{ "metadata": { "name": "", "signature": "sha256:4504e5e6afcec1f6614278d6f9600661d0ec5a3c47dfce6a30afddafccb88875" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 10 : Friction" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.1 Page No : 263" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "from numpy import linalg\n", "\n", "# Variables:\n", "theta = 30. \t\t\t#degrees\n", "P1 = 180. \t\t\t#Pulling force N\n", "P2 = 220. \t\t\t#Pushing force N\n", "\n", "#Solution:\n", "#Resolving the forces horizontally for the pull of 180N\n", "F1 = P1*math.cos(math.radians(theta)) \t\t\t#N\n", "#Resolving the forces for the push of 220 N\n", "F2 = P2*math.cos(math.radians(theta)) \t\t\t#N\n", "#Calculating the coefficient of friction\n", "#For the pull of 180N F1 = mu*W-90*mu or F1/mu-W = -90 .....(i)\n", "#For the push of 220N F2 = W*mu+110*mu or F2/mu-W = 110 .....(ii)\n", "A = [[F1, -1],[ F2, -1]]\n", "B = [-90., 110.]\n", "V = linalg.solve(A, B)\n", "mu = 1/V[0]\n", "W = V[1]\n", "\n", "#Results:\n", "print \" The weight of the body, W = %d N.\"%(round(W,-2))\n", "print \" The coefficient of friction, mu = %.4f.\"%(mu)\n", "\n", "# note : rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " The weight of the body, W = 1000 N.\n", " The coefficient of friction, mu = 0.1732.\n" ] } ], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.2 Page No : 268" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "from numpy import linalg\n", "\n", "# Variables:\n", "P1 = 1500.\n", "P2 = 1720. \t\t\t#N\n", "alpha1 = 12.\n", "alpha2 = 15. \t\t\t#degrees\n", "\n", "#Solution:\n", "#Refer Fig. 10.10\n", "#Effort applied parallel to the plane P1 = W*(math.sin(alpha1)+mu*math.cos(alpha1)) or P1/W-mu*math.cos(alpha1) = math.sin(alpha1) .....(i)\n", "#Effort applied parallel to the plane P2 = W*(math.sin(alpha2)+mu*math.cos(alpha2)) or P2/W-mu*math.cos(alpha2) = math.sin(alpha2) .....(ii)\n", "A = [[P1, -math.cos(math.radians(alpha1))],[ P2, -math.cos(math.radians(alpha2))]]\n", "B = [math.sin(math.radians(alpha1)), math.sin(math.radians(alpha2))]\n", "V = linalg.solve(A, B)\n", "W = 1.0/V[0]\n", "mu = V[1]\n", "\n", "#Results:\n", "print \" Coefficient of friction, mu = %.3f.\"%(mu)\n", "print \" Weight of the body, W = %d N.\"%(W)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Coefficient of friction, mu = 0.131.\n", " Weight of the body, W = 4462 N.\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.3 Page No : 272" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "W = 75.*1000 \t\t\t#W\n", "v = 300. \t\t\t#mm/min\n", "p = 6.\n", "d0 = 40. \t \t\t#mm\n", "mu = 0.1\n", "\n", "#Solution:\n", "#Calculating the mean diameter of the screw\n", "d = (d0-p/2)/1000 \t\t\t#m\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d*1000)) \t\t\t#radians\n", "#Calculating the force required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the torque required to overcome the friction\n", "T = P*d/2 \t\t\t#N-m\n", "#Calculating the speed of the screw\n", "N = v/p \t\t\t#rpm\n", "#Calculating the angular speed\n", "omega = round(2*math.pi*N/60,2) \t\t\t#rad/s\n", "#Calculating the power of the motor\n", "Power = T*omega/1000 \t\t\t#Power of the motor kW\n", "\n", "#Results:\n", "print \" Power of the motor required = %.3f kW.\"%(Power)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power of the motor required = 1.114 kW.\n" ] } ], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.4 Page No : 273" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "p = 12.\n", "d = 40. \t\t\t#mm\n", "mu = 0.16\n", "W = 2500. \t\t\t#N\n", "\n", "#Solutiom:\n", "#Work done in drawing the wagons together agianst a steady load of 2500 N:\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d)) \t\t\t#radians\n", "#Calculating the effort required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the torque required to overcome friction between the screw and nut\n", "T = P*d/(2*1000) \t\t\t#N-m\n", "#Calculating the number of turns required\n", "N = 240/(2*p)\n", "#Calculating the work done\n", "W1 = T*2*math.pi*N \t\t\t#Work done N-m\n", "#Work done in drawing the wagons together when the load increases from 2500 N to 6000 N:\n", "W2 = W1*(6000.-2500)/2500.0 \t\t\t#Work done N-m\n", "\n", "#Results:\n", "print \" Work done in drawing the wagons together agianst a steady load of 2500 N = %.1f N-m.\"%(W1)\n", "print \" Work done in drawing the wagons together when the load increases from 2500 N to 6000 N = %.1f N-m.\"%(W2)\n", "\n", "# note : answer in book is wrong. " ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Work done in drawing the wagons together agianst a steady load of 2500 N = 826.2 N-m.\n", " Work done in drawing the wagons together when the load increases from 2500 N to 6000 N = 1156.7 N-m.\n" ] } ], "prompt_number": 7 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.5 Page No : 274" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "D = 150./1000 \t\t\t#m\n", "ps = 2.*10**6 \t\t\t#N/m**2\n", "d0 = 50.\n", "p = 6. \t\t\t#mm\n", "mu = 0.12\n", "\n", "#Solution:\n", "#Calculating the load on the valve\n", "W = ps*math.pi/4*D**2 \t\t\t#N\n", "#Calculating the mean diameter of the screw\n", "d = (d0-p/2)/1000 \t\t\t#m\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d*1000))\n", "#Calculating the force required to turn the handle\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the torque required to turn the handle\n", "T = P*d/2 \t\t\t#N-m\n", "\n", "#Results:\n", "print \" The torque required to turn the handle, T = %.1f N-m.\"%(T)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " The torque required to turn the handle, T = 135.1 N-m.