{
"metadata": {
"name": "",
"signature": "sha256:13844e2491f60bc4c521c5dd5ea680eaf26c07059f7a5c77baf4e368f712f152"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 15 : Inertia Forces in Reciprocating Parts"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.1 Page No : 521"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"OC = 200./1000 #m\n",
"PC = 700./1000 \t\t\t#m\n",
"omega = 120. \t\t\t#rad/s\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.5\n",
"OM = 127./1000\n",
"CM = 173./1000\n",
"QN = 93./1000\n",
"NO = 200./1000 \t\t\t#m\n",
"\n",
"#Velocity and acceleration of the piston:\n",
"#Calculating the velocity of the piston P\n",
"vP = omega*OM \t\t\t#m/s\n",
"#Calculating the acceleration of the piston P\n",
"aP = omega**2*NO \t\t\t#m/s**2\n",
"#Velocity and acceleration of the mid-point of the connecting rod:\n",
"#By measurement\n",
"OD1 = 140./1000\n",
"OD2 = 193./1000 \t\t\t#m\n",
"#Calculating the velocity of D\n",
"vD = omega*OD1 \t\t\t#m/s\n",
"#Calculating the acceleration of D\n",
"aD = omega**2*OD2 \t\t\t#m/s**2\n",
"#Angular velocity and angular acceleration of the connecting rod:\n",
"#Calculating the velocity of the connecting rod PC\n",
"vPC = omega*CM \t\t\t#m/s\n",
"#Calculating the angular velocity of the connecting rod PC\n",
"omegaPC = vPC/PC \t\t\t#rad/s\n",
"#Calculating the math.tangential component of the acceleration of P with respect to C\n",
"atPC = omega**2*QN \t\t\t#m/s**2\n",
"#Calculating the angular acceleration of the connecting rod PC\n",
"alphaPC = atPC/PC \t\t\t#ra/s**2\n",
"\n",
"#Results:\n",
"print \" Velocity of the piston P, vP = %.2f m/s.\"%(vP)\n",
"print \" Acceleration of the piston P, aP = %d m/s**2.\"%( aP)\n",
"print \" Velocity of D, vD = %.1f m/s.\"%(vD)\n",
"print \" Acceleration of D, aD = %.1f m/s**2.\"%(aD)\n",
"print \" Angular velocity of the connecting rod PC, omegaPC = %.2f rad/s.\"%(omegaPC)\n",
"print \" Angular acceleration of the connecting rod PC alphaPC = %.2f rad/s**2.\"%(alphaPC)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Velocity of the piston P, vP = 15.24 m/s.\n",
" Acceleration of the piston P, aP = 2880 m/s**2.\n",
" Velocity of D, vD = 16.8 m/s.\n",
" Acceleration of D, aD = 2779.2 m/s**2.\n",
" Angular velocity of the connecting rod PC, omegaPC = 29.66 rad/s.\n",
" Angular acceleration of the connecting rod PC alphaPC = 1913.14 rad/s**2.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.2 Page No : 522"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"OC = 150./1000 #m\n",
"PC = 600./1000 #m\n",
"CD = 150./1000 \t\t#m\n",
"N = 450. \t\t\t#rpm\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.6\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#By measurement\n",
"OM = 145./1000\n",
"CM = 78./1000\n",
"QN = 130./1000\n",
"NO = 56./1000 \t\t\t#m\n",
"\n",
"#Velocity and acceleration of alider:\n",
"#Calculating the velocity of the slider P\n",
"vP = omega*OM \t\t\t#m/s\n",
"#Calculating the acceleration of the slider P\n",
"aP = omega**2*NO \t\t\t#m/s**2\n",
"#Velocity and acceleration of point D on the connecting rod:\n",
"#Calculating the length od CD1\n",
"CD1 = CD/PC*CM \t\t\t#m\n",
"#By measurement\n",
"OD1 = 145./1000\n",
"OD2 = 120./1000 \t\t\t#m\n",
"\n",
"#Calculating the velocity of point D\n",
"vD = omega*OD1 \t\t\t#m/s\n",
"#Calculating the acceleration of point D\n",
"aD = omega**2*OD2 \t\t\t#m/s**2\n",
"#Angular velocity and angular acceleration of the connecting rod:\n",
"#Calculating the velocity of the connecting rod PC\n",
"vPC = omega*CM \t\t\t#m/s\n",
"#Calculating the angular velocity of the connecting rod\n",
"omegaPC = vPC/PC \t\t\t#rad/s\n",
"#Calculating the tangential component of the acceleration of P with respect to C\n",
"atPC = omega**2*QN \t\t\t#m/s**2\n",
"#Calculating the angular acceleration of the connecting rod PC\n",
"alphaPC = atPC/PC \t\t\t#rad/s**2\n",
"\n",
"#Results:\n",
"print \" Velocity of the slider P vP = %.3f m/s.\"%(vP)\n",
"print \" Acceleration of the slider P aP = %.1f m/s**2.\"%(aP)\n",
"print \" Velocity of point D vD = %.3f m/s.\"%(vD)\n",
"print \" Acceleration of point D aD = %.2f m/s**2.\"%(aD)\n",
"print \" Angular velocity of the connecting rod omegaPC = %.3f rad/s.\"%(omegaPC)\n",
"print \" Angular acceleration of the connecting rod PC alphaPC = %.2f rad/s**2.\"%(alphaPC)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Velocity of the slider P vP = 6.833 m/s.\n",
" Acceleration of the slider P aP = 124.4 m/s**2.\n",
" Velocity of point D vD = 6.833 m/s.\n",
" Acceleration of point D aD = 266.48 m/s**2.\n",
" Angular velocity of the connecting rod omegaPC = 6.126 rad/s.\n",
" Angular acceleration of the connecting rod PC alphaPC = 481.14 rad/s**2.\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.3 Page No : 527"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"r = 300./1000\n",
"l = 1. \t\t\t#m\n",
"N = 200. \t\t\t#rpm\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Crank angle at which the maximum velocity occurs:\n",
"#Calculating the ratio of length of connecting rod to crank radius\n",
