{ "metadata": { "name": "", "signature": "sha256:844073ad5c928bfdaf4248256fc5562a2dc71b1c2156a1230e38f2c366a18bf7" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 12 : Toothed Gearing" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.1 Page No : 393" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "P = 120.*1000 \t\t\t#W\n", "d = 250./1000\n", "r = d/2 \t\t\t#m\n", "N = 650. \t\t\t#rpm\n", "phi = 20. \t\t\t#degrees\n", "\n", "#Solution:\n", "#Calculating the angular speed of the gear\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted\n", "T = P/omega \t\t\t#N-m\n", "#Calculating the math.tangential load on the pinion\n", "FT = T/r \t\t\t#N\n", "#Calculating the total load due to power transmitted\n", "F = FT/(math.cos(math.radians(phi))*1000) \t\t\t#kN\n", "\n", "#Results:\n", "print \" Total load due to power transmitted, F = %.2f kN.\"%(F)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Total load due to power transmitted, F = 15.01 kN.\n" ] } ], "prompt_number": 3 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.2 Page No : 397" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "T = 40.\n", "t = T\n", "phi = 20. \t\t\t#degrees\n", "m = 6. \t\t\t#mm\n", "\n", "#Solution:\n", "#Calculating the circular pitch\n", "pc = math.pi*m \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = 1.75*pc \t\t\t#Length of arc of contact mm\n", "#Calculating the length of path of contact\n", "Lpc = Lac*math.cos(phi) \t\t\t#Length of path of contact mm\n", "#Calculating the pitch circle radii of each wheel\n", "R = m*T/2 \t\t\t#mm\n", "r = R \t\t\t#mm\n", "#Calculating the radius of the addendum circle of each wheel\n", "RA = math.sqrt(R**2*(math.cos(phi))**2+(Lpc/2+R*math.sin(phi))**2) \t\t\t#mm\n", "#Calculating the addendum of the wheel\n", "Ad = RA-R \t\t\t#Addendum of the wheel mm\n", "\n", "#Results:\n", "print \" Addendum of the wheel = %.2f mm.\"%(Ad)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Addendum of the wheel = 6.17 mm.\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.3 Page No : 398" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "t = 30.\n", "T = 80.\n", "phi = 20. \t\t\t#degrees\n", "m = 12. \t\t\t#mm\n", "Addendum = 10. \t\t#mm\n", "\n", "#Solution:\n", "#Length of path of contact:\n", "#Calculating the pitch circle radius of pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of gear\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of pinion\n", "rA = r+Addendum \t\t\t#mm\n", "#Calculating the radius of addendum circle of gear\n", "RA = R+Addendum \t\t\t#mm\n", "#Calculating the length of path of approach\n", "#Refer Fig. 12.11\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#Length of arc of contact mm\n", "#Contact ratio:\n", "#Calculating the circular pitch\n", "Pc = math.pi*m \t\t\t#mm\n", "#Calculating the contact ratio\n", "CR = Lac/Pc \t\t\t#Contact ratio\n", "\n", "#Results:\n", "print \" Length of path of contact, KL = %.1f mm.\"%(KL)\n", "print \" Length of arc of contact = %.2f mm.\"%(Lac)\n", "print \" Contact ratio = %.1f.\"%(CR)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Length of path of contact, KL = 52.3 mm.\n", " Length of arc of contact = 55.61 mm.\n", " Contact ratio = 1.5.\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.4 Page No : 399" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "phi = 20. \t\t\t#degrees\n", "t = 20.\n", "G = 2.