\n" ] } ], "prompt_number": 3 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.6 Page No : 274" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "dc = 22.5 #mm\n", "p = 5. #mm\n", "D = 50. #mm\n", "R = D/2 #mm\n", "l = 500. \t #mm\n", "mu = 0.1\n", "mu1 = 0.16\n", "W = 10.*1000 \t\t\t#N\n", "\n", "#Solution:\n", "#Calculating the mean diameter of the screw\n", "d = dc+p/2 \t\t\t#mm\n", "#Calculating the helix angle\n", "alpha = p/(math.pi*d) \t\t\t#radians\n", "#Calculating the force required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the total torque required\n", "T = P*d/2+mu1*W*R \t\t\t#N-mm\n", "#Calculating the force required at the end of a spanner\n", "P1 = T/l \t\t\t#N\n", "\n", "#Results:\n", "print \" Force required at the end of a spanner, P1 = %.2f N.\"%(P1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Force required at the end of a spanner, P1 = 121.37 N.\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.7 Page No : 275" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 50. #mm\n", "p = 12.5 #mm\n", "D = 60. #mm\n", "R = D/2 \t\t\t#mm\n", "W = 10.*1000 #N\n", "P1 = 100. \t\t\t#N\n", "mu = 0.15\n", "mu1 = 0.18\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d)) \t\t\t#radians\n", "#Calculating the math.tangential force required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the total torque required to turn the hand wheel\n", "T = P*d/2+mu1*W*R \t\t\t#N-mm\n", "#Calculating the diameter of the hand wheel\n", "D1 = T/(2*P1*1000)*2 \t\t\t#m\n", "\n", "#Results:\n", "print \" Diameter of the hand wheel, D1 = %.3f m.\"%(D1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Diameter of the hand wheel, D1 = 1.128 m.\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.8 Page No : 276" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d0 = 55. #mm\n", "D2 = 60. #mm\n", "R2 = D2/2 #mm\n", "D1 = 90. #mm\n", "R1 = D1/2 \t\t #mm\n", "p = 10./1000 \t #m\n", "W = 400. \t\t #N\n", "mu = 0.15\n", "v = 6. \t\t\t#Cutting speed m/min\n", "\n", "#Solution:\n", "#Calculating the mean diameter of the screw\n", "d = d0-p/2 \t\t\t#mm\n", "#Calculating the helix angle\n", "alpha = p/(math.pi*d) \t\t\t#radians\n", "#Calculating the force required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the mean radius of the flat surface\n", "R = (R1+R2)/2 \t\t\t#mm\n", "#Calculating the torque required\n", "T = (P*d/2+mu1*W*R)/1000 \t\t\t#N-m\n", "#Calculating the speed of the screw\n", "N = v/p \t\t\t#rpm\n", "#Calculating the angular speed\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the power required to operate the nut\n", "Power = T*omega/1000 \t\t\t#Power required to operate the nut kW\n", "\n", "#Results:\n", "print \" Power required to operate the nut = %.3f kW.\"%(Power)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power required to operate the nut = 0.275 kW.\n" ] } ], "prompt_number": 11 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.9 Page No : 277" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 50./1000\n", "l = 0.7 \t\t\t#m\n", "p = 10. \t\t\t#mm\n", "mu = 0.15\n", "W = 20.*1000 \t\t\t#N\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d*1000)) \t\t\t#radians\n", "#Force required to raise the load:\n", "#Calculating the force required at the circumference of the screw\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P1 = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the force required at the end of the lever\n", "P11 = P1*d/(2*l) \t\t\t#N\n", "#Calculating the force required at the circumference of the screw\n", "P2 = W*(phi-alpha) \t\t\t#N\n", "#Foce rewuired to lower the load:\n", "#Calculating the force required at the end of the lever\n", "P21 = P2*d/(2*l) \t\t\t#N\n", "\n", "#Results:\n", "print \" Force required at the end of the lever to raise the load, P1 = %d N.\"%(P11)\n", "print \" Force required at the end of the lever to lower the load, P1 = %d N.\"%(P21)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Force required at the end of the lever to raise the load, P1 = 155 N.\n", " Force required at the end of the lever to lower the load, P1 = 62 N.\n" ] } ], "prompt_number": 12 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.10 Page No : 279" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 50.\n", "p = 12.5 \t\t\t#mm\n", "mu = 0.13\n", "W = 25.*1000 \t\t\t#N\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = round(math.tan(p/(math.pi*d)),2) \t\t\t#radians\n", "#Calculating the force required on the screw to raise the load\n", "phi = round(math.tan(mu),2) \t\t\t#Limiting angle of friction radians\n", "P1 = W*(alpha+phi)/(1-(alpha*phi)) \t\t\t#N\n", "#Calculating the torque required on the screw to raise the load\n", "T1 = P1*d/2 \t\t\t#N-mm\n", "#Calculating the force required on the screw to lower the load\n", "P2 = W*math.tan(phi-alpha) \t\t\t#N\n", "#Calculating the torque required to lower the load\n", "T2 = P2*d/2 \t\t\t#N\n", "#Calculating the ratio of the torques required\n", "r = T1/T2 \t\t\t#Ratio of the torques required N-mm\n", "#Calculating the efficiency of the machine\n", "eta = math.tan(alpha)/math.tan(alpha+phi)*100 \t\t\t#%\n", "\n", "#Results:\n", "print \" Torque required on the screw to raise the load, T1 = %d N-mm.\"%(T1)\n", "print \" Ratio of the torque required to raise the load to the torque required to lower the load = %.1f.\"%(r)\n", "print \" Efficiency of the machine, eta = %.1f %%.\"%(eta)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Torque required on the screw to raise the load, T1 = 132629 N-mm.