"n = l/r\n",
"#Velocity of the piston vP = omega*r*(math.sin(math.radians(theta)+math.sin(math.radians(2*theta)/(2*n)) .....(i)\n",
"#For maximum velocity d(vP)/d(theta) = 0 .....(ii)\n",
"#Substituting (i) in (ii) we get 2(math.cos(theta))**2+n*math.cos(theta)-1 = 0\n",
"a = 2.\n",
"b = n\n",
"c = -1.\n",
"costheta = (-b+math.sqrt(b**2-4*a*c))/(2*a)\n",
"#Calculating the crank angle from the inner dead centre at which the maximum velocity occurs\n",
"theta = round(math.degrees(math.acos(costheta))) \t\t\t#degrees\n",
"#Calculating the maximum velocity of the piston:\n",
"vPmax = omega*r*(math.sin(math.radians(theta))+math.sin(math.radians(2*theta))/(2*n)) \t\t\t#m/s\n",
"#Results:\n",
"print \" Crank angle from the inner dead centre at which the maximum velocity occurs theta = %.2f degrees.\"%(theta)\n",
"print \" Maximum velocity of the piston( vPmax) = %.2f m/s.\"%(vPmax)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Crank angle from the inner dead centre at which the maximum velocity occurs theta = 75.00 degrees.\n",
" Maximum velocity of the piston( vPmax) = 6.54 m/s.\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.4 Page No : 528"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"r = 0.3 #m\n",
"l = 1.5 \t\t\t#m\n",
"N = 180. \t\t\t#rpm\n",
"theta = 40. \t\t#degrees\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the piston\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Velocity of the piston:\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the velocity of the piston\n",
"vP = omega*r*(math.sin(math.radians(theta))+math.sin(math.radians(2*theta))/(2*n)) \t\t\t#m/s\n",
"#Calculating the acceleration of the piston\n",
"aP = omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#m/s**2\n",
"#Position of the crank for zero acceleration of the piston:\n",
"ap1 = 0\n",
"#Calculating the position of the crank from the inner dead centre for zero acceleration of the piston\n",
"#We have ap1 = omega**2*r*(math.cos(theta1)+math.cos(2*theta1)/n) or 2*(math.cos(theta1))**2+n*math.cos(theta1)-1 = 0\n",
"a = 2.\n",
"b = n\n",
"c = -1.\n",
"costheta1 = (-b+math.sqrt(b**2-4*a*c))/(2*a)\n",
"#Calculating the crank angle from the inner dead centre for zero acceleration of the piston\n",
"theta1 = math.degrees(math.acos(costheta1)) \t\t\t#degrees\n",
"\n",
"#Results:\n",
"print \" Velocity of the piston vP = %.2f m/s.\"%( vP)\n",
"print \" Acceleration of the piston aP = %.2f m/s**2.\"%(aP)\n",
"print \" Position of the crank for zero acceleration of the piston theta1 = %.2f degrees or\\\n",
" %.2f degrees.\"%(theta1,360-theta1)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Velocity of the piston vP = 4.19 m/s.\n",
" Acceleration of the piston aP = 85.36 m/s**2.\n",
" Position of the crank for zero acceleration of the piston theta1 = 79.27 degrees or 280.73 degrees.\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.5 Page No : 528"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"r = 150./1000\n",
"l = 600./1000 \t\t\t#m\n",
"theta = 60. \t\t\t#degrees\n",
"N = 450. \t\t\t#rpm\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Velocity and acceleration of the slider:\n",
"#Calculating the ratio of length of connecting rod and crank\n",
"n = l/r\n",
"#Calculating the velocity of the slider\n",
"vP = omega*r*(math.sin(math.radians(theta))+math.sin(math.radians(2*theta))/(2*n)) \t\t\t#m/s\n",
"#Calculating the acceleration of the slider\n",
"aP = omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#m/s**2\n",
"#Angular velocity and angular acceleration of the connecting rod:\n",
"#Calculating the angular velocity of the connecting rod\n",
"omegaPC = omega*math.cos(math.radians(theta))/n \t\t\t#rad/s\n",
"#Calculating the angular acceleration of the connecting rod\n",
"alphaPC = round(omega**2*math.sin(math.radians(theta))/n) \t\t\t#rad/s**2\n",
"\n",
"#Results:\n",
"print \" Velocity of the slider vP = %.1f m/s.\"%(vP)\n",
"print \" Acceleration of the slider aP = %.2f m/s**2.\"%(aP)\n",
"print \" Angular velocity of the connecting rod omegaPC = %.1f rad/s.\"%(omegaPC)\n",
"print \" Angular acceleration of the connecting rod alphaPC = %d rad/s**2.\"%(alphaPC)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Velocity of the slider vP = 6.9 m/s.\n",
" Acceleration of the slider aP = 124.91 m/s**2.\n",
" Angular velocity of the connecting rod omegaPC = 5.9 rad/s.\n",
" Angular acceleration of the connecting rod alphaPC = 481 rad/s**2.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.6 Page No : 532"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 175./1000 #m\n",
"L = 200./1000 #m\n",
"r = L/2 #m\n",
"l = 400./1000 \t #m\n",
"N = 500. \t\t #rpm\n",
"mR = 180. \t\t #kg\n",
"theta = 60 #degrees\n",
"#Solution:\n",
"omega = round(2*math.pi*N/60,1) \t\t\t#rad/s\n",
"\n",
"# Graphical method\n",
"ON = 0.038 # m\n",
"aR = omega**2 * ON\n",
"FI = mR * aR/1000\n",
"print \" Inertia force FI = %.2f kN.\"%(FI)\n",
"\n",
"#Calculating the angular speed of the crank\n",
"\n",
"#Analytical method:\n",
