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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 7 Application of friction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.1 Application of Friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The angle of lap for the larger pulley is 203.073918 degree\n",
+ "The angle of lap for the smaller pulley is 156.926082 degree\n",
+ "The length of pulley required is 117.748668 cm\n"
+ ]
+ }
+ ],
+ "source": [
+ "import math\n",
+ "# Initilization of variables\n",
+ "d1=24 # cm # diameter of larger pulley\n",
+ "d2=12 # cm # diameter of smaller pulley\n",
+ "d=30 #cm # seperation betweem 1st & the 2nd pulley\n",
+ "# Calcuations\n",
+ "r1=d1/2 #cm # radius of 1st pulley\n",
+ "r2=d2/2 #cm # radius of 2nd pulley\n",
+ "theta=math.degrees(math.asin((r1-r2)/d)) #degrees \n",
+ "# Angle of lap\n",
+ "beta_1=180+(2*theta) #degree # for larger pulley\n",
+ "beta_2=180-(2*theta) #degree #for smaller pulley\n",
+ "L=math.pi*(r1+r2)+(2*d)+((r1-r2)**2/d) #cm # Length of the belt\n",
+ "# Results\n",
+ "print('The angle of lap for the larger pulley is %f degree'%beta_1)\n",
+ "print('The angle of lap for the smaller pulley is %f degree'%beta_2)\n",
+ "print('The length of pulley required is %f cm'%L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.2 Application of Friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The angle of lap for the pulley is 194.774097 degree\n",
+ "The length of pulley required is 8.420145 m\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "d1=0.6 #m # diameter of larger pulley\n",
+ "d2=0.3 #m # diameter of smaller pulley\n",
+ "d=3.5 #m # separation between the pulleys\n",
+ "# Calculations\n",
+ "r1=d1/2 #m # radius of larger pulley\n",
+ "r2=d2/2 #m # radius of smaller pulley\n",
+ "theta=math.degrees(math.asin((r1+r2)/d)) #degree\n",
+ "# Angle of lap for both the pulleys is same, i.e\n",
+ "beta=180+(2*theta) # degree\n",
+ "L=((math.pi*(r1+r2))+(2*d)+((r1-r2)**2/d)) #cm # Length of the belt\n",
+ "# Results\n",
+ "print('The angle of lap for the pulley is %f degree'%beta)\n",
+ "print('The length of pulley required is %f m'%L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.4 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The minimum weight W2 to keep W1 in equilibrium is 455.975258 N\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "W1=1000 #N\n",
+ "mu=0.25 #coefficient of friction\n",
+ "T1=W1 # Tension in the 1st belt carrying W1\n",
+ "e=2.718 #constant\n",
+ "# Calculations\n",
+ "T2=T1/(e**(mu*math.pi)) #N # Tension in the 2nd belt\n",
+ "W2=T2 #N\n",
+ "# Results\n",
+ "print('The minimum weight W2 to keep W1 in equilibrium is %f N'%W2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.5 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The braking moment (M) exerted by the vertical weight W is 42.980225 N-m\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "mu=0.5 # coefficient of friction between the belt and the wheel\n",
+ "W=100 #N\n",
+ "theta=45 #degree\n",
+ "e=2.718\n",
+ "Lac=0.75 #m # ength of the lever\n",
+ "Lab=0.25 #m\n",
+ "Lbc=0.50 #m\n",
+ "r=0.25 #m\n",
+ "# Calculations\n",
+ "beta=((180+theta)*math.pi)/180 # radian # angle of lap\n",
+ "# from eq'n 2\n",
+ "T1=(W*Lbc)/Lab #N \n",
+ "T2=T1/(e**(mu*beta)) #N # from eq'n 1\n",
+ "# consider the F.B.D of the pulley and take moment about its center, we get Braking Moment (M)\n",
+ "M=r*(T1-T2) #N-m\n",
+ "# Results\n",