\n", " Ratio of the torque required to raise the load to the torque required to lower the load = 4.2.\n", " Efficiency of the machine, eta = 37.6 %.\n" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.11 Page No : 280" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "p = 10. #mm\n", "d = 50. #mm\n", "D2 = 60. #mm\n", "R2 = D2/2 #mm\n", "D1 = 10. #mm\n", "R1 = D1/2 \t\t#mm\n", "W = 20.*1000 \t#N\n", "mu = 0.08\n", "mu1 = mu\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = round(math.tan(p/(math.pi*d)),4) \t\t\t#radians\n", "#Calculating the force required at the circumference of the screw to lift the load\n", "phi = round(math.tan(mu),2) \t\t\t#Limiting angle of friction radians\n", "P = round(W*(alpha+phi)/(1 - (alpha*phi)),-1) #math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the torque required to overcome friction at the screw\n", "T = P*d/(2*1000) \t\t\t#N-m\n", "#Calculating the number of rotations made by the screw\n", "N = 170/p\n", "#When the load rotates with the screw:\n", "#Calculating the work done in lifting the load\n", "W1 = T*2*math.pi*N \t\t\t#Work done in lifting the load N-m\n", "#Calculating the efficiency of the screw jack\n", "eta1 = math.tan(alpha)/math.tan(alpha+phi)*100 \t\t\t#%\n", "#When the load does not rotate with the screw:\n", "#Calculating the mean radius of the bearing surface\n", "R = (R1+R2)/2 \t\t\t#mm\n", "#Calculating the torque required to overcome friction at the screw and the collar\n", "T = (P*d/2+mu1*W*R)/1000 \t\t\t#N-m\n", "#Calculating the work done by the torque in lifting the load\n", "W2 = T*2*math.pi*N \t\t\t#Work done by the torque in lifting the load N-m\n", "#Calculating the torque required to lift the load neglecting frition\n", "T0 = (W*math.tan(alpha)*d/2)/1000 \t\t\t#N-m\n", "#Calculating the efficiency of the screw jack\n", "eta2 = T0/T*100 \t\t\t#%\n", "\n", "#Results:\n", "print \" When the load rotates with the screw, work done in lifting the load = %.f N-m.\"%(W1)\n", "print \" Efficiency of the screw jack, eta = %.1f %%.\"%(eta1)\n", "print \" When the load does not rotate with the screw, work done in lifting the load = %d N-m.\"%(round(W2,-1))\n", "print \" Efficiency of the screw jack, eta = %.1f %%.\"%(eta2)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " When the load rotates with the screw, work done in lifting the load = 7717 N-m.\n", " Efficiency of the screw jack, eta = 44.1 %.\n", " When the load does not rotate with the screw, work done in lifting the load = 10710 N-m.\n", " Efficiency of the screw jack, eta = 31.8 %.\n" ] } ], "prompt_number": 24 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.12 Page No : 282" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "# Variables:\n", "W = 10.*1000 #N\n", "P1 = 100. \t\t\t#N\n", "p = 12. #mm\n", "d = 50. \t\t\t#mm\n", "mu = 0.15\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d)) \t\t\t#radians\n", "#Calculating the effort required at the circumference of the screw to raise the load\n", "phi = math.tan(mu) \t\t\t#Limiting angle of friction radians\n", "P = W*math.tan(alpha+phi) \t\t\t#N\n", "#Calculating the torque required to overcome friction\n", "T = P*d/2 \t\t\t#N-mm\n", "#Calculating the length of the lever\n", "l = T/P1 \t\t\t#mm\n", "#Calculating the mechanical advantage\n", "MA = W/P1\n", "#Calculating the efficiency of the screw jack\n", "eta = math.tan(alpha)/math.tan(alpha+phi)*100 \t\t\t#%\n", "\n", "#Results:\n", "print \" The length of the lever to be used, l = %.1f mm.\"%(l)\n", "print \" Mechanical advantage obtained, M.A. = %d.\"%(MA)\n", "if eta<50:\n", " print \" The screw is a self locking screw.\";\n", "else:\n", " print \" The screw is not a self locking screw.\";\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " The length of the lever to be used, l = 579.2 mm.\n", " Mechanical advantage obtained, M.A. = 100.\n", " The screw is a self locking screw.\n" ] } ], "prompt_number": 17 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.13 Page No : 284" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 22. #mm\n", "p = 3. \t\t\t#mm\n", "beta = 60./2 \t\t\t#degrees\n", "W = 40.*1000 \t\t\t#N\n", "mu = 0.15\n", "\n", "#Solution:\n", "#Calculating the helix angle\n", "alpha = round(math.tan(p/(math.pi*d)),4) \t\t\t#radians\n", "#Calculating the virtual coefficient of friction\n", "mu1 = round(mu/math.cos(math.radians(beta)),3)\n", "#Calculating the force required at the circumference of the screw\n", "P = W*((alpha+mu1)/(1 - alpha * mu1))\n", "#Calculating the torque on one rod\n", "T = P*d/(2.*1000) \t\t\t#N-m\n", "#Calculating the torque required on the nut\n", "T1 = 2*T \t\t\t#N-m\n", "\n", "\n", "#Results:\n", "print \" The torque required on the nut, T1 = %.2f N-m.\"%(T1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " The torque required on the nut, T1 = 191.87 N-m.\n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.14 Page No : 284 " ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 25. #mm\n", "p = 5. #mm\n", "R = 25. \t\t\t#mm\n", "beta = 27.5 \t\t#degrees\n", "mu = 0.1\n", "mu2 = 0.16\n", "l = 0.5 \t\t\t#m\n", "W = 10.*1000 \t\t#N\n", "\n", "#Solution:\n", "#Calculating the virtual coefficient of friction\n", "mu1 = mu/math.cos(math.radians(beta))\n", "#Calculating the helix angle\n", "alpha = math.tan(p/(math.pi*d)) \t\t\t#radians\n", "#Calculating the force on the screw\n", "phi1 = math.tan(mu1) \t\t\t#Virtual limiting angle of frcition radians\n", "P = W*math.tan(alpha+phi1) \t\t\t#N\n", "#Calculating the total torque transmitted\n", "T = (P*d/2+mu2*W*R)/1000 \t\t\t#N-m\n", "#Calculating the force required at the end of a spanner\n", "P1 = T/l \t\t\t#N\n", "\n", "#Results:\n", "print \" Force required at the end of a spanner, P1 = %.1f N.