"#Calculating the ratio of lengths of connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n)/1000 \t\t\t#kN\n",
"\n",
"#Results:\n",
"print \" Inertia force FI = %.2f kN.\"%(FI)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Inertia force FI = 18.78 kN.\n",
" Inertia force FI = 18.53 kN.\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.7 Page No : 533"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import numpy\n",
"\n",
"# Variables:\n",
"r = 300./1000 #m\n",
"l = 1.2 #m\n",
"D = 0.5 \t\t\t#m\n",
"mR = 250. \t\t\t#kg\n",
"theta = 60. \t\t#degrees\n",
"dp = 0.35 \t\t\t#p1-p2 N/mm**2\n",
"N = 250. \t\t\t#rpm\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Calculating the net load on the piston\n",
"FL = (dp)*math.pi/4*(D*1000)**2 \t\t\t#N\n",
"#Calculating the ratio of length of connecting rod and crank\n",
"n = l/r\n",
"#Calculating the accelerating or inertia force on reciprocating parts\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#N\n",
"#Calculating the piston effort\n",
"FP = (FL-FI)/1000 \t\t\t#kN\n",
"#Pressure on slide bars:\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the pressure on the slide bars\n",
"FN = FP*math.tan(math.radians(phi)) \t\t\t#kN\n",
"#Calculating the thrust in the connecting rod\n",
"FQ = FP/math.cos(math.radians(phi)) \t\t\t#kN\n",
"#Calculating the tangential force on the crank pin\n",
"FT = FQ*math.sin(math.radians(theta+phi)) \t\t\t#kN\n",
"#Calculating the turning moment on the crank shaft\n",
"T = FT*r \t\t\t#kN-m\n",
"\n",
"#Results:\n",
"print \" Pressure on the slide bars FN = %.2f kN.\"%(FN)\n",
"print \" Thrust in the connecting rod FQ = %.2f kN.\"%(FQ)\n",
"print \" Tangential force on the crank-pin FT = %.2f kN.\"%(FT)\n",
"print \" Turning moment on the crank shaft T = %.3f kN-m.\"%(T)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Pressure on the slide bars FN = 10.97 kN.\n",
" Thrust in the connecting rod FQ = 50.65 kN.\n",
" Tangential force on the crank-pin FT = 48.30 kN.\n",
" Turning moment on the crank shaft T = 14.491 kN-m.\n"
]
}
],
"prompt_number": 28
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.8 Page No : 534"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 300./1000 #m\n",
"L = 450./1000 #m\n",
"r = L/2 #m\n",
"d = 50./1000 #m\n",
"l = 1.2 \t\t\t#m\n",
"N = 200. \t\t\t#rpm\n",
"mR = 225. \t\t\t#kg\n",
"theta = 125. \t\t\t#degrees\n",
"p1 = 30*1000. #N/m**2\n",
"p2 = 1.5*1000. \t\t\t#N/m**2\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Calculating the area of the piston\n",
"A1 = math.pi/4*D**2 \t\t\t#m**2\n",
"#Calculating the area of the piston rod\n",
"a = math.pi/4*d**2 \t\t\t#m**2\n",
"#Calculating the force on the piston due to steam pressure\n",
"FL = round(p1*A1-p2*(A1-a)) \t\t\t#N\n",
"#Calculating the ratio of lengths of connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force on the reciprocating parts\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#N\n",
"#Calculating the net force on the piston or piston effort\n",
"FP = FL-FI+mR*9.81 \t\t\t#N\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the effective turning moment on the crank shaft\n",
"T = FP*math.sin(math.radians(theta+phi))/math.cos(math.radians(phi))*r \t\t\t#N-m\n",
"\n",
"#Results:\n",
"print \" Effective turning moment of the crank shaft T = %.f N-m.\"%(T)\n",
"\n",
"# rounding off error"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Effective turning moment of the crank shaft T = 3020 N-m.\n"
]
}
],
"prompt_number": 35
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.9 Page No : 534"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"N = 1800. \t\t\t#rpm\n",
"r = 50./1000 #m\n",
"l = 200./1000 #m\n",
"D = 80./1000 #m\n",
"x = 10./1000 \t\t#m\n",
"mR = 1. \t\t\t#kg\n",
"p = 0.7 \t\t\t#N/mm**2\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Net load on the gudgeon pin:\n",
"#Calculating the load on the piston\n",
"FL = round(math.pi/4*(D*1000)**2*p) \t\t\t#N\n",
"#Refer Fig. 15.10\n",
"#By measurement\n",
"theta = 33. \t\t\t#degrees\n",
"#Calculating the ratio of lengths of connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force on the reciprocating parts\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#N\n",
"#Calculating the net load on the gudgeon pin\n",
"FP = FL-FI \t\t\t#N\n",
"#Thrust in the connecting rod:\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the thrust in the connecting rod\n",
"FQ = FP/math.cos(math.radians(phi)) \t\t\t#N\n",
"#Calculating the reaction between the piston and cylinder\n",
"FN = FP*math.tan(math.radians(phi)) \t\t\t#N\n",
"#Engine speed at which the abov values will become zero:\n",
"#Calculating the speed at which FI = FL\n",
"omega1 = math.sqrt((math.pi/4*(D*1000)**2*p)/(mR*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n))) \t\t\t#rad/s\n",
"#Calculating the corresponding speed in rpm\n",
"N1 = omega1*60/(2*math.pi) \t\t\t#rpm\n",
"\n",
"print phi\n",
"#Results:\n",
"print \" Net load on the gudgeon pin FP = %.f N.\"%(FP)\n",
"print \" Thrust in the connecting rod FQ = %.1f N.\"%(FQ)\n",
