t\t\t#mm\n", + "#Calculating the pitch circle diameter of the second gear\n", + "d2dash = T2*pc/math.pi \t\t\t#mm\n", + "#Calculating the exact distance between the two shafts\n", + "xdash = (d1dash+d2dash)/2 \t\t\t#mm\n", + "\n", + "#Results:\n", + "print \" The number of teeth on the first and second gear must be %d and %d and their pitch\\\n", + " circle diameters must be %.2f mm and %.1f mm respectively.\"%(T1,T2,d1dash,d2dash)\n", + "print \" The exact distance between the two shafts must be %.2f mm.\"%(xdash)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " The number of teeth on the first and second gear must be 38 and 114 and their pitch circle diameters must be 302.39 mm and 907.2 mm respectively.\n", + " The exact distance between the two shafts must be 604.79 mm.\n" + ] + } + ], + "prompt_number": 2 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.3 Page No : 435" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "rAD = 12. \t\t\t#Speed ratio NA/ND\n", + "mA = 3.125 #mm\n", + "mB = mA #mm\n", + "mC = 2.5 #mm\n", + "mD = mC #mm\n", + "x = 200. \t\t\t#mm\n", + "\n", + "#Solution:\n", + "#Calculating the speed ratio between the gears A and B and C and D \n", + "rAB = math.sqrt(rAD) \t\t\t#Speed ratio between the gears A and B\n", + "rCD = math.sqrt(rAB) \t\t\t#Speed ratio between the gears C and D\n", + "#Calculating the ratio of teeth on gear B to gear A\n", + "rtBA = rAB \t\t\t#Ratio of teeth on gear B to gear A\n", + "#Calculating the ratio of teeth on gear D to gear C\n", + "rtDC = rCD \t\t\t#Ratio of teeth on gear D to gear C\n", + "#Calculating the number of teeth on the gears A and B\n", + "#Distance between the shafts x = mA*TA/2+mB*TB/2 or (mA/2)*TA+(mB/2)*TB = x .....(i)\n", + "#Ratio of teeth on gear B to gear A TB/TA = math.sqrt(12) or math.sqrt(12)*TA-TB = 0 .....(ii)\n", + "A = [[mA/2, mB/2],[math.sqrt(12) ,-1]]\n", + "B = [x, 0]\n", + "V = linalg.solve(A,B)\n", + "TA = int(V[0])\n", + "TB = round(V[1])\n", + "#Calculating the number of teeth on the gears C and D\n", + "#Dismath.tance between the shafts x = mC*TC/2+mD*TD/2 or (mC/2)*TC+(mD/2)*TD = x .....(iii)\n", + "#Ratio of teeth on gear D to gear C TD/TC = math.sqrt(12) or math.sqrt(12)*TC-TD = 0 .....(iv)\n", + "A = [[mC/2, mD/2],[ math.sqrt(12) ,-1]]\n", + "B = [x, 0]\n", + "V = linalg.solve(A,B)\n", + "TC = round(V[0])\n", + "TD = int(V[1])\n", + "\n", + "#Results:\n", + "print \" Number of teeth on gear A, TA = %d.\"%(TA)\n", + "print \" Number of teeth on gear B, TB = %d.\"%(TB)\n", + "print \" Number of teeth on gear C, TC = %d.\"%(TC)\n", + "print \" Number of teeth on gear D, TD = %d.\"%(TD)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on gear A, TA = 28.\n", + " Number of teeth on gear B, TB = 99.\n", + " Number of teeth on gear C, TC = 36.\n", + " Number of teeth on gear D, TD = 124.\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.4 Page No : 438" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TA = 36.\n", + "TB = 45.\n", + "NC = 150. \t\t\t#rpm anticlockwise\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.7\n", + "#Algebraic method:\n", + "#Calculating the speed of gear B when gear A is fixed\n", + "NA = 0.\n", + "NC = 150. \t\t\t#rpm\n", + "NB1 = (-TA/TB)*(NA-NC)+NC \t\t\t#rpm\n", + "#Calculating the speed of gear B when gear A makes 300 rpm clockwise\n", + "NA = -300. \t\t\t#rpm\n", + "NB2 = (-TA/TB)*(NA-NC)+NC \t\t\t#rpm\n", + "\n", + "#Results:\n", + "print \" Speed of gear B when gear A is fixed, NB = %d rpm.\"%(NB1)\n", + "print \" Speed of gear B when gear A makes 300 rpm clockwise, NB = %d rpm.\"%(NB2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of gear B when gear A is fixed, NB = 270 rpm.\n", + " Speed of gear B when gear A makes 300 rpm clockwise, NB = 510 rpm.