+ "print('The braking moment (M) exerted by the vertical weight W is %f N-m'%M)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.6 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The force required to suport the weight of 1000 N i.e 1kN is 3.502492 N\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initiization of variables\n",
+ "W= 1000 #N # or 1kN\n",
+ "mu=0.3 # coefficient of friction between the rope and the cylinder\n",
+ "e=2.718 # constant\n",
+ "alpha=90 # degree # since 2*alpha=180 egree\n",
+ "# Calculations\n",
+ "beta=2*math.pi*3 # radian # for 3 turn of the rope\n",
+ "# Here T1 is the larger tension in that section of the rope which is about to slip\n",
+ "T1=W #N\n",
+ "F=W/e**(mu*(1/(math.sin(alpha*math.pi/180)))*(beta)) #N Here T2=F\n",
+ "# Results\n",
+ "print('The force required to suport the weight of 1000 N i.e 1kN is %f N'%F)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.7 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The number of ropes required to transmit 50 kW is 4.789906 or ~ 5\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "Pw=50 #kW\n",
+ "T_max=1500 #N\n",
+ "v=10 # m/s # velocity of rope\n",
+ "w=4 # N/m # weight of rope\n",
+ "mu=0.2 # coefficient of friction \n",
+ "g=9.81 # m/s**2 # acceleration due to gravity\n",
+ "e=2.718 # constant\n",
+ "alpha=30 # degree # since 2*alpha=60 \n",
+ "# Calcuations\n",
+ "T_e=(w*v**2)/g # N # where T_e is the centrifugal tension\n",
+ "T1=(T_max)-(T_e) #N\n",
+ "T2=T1/(e**(mu*(1/math.sin(alpha*math.pi/180))*(math.pi))) #N # From eq'n T1/T2=e^(mu*cosec(alpha)*beta)\n",
+ "P=(T1-T2)*v*(10**-3) #kW # power transmitted by a single rope\n",
+ "N=Pw/P # Number of ropes required\n",
+ "# Results\n",
+ "print('The number of ropes required to transmit 50 kW is %f or ~ %.0f'%(N,N))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.8 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The power transmitted by the cross belt drive is 1.180543 kW\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "d1=0.45 #m # diameter of larger pulley\n",
+ "d2=0.20 #m # diameter of smaller pulley\n",
+ "d=1.95 #m # separation between the pulley's\n",
+ "T_max=1000 #N # or 1kN which is the maximum permissible tension\n",
+ "mu=0.20 # coefficient of friction\n",
+ "N=100 # r.p.m # speed of larger pulley\n",
+ "e=2.718 # constant\n",
+ "T_e=0 #N # as the data for calculating T_e is not given we assume T_e=0\n",
+ "# Calculations\n",
+ "r1=d1/2 #m # radius of larger pulley\n",
+ "r2=d2/2 #m # radius of smaller pulley\n",
+ "theta=math.degrees(math.asin((r1+r2)/d)) # degree\n",
+ "# for cross drive the angle of lap for both the pulleys is same\n",
+ "beta=((180+(2*(theta)))*math.pi)/180 #radian\n",
+ "T1=T_max-T_e #N\n",
+ "T2=T1/(e**(mu*(beta))) #N # from formulae, T1/T2=e**(mu*beta)\n",
+ "v=((2*math.pi)*N*r1)/60 # m/s # where v=velocity of belt which is given as, v=wr=2*pie*N*r/60\n",
+ "P=(T1-T2)*v*(10**-3) #kW # Power\n",
+ "# Results\n",
+ "print('The power transmitted by the cross belt drive is %f kW'%P)\n",
+ "#answer given in the textbook is incorrect"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.9 Belt friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The initial power transmitted is 14.808043 kW\n",
+ "The initial tension in the belt is 682.223893 N\n",
+ "The maximum power that can be transmitted is 15.020439 kW\n",
+ "The maximum power is transmitted at a belt speed of 24.343225 m/s\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variabes\n",
+ "b=0.1 #m #width of the belt\n",
+ "t=0.008 #m #thickness of the belt\n",
+ "v=26.67 # m/s # belt speed\n",