\"%(P1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Force required at the end of a spanner, P1 = 124.7 N.\n" ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.15 Page No : 286" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d = 60.\n", "r = d/2. \t\t\t#mm\n", "W = 2000. \t\t\t#N\n", "mu = 0.03\n", "N = 1440. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted\n", "T = mu*W*(r/1000) \t\t\t#N-m\n", "#Calculating the power transmitted\n", "P = T*omega \t\t\t#W\n", "\n", "#Results:\n", "print \" The power transmitted, P = %.1f W.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " The power transmitted, P = 271.4 W.\n" ] } ], "prompt_number": 20 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.16 Page No : 288" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "D = 150./1000 #m\n", "R = D/2 \t\t\t#m\n", "N = 100. \t\t\t#rpm\n", "W = 20.*1000 \t\t\t#N\n", "mu = 0.05\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the total frictional torque for uniform pressure distribution\n", "T = 2./3*mu*W*R \t\t\t#N-m\n", "#Calculating the power lost in friction\n", "P = T*omega \t\t\t#W\n", "\n", "#Results:\n", "print \" Power lost in friction, P = %.1f W.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power lost in friction, P = 523.6 W.\n" ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.17 Page No : 292" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "W = 20.*1000 \t\t\t#N\n", "alpha = 120./2 \t\t\t#degrees\n", "Pn = 0.3 \t\t\t#N/mm**2\n", "N = 200. \t \t\t#rpm\n", "mu = 0.1\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the inner radius of the bearing surface\n", "r2 = math.sqrt(W/(3*math.pi*Pn)) \t\t\t#mm\n", "#Calculating the outer radius of the bearing surface\n", "r1 = 2*r2 \t\t\t#mm\n", "#Calculating the total frictional torque assuming uniform pressure\n", "T = 2./3*mu*W*(1/math.sin(math.radians(alpha)))*(r1**3-r2**3)/(r1**2-r2**2)/1000.0 \t\t\t#N-m\n", "#Calculating the power absorbed in friction\n", "P = T*omega/1000.0 \t\t\t#kW\n", "\n", "#Results:\n", "print \"External Diameters r1= %.f mm\"%r1\n", "print \"Internal Diameters r2= %.f mm\"%r2\n", "print \" Power absorbed in friction, P = %.3f kW.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "External Diameters r1= 168 mm\n", "Internal Diameters r2= 84 mm\n", " Power absorbed in friction, P = 6.328 kW.\n" ] } ], "prompt_number": 27 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.18 Page No : 292" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "D = 200./1000\n", "R = D/2 \t\t\t#m\n", "W = 30.*1000 \t\t\t#N\n", "alpha = 120./2 \t\t\t#degrees\n", "mu = 0.025\n", "N = 140. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Power lost in friction assuming uniform pressure:\n", "#Calculating the total frictional torque\n", "T = 2./3*mu*W*R*(1/math.sin(math.radians(alpha))) \t\t\t#N-m\n", "#Calculating the power lost in friction\n", "P1 = T*omega \t\t\t#Power lost in friction W\n", "#Power lost in friction assuming uniform wear:\n", "#Calculating the total frictional torque\n", "T = 1./2*mu*W*R*(1./math.sin(math.radians(alpha))) \t\t\t#N-m\n", "#Calculating the power lost in friction\n", "P2 = T*omega \t\t\t#Power lost in friction W\n", "\n", "#Resluts:\n", "print \" Power lost in friction assuming uniform pressure, P = %d W.\"%(P1)\n", "print \" Power lost in friction assuming uniform wear, P = %.1f W.\"%(P2)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power lost in friction assuming uniform pressure, P = 846 W.\n", " Power lost in friction assuming uniform wear, P = 634.8 W.\n" ] } ], "prompt_number": 19 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.19 Page No : 295" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n = 6.\n", "d1 = 600. #mm\n", "r1 = d1/2 #mm\n", "d2 = 300. #mm\n", "r2 = d2/2 \t\t #mm\n", "W = 100.*1000 \t #N\n", "mu = 0.12\n", "N = 90. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the engine\n", "omega = 2*math.pi*N/60.0 \t\t\t#rad/s\n", "#Power absorbed in friction assuming uniform pressure:\n", "#Calculating the total frictional torque transmitted\n", "T = 2./3*mu*W*(r1**3-r2**3)/(r1**2-r2**2)/1000.0 \t\t\t#N-m\n", "#Calculating the power absorbed in friction\n", "P1 = T*omega/1000.0 \t\t\t#Power absorbed in friction assuming uniform pressure kW\n", "#Power absorbed in friction assuming uniform wear:\n", "#Calculating the total frictional torque transmitted\n", "T = 1./2*mu*W*(r1+r2)/1000.0 \t\t\t#N-m\n", "#Calculating the power absorbed in friction\n", "P2 = T*omega/1000.0 \t\t\t#Power absorbed in friction assuming uniform wear kW\n", "\n", "#Results:\n", "print \" Power absorbed in friction assuming uniform pressure, P = %.1f kW.\"%(P1)\n", "print \" Power absorbed in friction assuming uniform wear, P = %.2f kW.\"%(P2)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power absorbed in friction assuming uniform pressure, P = 26.4 kW.\n", " Power absorbed in friction assuming uniform wear, P = 25.45 kW.\n" ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.20 Page No : 296" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d1 = 400. #mm\n", "r1 = d1/2 #mm\n", "d2 = 250. #mm\n", "r2 = d2/2 \t\t\t#mm\n", "p = 0.35 \t\t\t#N/mm**2\n", "mu = 0.05\n", "N = 105. \t\t\t#rpm\n", "W = 150.*1000 \t\t\t#N\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the total frictional torque transmitted for uniform pressure\n", "T = 2./3*mu*W*(r1**3-r2**3)/(r1**2-r2**2)/1000 \t\t\t#N-m\n", "#Calculating the power absorbed\n", "P = T*omega/1000 \t\t\t#kW\n", "#Calculating the number of collars required\n", "n = W/(p*math.pi*(r1**2-r2**2))\n", "\n", "#Results:\n", "print \" Power absorbed, P = %.2f kW.