\n" + ] + } + ], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.5 Page No : 440" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TB = 75.\n", + "TC = 30.\n", + "TD = 90.\n", + "NA = 100. \t\t\t#rpm clockwise\n", + "\n", + "#Solution:\n", + "#Refer Table 13.3\n", + "#Calculating the number of teeth on gear E\n", + "TE = TC+TD-TB\n", + "#Calculating the speed of gear C\n", + "y = -100.\n", + "x = y*(TB/TE)\n", + "NC = y-x*(TD/TC) \t\t\t#rpm\n", + "\n", + "#Results:\n", + "print \" Speed of gear C, NC = %d rpm, anticlockwise.\"%(NC)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of gear C, NC = 400 rpm, anticlockwise.\n" + ] + } + ], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.6 Page No : 443" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TA = 72.\n", + "TC = 32.\n", + "NEF = 18. \t\t\t#Speed of arm EF rpm\n", + "\n", + "#Solution:\n", + "#Refer Table 13.5\n", + "#Speed of gear C:\n", + "y = 18. \t\t\t#rpm\n", + "x = y*(TA/TC)\n", + "NC = x+y \t\t\t#Speed of gear C rpm\n", + "#Speed of gear B:\n", + "#Calculating the number of teeth on gear B\n", + "TB = (TA-TC)/2\n", + "#Calculating the speed of gear B\n", + "NB = y-x*(TC/TB) \t\t\t#Speed of gear B rpm\n", + "\n", + "#Solution:\n", + "print \" Speed of gear C = %.1f rpm.\"%(NC)\n", + "print \" Speed of gear B = %.1f rpm in the opposite direction of arm.\"%(-NB)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of gear C = 58.5 rpm.\n", + " Speed of gear B = 46.8 rpm in the opposite direction of arm.\n" + ] + } + ], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.7 Page No : 444" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TA = 40.\n", + "TD = 90.\n", + "\n", + "#Solution:\n", + "#Calculating the number of teeth on gears B and C\n", + "#From geometry of the Fig. 13.11 dA+2*dB = dD\n", + "#Since the number of teeth are proportional to their pitch circle diameters\n", + "TB = (TD-TA)/2\n", + "TC = TB\n", + "#Refer Table 13.6\n", + "#Speed of arm when A makes 1 revolution clockwise and D makes half revolution anticlockwise:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table -x-y = -1 or x+y = 1 .....(i)\n", + "#The gear D makes half revolution anticlockwise i.e. x*(TA/TD)-y = 1/2 .....(ii)\n", + "A = [[1, 1],[TA/TD, -1]]\n", + "B = [1, 1./2]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the speed of arm\n", + "varm = -y \t\t\t#Speed of arm revolutions\n", + "\n", + "#Results:\n", + "print \" Speed of arm when A makes 1 revolution clockwise and D makes half revolution\\\n", + " anticlockwise = %.2f revolution anticlockwise.\"%(varm)\n", + "#Speed of arm when A makes 1 revolution clockwise and D is stationary:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table -x-y = -1 or x+y = 1 .....(iii)\n", + "#The gear D is stationary i.e. x*(TA/TD)-y = 0 .....(iv)\n", + "A = [[1, 1],[ TA/TD, -1]]\n", + "B = [1, 0]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the speed of arm\n", + "varm = -y \t\t\t#Speed of arm revolutions\n", + "\n", + "#Results:\n", + "print \" Speed of arm when A makes 1 revolution clockwise and D is stationary = %.3f revolution\\\n", + " clockwise.\"%(-varm)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of arm when A makes 1 revolution clockwise and D makes half revolution anticlockwise = 0.04 revolution anticlockwise.\n", + " Speed of arm when A makes 1 revolution clockwise and D is stationary = 0.308 revolution clockwise.\n" + ] + } + ], + "prompt_number": 9 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.8 Page No : 446" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TC = 28.\n", + "TD = 26.\n", + "TE = 18.