"print \" Reaction between the piston and cylinder FN = %d N.\"%(FN)\n",
"print \" Engine speed at which the above values will become zero N1 = %d rpm.\"%(N1)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"7.82568845753\n",
" Net load on the gudgeon pin FP = 1848 N.\n",
" Thrust in the connecting rod FQ = 1865.8 N.\n",
" Reaction between the piston and cylinder FN = 254 N.\n",
" Engine speed at which the above values will become zero N1 = 2612 rpm.\n"
]
}
],
"prompt_number": 50
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.10 Page No : 536"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"aP = 36. \t\t\t#m/s**2\n",
"theta = 30. \t\t#degrees\n",
"p = 0.5 \t\t\t#N/mm**2\n",
"RF = 600. \t\t\t#N\n",
"D = 300./1000 #m\n",
"r = 300./1000 \t\t#m\n",
"mR = 180. \t\t\t#kg\n",
"n = 4.5\n",
"\n",
"#Solution:\n",
"#Reaction on the guide bars:\n",
"#Calculating the load on the piston\n",
"FL = round(p*math.pi/4*(D*1000)**2) \t\t\t#N\n",
"#Calculating the inertia force due to reciprocating parts\n",
"FI = mR*aP \t\t\t#N\n",
"#Calculating the piston effort\n",
"FP = (FL-FI-RF)/1000 \t\t\t#kN\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the reaction on the guide bars\n",
"FN = FP*math.tan(phi) \t\t\t#kN\n",
"#Calculating the thrust on the crank shaft bearing\n",
"FB = (FP*math.cos(math.radians(phi+theta)))/math.cos(math.radians(phi)) \t\t\t#kN\n",
"#Calculating the turning moment on the crank shaft\n",
"T = (FP*math.sin(math.radians(theta+phi)))/math.cos(math.radians(phi))*r \t\t\t#kN-m\n",
"\n",
"\n",
"#Results:\n",
"print \" Reaction on the guide bars FN = %.f kN.\"%(FN)\n",
"print \" Thrust on the crank shaft bearing FB = %.1f kN.\"%(FB)\n",
"print \" Turning moment on the crank shaft T = %.2f kN-m.\"%(T)\n",
"\n",
"# rounding off error"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Reaction on the guide bars FN = 3 kN.\n",
" Thrust on the crank shaft bearing FB = 22.9 kN.\n",
" Turning moment on the crank shaft T = 5.06 kN-m.\n"
]
}
],
"prompt_number": 60
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.11 Page No : 537"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 100./1000 #m\n",
"L = 120./1000 #m\n",
"r = L/2 #m\n",
"l = 250./1000 \t\t\t#m\n",
"mR = 1.1 \t\t\t#kg\n",
"N = 2000. \t \t\t#rpm\n",
"theta = 20. \t\t\t#degrees\n",
"p = 700. \t\t \t#kN/m**2\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Net force on the piston:\n",
"#Calculating the force due to gas pressure\n",
"FL = p*math.pi/4*D**2 \t\t\t#kN\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force on the piston\n",
"FI = round(mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n)) \t\t\t#N\n",
"#Calculating the net force on the piston\n",
"FP = (FL*1000)-FI+mR*9.81 \t\t\t#N\n",
"#Resulmath.tant force on the gudgeon pin:\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(theta)/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the resultant load on the gudgeon pin\n",
"FQ = round(FP/math.cos(phi)) \t\t\t#N\n",
"#Calculating the thrust on the cylinder walls\n",
"FN = FP*math.tan(math.radians(4.7)) \t\t\t#N \n",
"#Speed above which the gudgeon pin load would be reversed in direction:\n",
"#Calculating the minimum speed for FP to be negative\n",
"omega1 = math.sqrt((FL*1000+mR*9.81)/(mR*r*(math.cos(theta)+math.cos(2*theta)/n))) \t\t\t#rad/s\n",
"#Calculating the corresponding speed in rpm\n",
"N1 = 273*60/(2*math.pi) \t\t\t#rpm\n",
"\n",
"#Results:\n",
"print \" Net force on the piston FP = %.1f N.\"%(FP)\n",
"print \" Resultant load on the gudgeon pin FQ = %d N.\"%(FQ)\n",
"print \" Thrust on the cylinder walls FN = %.1f N.\"%(FN)\n",
"print \" Speed above which the gudgeon pin load would be reversed in direction N1 > %d rpm.\"%(N1)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Net force on the piston FP = 2255.6 N.\n",
" Resultant load on the gudgeon pin FQ = 2265 N.\n",
" Thrust on the cylinder walls FN = 185.4 N.\n",
" Speed above which the gudgeon pin load would be reversed in direction N1 > 2606 rpm.\n"
]
}
],
"prompt_number": 71
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.12 Page No : 538"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"N = 120. \t\t\t#rpm\n",
"D = 250./1000 #m\n",
"L = 400./1000 #m\n",
"r = L/2 #m\n",
"l = 0.6 #m\n",
"d = 50./1000 \t\t#m\n",
"mR = 60. \t\t\t#kg\n",
"theta = 45. \t\t#degrees\n",
"p1 = 550.*1000 #N/m**2\n",
"p2 = 70.*1000 \t\t\t#N/m**2\n",
"\n",
"#Solution:\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Turning moment on the crankshaft:\n",
"#Calculating the area of the piston on the cover end side\n",
"A1 = math.pi/4*D**2 \t\t\t#m**2\n",
"#Calculating the area of the piston rod\n",
"a = math.pi/4*d**2 \t\t\t#m**2\n",
"#Calculating the net load on the piston\n",
"FL = p1*A1-p2*(A1-a) \t\t\t#N\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force on the reciprocating parts\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#N\n",
"#Calculating the net force on the piston or piston effort\n",