\n", "m = 5. \t\t\t#mm\n", "v = 1.2 \t\t\t#m/s\n", "addendum = 1*m \t\t\t#mm\n", "\n", "#Solution:\n", "#Angle turned through by pinion when one pair of teeth is in mesh:\n", "#Calculating the pitch circle radius of pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of wheel\n", "R = m*G*t/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of pinion\n", "rA = r+addendum \t\t\t#mm\n", "#Calculating the radius of addendum circle of wheel\n", "RA = R+addendum \t\t\t#mm\n", "#Calculating the length of path of approach\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#mm\n", "#Calculating the angle turned by the pinion\n", "angle = Lac*360/(2*math.pi*r) \t\t\t#Angle turned by the pinion degrees\n", "#Maximum velocity of sliding:\n", "#Calculating the angular speed of pinion\n", "omega1 = v*1000/r \t\t\t#rad/s\n", "#Calculating the angular speed of wheel\n", "omega2 = v*1000/R \t\t\t#rad/s\n", "#Calculating the maximum velocity of sliding\n", "vS = (omega1+omega2)*KP \t\t\t#mm/s\n", "\n", "#Results:\n", "print \" Angle turned through by pinion when one pair of teeth is in mesh = %.2f degrees.\"%(angle)\n", "print \" Maximum velocity of sliding, vS = %.1f mm/s.\"%(vS)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Angle turned through by pinion when one pair of teeth is in mesh = 29.43 degrees.\n", " Maximum velocity of sliding, vS = 455.3 mm/s.\n" ] } ], "prompt_number": 11 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.5 Page No : 400" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "T = 40.\n", "t = 20.\n", "N1 = 2000. \t\t\t#rpm\n", "phi = 20. \t\t\t#degrees\n", "addendum = 5.\n", "m = 5. \t\t\t#mm\n", "\n", "#Solution:\n", "#Calculating the angular velocity of the smaller gear\n", "omega1 = 2*math.pi*N1/60 \t\t\t#rad/s\n", "#Calculating the angular velocity of the larger gear\n", "omega2 = omega1*t/T \t\t\t#rad/s\n", "#Calculating the pitch circle radius of the smaller gear\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of the larger gear\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of aaddendum circle of smaller gear\n", "rA = r+addendum \t\t\t#mm\n", "#Calculating the radius of addendum circle of larger gear\n", "RA = R+addendum \t\t\t#mm\n", "#Calculating the length of path of approach\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the velocity of sliding at the point of engagement\n", "vSK = (omega1+omega2)*KP \t\t\t#mm/s\n", "#Calculating the velocity of sliding at the point of disengagement\n", "vSL = (omega1+omega2)*PL \t\t\t#mm/s\n", "#Angle through which the pinion turns:\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#Length of arc of contact mm\n", "#Calculating the circumference of pinion\n", "C = 2*math.pi*r \t\t\t#Circumference of pinion mm\n", "#Calculating the angle through which the pinion turns\n", "angle = Lac*360/C \t\t\t#Angle through which the pinion turns degrees\n", "\n", "#Results:\n", "print \" Velocity of sliding at the point of engagement, vSK = %.f mm/s.\"%(vSK)\n", "print \" Velocity of sliding at the point of disengagement, vsL = %.f mm/s.\"%(vSL)\n", "print \" Angle through which the pinion turns = %.2f degrees.\"%(angle)\n", "\n", "# answers differ due to rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Velocity of sliding at the point of engagement, vSK = 3973 mm/s.\n", " Velocity of sliding at the point of disengagement, vsL = 3610 mm/s.\n", " Angle through which the pinion turns = 29.43 degrees.\n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.6 Page No : 401" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "# Variables:\n", "phi = 20. \t\t\t#degrees\n", "m = 6.\n", "addendum = 1*m \t\t\t#mm\n", "t = 17.\n", "T = 49.