\n", + "TF = TE\n", + "\n", + "#Solution:\n", + "#The sketch is as in Fig. 13.12\n", + "#Number of teeth on wheels A and B:\n", + "#From geometry dA = dC+2*dE and dB = dD+2*dF\n", + "#Since the number of teeth are proportional to their pitch circle diameters\n", + "TA = TC+2*TE\n", + "TB = TD+2*TF\n", + "#Speed of wheel B when arm G makes 100 rpm clockwise and wheel A is fixed:\n", + "#Since the arm G makes 100 rpm clockwise therefore from the fourth row of Table 13.7\n", + "y = -100\n", + "x = -y\n", + "#Calculating the speed of wheel B\n", + "NB1 = y+x*(TA/TC)*(TD/TB) \t\t\t#Speed of wheel B when arm G makes 100 rpm clockwise and wheel A is fixed rpm\n", + "#Speed of wheel B when arm G makes 100 rpm clockwise and wheel A makes 10 rpm counter clockwise:\n", + "#Since the arm G makes 100 rpm clockwise therefore from the fourth row of Table 13.7\n", + "y = -100\n", + "x = 10-y\n", + "#Calculating the speed of wheel B\n", + "NB2 = y+x*(TA/TC)*(TD/TB) \t\t\t#Speed of wheel B when arm G makes 100 rpm clockwise and wheel A makes 10 rpm counter clockwise rpm\n", + "\n", + "#Solution:\n", + "print \" Number of teeth on wheel A, TA = %d.\"%(TA)\n", + "print \" Number of teeth on wheel B, TB = %d.\"%(TB)\n", + "print \" Speed of wheel B when arm G makes 100 rpm clockwise and wheel A is fixed = %.1f rpm, clockwise.\"%(-NB1)\n", + "print \" Speed of wheel B when arm G makes 100 rpm clockwise and wheel A makes 10 rpm counter\\\n", + " clockwise = %.1f rpm, counter clockwise.\"%(NB2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on wheel A, TA = 64.\n", + " Number of teeth on wheel B, TB = 62.\n", + " Speed of wheel B when arm G makes 100 rpm clockwise and wheel A is fixed = 4.1 rpm, clockwise.\n", + " Speed of wheel B when arm G makes 100 rpm clockwise and wheel A makes 10 rpm counter clockwise = 5.4 rpm, counter clockwise.\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.9 Page No : 447" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "dD = 224.\n", + "m = 4. \t\t\t#mm\n", + "\n", + "#Solution:\n", + "#Refer Table 13.8\n", + "#Calculating the values of x and y\n", + "y = 1.\n", + "x = 5-y\n", + "#Calculating the number of teeth on gear D\n", + "TD = dD/m\n", + "#Calculating the number of teeth on gear B\n", + "TB = y/x*TD\n", + "#Calculating the number of teeth on gear C\n", + "TC = (TD-TB)/2\n", + "\n", + "#Results:\n", + "print \" Number of teeth on gear D, TD = %d.\"%(TD)\n", + "print \" Number of teeth on gear B, TB = %d.\"%(TB)\n", + "print \" Number of teeth on gear C, TC = %d.\"%(TC)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on gear D, TD = 56.\n", + " Number of teeth on gear B, TB = 14.\n", + " Number of teeth on gear C, TC = 21.\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.10 Page No : 448" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TC = 50.\n", + "TD = 20.\n", + "TE = 35.\n", + "NA = 110. \t\t\t#rpm\n", + "\n", + "#Solution:\n", + "#Calculating the number of teeth on internal gear G\n", + "TG = TC+TD+TE\n", + "#Speed of shaft B:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of Table 13.9\n", + "#y-x*(TC/TD)*(TE/TG) = 0 .....(i)\n", + "#Also x+y = 110 or y+x = 110 .....(ii)\n", + "A = [[1, -(TC/TD)*(TE/TG)],[ 1, 1]]\n", + "B = [0, 110]\n", + "V = linalg.solve(A,B)\n", + "x = V[1]\n", + "y = V[0]\n", + "#Calculating the speed of shaft B\n", + "NB = round(+y) \t\t\t#Speed of shaft B rpm\n", + "\n", + "#Results:\n", + "print \" Number of teeth on internal gear G, TG = %d.\"%(TG)\n", + "print \" Speed of shaft B = %d rpm, anticlockwise.\"%(NB)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on internal gear G, TG = 105.\n", + " Speed of shaft B = 50 rpm, anticlockwise.\n" + ] + } + ], + "prompt_number": 13 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.11 Page No : 450" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TA = 12.\n", + "TB = 30.\n", + "TC = 14.\n", + "NA = 1.