\"%(P)\n", "print \" Number of collars required, n = %d.\"%(n+1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power absorbed, P = 13.64 kW.\n", " Number of collars required, n = 6.\n" ] } ], "prompt_number": 27 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.21 Page No : 296" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d2 = 300./1000 #mm\n", "r2 = d2/2 \t\t\t#m\n", "W = 200.*1000 \t\t#N\n", "N = 75. \t\t\t#rpm\n", "mu = 0.05\n", "p = 0.3 \t\t\t#N/mm**2\n", "P = 16.*1000 \t\t#W\n", "\n", "#Solution:\n", "#Calculating the angular velocity of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the total frictional torque transmitted\n", "T = P/omega \t\t\t#N-m\n", "#Calculating the external diameter of the collar\n", "#We have T = 2/3*mu*W*(r1**3-r2**3)/(r1**2-r2**2) or (2*mu*W)*r1**2-(3*T-2*mu*W*r2)*r1+(2*mu*W*r2**2-3*T*r2) = 0\n", "A = 2*mu*W\n", "B = -(3*T-2*mu*W*r2)\n", "C = 2*mu*W*r2**2-3*T*r2\n", "r1 = (-B+math.sqrt(B**2-4*A*C))/(2*A)*1000 \t\t\t#mm\n", "d1 = 2*r1 \t\t\t#mm\n", "#Calculating the number of collars\n", "n = W/(p*math.pi*(r1**2-(r2*1000)**2))\n", "\n", "#Results:\n", "print \" External diameter of the collar, d1 = %d mm.\"%(d1)\n", "print \" Number of collars, n = %d.\"%(n+1)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " External diameter of the collar, d1 = 498 mm.\n", " Number of collars, n = 6.\n" ] } ], "prompt_number": 28 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.22 Page No : 302" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "W = 4.*1000 \t\t#N\n", "r2 = 50.\n", "r1 = 100. \t\t\t#mm\n", "\n", "#Solution:\n", "#Calculating the maximum pressure\n", "pmax = W/(2*math.pi*r2*(r1-r2)) \t\t\t#N/mm**2\n", "#Calculating the minimum pressure\n", "pmin = W/(2*math.pi*r1*(r1-r2)) \t\t\t#N/mm**2\n", "#Calculating the average pressure\n", "pav = W/(math.pi*(r1**2-r2**2)) \t\t\t#N/mm**2\n", "\n", "#Results:\n", "print \" Maximum pressure, pmax = %.4f N/mm**2.\"%(pmax)\n", "print \" Minimum pressure, pmin = %.4f N/mm**2.\"%(pmin)\n", "print \" Average pressure, pav = %.2f N/mm**2.\"%(pav)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Maximum pressure, pmax = 0.2546 N/mm**2.\n", " Minimum pressure, pmin = 0.1273 N/mm**2.\n", " Average pressure, pav = 0.17 N/mm**2.\n" ] } ], "prompt_number": 29 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.23 Page No : 303" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "d1 = 300. #mm\n", "r1 = d1/2 #mm\n", "d2 = 200. #mm\n", "r2 = d2/2 \t\t\t#mm\n", "p = 0.1 \t\t\t#N/mm**2\n", "mu = 0.3\n", "N = 2500. \t\t\t#rpm\n", "n = 2.\n", "\n", "#Solution:\n", "#Calculating the radial speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the intensity of pressure\n", "C = p*r2 \t\t\t#N/mm\n", "#Calculating the axial thrust\n", "W = 2*math.pi*C*(r1-r2) \t\t\t#N\n", "#Calculating the mean radius of the friction surfaces for uniform wear\n", "R = (r1+r2)/(2*1000) \t\t\t#m\n", "#Calculating the torque transmitted\n", "T = n*mu*W*R \t\t\t#N-m\n", "#Calculating the power transmitted by a clutch\n", "P = T*omega/1000 \t\t\t#kW\n", "\n", "#Results:\n", "print \" Power transmitted by a clutch, P = %.3f kW.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power transmitted by a clutch, P = 61.685 kW.\n" ] } ], "prompt_number": 30 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.24 Page No : 303" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n = 2.\n", "mu = 0.255\n", "P = 25.*1000 \t\t#W\n", "N = 3000. \t\t\t#rpm\n", "r = 1.25 \t\t\t#Ratio of radii r1/r2\n", "p = 0.1 \t\t\t#N/mm**2\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted\n", "T = P/omega*1000 \t\t\t#N-mm\n", "#Calculating the inner radius\n", "r2 = (T/(n*mu*2*math.pi*0.1*(1.25-1)*(1.25+1)/2))**(1./3) \t\t\t#mm\n", "#Calculating the outer radius\n", "r1 = r*r2 \t\t\t#mm\n", "#Calculating the axial thrust to be provided by springs\n", "C = 0.1*r2 \t\t\t#Intensity of pressure N/mm\n", "W = 2*math.pi*C*(r1-r2) \t\t\t#N\n", "\n", "#Results:\n", "print \" Outer radius of the frictional surface, r1 = %.f mm.\"%(r1)\n", "print \" Inner radius of the frictional surface, r2 = %.f mm.\"%(r2)\n", "print \" Axial thrust to be provided by springs, W = %.f N.\"%(W)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Outer radius of the frictional surface, r1 = 120 mm.\n", " Inner radius of the frictional surface, r2 = 96 mm.\n", " Axial thrust to be provided by springs, W = 1446 N.\n" ] } ], "prompt_number": 26 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.25 Page No : 304" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linalg\n", "import math \n", "\n", "# Variables:\n", "P = 7.5*1000 \t\t#W\n", "N = 900. \t\t\t#rpm\n", "p = 0.07 \t\t\t#N/mm**2\n", "mu = 0.25\n", "n = 2.\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted\n", "T = P/omega*1000 \t\t\t#N-mm\n", "#Calculating the mean radius of the friction lining\n", "R = (T/(math.pi/2*n*mu*p))**(1./3) \t\t\t#mm\n", "#Calculating the face width of the friction lining\n", "w = R/4 \t\t\t#mm\n", "#Calculating the outer and inner radii of the clutch plate\n", "#We have w = r1-r2 or r1-r2 = w .....(i) \n", "#Also R = (r1+r2)/2 or r1+r2 = 2*R .....(ii) \n", "A = [[1, -1],[ 1, 1]]\n", "B = [w,2*R]\n", "V = linalg.solve(A,B)\n", "r1 = V[0]\n", "r2 = V[1]\n", "\n", "#Results:\n", "print \" Mean radius of the friction lining, R = %d mm.\"%(R)\n", "print \" Face width of the friction lining, w = %.2f mm.\"%(w)\n", "print \" Outer radius of the clutch plate, r1 = %.3f mm.