\n", "\n", "#Solution:\n", "#Number of pairs of teeth in contact:\n", "#Calculating the pitch circle radius of pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of gear\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of pinion\n", "rA = r+addendum \t\t\t#mm\n", "#Calculating the radius of addendum circle of gear\n", "RA = R+addendum \t\t\t#mm\n", "#Calculating the length of path of approach\n", "#Refer Fig. 12.11\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#Length of arc of contact mm\n", "#Calculating the circular pitch\n", "pc = math.pi*m \t\t\t#mm\n", "#Calculating the number of pairs of teeth in contact\n", "n = Lac/pc \t\t\t#Number of pairs of teeth in contact\n", "#Angle turned by the pinion and gear wheel when one pair of teeth is in contact:\n", "#Calculating the angle turned through by the pinion\n", "anglep = Lac*360/(2*math.pi*r) \t\t\t#Angle turned through by the pinion degrees\n", "#Calculating the angle turned through by the wheel\n", "angleg = Lac*360/(2*math.pi*R) \t\t\t#Angle turned through by the gear wheel degrees\n", "#Ratio of sliding to rolling motion:\n", "#At the instant when the tip of a tooth on the larger wheel is just making contact with its mating teeth\n", "r1 = ((1+t/T)*KP)/r \t\t\t#Ratio of sliding velocity to rolling velocity\n", "#At the instant when the tip of a tooth on a larger wheel is just leaving contact with its mating teeth\n", "r2 = ((1+t/T)*PL)/r \t\t\t#Ratio of sliding velocity to rolling velocity\n", "\n", "\n", "#Results:\n", "print \" Number of pairs of teeth in contact = %.f.\"%(n)\n", "print \" Angle turned through by the pinion = %.1f degrees.\"%(anglep)\n", "print \" Angle turned through by the gear wheel = %.f degrees.\"%(angleg)\n", "print \" At the instant when the tip of a tooth on the larger wheel is just\\\n", " making contact with its mating teeth, ratio of sliding \\nvelocity to rolling velocity = %.2f.\"%(r1)\n", "print \" At the instant when the tip of a tooth on a larger wheel is just leaving contact\\\n", " with its mating teeth, ratio of sliding velocity to rolling velocity = %.3f.\"%(r2)\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Number of pairs of teeth in contact = 2.\n", " Angle turned through by the pinion = 34.6 degrees.\n", " Angle turned through by the gear wheel = 12 degrees.\n", " At the instant when the tip of a tooth on the larger wheel is just making contact with its mating teeth, ratio of sliding \n", "velocity to rolling velocity = 0.41.\n", " At the instant when the tip of a tooth on a larger wheel is just leaving contact with its mating teeth, ratio of sliding velocity to rolling velocity = 0.354.\n" ] } ], "prompt_number": 3 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.7 Page No : 403" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "t = 18.\n", "T = 72.\n", "phi = 20. \t\t\t#degrees\n", "m = 4. \t\t\t #mm\n", "addendump = 8.5 \t\t\t#Addendum on pinion mm\n", "addendumg = 3.5 \t\t\t#Addendum on gear mm\n", "\n", "#SOlution:\n", "#Refer Fig. 12.12\n", "#Calculating the pitch circle radius of the pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of the gear\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of the pinion\n", "rA = r+addendump \t\t\t#mm\n", "#Calculating the radius of addendum circle of the gear\n", "RA = R-addendumg \t\t\t#mm\n", "#Calculating the radius of the base circle of the pinion\n", "O1M = r*math.cos(math.radians(phi)) \t\t\t#mm\n", "#Calculating the radius of the base circle of the gear\n", "O2N = R*math.cos(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of approach\n", "KP = R*math.sin(math.radians(phi))-math.sqrt(RA**2-O2N**2) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-O1M**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of the path of contact\n", "KL = KP+PL \t\t\t#mm\n", "\n", "#Results:\n", "print \" Length of the path of contact, KL = %.2f mm.\"%(KL)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Length of the path of contact, KL = 28.04 mm.\n" ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.8 Page No : 406" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "t = 20.\n", "T = 40.