\n", + "ND = 5. \t\t\t#rps\n", + "\n", + "#Solution:\n", + "#Number of teeth on wheels D and E:\n", + "#Calculating the number of teeth on wheel E\n", + "TE = TA+2*TB\n", + "#Calculating the number of teeth on wheel E\n", + "TD = TE-(TB-TC)\n", + "#Magnitude and direction of angular velocities of arm OP and wheel E:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of Table 13.10 -x-y = -1 or x+y = 1 .....(i)\n", + "#Also x*(TA/TB)*(TC/TD)-y = 5 .....(ii)\n", + "A = [[1, 1],[(TA/TB)*(TC/TD) ,-1]]\n", + "B = [1, 5]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the angular velocity of arm OP\n", + "omegaOP = -y*2*math.pi \t\t\t#Angular velocity of arm OP rad/s\n", + "#Calculating the angular velocity of wheel E\n", + "omegaE = (x*TA/TE-y)*2*math.pi \t\t\t#Angular velocity of wheel E rad/s\n", + "\n", + "#Results:\n", + "print \" Number of teeth on wheel E, TE = %d.\"%(TE)\n", + "print \" Number of teeth on wheel D, TD = %d.\"%(TD)\n", + "print \" Angular velocity of arm OP = %.3f rad/s, counter clockwise.\"%(omegaOP)\n", + "print \" Angular velocity of wheel E = %.2f rad/s, counter clockwise.\"%(omegaE)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on wheel E, TE = 72.\n", + " Number of teeth on wheel D, TD = 56.\n", + " Angular velocity of arm OP = 27.989 rad/s, counter clockwise.\n", + " Angular velocity of wheel E = 33.70 rad/s, counter clockwise.\n" + ] + } + ], + "prompt_number": 14 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.12 Page No : 451" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TB = 80.\n", + "TC = 82.\n", + "TD = 28.\n", + "NA = 500. \t\t\t#rpm\n", + "\n", + "#Solution:\n", + "#Calculating the number of teeth on wheel E\n", + "TE = TB+TD-TC\n", + "#Calculating the values of x and y\n", + "y = 800.\n", + "x = -y*(TE/TB)*(TC/TD)\n", + "#Calculating the speed of shaft F\n", + "NF = x+y \t\t\t#Speed of shaft F rpm\n", + "\n", + "#Results:\n", + "print \" Speed of shaft F = %d rpm, anticlockwise.\"%(NF)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of shaft F = 38 rpm, anticlockwise.\n" + ] + } + ], + "prompt_number": 15 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.13 Page No : 452" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# variables\n", + "\n", + "TA = 100. ; # Gear A teeth\n", + "TC = 101. ; # Gear C teeth\n", + "TD = 99. ; # Gear D teeth\n", + "TP = 20. # Gear planet teeth\n", + "y = 1\n", + "x = 0 - y\n", + "\n", + "# calculations\n", + "NC = y + x * TA/TC\n", + "ND = y + x * TA/TD\n", + "\n", + "# results\n", + "print \"revolutions of gear C : %.4f\"%NC\n", + "print \"revolutions of gear D : %.4f \"%ND\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "revolutions of gear C : 0.0099\n", + "revolutions of gear D : -0.0101 \n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.14 Page No : 453" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "NA = 300. \t\t\t#rpm\n", + "TD = 40.\n", + "TE = 30.\n", + "TF = 50.\n", + "TG = 80.\n", + "TH = 40.\n", + "TK = 20.\n", + "TL = 30.\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.18 and Table 13.13\n", + "#Calculating the speed of wheel E\n", + "NE = NA*(TD/TE) \t\t\t#rpm\n", + "#Calculating the number of teeth on wheel C\n", + "TC = TH+TK+TL\n", + "#Speed and direction of rotation of shaft B:\n", + "#Calculating the values of x and y\n", + "#We have -x-y = -400 or x+y = 400 .....(i)\n", + "#Also x*(TH/TK)*(TL/TC)-y = 0 .....(ii) \n", + "A = [[1, 1],[ (TH/TK)*(TL/TC), -1]]\n", + "B = [400, 0]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the speed of wheel F\n", + "NF = -y \t\t\t#rpm\n", + "#Calculating the speed of shaft B\n", + "NB = -NF*(TF/TG) \t\t\t#Speed of shaft B rpm\n", + "\n", + "#Results:\n", + "print \" Number of teeth on wheel C, TC = %.1f.