"FP = (FL-FI)/1000 \t\t\t#kN\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the turning moment on the crank shaft\n",
"T = (FP*math.sin(math.radians(theta+phi)))/math.cos(math.radians(phi))*r*1000\t\t\t#N-m\n",
"#Calculating the thrust on the bearings\n",
"FB = (FP*math.cos(math.radians(theta+phi)))/math.cos(math.radians(phi)) \t\t\t#kN\n",
"#Acceleration of the flywheel:\n",
"P = 20.*1000 \t\t\t#W\n",
"m = 60. \t\t\t#kg\n",
"k = 0.6 \t\t\t#m\n",
"#Calculating the mass moment of inertia of the flywheel\n",
"I = m*k**2 \t\t\t#kg-m**2\n",
"#Calculating the resisting torque\n",
"TR = P*60/(2*math.pi*N) \t\t\t#N-m\n",
"#Calculating the acceleration of the flywheel\n",
"alpha = (T-TR)/I \t\t\t#rad/s**2\n",
"\n",
"\n",
"#Results:\n",
"print \" Turning moment on the crank shaft T = %f N-m.\"%(T) # rounding off error \n",
"print \" Thrust on the bearings FB = %.2f kN.\"%(FB)\n",
"print \" Acceleration of the flywheel alpha = %.1f rad/s**2.\"%(alpha)\n",
"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Turning moment on the crank shaft T = 3929.026139 N-m.\n",
" Thrust on the bearings FB = 11.98 kN.\n",
" Acceleration of the flywheel alpha = 108.2 rad/s**2.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.13 Page No : 540"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 300./1000 #m\n",
"L = 500./1000 #m\n",
"r = L/2 \t\t\t#m\n",
"n = 4.5\n",
"N = 180. \t\t\t#rpm\n",
"mR = 280. \t\t\t#kg\n",
"theta = 45. \t\t#degrees\n",
"p1 = 0.1 \t\t\t#N/mm**2\n",
"CR = 14. \t\t\t#Compression ration V1/V2\n",
"p4 = 2.2\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.12\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Calculating the pressure corresponding to point 2\n",
"p2 = p1*(CR)**1.35 \t\t\t#N/mm**2\n",
"#Calculating the swept volume\n",
"VS = math.pi/4*D**2*L \t\t\t#m**3\n",
"VC = VS/(CR-1) \t\t\t#m**3\n",
"#Calculating the volume corresponding to point 3\n",
"V3 = VC+(1/10*VS) \t\t\t#m**3\n",
"#Calculating the print lacement of the piston corresponding to crank print lacement of 45 degrees\n",
"x = r*((1-math.cos(math.radians(theta)))+(math.sin(math.radians(theta)))**2/(2*n)) \t\t\t#m\n",
"#Calculating the volume corresponding to point 4'\n",
"V4dash = VC+(math.pi/4*D**2*x) \t\t\t#m**2\n",
"#Calculating the pressure corresponding to point 4'\n",
"p3 = p2\n",
"p4dash = p3*(V3/V4dash)**1.35 \t\t\t#N/mm**2\n",
"\n",
"#Calculating the difference of pressures on two sides of the piston\n",
"p = (p4-p1)*10**6 \t\t\t#N/m**2\n",
"#Calculating the net load on the piston\n",
"FL = p*math.pi/4*D**2 \t\t\t#N\n",
"#Calculating the inertia force on the reciprocating parts\n",
"FI = mR*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#N\n",
"#Calculating the net force on the piston or piston effort\n",
"FP = FL-FI+mR*9.81 \t\t\t#N\n",
"#Crank-pin effort:\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = math.degrees(math.asin(sinphi))\n",
"#Calculating the crank-pin effort\n",
"FT = (FP*math.sin(math.radians(theta+phi)))/(math.cos(math.radians(phi))*1000) \t\t\t#kN\n",
"#Calculating the thrust on the bearings\n",
"FB = (FP*math.cos(math.radians(theta+phi)))/(math.cos(math.radians(phi))*1000) \t\t\t#kN\n",
"#Calculating the turning moment on the crankshaft\n",
"T = FT*r \t\t\t#kN-m\n",
"\n",
"#Results:\n",
"print \" Crank-pin effort FT = %.3f kN.\"%(FT)\n",
"print \" Thrust on the bearings FB = %.3f kN.\"%(FB)\n",
"print \" Turning moment on the crankshaft T = %.2f kN-m.\"%(T)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Crank-pin effort FT = 109.501 kN.\n",
" Thrust on the bearings FB = 79.438 kN.\n",
" Turning moment on the crankshaft T = 27.38 kN-m.\n"
]
}
],
"prompt_number": 94
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.14 Page No : 542"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 240./1000 #m\n",
"L = 360./1000 #m\n",
"r = L/2 #m\n",
"l = 0.6 \t\t\t#m\n",
"N = 300. \t\t\t#rpm\n",
"mR = 160. \t\t\t#kg\n",
"pA = (8+1.03)*10**5\n",
"pE = (-0.75+1.03)*10**5 \t\t\t#N/m**2\n",
"FR = 500. \t \t\t#N\n",
"theta = 75. \t\t\t#degrees\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.13\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Calculating the stroke volume\n",
"VS = math.pi/4*D**2*L \t\t\t#m**3\n",
"#Calculating the volume of steam at cut-off\n",
"VB = VS/3 \t\t\t#m**3\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the print lacement of the piston when the crank position is 75 degrees from the top dead centre\n",
"x = r*((1-math.cos(math.radians(theta)))+(math.sin(math.radians(theta)))**2/(2*n)) \t\t\t#m**3\n",
"#Calculating the volume corresponding to point C'\n",
"VCdash = VS*x/L \t\t\t#m**3\n",
"#Calculating the pressure corresponding to point C'\n",
"pB = pA\n",
"pCdash = round((pB*VB)/VCdash) \t\t\t#N/m**2\n",
"#Calculating the difference of pressures on the two sides of the piston\n",
"p = round(pCdash-pE) \t\t\t#N/m**2\n",
"#Calculating the net load on the piston\n",
"FL = round(math.pi/4*D**2*p) \t\t\t#N\n",
"#Calculating the inertia force on the reciprocating parts\n",
"FI = round(mR*omega**2*r*(math.cos(math.radians(theta))+(math.cos(math.radians(2*theta))/n))) \t\t\t#N\n",