\n", "m = 10. \t\t\t#mm\n", "phi = 20. \t\t\t#degrees\n", "\n", "#Solution:\n", "#Addendum height for each gear wheel:\n", "#Calculating the pitch circle radius of the smaller gear wheel\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of the larger wheel\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle for the larger gear wheel\n", "RA = math.sqrt((r*math.sin(math.radians(phi))/2+R*math.sin(math.radians(phi)))**2+R**2*(math.cos(math.radians(phi)))**2) \t\t\t#mm\n", "#Calculating the addendum height for larger gear wheel\n", "addendumg = RA-R \t\t\t#mm\n", "#Calculating the radius of addendum circle for the smaller gear wheel\n", "rA = math.sqrt((R*math.sin(math.radians(phi))/2+r*math.sin(math.radians(phi)))**2+r**2*(math.cos(math.radians(phi)))**2) \t\t\t#mm\n", "#Calculating the addendum height for smaller gear wheel\n", "addendump = rA-r \t\t\t#mm\n", "#Calculating the length of the path of contact\n", "Lpc = (r+R)*math.sin(math.radians(phi))/2 \t\t\t#Length of the path of contact mm\n", "#Calculating the length of the arc of contact\n", "Lac = Lpc/math.cos(math.radians(phi)) \t\t\t#Length of the arc of contact mm\n", "#Contact ratio:\n", "#Calculating the circular pitch\n", "pc = math.pi*m \t\t\t#mm\n", "#Calculating the contact ratio\n", "CR = Lpc/pc \t\t\t#Contact ratio\n", "\n", "#Results:\n", "print \" Addendum height for larger gear wheel = %.1f mm.\"%(addendumg)\n", "print \" Addendum height for smaller gear wheel = %.1f mm.\"%(addendump)\n", "print \" Length of the path of contact = %.1f mm.\"%(Lpc)\n", "print \" Length of the arc of contact = %.1f mm.\"%(Lac)\n", "print \" Contact ratio = %d.\"%(CR+1)\n", "\n", "# book answer is wrong for 2nd " ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Addendum height for larger gear wheel = 6.5 mm.\n", " Addendum height for smaller gear wheel = 16.2 mm.\n", " Length of the path of contact = 51.3 mm.\n", " Length of the arc of contact = 54.6 mm.\n", " Contact ratio = 2.\n" ] } ], "prompt_number": 18 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.9 Page No : 410" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "G = 3.\n", "phi = 20. \t\t\t#degrees\n", "Aw = 1. \t\t\t#module\n", "\n", "#Solution:\n", "#Calculating the minimum number of teeth for a gear ratio of 3:1\n", "t1 = (2*Aw)/(G*(math.sqrt(1+1/G*(1/G+2)*(math.sin(math.radians(phi)))**2)-1))\n", "#Calculating the minimum number of teeth for equal wheel\n", "t2 = (2*Aw)/(math.sqrt(1+3*(math.sin(math.radians(phi)))**2)-1)\n", "\n", "#Results:\n", "print \" Minimum number of teeth for a gear ratio of 3:1, t = %.f.\"%(t1+1)\n", "print \" Minimum number of teeth for equal wheel, t = %d.\"%(t2+1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Minimum number of teeth for a gear ratio of 3:1, t = 16.\n", " Minimum number of teeth for equal wheel, t = 13.\n" ] } ], "prompt_number": 20 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.10 Page No : 410" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "import numpy\n", "\n", "# Variables:\n", "G = 4.\n", "phi = 14.5 \t\t\t#degrees\n", "\n", "#Solution:\n", "#Least number of teeth on each wheel:\n", "#Calculating the least number of teeth on the pinion\n", "t = 2*math.pi/(math.tan(math.radians(phi)))\n", "#Calculating the least number of teeth on the gear\n", "T = G*t\n", "\n", "#Results:\n", "print \" Least number of teeth on the pinion, t = %.1f.\"%(t)\n", "print \" Least number of teeth on the gear, T = %.f.\"%(round(T,-1))\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Least number of teeth on the pinion, t = 24.3.\n", " Least number of teeth on the gear, T = 100.\n" ] } ], "prompt_number": 26 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.11 Page No : 411" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "phi = 16. \t\t\t#degrees\n", "m = 6. \t\t\t #mm\n", "t = 16.\n", "G = 1.75\n", "T = G*t\n", "N1 = 240. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the pinion\n", "omega1 = 2*math.pi*N1/60 \t\t\t#rad/s\n", "#Addenda on pinion and gear wheel:\n", "#Calculating the addendum on pinion\n", "addendump = m*t/2*(math.sqrt(1+T/t*(T/t+2)*(math.sin(math.radians(phi)))**2)-1) \t\t\t#Addendum on pinion mm\n", "#Calculating the addendum on wheel\n", "addendumg = m*T/2*(math.sqrt(1+t/T*(t/T+2)*(math.sin(math.radians(phi)))**2)-1) \t\t\t#Addendum on wheel mm\n", "#Length of path of contact:\n", "#Calculating the pitch circle radius of wheel\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the pitch circle radius of pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the addendum circle radius of wheel\n", "RA = R+addendump \t\t\t#mm\n", "#Calculating the addendum circle radius of pinion\n", "rA = r+addendumg \t\t\t#mm\n", "#Calculating the length of path of approach\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Maximum velocity of sliding of teeth on either side of pitch point:\n", "#Calculating the angular speed of gear wheel\n", "omega2 = omega1/G \t\t\t#rad/s\n", "#Calculating the maximum velocity of sliding of teeth on the left side of pitch point\n", "vmaxl = (omega1+omega2)*KP \t\t\t#Maximum velocity of sliding of teeth on the left side of pitch point mm/s\n", "#Calculating the maximum velocity of sliding of teeth on the right side of pitch point\n", "vmaxr = (omega1+omega2)*PL \t\t\t#Maximum velocity of sliding of teeth on the right side of pitch point mm/s\n", "\n", "#Results:\n", "print \" Addendum on pinion = %.2f mm.\"%(addendump)\n", "print \" Addendum on wheel = %.2f mm.\"%(addendumg)\n", "print \" Length of path of contact, KL = %.2f mm.\"%(KL)\n", "print \" Maximum velocity of sliding of teeth on the left side of pitch point = %d mm/s.\"%(vmaxl)\n", "print \" Maximum velocity of sliding of teeth on the right side of pitch point = %d mm/s.\"%(vmaxr)\n", "\n", "# rounding error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Addendum on pinion = 10.76 mm.\n", " Addendum on wheel = 4.56 mm.\n", " Length of path of contact, KL = 38.39 mm.\n", " Maximum velocity of sliding of teeth on the left side of pitch point = 1044 mm/s.\n", " Maximum velocity of sliding of teeth on the right side of pitch point = 471 mm/s.\n" ] } ], "prompt_number": 28 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.12 Page No : 412" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "phi = 20. \t\t\t#degrees\n", "t = 30.\n", "T = 50.\n", "m = 4.\n", "N1 = 1000. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of thr pinion\n", "omega1 = 2*math.pi*N1/60 \t\t\t#rad/s\n", "#Sliding velocities at engagement and at disengagement of a pair of teeth:\n", "#Calculating the addendum of the smaller gear\n", "addendump = m*t/2*(math.sqrt(1+T/t*(T/t+2)*(math.sin(math.radians(phi)))**2)-1) \t\t\t#Addendum of the smaller gear mm\n", "#Calculating the addendum of the larger gear\n", "addendumg = m*T/2*(math.sqrt(1+t/T*(t/T+2)*(math.sin(math.radians(phi)))**2)-1) \t\t\t#Addendum of the larger gear mm\n", "#Calculating the pitch circle radius of the smaller gear\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of the smaller gear\n", "rA = r+addendump \t\t\t#mm\n", "#Calculating the pitch circle radius of the larer gear\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of the larger gear\n", "RA = R+addendumg \t\t\t#mm\n", "#Calculating the path of approach\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the angular speed of the larger gear\n", "omega2 = omega1*t/T \t\t\t#rad/s\n", "#Calculating the sliding velocity at engagement of a pair of teeth\n", "v1 = (omega1+omega2)*KP \t\t\t#Sliding velocity at engagement of a pair of teeth mm/s\n", "#Calculating the sliding velocity at disengagement of a pair of teeth\n", "v2 = (omega1+omega2)*PL \t\t\t#Sliding velocity at disengagement of a pair of teeth mm/s\n", "#Contact ratio:\n", "#Calculating the length of the arc of contact\n", "Lac = (KP+PL)/math.cos(math.radians(phi)) \t\t\t#mm\n", "#Calculating the circular pitch\n", "pc = math.pi*m \t\t\t#Circular pitch mm\n", "#Calculating the contact ratio\n", "CR = Lac/pc \t\t\t#Contact ratio\n", "\n", "#Results:\n", "print \" Sliding velocity at engagement of a pair of teeth = %.3f m/s.