\"%(TC)\n", + "print \" Speed of shaft B = %.1f rpm, anticlockwise.\"%(NB)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on wheel C, TC = 90.0.\n", + " Speed of shaft B = 100.0 rpm, anticlockwise.\n" + ] + } + ], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.15 Page No : 455" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "T1 = 80.\n", + "T8 = 160.\n", + "T4 = 100.\n", + "T3 = 120.\n", + "T6 = 20.\n", + "T7 = 66.\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.19 and Table 13.14\n", + "#Calculating the number of teeth on wheel 2\n", + "T2 = (T3-T1)/2\n", + "#Calculating the values of x and y\n", + "#Assuming that wheel 1 makes 1 rps anticlockwise x+y = 1 .....(i)\n", + "#Also y-x*(T1/T3) = 0 or x*(T1/T3)-y = 0 .....(ii)\n", + "A = [[1, 1],[ 1, T1/T3]]\n", + "B = [1, 0]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the speed of casing C\n", + "NC = y \t\t\t#Speed of casing C rps\n", + "#Calculating the speed of wheel 2\n", + "N2 = y-x*(T1/T2) \t\t\t#Speed of wheel 2 rps\n", + "#Calculating the number of teeth on wheel 5\n", + "T5 = (T4-T6)/2\n", + "#Calculating the values of x1 and y1\n", + "y1 = -2\n", + "x1 = (y1-0.4)*(T4/T6)\n", + "#Calculating the speed of wheel 6\n", + "N6 = x1+y1 \t\t\t#Speed of wheel 6 rps\n", + "#Calculating the values of x2 and y2\n", + "y2 = 0.4\n", + "x2 = -(14+y2)*(T7/T8)\n", + "#Calculating the speed of wheel 8\n", + "N8 = x2+y2 \t\t\t#Speed of wheel 8 rps\n", + "#Calculating the velocity ratio of the output shaft B to the input shaft A\n", + "vr = N8/1 \t\t\t#Velocity ratio\n", + "\n", + "#Results:\n", + "print \" Velocity ratio of the output shaft B to the input shaft A = %.2f.\"%(vr)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Velocity ratio of the output shaft B to the input shaft A = -5.54.\n" + ] + } + ], + "prompt_number": 17 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.16 Page No : 459" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TA = 40.\n", + "TB = 30.\n", + "TC = 50.\n", + "NX = 100.\n", + "NA = NX \t\t\t #rpm\n", + "Narm = 100. \t\t\t#Speed of armrpm\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.22 and Table 13.18\n", + "#Calculating the values of x and y\n", + "y = +100\n", + "x = -100-y\n", + "#Calculating the speed of the driven shaft\n", + "NY = y-x*(TA/TB) \t\t\t#rpm\n", + "\n", + "#Results:\n", + "print \" Speed of the driven shaft, NY = %.1f rpm, anticlockwise.\"%(NY)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of the driven shaft, NY = 366.7 rpm, anticlockwise.\n" + ] + } + ], + "prompt_number": 18 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.17 Page No : 460" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TB = 20.\n", + "TC = 80.\n", + "TD = 80.\n", + "TE = 30.\n", + "TF = 32.\n", + "NB = 1000. \t\t\t#rpm\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.23 and Table 13.19\n", + "#Speed of the output shaft when gear C is fixed:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table y-x*(TB/TC) = 0 .....(i) \n", + "#Also x+y = +1000 or y+x = 1000 .....(ii)\n", + "A = [[1, -TB/TC],[ 1, 1]]\n", + "B = [0, 1000]\n", + "V = linalg.solve(A,B)\n", + "x = V[1]\n", + "y = V[0]\n", + "#Calculating the speed of output shaft\n", + "NF1 = y-x*(TB/TD)*(TE/TF) \t\t\t#Speed of the output shaft when gear C is fixed rpm\n", + "#Speed of the output shaft when gear C is rotated at 10 rpm counter clockwise:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of te table y-x*(TB/TC) = +10 .....(iii)\n", + "#Also x+y = +1000 or y+x = 1000 .....(iv)\n", + "A = [[1, -TB/TC],[1, 1]]\n", + "B = [10, 1000]\n", + "V = linalg.solve(A,B)\n", + "x = V[1]\n", + "y = V[0]\n", + "#Calculating the speed of output shaft\n", + "NF2 = y-x*(TB/TD)*(TE/TF) \t\t\t#Speed of the output shaft when gear C is rotated at 10 rpm counter clockwise rpm\n", + "\n", + "#Results:\n", + "print \" Speed of the output shaft when gear C is fixed = %.1f rpm, counter clockwise.