"#Calculating the piston effort\n",
"FP = FL-FI+mR*9.81-FR \t\t\t#N\n",
"#Turning moment on the crankshaft:\n",
"#Calculating the angle of inclination of the connecting rod to the line of stroke\n",
"sinphi = math.sin(math.radians(theta))/n \t\t\t#degrees\n",
"phi = round(math.degrees(math.asin(sinphi)),3)\n",
"\n",
"#Calculating the turning moment on the crankshaft\n",
"T = (FP*math.sin(math.radians(theta+phi)))/math.cos(math.radians(phi))*r \t\t\t#N-m\n",
"\n",
"#Results:\n",
"print \" Turning moment on the crankshaft T = %d N-m.\"%(T)\n",
"\n",
"# note : rounding error. please check using calculator"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Turning moment on the crankshaft T = 5778 N-m.\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.15 Page No : 545"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"l = 300. #mm\n",
"l1 = 200. \t\t\t#mm\n",
"m = 15. \t\t\t#kg\n",
"I = 7000. \t\t\t#kg-mm**2\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.16 and Fig. 15.17\n",
"#Calculating the radius of gyration of the connecting rod about an axis pasmath.sing through its centre of gravity\n",
"kG = math.sqrt(I/m) \t\t\t#mm\n",
"#Calculating the distance of other mass from the centre of gravity\n",
"l2 = (kG)**2/l1 \t\t\t#mm\n",
"#Calculating the magnitude of mass placed at the small end centre\n",
"m1 = (l2*m)/(l1+l2) \t\t\t#kg\n",
"#Calculating the magnitude of the mass placed at a dismath.tance l2 from the centre of gravity\n",
"m2 = (l1*m)/(l1+l2) \t\t\t#kg\n",
"\n",
"#Results:\n",
"print \" Mass placed at the small end centre m1 = %.2f kg.\"%(m1)\n",
"print \" Mass placed at a distance %.2f mm from the centre of gravity m2 = %.2f kg.\"%(l2,m2)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Mass placed at the small end centre m1 = 0.17 kg.\n",
" Mass placed at a distance 2.33 mm from the centre of gravity m2 = 14.83 kg.\n"
]
}
],
"prompt_number": 26
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.16 Page No : 546"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables: \n",
"h = 650./1000 #m\n",
"l1 = (650.-25)/1000 #m\n",
"m = 37.5 \t\t\t#kg\n",
"tp = 1.87 \t\t\t#seconds\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.18 and Fig. 15.19\n",
"#Calculating the radius of gyration of the connecting rod about an axis pasmath.sing through its centre of gravity\n",
"kG = math.sqrt((tp/(2*math.pi))**2*(9.81*h)-h**2) \t\t\t#m\n",
"#Calculating the dismath.tance of mass m2 from the centre of gravity\n",
"l2 = (kG)**2/l1 \t\t\t#m\n",
"#Calculating the magnitude of mass placed at the small end centre\n",
"m1 = (l2*m)/(l1+l2) \t\t\t#kg\n",
"#Calculating the magnitude of mass placed at a dismath.tance l2 from centre of gravity\n",
"m2 = (l1*m)/(l1+l2) \t\t\t#kg\n",
"\n",
"#Results:\n",
"print \" Mass placed at the small end centre A m1 = %d kg.\"%(m1)\n",
"print \" Mass placed at a distance %.3f m from G m2 = %.1f kg.\"%(l2,m2)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Mass placed at the small end centre A m1 = 10 kg.\n",
" Mass placed at a distance 0.228 m from G m2 = 27.5 kg.\n"
]
}
],
"prompt_number": 27
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.17 Page No : 547"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from scipy.optimize import fsolve \n",
"import math \n",
"\n",
"# Variables:\n",
"m = 55. \t\t\t#kg\n",
"l = 850./1000 #m\n",
"d1 = 75./1000 #m\n",
"d2 = 100./1000 \t\t#m\n",
"tp1 = 1.83 #sec\n",
"tp2 = 1.68 \t\t\t#seconds\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.20\n",
"#Calculating the length of equivalent simple pendulum when suspended from the top of small end bearing\n",
"L1 = 9.81*(tp1/(2*math.pi))**2 \t\t\t#m\n",
"#Calculating the length of equivalent simple pendulum when suspended from the top of big end bearing\n",
"L2 = 9.81*(tp2/(2*math.pi))**2 \t\t\t#m\n",
"#Radius of gyration of the rod about an axis pasmath.sing through the centre of gravity and perpendicular to the plane of oscillation:\n",
"#Calculating the distances of centre of gravity from the top of small end and big end bearings\n",
"#We have h1*(L1-h1) = h2*(L2-h2) or h1**2-h2**2+h2*L2-h1*L1 = 0 .....(i)\n",
"#Also h1+h2 = d1/2+l+d2/2 or h1+h2-d1/2-l-d2/2 = 0 .....(ii)\n",
"def f(x):\n",
" y = [0,0]\n",
" h1 = x[0]\n",
" h2 = x[1]\n",
" y[0] = h1**2-h2**2+h2*L2-h1*L1\n",
" y[1] = h1+h2-d1/2-l-d2/2\n",
" return y\n",
"\n",
"z = fsolve(f,[1,1])\n",
"h1 = z[0]\n",
"h2 = z[1] \t\t\t#m\n",
"\n",
"#Calculating the required radius of gyration of the rod\n",
"kG = math.sqrt(h1*(L1-h1)) \t\t\t#m\n",
"#Calculating the moment of inertia of the rod\n",
"I = m*(kG)**2 \t\t\t#kg-m**2\n",
"#Dynamically equivalent system for the rod:\n",
"#Calculating the distance of the mass situated at the centre of small end bearing from the centre of gravity\n",
"l1 = h1-d1/2 \t\t\t#m\n",
"#Calculating the distance of the second mass from the centre of gravity towards big end bearing\n",
"l2 = (kG)**2/l1 \t\t\t#m\n",
"#Calculating the magnitude of the mass situated at the centre of small end bearing\n",