\"%(v1/1000)\n", "print \" Sliding velocity at disengagement of a pair of teeth = %.3f m/s.\"%(v2/1000)\n", "print \" Contact ratio = %d.\"%(CR+1)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Sliding velocity at engagement of a pair of teeth = 3.438 m/s.\n", " Sliding velocity at disengagement of a pair of teeth = 5.731 m/s.\n", " Contact ratio = 5.\n" ] } ], "prompt_number": 30 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.13 Page No : 414" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "G = 3.\n", "m = 6.\n", "AP = 1*m\n", "AW = AP \t\t\t#mm\n", "phi = 20. \t\t\t#degrees\n", "N1 = 90. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the pinion\n", "omega1 = 2*math.pi*N1/60 \t\t\t#rad/s\n", "#Calculating the number of teeth on the pinion to avoid interference on it\n", "t = 2*AP/(math.sqrt(1+G*(G+2)*(math.sin(math.radians(phi)))**2)-1)\n", "#Calculating the corresponding number of teeth on the wheel\n", "T = G*t\n", "#Length of path and arc of contact:\n", "#Calculating the pitch circle radius of pinion\n", "r = m*t/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of pinion\n", "rA = r+AP \t\t\t#mm\n", "#Calculating the pitch circle radius of wheel\n", "R = m*T/2 \t\t\t#mm\n", "#Calculating the radius of addendum circle of wheel\n", "RA = R+AW \t\t\t#mm\n", "#Calculating the path of approach\n", "KP = math.sqrt(RA**2-R**2*(math.cos(math.radians(phi)))**2)-R*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the path of recess\n", "PL = math.sqrt(rA**2-r**2*(math.cos(math.radians(phi)))**2)-r*math.sin(math.radians(phi)) \t\t\t#mm\n", "#Calculating the length of path of contact\n", "KL = KP+PL \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#Length of arc of contact mm\n", "#Number of pairs of teeth in contact:\n", "#Calculating the circular pitch\n", "pc = math.pi*m \t\t\t#mm\n", "#Calculating the number of pairs of teeth in contact\n", "n = Lac/pc \t\t\t#Number of pairs of teeth in contact\n", "#Maximum velocity of sliding:\n", "#Calculating the angular speed of wheel\n", "omega2 = omega1*t/T \t\t\t#rad/s\n", "#Calculating the maximum velocity of sliding\n", "vs = (omega1+omega2)*KP \t\t\t#mm/s\n", "\n", "#Results:\n", "print \" Number of teeth on the pinion to avoid interference, t = %d.\"%(t+1)\n", "print \" Corresponding number of teeth on the wheel, T = %.F.\"%(T+1)\n", "print \" Length of path of contact, KL = %.2f mm.\"%(KL)\n", "print \" Length of arc of contact = %.2f mm.\"%(Lac)\n", "print \" Number of pairs of teeth in contact = %d.\"%(n+1)\n", "print \" Maximum velocity of sliding, vs = %.f mm/s.\"%(vs)\n", "\n", "# ROUNDING ERROR" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Number of teeth on the pinion to avoid interference, t = 19.\n", " Corresponding number of teeth on the wheel, T = 56.\n", " Length of path of contact, KL = 29.24 mm.\n", " Length of arc of contact = 31.12 mm.\n", " Number of pairs of teeth in contact = 2.\n", " Maximum velocity of sliding, vs = 197 mm/s.\n" ] } ], "prompt_number": 34 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.14 Page No : 416" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "T = 20.