\"%(NF1)\n", + "print \" Speed of the output shaft when gear C is rotated at 10 rpm counter clockwise = %.1f rpm, \\\n", + " counter clockwise.\"%(NF2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of the output shaft when gear C is fixed = 12.5 rpm, counter clockwise.\n", + " Speed of the output shaft when gear C is rotated at 10 rpm counter clockwise = 22.4 rpm, counter clockwise.\n" + ] + } + ], + "prompt_number": 19 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.18 Page No : 461" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "TA = 10.\n", + "TB = 60.\n", + "NA = 1000.\n", + "NQ = 210.\n", + "ND = NQ \t\t\t#rpm\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.24 and Table 13.20\n", + "#Calculating the speed of crown gear B\n", + "NB = NA*(TA/TB) \t\t\t#rpm\n", + "#Calculating the values of x and y\n", + "y = 200.\n", + "x = y-210.\n", + "#Calculating the speed of road wheel attached to axle P\n", + "NC = x+y \t\t\t#Speed of road wheel attached to axle P rpm\n", + "\n", + "#Results:\n", + "print \" Speed of road wheel attached to axle P = %d rpm.\"%(NC)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of road wheel attached to axle P = 190 rpm.\n" + ] + } + ], + "prompt_number": 20 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.19 Page No : 463" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TA = 15.\n", + "TB = 20.\n", + "TC = 15.\n", + "NA = 1000. \t\t\t#rpm\n", + "Tm = 100. \t\t\t#Torque developed by motor N-m\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.26 and Table 13.21\n", + "#Calculating the number of teeth on gears E and D\n", + "TE = TA+2*TB\n", + "TD = TE-(TB-TC)\n", + "#Speed of the machine shaft:\n", + "#From the fourth row of the table x+y = 1000 or y+x = 1000 .....(i)\n", + "#Also y-x*(TA/TE) = 0 .....(ii) \n", + "A = [[1, 1],[1, -TA/TE]]\n", + "B = [1000, 0]\n", + "V = linalg.solve(A,B)\n", + "y = round(V[0])\n", + "x = round(V[1])\n", + "#Calculating the speed of machine shaft\n", + "ND = y-x*(TA/TB)*(TC/TD) \t\t\t#rpm\n", + "#Calculating the torque exerted on the machine shaft\n", + "Ts = Tm*NA/ND \t\t\t#Torque exerted on the machine shaft N-m\n", + "\n", + "#Results:\n", + "print \" Speed of machine shaft, ND = %.2f rpm, anticlockwise.\"%(ND)\n", + "print \" Torque exerted on the machine shaft = %.f N-m.\"%(Ts)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of machine shaft, ND = 37.15 rpm, anticlockwise.\n", + " Torque exerted on the machine shaft = 2692 N-m.\n" + ] + } + ], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.20 Page No : 465" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "\n", + "# Variables:\n", + "Ts = 100 \t\t\t#Torque on the sun wheel N-m\n", + "r = 5 \t\t\t#Ratio of speeds of gear S to C NS/NC\n", + "#Refer Fig. 13.27 and Table 13.22\n", + "#Number of teeth on different wheels:\n", + "#Calculating the values of x and y\n", + "y = 1.\n", + "x = 5-y\n", + "\n", + "#Calculating the number of teeth on wheel E\n", + "TS = 16.\n", + "TE = 4*TS\n", + "#Calculating the number of teeth on wheel P\n", + "TP = (TE-TS)/2\n", + "#Torque necessary to keep the internal gear stationary:\n", + "Tc = Ts*r \t\t\t#Torque on CN-m\n", + "#Caluclating the torque necessary to keep the internal gear stationary\n", + "Ti = Tc-Ts \t\t\t#Torque necessary to keep the internal gear stationary N-m\n", + "\n", + "#Results:\n", + "print \" Number of teeth on different wheels, TE = %d.\"%(TE)\n", + "print \" Torque necessary to keep the internal gear stationary = %d N-m.\"%(Ti)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Number of teeth on different wheels, TE = 64.