\"%(r1)\n", "print \" Inner radius of the clutch plate, r2 = %.3f mm.\"%(r2)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Mean radius of the friction lining, R = 113 mm.\n", " Face width of the friction lining, w = 28.28 mm.\n", " Outer radius of the clutch plate, r1 = 127.258 mm.\n", " Inner radius of the clutch plate, r2 = 98.979 mm.\n" ] } ], "prompt_number": 28 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.26 Page No : 305" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "P = 100. \t\t\t#kW\n", "N = 2400. \t\t\t#rpm\n", "T = 500.*1000 \t\t#N-mm\n", "p = 0.07 \t\t\t#N/mm**2\n", "mu = 0.3\n", "Ns = 8. \t\t\t#Number of springs\n", "k = 40. \t\t\t#Stiffness N/mm\n", "n = 2.\n", "\n", "#Solution:\n", "#Calculating the inner radius of the friction plate\n", "r2 = round((T/(n*mu*2*math.pi*p*(1.25-1)*(1.25+1)/2))**(1./3),-1) \t\t\t#mm\n", "#Calculating the outer radius of the friction plate\n", "r1 = 1.25*r2 \t\t\t#mm\n", "#Calculating the total stiffness of the springs\n", "s = k*Ns \t\t\t#N/mm\n", "#Calculating the intensity of pressure\n", "C = p*r2 \t\t\t#N/mm\n", "#Calculating the axial force required to engage the clutch\n", "W = 2*math.pi*C*(r1-r2) \t\t\t#N\n", "#Calculating the initial compression in the springs\n", "IC = W/s \t\t\t#Initial compression in the springs mm\n", "\n", "#Results:\n", "print \" Outer radius of the friction plate, r1 = %.1f mm.\"%(r1)\n", "print \" Inner radius of the friction plate, r2 = %.f mm.\"%(r2)\n", "print \" Initial compression in the springs = %.1f mm.\"%(IC)\n", "\n", "# book answer is wrong." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Outer radius of the friction plate, r1 = 237.5 mm.\n", " Inner radius of the friction plate, r2 = 190 mm.\n", " Initial compression in the springs = 12.4 mm.\n" ] } ], "prompt_number": 33 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.27 Page No : 306" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "from numpy import linalg\n", "\n", "# Variables:\n", "d1 = 220. #mm\n", "r1 = d1/2 #mm\n", "d2 = 160. #mm\n", "r2 = d2/2 \t\t\t#mm\n", "W = 570. \t\t\t#N\n", "m1 = 800. #kg\n", "m2 = 1300. \t\t\t#kg\n", "k1 = 200./1000 #m\n", "k2 = 180./1000 \t\t#m\n", "mu = 0.35\n", "N1 = 1250. \t\t\t #rpm\n", "n = 2.\n", "\n", "#Solution:\n", "#Calculating the initial angular speed of the motor shaft\n", "omega1 = 2*math.pi*N1/60 \t\t\t#rad/s\n", "#Calculating the moment of inertia for the motor armature and shaft\n", "I1 = m1*k1**2 \t\t\t#kg-m**2\n", "#Calculating the moment of inertia for the rotor\n", "I2 = m2*k2**2 \t\t\t#kg-m**2\n", "#Calculating the final speed of the motor and rotor\n", "omega2 = 0\n", "omega3 = (I1*omega1+I2*omega2)/(I1+I2) \t\t\t#rad/s\n", "#Calculating the mean radius of the friction plate\n", "R = (r1+r2)/(2*1000) \t\t\t#m\n", "#Calculating the frictional torque\n", "T = n*mu*W*R \t\t\t#N-m\n", "#Calculating the angular acceleration of the rotor\n", "alpha2 = T/I2 \t\t\t#rad/s**2\n", "#Calculating the time to reach the speed of omega3\n", "omegaF = omega3\n", "omegaI = omega2\n", "t = (omegaF-omegaI)/alpha2 \t\t\t#seconds\n", "#Calculating the angular kinetic energy before impact\n", "E1 = 1./2*I1*omega1**2+1./2*I2*omega2**2 \t\t\t#N-m\n", "#Calculating the angular kinetic energy after impact\n", "E2 = 1./2*(I1+I2)*omega3**2 \t\t\t#N-m\n", "#Calculating the kinetic energy lost during the period of slipping\n", "E = round(E1-E2,-3) \t\t\t#N-m\n", "#Calculating the torque on armature shaft\n", "T1 = -60-T \t\t\t#N-m\n", "#Calculating the torque on rotor shaft\n", "T2 = T \t\t\t#N-m\n", "#Calculating the time of slipping assuming constant resisting torque:\n", "#Considering armature shaft omega3 = omega1+alpha1*t1 or omega3-(T1/I1)*t1 = omega1 .....(i)\n", "#Considering rotor shaft omega3 = alpha2*t1 or omega3-(T2/I2)*t1 = 0 .....(ii)\n", "A = [[1, -T1/I1],[ 1, -T2/I2]]\n", "B = [omega1, 0]\n", "V = linalg.solve(A,B)\n", "t11 = V[1] \t\t\t#Time of slipping assuming constant resisting torque seconds\n", "#Calculating the time of slipping assuming constant driving torque:\n", "#Calculating the torque on armature shaft\n", "T1 = 60-T \t\t\t#N-m\n", "t12 = (omega2-omega1)/(T1/I1-T2/I2) \t\t\t#Time of slipping assuming constant driving torque seconds\n", "\n", "#Results:\n", "print \" Final speed of the motor and rotor, omega3 = %.2f rad/s.\"%(omega3)\n", "print \" Time to reach the speed of %.2f rad/s, t = %.1f s.\"%(omega3,t)\n", "print \" Kinetic energy lost during the period of slipping = %d N-m.\"%(E)\n", "print \" Time of slipping assuming constant resisting torque, t1 = %.1f s.\"%(t11)\n", "print \" Time of slipping assuming constant driving torque, t1 = %d s.\"%(t12)\n", "\n", "# rounding off error." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Final speed of the motor and rotor, omega3 = 56.51 rad/s.\n", " Time to reach the speed of 56.51 rad/s, t = 62.8 s.\n", " Kinetic energy lost during the period of slipping = 156000 N-m.\n", " Time of slipping assuming constant resisting torque, t1 = 33.1 s.\n", " Time of slipping assuming constant driving torque, t1 = 624 s.\n" ] } ], "prompt_number": 37 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.28 Page No : 308" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n = 4.\n", "mu = 0.3\n", "p = 0.127 \t\t\t#N/mm**2\n", "N = 500. \t\t\t#rpm\n", "r1 = 125. #mm\n", "r2 = 75. \t\t\t#mm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the maximum intensity of pressure\n", "C = p*r2 \t\t\t#N/mm\n", "#Calculating the axial force required to engage the clutch\n", "W = 2*math.pi*C*(r1-r2) \t\t\t#N\n", "#Calculating the mean radius of the friction surfaces\n", "R = (r1+r2)/(2*1000) \t\t\t#m\n", "#Calculating the torque transmitted\n", "T = n*mu*W*R \t\t\t#N-m\n", "#Calculating the power transmitted\n", "P = T*omega/1000 \t\t\t#kW\n", "\n", "#Results:\n", "print \" Power transmitted, P = %.1f kW.