\n", "d = 125. #mm\n", "r = d/2\n", "OP = r\n", "LH = 6.25 \t\t\t#mm\n", "#Calculating the least pressure angle to avoid interference\n", "phi = math.sin(math.sqrt(LH/r))*180/math.pi \t\t\t#degrees\n", "#Length of arc of contact:\n", "#Calculating the length of path of contact\n", "KL = math.sqrt((OP+LH)**2-(OP*math.cos(math.radians(phi)))**2) \t\t\t#mm\n", "#Calculating the length of arc of contact\n", "Lac = KL/math.cos(math.radians(phi)) \t\t\t#Length of arc of contact mm\n", "#Minimum number of teeth:\n", "#Calculating the circular pitch\n", "pc = math.pi*d/T \t\t\t#mm\n", "#Calculating the number of pairs of teeth in contact\n", "n = Lac/pc \t\t\t#Number of pairs of teeth in contact\n", "#Calculating the minimum number of teeth in contact\n", "nmin = n \t\t\t#Mimimum number of teeth in contact\n", "\n", "#Results:\n", "print \" Least pressure angle to avoid interference, phi = %.3f degrees.\"%(phi)\n", "print \" Length of arc of contact = %.2f mm.\"%(Lac)\n", "print \" Minimum number of teeth in contact = %d or %d pair.\"%(nmin+1,(nmin+1)/2)\n", "\n", "# rounding error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Least pressure angle to avoid interference, phi = 17.818 degrees.\n", " Length of arc of contact = 36.17 mm.\n", " Minimum number of teeth in contact = 2 or 1 pair.\n" ] } ], "prompt_number": 41 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.15 Page No : 421" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linalg\n", "from scipy.optimize import fsolve \n", "import math \n", "\n", "# Variables:\n", "L = 175./1000\n", "d2 = 100./1000 #m\n", "r2 = d2/2 \t\t\t #m\n", "theta = 70. \t\t\t#degrees\n", "G = 1.5\n", "T2 = 80.\n", "Tf = 75. \t\t\t#Torque on faster wheel N-m\n", "\n", "#Solution:\n", "#Spiral angles for each wheel:\n", "#Calculating the number of teeth on slower wheel\n", "T1 = T2*G\n", "#Calculating the pitch circle diameter of the slower wheel\n", "d1 = (L*2)-d2 \t\t\t#m\n", "#Calculating the spiral angles\n", "#We have d2/d1 = (T2*math.cos(alpha1))/(T1*math.cos(alpha2)) or T2*d1*math.cos(alpha1)-T1*d2*math.cos(alpha2) = 0 .....(i)\n", "#Also alpha1+alpha2 = theta or alpha1+alpha2-theta = 0 .....(ii)\n", "def f(x):\n", " alpha1 = x[0]\n", " alpha2 = x[1]\n", " y = [0,0]\n", " y[0] = T2*d1*math.cos(alpha1)-T1*d2*math.cos(alpha2)\n", " y[1] = alpha1+alpha2-theta*math.pi/180\n", " return y\n", " \n", "z = fsolve(f,[1,1])\n", "alpha1 = z[0]*180/math.pi \t\t\t#Spiral angle for slower wheel degrees \n", "alpha2 = z[1]*180/math.pi \t\t\t#Spiral angle for faster wheel degrees\n", "#Axial thrust on each shaft:\n", "#Calculating the math.tangential force at faster wheel\n", "F2 = Tf/r2 \t\t\t#N\n", "#Calculating the normal reaction at the point of contact\n", "RN = F2/math.cos(math.radians(alpha2)) \t\t\t#N\n", "#Calculating the axial thrust on the shaft of slower wheel\n", "Fa1 = RN*math.sin(math.radians(alpha1)) \t\t\t#N\n", "#Calculating the axial thrust on the shaft of faster wheel\n", "Fa2 = RN*math.sin(math.radians(alpha2)) \t\t\t#N\n", "\n", "#Results:\n", "print \" Spiral angle for slower wheel, alpha1 = %.2f degrees.\"%(alpha1)\n", "print \" Spiral angle for faster wheel, alpha2 = %.2f degrees.\"%(alpha2)\n", "print \" Axial thrust on the shaft of slower wheel, Fa1 = %d N.\"%(Fa1+1)\n", "print \" Axial thrust on the shaft of faster wheel, Fa2 = %d N.\"%(Fa2+1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Spiral angle for slower wheel, alpha1 = 54.65 degrees.\n", " Spiral angle for faster wheel, alpha2 = 15.35 degrees.\n", " Axial thrust on the shaft of slower wheel, Fa1 = 1269 N.\n", " Axial thrust on the shaft of faster wheel, Fa2 = 412 N.\n" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.16 Page No : 422" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "L = 400./1000 \t\t\t#m\n", "G = 3.\n", "theta = 50.