\n", + " Torque necessary to keep the internal gear stationary = 400 N-m.\n" + ] + } + ], + "prompt_number": 23 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.21 Page No : 466" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math \n", + "from numpy import linalg\n", + "\n", + "# Variables:\n", + "TA = 14.\n", + "TC = 100.\n", + "r = 98./41 \t\t\t#TE/TD\n", + "PA = 1.85*1000 \t\t\t#W\n", + "NA = 1200. \t\t\t#rpm\n", + "TB = 43\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.28 and Table 13.23\n", + "#Calculating the number of teeth on wheel B TB = (TC-TA)/2\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table -y+x*(TA/TC) = 0 or x*(TA/TC)-y = 0 .....(i)\n", + "#Also x-y = 1200 or x+y = -1200 .....(ii)\n", + "A = [[TA/TC, -1],[ 1, 1]]\n", + "B = [0, -1200]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "#Calculating the speed of gear E\n", + "NE = round(-y+x*(TA/TB)*(1./r)) \t\t\t#rpm\n", + "#Fixing torque required at C:\n", + "#Calculating the torque on A\n", + "Ta = PA*60./(2*math.pi*NA) \t\t\t#Torque on A N-m\n", + "#Calculating the torque on E\n", + "Te = PA*60./(2*math.pi*NE) \t\t\t#Torque on E\n", + "#Calculating the fixing torque required at C\n", + "Tc = Te-Ta \t\t\t#Fixing torque at C N-m\n", + "\n", + "#Results:\n", + "print \" Speed and direction of rotation of gear E, NE = %d rpm, anticlockwise.\"%(NE)\n", + "print \" Fixing torque required at C = %.1f N-m.\"%(Tc)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed and direction of rotation of gear E, NE = 4 rpm, anticlockwise.\n", + " Fixing torque required at C = 4401.8 N-m.\n" + ] + } + ], + "prompt_number": 9 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.22 Page No : 468" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TB = 15.\n", + "TA = 60.\n", + "TC = 20.\n", + "omegaY = 740.\n", + "omegaA = omegaY \t\t\t#rad/s\n", + "P = 130*1000. \t\t\t#W\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.29 and Table 13.24\n", + "#Calculating the number of teeth on wheel D\n", + "TD = TA-(TC+TB)\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table y-x*(TD/TC)*(TB/TA) = 740 .....(i)\n", + "#Also x+y = 0 or y+x = 0 .....(ii)\n", + "A = [[1, -(TD/TC)*(TB/TA)],[ 1, 1]]\n", + "B = [740, 0]\n", + "V = linalg.solve(A,B)\n", + "x = V[1]\n", + "y = V[0]\n", + "#Calculating the speed of shaft X\n", + "omegaX = y \t\t\t#rad/s\n", + "#Holding torque on wheel D:\n", + "#Calculating the torque on A\n", + "Ta = P/omegaA \t\t\t#Torque on A N-m\n", + "#Calculating the torque on X\n", + "Tx = P/omegaX \t\t\t#Torque on X N-m\n", + "#Calculating the holding torque on wheel D\n", + "Td = Tx-Ta \t\t\t#Holding torque on wheel D N-m\n", + "\n", + "#Results:\n", + "print \" Speed of shaft X, omegaX = %.1f rad/s.\"%(omegaX)\n", + "print \" Holding torque on wheel D = %.1f N-m.\"%(Td)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of shaft X, omegaX = 563.8 rad/s.\n", + " Holding torque on wheel D = 54.9 N-m.\n" + ] + } + ], + "prompt_number": 25 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.23 Page No : 469" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TP = 144.\n", + "TQ = 120.\n", + "TR = 120.\n", + "TX = 36.\n", + "TY = 24.\n", + "TZ = 30.\n", + "NI = 1500. \t\t\t#rpm\n", + "P = 7.5*1000 \t\t\t#W\n", + "eta = 0.8\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.30 and Table 13.25\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table x+y = -1500 .....(i) \n", + "#Also y-x*(TZ/TR) = 0 or -x*(TZ/TR)+y = 0 .....(ii)\n", + "A = [[1, 1],[-TZ/TR, 1]]\n", + "B = [-1500, 0]\n", + "V = linalg.solve(A,B)\n", + "x = V[0]\n", + "y = V[1]\n", + "\n", + "#Calculating the values of x1 and y1\n", + "#We have y1-x1*(TY/TQ) = y .....(iii)\n", + "#Also x1+y1 = x+y or y1+x1 = x+y .....