+ "beta=165 # radian # angle of lap for the smaller belt\n",
+ "mu=0.3 # coefficient of friction\n",
+ "sigma_max=2 # MN/m**2 # maximum permissible stress in the belt\n",
+ "m=0.9 # kg/m # mass of the belt\n",
+ "g=9.81 # m/s**2\n",
+ "e=2.718 # constant\n",
+ "# Calculations\n",
+ "A=b*t # m**2 # cross-sectional area of the belt\n",
+ "T_e=m*v**2 # N # where T_e is the Centrifugal tension\n",
+ "T_max=(sigma_max)*(A)*(10**6) # N # maximum tension in the belt\n",
+ "T1=(T_max)-(T_e) # N \n",
+ "T2=T1/(e**((mu*math.pi*beta)/180)) #N # from formulae T1/T2=e**(mu*beta)\n",
+ "P=(T1-T2)*v*(10**-3) #kW # Power transmitted\n",
+ "T_o=(T1+T2)/2 # N # Initial tension\n",
+ "# Now calculations to transmit maximum power\n",
+ "Te=T_max/3 # N # max tension\n",
+ "u=math.sqrt(T_max/(3*m)) # m/s # belt speed for max power\n",
+ "T_1=T_max-Te # N # T1 for case 2\n",
+ "T_2=T_1/(e**((mu*math.pi*beta)/180)) # N \n",
+ "P_max=(T_1-T_2)*u*(10**-3) # kW # Max power transmitted\n",
+ "# Results\n",
+ "print('The initial power transmitted is %f kW'%P)\n",
+ "print('The initial tension in the belt is %f N'%T_o)\n",
+ "print('The maximum power that can be transmitted is %f kW'%P_max)\n",
+ "print('The maximum power is transmitted at a belt speed of %f m/s'%u)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.10 Friction in a square threaded screw"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The force required (i.e F1) to raise the weight is 1.208561 kN\n",
+ "The force required (i.e F2) to lower the weight is -0.794732 kN\n",
+ "The efficiency of the jack is 16.461202 percent\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variables\n",
+ "p=0.0125 # m # pitch of screw\n",
+ "d=0.1 #m # diameter of the screw\n",
+ "r=0.05 #m # radius of the screw\n",
+ "l=0.5 #m # length of the lever\n",
+ "W=50 #kN # load on the lever\n",
+ "mu=0.20 # coefficient of friction \n",
+ "# Calculations\n",
+ "theta=math.degrees(math.atan(p/(2*math.pi*r))) #degree # theta is the Helix angle\n",
+ "phi=math.degrees(math.atan(mu)) # degree # phi is the angle of friction\n",
+ "# Taking the leverage due to handle into account,force F1 required is,\n",
+ "a=theta+phi\n",
+ "b=theta-phi\n",
+ "F1=(W*(math.tan(a*math.pi/180)))*(r/l) #kN\n",
+ "# To lower the load\n",
+ "F2=(W*(math.tan(b*math.pi/180)))*(r/l) #kN # -ve sign of F2 indicates force required is in opposite sense\n",
+ "E=(math.tan(theta*math.pi/180)/math.tan((theta+phi)*math.pi/180))*100 # % # here E=eata=efficiency in %\n",
+ "# Results\n",
+ "print('The force required (i.e F1) to raise the weight is %f kN'%F1)\n",
+ "print('The force required (i.e F2) to lower the weight is %f kN'%F2)\n",
+ "print('The efficiency of the jack is %f percent'%E)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Example 7.11 Application of Friction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The frictional torque is 240.000000 N-m\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Initilization of variabes\n",
+ "P=20000 #N #Weight of the shaft\n",
+ "D=0.30 #m #diameter of the shaft\n",
+ "R=0.15 #m #radius of the shaft\n",
+ "mu=0.12 # coefficient of friction\n",
+ "# Calculations\n",
+ "# Friction torque T is given by formulae,\n",
+ "T=(2/3)*P*R*mu #N-m\n",
+ "M=T #N-m\n",
+ "# Results\n",
+ "print('The frictional torque is %f N-m'%M)"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.5.1"
+ },
+ "widgets": {
+ "state": {},
+ "version": "1.1.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
generated by cgit (git 2.34.1) at 2024-12-21 08:34:53 +0000