\n", "phi = 6. \t\t\t#degrees\n", "pN = 18. \t\t\t#mm\n", "\n", "#Solution:\n", "#Number of teeth on each wheel:\n", "#Calculating the spiral angles of the driving and driven wheels\n", "alpha1 = theta/2 \t\t\t#degrees\n", "alpha2 = alpha1 \t\t\t#degrees\n", "#Calculating the number of teeth on driver wheel\n", "T1 = L*1000*2*math.pi/(pN*(1/math.cos(math.radians(alpha1))+G/math.cos(math.radians(alpha2))))\n", "#Calculating the number of teeth on driven wheel\n", "T2 = G*T1\n", "#Calculating the exact centre distance\n", "#L1 = pN*T1/(2*math.pi)*(1/math.cos(math.radians(alpha1))+G/math.cos(math.radians(alpha2))) \t\t\t#mm\n", "L1 = pN*T1/(2*math.pi)*((1+G)/math.cos(math.radians(alpha1))) \t\t\t#mm\n", "#Calculating the efficiency of the drive\n", "eta = (math.cos(math.radians(alpha2+phi))*math.cos(math.radians(alpha1)))/(math.cos(math.radians(alpha1-phi))*math.cos(math.radians(alpha2)))*100 \t\t\t#%\n", "\n", "#Results:\n", "print \" Number of teeth on driver wheel, T1 = %d.\"%(T1+1)\n", "print \" Number of teeth on driven wheel, T2 = %.f.\"%(T2+1)\n", "print \" Exact centre distance, L1 = %.1f mm.\"%(L1)\n", "print \" Efficiency of the drive, eta = %.1f %%.\"%(eta)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Number of teeth on driver wheel, T1 = 32.\n", " Number of teeth on driven wheel, T2 = 96.\n", " Exact centre distance, L1 = 400.0 mm.\n", " Efficiency of the drive, eta = 90.7 %.\n" ] } ], "prompt_number": 51 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 12.17 Page No : 423" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linalg\n", "from scipy.optimize import fsolve \n", "import math \n", "\n", "# Variables:\n", "pN = 12.5\n", "L = 134. \t\t\t#mm\n", "theta = 80.\n", "phi = 6. \t\t\t#degrees\n", "G = 1.25\n", "\n", "#Solution:\n", "#Spiral angle of each wheel:\n", "#Calculating the spiral angles of wheels 1 and 2\n", "#We have d2/d1 = (T2*math.cos(alpha1))/(T1*math.cos(alpha2)) or math.cos(alpha1)-G*math.cos(alpha2) = 0 .....(i)\n", "#Also alpha1+alpha2 = theta or alpha1+alpha2-theta = 0 .....(ii)\n", "def f(x):\n", " alpha1 = x[0]\n", " alpha2 = x[1]\n", " y = [0,0]\n", " y[0] = math.cos(alpha1)-G*math.cos(alpha2)\n", " y[1] = alpha1+alpha2-theta*math.pi/180\n", " return y\n", "\n", "z = fsolve(f,[1,1])\n", "alpha1 = z[0]*180/math.pi \t\t\t#Spiral angle for slower wheel degrees\n", "alpha2 = z[1]*180/math.pi \t\t\t#Spiral angle for faster wheel degrees\n", "#Number of teeth on each wheel:\n", "#Calculating the diameters of the wheels\n", "d1 = L\n", "d2 = d1 \t\t\t#mm\n", "#Calculating the number of teeth on wheel 1\n", "T1 = d1*math.pi*math.cos(math.radians(alpha1))/pN\n", "#Calculating the number of teeth on wheel 2\n", "T2 = T1/G\n", "#Calculating the efficiency of the drive\n", "eta = (math.cos(math.radians(alpha2+phi))*math.cos(math.radians(alpha1)))/(math.cos(math.radians(alpha1-phi))*math.cos(math.radians(alpha2)))*100 \t\t\t#%\n", "#Calculating the maximum efficiency\n", "etamax = (math.cos(math.radians(theta+phi))+1)/(math.cos(math.radians(theta-phi))+1)*100 \t\t\t#%\n", "\n", "#Results:\n", "print \" Spiral angle for slower wheel, alpha1 = %.2f degrees.\"%(alpha1)\n", "print \" Spiral angle for faster wheel, alpha2 = %.2f degrees.\"%(alpha2)\n", "print \" Number of teeth on wheel 1, T1 = %.1f.\"%(T1)\n", "print \" Number of teeth on wheel 2, T2 = %.f.\"%(T2+1)\n", "print \" Efficiency of the drive, eta = %d %%.\"%(eta+1)\n", "print \" Maximum efficiency, etamax = %.1f %%.\"%(etamax)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Spiral angle for slower wheel, alpha1 = 32.46 degrees.\n", " Spiral angle for faster wheel, alpha2 = 47.54 degrees.\n", " Number of teeth on wheel 1, T1 = 28.4.\n", " Number of teeth on wheel 2, T2 = 24.\n", " Efficiency of the drive, eta = 83 %.\n", " Maximum efficiency, etamax = 83.9 %.\n" ] } ], "prompt_number": 57 } ], "metadata": {} } ] }