(iv)\n", + "A = [[1, -TY/TQ],[ 1, 1]]\n", + "B = [y, x+y]\n", + "V = linalg.solve(A , B)\n", + "x1 = V[1]\n", + "y1 = V[0]\n", + "#Speed and direction of the driven shaft O and the wheel P:\n", + "#Calculating the speed of shaft O\n", + "NO = y1 \t\t\t#rpm\n", + "#Calculating the speed of wheel P\n", + "NP = y1+x1*(TY/TQ)*(TX/TP) \t\t\t#rpm\n", + "#Torque tending to rotate the fixed wheel R:\n", + "#Calculating the torque on shaft I\n", + "T1 = P*60/(2*math.pi*NI) \t\t\t#N-m\n", + "#Calculating the torque on shaft O\n", + "T2 = eta*P*60/(2*math.pi*(-NO)) \t\t\t#N-m\n", + "#Calculating the torque tending to rotate the fixed wheel R\n", + "T = T2-T1 \t\t\t#Torque tending to rotate the fixed wheel R N-m\n", + "\n", + "#Results:\n", + "print \" Speed of the driven shaft O, NO = %d rpm, clockwise.\"%(-NO)\n", + "print \" Speed of the wheel P, NP = %d rpm, clockwise.\"%(-NP)\n", + "print \" Torque tending to rotate the fixed wheel R = %.2f N-m.\"%(T)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Speed of the driven shaft O, NO = 500 rpm, clockwise.\n", + " Speed of the wheel P, NP = 550 rpm, clockwise.\n", + " Torque tending to rotate the fixed wheel R = 66.85 N-m.\n" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 13.24 Page No : 471" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from numpy import linalg\n", + "import math \n", + "\n", + "# Variables:\n", + "TA = 34.\n", + "TB = 120.\n", + "TC = 150.\n", + "TD = 38.\n", + "TE = 50.\n", + "PX = 7.5*1000 \t\t\t#W\n", + "NX = 500. \t\t\t#rpm\n", + "m = 3.5 \t\t\t#mm\n", + "\n", + "#Solution:\n", + "#Refer Fig. 13.31 and Table 13.27\n", + "#Output torque of shaft Y:\n", + "#Calculating the values of x and y\n", + "#From the fourth row of the table x+y = 500 or y+x = 500 .....(i)\n", + "#Alsoy-x*(TA/TC) = 0 .....(ii)\n", + "A = [[1, 1],[ 1, -TA/TC]]\n", + "B = [500, 0]\n", + "V = linalg.solve(A, B)\n", + "y = round(V[0],1) \t\t\t#rpm\n", + "x = round(V[1],1) \t\t\t#rpm\n", + "#Calculating the speed of output shaft Y\n", + "NY = y-x*(TA/TB)*(TD/TE) \t\t\t#rpm\n", + "#Calculating the speed of wheel E\n", + "NE = NY \t\t\t#rpm\n", + "#Calculating the input power assuming 100 per cent efficiency\n", + "PY = PX \t\t\t#W\n", + "#Calculating the output torque of shaft Y\n", + "Ty = PY*60/(2*math.pi*NY*1000) \t\t\t#Output torque on shaft Y kN-m\n", + "#Tangential force between wheels D and E:\n", + "#Calculating the pitch circle radius of wheel E\n", + "rE = m*TE/(2*1000) \t\t\t#m\n", + "#Calculating the tangential force between wheels D and E\n", + "FtDE = Ty/rE \t\t\t#Tangential force between wheels D and E kN\n", + "#Tangential force between wheels B and C:\n", + "#Calculating the input torque on shaft X\n", + "Tx = PX*60/(2*math.pi*NX) \t\t\t#Input torque on shaft X N-m\n", + "#Calculating the fixing torque on the fixed wheel C\n", + "Tf = Ty-Tx/1000 \t\t\t#Fixing torque on the fixed wheelC kN-m\n", + "#Calculating the pitch circle radius of wheel C\n", + "rC = m*TC/(2*1000) \t\t\t#m\n", + "#Calculating the tangential forces between wheels B and C\n", + "FtBC = Tf/rC \t\t\t#kN\n", + "\n", + "#Results:\n", + "print \" Output torque of shaft Y = %.3f kN-m.\"%(Ty)\n", + "print \" Tangential force between wheels D and E = %.1f kN.\"%(FtDE)\n", + "print \" Tangential force between wheels B and C = %.f kN.\"%(FtBC)\n", + "\n", + "# note : answers are slightly different because of solve function and rounding off errors." + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " Output torque of shaft Y = 15.468 kN-m.\n", + " Tangential force between wheels D and E = 176.8 kN.\n", + " Tangential force between wheels B and C = 58 kN.\n" + ] + } + ], + "prompt_number": 6 + } + ], + "metadata": {} + } + ] +}
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