"m1 = (l2*m)/(l1+l2) \t\t\t#kg\n",
"#Calculating the magnitude of the second mass\n",
"m2 = (l1*m)/(l1+l2) \t\t\t#kg\n",
"\n",
"#Results:\n",
"print \" Radius of gyration of the rod about an axis passing through the centre of\\\n",
" gravity and perpendicular to the plane of oscillation, kG = %.3f m.\"%(kG)\n",
"print \" Moment of inertia of the rod, I = %.2f kg-m**2.\"%(I)\n",
"print \" Magnitude of the mass situated at the centre of small end bearing, m1 = %.2f kg.\"%(m1)\n",
"print \" Magnitude of the second mass, m2 = %.2f kg.\"%(m2)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Radius of gyration of the rod about an axis passing through the centre of gravity and perpendicular to the plane of oscillation, kG = 0.345 m.\n",
" Moment of inertia of the rod, I = 6.56 kg-m**2.\n",
" Magnitude of the mass situated at the centre of small end bearing, m1 = 13.32 kg.\n",
" Magnitude of the second mass, m2 = 41.68 kg.\n"
]
}
],
"prompt_number": 28
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.18 Page No : 550"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"m = 2. \t\t\t#kg\n",
"l = 250./1000 #m\n",
"l1 = 100./1000 #m\n",
"kG = 110./1000 \t\t\t#m\n",
"alpha = 23000. \t\t\t#rad/s**2\n",
"\n",
"#Solution:\n",
"#Equivalent dynamical system:\n",
"#Calculating the distance of the second mass from the centre of gravity\n",
"l2 = (kG)**2/l1 \t\t\t#m\n",
"#Calculating the magnitude of the mass placed at the gudgeon pin\n",
"m1 = (l2*m)/(l1+l2) \t\t\t#kg\n",
"#Calculating the magnitude of the mass placed at a distance l2 from centre of gravity\n",
"m2 = (l1*m)/(l1+l2) \t\t\t#kg\n",
"#Correction couple:\n",
"#Calculating the magnitude of l3\n",
"l3 = l-l1 \t\t\t#m\n",
"#Calculating the new radius of gyration\n",
"k1 = math.sqrt(l1*l3) \t\t\t#m**2\n",
"#Calculating the correction couple\n",
"Tdash = m*(k1**2-kG**2)*alpha \t\t\t#N-m\n",
"\n",
"#Results:\n",
"print \" Mass placed at the gudgeon pin, m1 = %.1f kg.\"%(m1)\n",
"print \" Mass placed at a distance %.3f m from the centre of gravity m2 = %.1f kg.\"%(l2,m2)\n",
"print \" Correction couple Tdash = %.1f N-m.\"%(Tdash)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Mass placed at the gudgeon pin, m1 = 1.1 kg.\n",
" Mass placed at a distance 0.121 m from the centre of gravity m2 = 0.9 kg.\n",
" Correction couple Tdash = 133.4 N-m.\n"
]
}
],
"prompt_number": 30
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.19 Page No : 554"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"r = 125. #mm\n",
"OC = r #mm\n",
"l = 500. #mm\n",
"PC = l #mm\n",
"PG = 275. #mm\n",
"kG = 150. \t\t\t#mm\n",
"mC = 60. \t\t\t#kg\n",
"N = 600. \t\t\t#rpm\n",
"theta = 45. \t\t\t#degrees\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.24\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Acceleration of the piston:\n",
"#By measurement\n",
"NO = 90./1000 \t\t\t#m\n",
"#Calculating the acceleration of the piston\n",
"aP = omega**2*NO \t\t\t#m/s**2\n",
"#The magnitude position and direction of inertia force due to the mass of the connecting rod:\n",
"#By measurement\n",
"gO = 103./1000 \t\t\t#m\n",
"#Calculating the magnitude of the inertia force of the connecting rod\n",
"FC = mC*omega**2*gO/1000 \t\t\t#kN\n",
"\n",
"#Results:\n",
"print \" Acceleration of the piston aP = %.1f m/s**2.\"%(aP)\n",
"print \" The magnitude of inertia force due to the mass of the connecting rod FC = %.1f kN.\"%(FC)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Acceleration of the piston aP = 355.3 m/s**2.\n",
" The magnitude of inertia force due to the mass of the connecting rod FC = 24.4 kN.\n"
]
}
],
"prompt_number": 32
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.20 Page No : 555"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"D = 240./1000 #m\n",
"L = 600./1000 #m\n",
"r = L/2 #m\n",
"l = 1.5 #m\n",
"GC = 500./1000 #m\n",
"kG = 650./1000 \t #m\n",
"mR = 300.\n",
"mC = 250. \t\t\t#kg\n",
"N = 125. \t\t\t#rpm\n",
"theta = 30. \t\t\t#degrees\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.25\n",
"#Calculating the angular speed of the crank\n",
"omega = round(2*math.pi*N/60,1) \t\t\t#rad/s\n",
"#Analytical method:\n",
"#Calculating the distance of centre of gravity of the connecting rod from P\n",
"l1 = l-GC \t\t\t#m\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the inertia force due to total mass of the reciprocating parts at P\n",
"FI = int((mR+(l-l1)/l*mC)*omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n)) \t\t\t#N\n",
"#Calculating the corresponding torque due to FI\n",
"TI = int(FI*r*(math.sin(math.radians(theta))+math.sin(math.radians(2*theta))/ \\\n",
"(2*math.sqrt(n**2-(math.sin(math.radians(theta)))**2)))) \t\t\t#N-m\n",
"#Calculating the equivalent length of a simple pendulum when swung about an axis through P\n",
"L = ((kG)**2+(l1)**2)/l1 \t\t\t#m\n",
"#Calculating the correcting torque\n",
"TC = mC*l1*(l-L) *(omega**2 * math.sin(math.radians(60)))/(2*n**2)\t\t\t#N-m\n",
"#Calculating the torque due to the weight of the connecting rod at C\n",
"TW = mC*9.81*(l1/n)*math.cos(math.radians(theta)) \t\t\t#N-m\n",