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power transmitted, P = 18.8 kW.\n" ] } ], "prompt_number": 35 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.29 Page No : 308" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n1 = 3.\n", "n2 = 2.\n", "mu = 0.3\n", "d1 = 240. #mm\n", "r1 = d1/2 #mm\n", "d2 = 120. #mm\n", "r2 = d2/2 \t\t\t#mm\n", "P = 25.*1000 \t\t#W\n", "N = 1575. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\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 number of pairs of friction surfaces\n", "n = n1+n2-1\n", "#Calculating the mean radius of friction surfaces for uniform wear\n", "R = (r1+r2)/(2*1000) \t\t\t#m\n", "#Calculating the axial force on each friction surface\n", "W = T/(n*mu*R) \t\t\t#N\n", "#Calculating the maximum axial intensity of pressure\n", "p = W/(2*math.pi*r2*(r1-r2)) \t\t\t#N/mm**2\n", "\n", "#Results:\n", "print \" Maximum axial intensity of pressure, p = %.3f N/mm**2.\"%(p)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Maximum axial intensity of pressure, p = 0.062 N/mm**2.\n" ] } ], "prompt_number": 36 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.30 Page No : 309" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n1 = 3.\n", "n2 = 2.\n", "n = 4.\n", "mu = 0.3\n", "d1 = 240. #mm\n", "r1 = d1/2 #mm\n", "d2 = 120. #mm\n", "r2 = d2/2 \t\t\t#mm\n", "P = 25.*1000 \t\t#W\n", "N = 1575. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\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 mean radius of the contact surface for uniform pressure\n", "R = 2./3*(r1**3-r2**3)/(r1**2-r2**2)/1000 \t\t\t#m\n", "#Calculating the total spring load\n", "W1 = T/(n*mu*R) \t\t\t#N\n", "#Calculating the maximum power transmitted:\n", "\n", "# Variables:\n", "ns = 6. \t\t\t#Number of springs\n", "c = 8. \t\t\t#Contact surfaces of the spring\n", "w = 1.25 \t\t\t#Wear on each contact surface mm\n", "k = 13.*1000 \t\t\t#Stiffness of each spring N/m\n", "#Calculating the total wear\n", "Tw = c*w/1000 \t\t\t#Total wear m\n", "#Calculating the reduction in spring force\n", "Rs = Tw*k*ns \t\t\t#N\n", "#Calculating the new axial load\n", "W2 = W1-Rs \t\t\t#N\n", "#Calculating the mean radius of the contact surfaces for uniform wear\n", "R = (r1+r2)/(2*1000) \t\t\t#m\n", "#Calculating the torque transmitted\n", "T = n*mu*W2*R \t\t\t#N-m\n", "#Calculating the maximum power transmitted\n", "P = T*omega/1000 \t\t\t#kw\n", "\n", "#Results:\n", "print \" Total spring load, W = %d N.\"%(W1)\n", "print \" Maximum power that can be transmitted, P = %.2f kW.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Total spring load, W = 1353 N.\n", " Maximum power that can be transmitted, P = 10.21 kW.\n" ] } ], "prompt_number": 37 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.31 Page No : 314" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linalg\n", "import math \n", "\n", "# Variables:\n", "P = 90.*1000 \t\t#W\n", "N = 1500. \t\t\t#rpm\n", "alpha = 20. \t\t#degrees\n", "mu = 0.2\n", "D = 375.\n", "R = D/2. \t\t\t#mm\n", "pn = 0.25 \t\t\t#N/mm**2\n", "\n", "#SOlution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted\n", "T = P*1000/156 \t\t\t#N-mm\n", "#Calculating the width of the bearing surface\n", "b = T/(2*math.pi*mu*pn*R**2) \t\t\t#mm\n", "#Calculating the external and internal radii of the bearing surface\n", "#We know that r1+r2 = 2*R and r1-r2 = b*math.sin(math.radians(alpha)\n", "A = [[1, 1],[1, -1]]\n", "B = [2*R, b*math.sin(math.radians(alpha))]\n", "V = linalg.solve(A,B)\n", "r1 = V[0] \t\t\t#mm\n", "r2 = V[1] \t\t\t#mm\n", "#Calculating the intensity of pressure\n", "C = round(pn*r2,1) \t\t\t#N/mm\n", "#Calculating the axial load required\n", "W = 2*math.pi*C*(r1-r2) \t\t\t#N\n", "\n", "#Results:\n", "print \" Width of the bearing surface, b = %.1f mm.\"%(b)\n", "print \" External radius of the bearing surface, r1 = %.1f mm.\"%(r1)\n", "print \" Internal radius of the bearing surface, r2 = %.1f mm.\"%(r2)\n", "print \" Axial load required, W = %.f N.\"%(W)\n", "\n", "# rounding off error. Please check." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Width of the bearing surface, b = 52.2 mm.\n", " External radius of the bearing surface, r1 = 196.4 mm.\n", " Internal radius of the bearing surface, r2 = 178.6 mm.\n", " Axial load required, W = 5006 N.\n" ] } ], "prompt_number": 30 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.32 Page No : 315" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "P = 45.*1000 \t\t#W\n", "N = 1000. \t\t\t#rpm\n", "alpha = 12.5 \t\t#degrees\n", "D = 500./1000\n", "R = D/2 \t\t\t#m\n", "mu = 0.2\n", "pn = 0.1 \t\t\t#N/mm**2\n", "\n", "#Solution:\n", "#Calculating the angular speed of the shaft\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque developed by the clutch\n", "T = P/omega \t\t\t#N-m\n", "#Calculating the normal load acting on the friction surface\n", "Wn = T/(mu*R) \t\t\t#N\n", "#Calculating the axial spring force necessary to engage the clutch\n", "We = Wn*(math.sin(math.radians(alpha))+mu*math.cos(math.radians(alpha))) \t\t\t#N\n", "#Calculating the face width required\n", "b = Wn/(pn*2*math.pi*R*1000) \t\t\t#mm\n", "\n", "#Results:\n", "print \" Axial force necessary to engage the clutch, We = %d N.\"%(round(We,-1))\n", "print \" Face width required, b = %.1f mm.