"#Calculating the total torque exerted on the crankshaft\n",
"Tt = TI+TC+TW \t\t\t#Total torque exerted on the crankshaft N-m\n",
"\n",
"\n",
"#Results:\n",
"print \" Total torque exerted on the crankshaft = %.1f N-m.\"%(round(Tt,-1))\n",
"\n",
"# rounding off error"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Total torque exerted on the crankshaft = 3840.0 N-m.\n"
]
}
],
"prompt_number": 110
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.21 Page No : 558"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"N = 1200. \t\t\t#rpm\n",
"L = 110./1000\n",
"r = L/2\n",
"l = 250./1000\n",
"PC = l\n",
"CG = 75./1000 \t\t\t#m\n",
"mC = 1.25 \t\t\t#kg\n",
"theta = 40. \t\t\t#degrees\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.26\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#Radius of gyration of the connecting rod about an axis through its mass centre:\n",
"#Calculating the distance of the centre of gravity from the point of suspension\n",
"l1 = l-CG \t\t\t#m\n",
"PG = l1\n",
"#Calculating the frequency of oscillation\n",
"n = 21./20 \t\t\t#Hz\n",
"#Calculating the radius of gyration of the connecting rod about an axis through its mass centre\n",
"kG = round(math.sqrt((9.81*l1/(2*math.pi*n)**2)-l1**2)*1000) \t\t\t#mm\n",
"#Acceleration of the piston:\n",
"#Calculating the ratio of lengths of the connecting rod and crank\n",
"n = l/r\n",
"#Calculating the acceleration of the piston\n",
"aP = omega**2*r*(math.cos(math.radians(theta))+math.cos(math.radians(2*theta))/n) \t\t\t#m/s**2\n",
"#Calculating the angular acceleration of the connecting rod\n",
"alphaPC = (-omega**2*math.sin(math.radians(theta)))/n \t\t\t#rad/s**2\n",
"#Inertia torque exerted on the crankshaft:\n",
"#Calculating the mass of the connecting rod at P\n",
"m1 = (l-l1)/l*mC \t\t\t#kg\n",
"#Calculating the vertical inertia force\n",
"FI = round(m1*aP) \t\t\t#N\n",
"#By measurement\n",
"OM = 0.0425\n",
"NC = 0.035 \t\t\t#m\n",
"#Calculating the corresponding torque due to FI\n",
"TI = FI*OM \t\t\t#N-m\n",
"#Calculating the equivalent length of a simple pendulum when swung about an axis pasmath.sing through P\n",
"L = ((kG/1000)**2+(l1)**2)/l1 \t\t\t#m\n",
"#Calculating the correction couple\n",
"Tdash = mC*l1*(l-L)*alphaPC \t\t\t#N-m\n",
"#Calculating the corresponding torque on the crankshaft\n",
"TC = -Tdash*math.cos(math.radians(theta))/n \t\t\t#N-m\n",
"#Calculating the torque due to mass at P\n",
"TP = m1*9.81*OM \t\t\t#N-m\n",
"#Calculating the equivalent mass of the connecting rod at C\n",
"m2 = mC*(l1/l) \t\t\t#kg\n",
"#Calculating the torque due to mass at C\n",
"TW = m2*9.81*NC \t\t\t#N-m\n",
"#Calculating the inertia force exerted on the crankshaft\n",
"Ti = TI+TC-TP-TW \t\t\t#Inertia torque exerted on the crankshaft N-m\n",
"\n",
"#Results:\n",
"print \" Radius of gyration of the connecting rod about an axis through its mass centre kG = %d mm.\"%(kG)\n",
"print \" Acceleration of the piston aP = %.1f m/s**2.\"%(aP)\n",
"print \" Angular acceleration of the connecting rod alphaPC = %.1f rad/s**2.\"%(alphaPC)\n",
"print \" Inertia torque exerted on the crankshaft = %.3f N-m.\"%(Ti)\n",
"\n",
"# rounding off error"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Radius of gyration of the connecting rod about an axis through its mass centre kG = 94 mm.\n",
" Acceleration of the piston aP = 698.5 m/s**2.\n",
" Angular acceleration of the connecting rod alphaPC = -2233.1 rad/s**2.\n",
" Inertia torque exerted on the crankshaft = 12.696 N-m.\n"
]
}
],
"prompt_number": 112
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 15.22 Page No : 559"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math \n",
"\n",
"# Variables:\n",
"l = 225./1000 #m\n",
"PC = l #m\n",
"L = 150./1000 #m\n",
"r = L/2 #m\n",
"D = 112.5/1000 #m\n",
"PG = 150./1000 #m\n",
"kG = 87.5/1000 \t #m\n",
"mC = 1.6\n",
"mR = 2.4 \t\t\t#kg\n",
"theta = 40 \t\t\t#degrees\n",
"p = 1.8*10**6 \t\t#N/m**2\n",
"N = 2000. \t\t\t#rpm\n",
"\n",
"#Solution:\n",
"#Refer Fig. 15.27\n",
"#Calculating the angular speed of the crank\n",
"omega = 2*math.pi*N/60 \t\t\t#rad/s\n",
"#By measurement\n",
"NO = 0.0625\n",
"gO = 0.0685\n",
"IC = 0.29\n",
"IP = 0.24\n",
"IY = 0.148\n",
"IX = 0.08 \t\t\t#m\n",
"#Calculating the force due to gas pressure\n",
"FL = math.pi/4*D**2*p \t\t\t#N\n",
"#Calculating the inertia force due to mass of the reciprocating parts\n",
"FI = mR*omega**2*NO \t\t\t#N\n",
"#Calculating the net force on the piston\n",
"FP = FL-FI \t\t\t#N\n",
"#Calculating the inertia force due to mass of the connecting rod\n",
"FC = mC*omega**2*gO \t\t\t#N\n",
"#Calculating the force acting perpendicular to the crank OC\n",
"FT = ((FP*IP)-((mC*9.81*IY)+(FC*IX)))/IC \t\t\t#N\n",
"#By measurement\n",
"FN = 3550.\n",
"FR = 7550.\n",
"FQ = 13750. \t\t\t#N\n",
"\n",
"#Results:\n",
"print \" Resultant force on the crank pin FQ = %d N.\"%(FQ)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" Resultant force on the crank pin FQ = 13750 N.\n"
]
}
],
"prompt_number": 36
}
],
"metadata": {}
}
]
}