\"%(b)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Axial force necessary to engage the clutch, We = 3540 N.\n", " Face width required, b = 54.7 mm.\n" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.33 Page No : 315" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linalg\n", "import math \n", "\n", "# Variables:\n", "alpha = 30./2 \t\t#degrees\n", "pn = 0.35 \t\t\t#N/mm**2\n", "P = 22.5*1000 \t\t#W\n", "N = 2000. \t\t\t#rpm\n", "mu = 0.15\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the torque transmitted by the clutch\n", "T = P/omega*1000 \t\t\t#N-mm\n", "#Calculating the mean radius of the contact surface\n", "R = (T/(2*math.pi*mu*pn/3))**(1./3) \t\t\t#mm\n", "#Calculating the face width of the contact surface\n", "b = R/3\n", "#Calculating the outer and inner radii of the contact surface\n", "#Refer Fig. 10.27\n", "#We have r1-r2 = b*math.sin(math.radians(alpha) and r1+r2 = 2*R\n", "A = [[1, -1],[ 1, 1]]\n", "B = [b*math.sin(math.radians(alpha)), 2*R]\n", "V = linalg.solve(A, B)\n", "r1 = V[0] \t\t\t#mm\n", "r2 = V[1] \t\t\t#mm\n", "\n", "#Results:\n", "print \" Mean radius of the contact surface, R = %d mm.\"%(R)\n", "print \" Outer radius of the contact surface, r1 = %.2f mm.\"%(r1)\n", "print \" Inner radius of the contact surface, r2 = %.2f mm.\"%(r2)\n", "\n", "# rounding off error" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Mean radius of the contact surface, R = 99 mm.\n", " Outer radius of the contact surface, r1 = 103.51 mm.\n", " Inner radius of the contact surface, r2 = 94.95 mm.\n" ] } ], "prompt_number": 48 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.34 Page No : 316" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "D = 75./1000 #mm\n", "R = D/2 \t\t\t#m\n", "alpha = 15 \t\t\t#degrees\n", "mu = 0.3\n", "W = 180. \t\t\t#N\n", "NF = 1000. \t\t\t#rpm\n", "m = 13.5 \t\t\t#kg\n", "k = 150./1000 \t\t#m\n", "\n", "#Solution:\n", "#Calculating the angular speed of the flywheel\n", "omegaF = 2*math.pi*NF/60 \t\t\t#rad/s\n", "#Calculating the torque required to produce slipping\n", "T = round(mu*W*R*(1/math.sin(math.radians(alpha))),1) \t\t\t#N-m\n", "#Calculating the mass moment of inertia of the flywheel\n", "IF = m*k**2 \t\t\t#kg-m**2\n", "#Calculating the angular acceleration of the flywheel\n", "alphaF = T/IF \t\t\t#rad/s**2\n", "#Calculating the time required for the flywheel to attain full speed\n", "tF = round(omegaF/alphaF,1) \t\t\t#seconds\n", "#Calculating the angle turned through by the motor and flywheel in time tF\n", "theta = 1./2*omegaF*tF \t\t\t#rad\n", "#Calculating the energy lost in slipping of the clutch\n", "E = T*theta \t\t\t#Energy lost in slipping of the clutch N-m\n", "\n", "#Results:\n", "print \" Torque required to produce slipping, T = %.1f N-m.\"%(T)\n", "print \" Time required for the flywheel to attain full speed, tF = %.1f s.\"%(tF)\n", "print \" Energy lost in slipping of the clutch = %d N-m.\"%(E)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Torque required to produce slipping, T = 7.8 N-m.\n", " Time required for the flywheel to attain full speed, tF = 4.1 s.\n", " Energy lost in slipping of the clutch = 1674 N-m.\n" ] } ], "prompt_number": 55 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.35 Page No : 319" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "P = 15.*1000 \t\t\t#W\n", "N = 900. \t\t\t #rpm\n", "n = 4.\n", "mu = 0.25\n", "R = 150./1000\n", "r = 120./1000 \t\t\t#m\n", "theta = 60. \t\t\t#degrees\n", "p = 0.1 \t\t\t #N/mm**2\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the speed at which the engagement begins\n", "omega1 = 3./4*omega \t\t\t#rad/s\n", "#Calculating the torque transmitted at the running speed\n", "T = P/omega \t\t\t#N-m\n", "#Calculating the mass of the shoes\n", "m = T/(n*mu*(omega**2*r-omega1**2*r)*R) \t\t\t#kg\n", "#Calculating the contact length of shoes\n", "l = (theta*math.pi/180)*R*1000 \t\t\t#mm\n", "#Calculating the centrifugal force acting on each shoe\n", "Pc = m*omega**2*r \t\t\t#N\n", "#Calculating the inward force on each shoe exerted by the spring\n", "Ps = m*omega1**2*r \t\t\t#N\n", "#Calculating the width of the shoes\n", "b = (Pc-Ps)/(l*p) \t\t\t#mm\n", "\n", "#Results:\n", "print \" Mass of the shoes, m = %.2f kg.\"%(m)\n", "print \" Width of the shoes, b = %.1f mm.\"%(b)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Mass of the shoes, m = 2.28 kg.\n", " Width of the shoes, b = 67.5 mm.\n" ] } ], "prompt_number": 56 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 10.36 Page No : 320" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables:\n", "n = 4.\n", "mu = 0.3\n", "c = 5.\n", "r = 160. \t\t\t#mm\n", "S = 500. \t\t\t#N\n", "D = 400./1000\n", "R = D/2 \t\t\t#m\n", "m = 8. \t\t\t #kg\n", "s = 50. \t\t\t#N/mm\n", "N = 500. \t\t\t#rpm\n", "\n", "#Solution:\n", "#Calculating the angular speed of the clutch\n", "omega = 2*math.pi*N/60 \t\t\t#rad/s\n", "#Calculating the operating radius\n", "r1 = (r+c)/1000 \t\t\t#m\n", "#Calculating the centrifugal force on each shoe\n", "Pc = m*omega**2*r1 \t\t\t#N\n", "#Calculating the inward force exerted by the spring\n", "Ps = S+c*s \t\t\t#N\n", "#Calculating the frictional force acting math.tangentially on each shoe\n", "F = mu*(Pc-Ps) \t\t\t#N\n", "#Calculating the total frictional torque transmitted by the clutch\n", "T = n*F*R \t\t\t#N-m\n", "#Calculating the power transmitted\n", "P = T*omega/1000 \t\t\t#kW\n", "\n", "#Results:\n", "print \" Power transmitted, P = %.1f kW.\"%(P)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " Power transmitted, P = 36.1 kW.\n" ] } ], "prompt_number": 45 } ], "metadata": {} } ] }