# Chapter4-Gears and Gear Drivers¶

## Ex1-pg133¶

In [2]:
##Chapter-4, Illustration 1, Page 133
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
TA=48.;##Wheel A teeth
TB=30.;##Wheel B teeth
m=5.;##Module pitch in mm
phi=20.;##Pressure angle in degrees

##CALCULATIONS
R=(m*TA)/2.;##Pitch circle radius of wheel A in mm
r=(m*TB)/2.;##Pitch circle radius of wheel B in mm
lp=(math.sqrt((RA**2.)-((R**2.)*(math.cos(phi/57.3)**2.))))+(math.sqrt((rA**2.)-((r**2.)*(math.cos(phi/57.3)**2.))))-((R+r)*math.sin(phi/57.3));##Length of path of contact in mm
la=lp/math.cos(phi/57.3);##Length of arc of contact in mm

##OUTPUT
print'%s %.1f %s'%('Length of arc of contact is ',la,' mm')

##================================END OF PROGRAM=============================================

Length of arc of contact is  26.7  mm


## Ex2-pg133¶

In [4]:
##Chapter-4, Illustration 2, Page 133
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
TA=40.;##Wheel A teeth
TB=TA;##Wheel B teeth
m=6.;##Module pitch in mm
phi=20.;##Pressure angle in degrees
pi=3.141
x=1.75;##Ratio of length of arc of contact to circular pitch

##CALCULATIONS
Cp=m*pi;##Circular pitch in mm
R=(m*TA)/2.;##Pitch circle radius of wheel A in mm
r=R;##Pitch circle radius of wheel B in mm
la=x*Cp;##Length of arc of contact in mm
lp=la*math.cos(phi/57.3);##Length of path of contact in mm

##OUTPUT

##================================END OF PROGRAM=============================================

Addendum of wheel is  6.1  mm


## Ex3-pg134¶

In [5]:
##Chapter-4, Illustration 3, Page 134
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
TA=48.;##Gear teeth
TB=24.;##Pinion teeth
m=6.;##Module in mm
phi=20.;##Pressure angle in degrees

##CALCULATIONS
r=(m*TB)/2.;##Pitch circle radius of pinion in mm
R=(m*TA)/2.;##Pitch circle radius of gear in mm
lp=((R+r)*math.sin(phi/57.3))/2.;##Length of path of contact in mm
la=lp/math.cos(phi/57.3);##Length of arc of contact in mm

##OUTPUT
print'%s %.1f %s  %.1f %s   %.1f %s '%('Addendum for pinion is',addp,' mm' ' Addendum for gear is ',addg,' mm' ' Length of arc of contact is ',la,' mm')

##================================END OF PROGRAM=============================================

Addendum for pinion is 11.7  mm Addendum for gear is   4.7  mm Length of arc of contact is    39.3  mm


## Ex4-pg135¶

In [6]:
##Chapter-4, Illustration 4, Page 135
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
x=3.5;##Ratio of teeth of wheels
C=1.2;##Centre distance between axes in m
DP=4.4;##Diametrical pitch in cm

##CALCULATIONS
D=2*C*100.;##Sum of diameters of wheels in cm
T=D*DP;##Sum of teeth of wheels
TB1=T/(x+1);##Teeth of wheel B
TB=math.floor(TB1);##Teeth of whhel B
TA=x*TB;##Teeth of wheel A
DA=TA/DP;##Diametral pitch of gear A in cm
DB=TB/DP;##Diametral pitch of gear B in cm
Ce=(DA+DB)/2.;##Exact centre distance between shafts in cm
TB2=math.ceil(TB1);##Teeth of wheel B
TA2=T-TB2;##Teeth of wheel A
VR=TA2/TB2;##Velocity ratio

##OUTPUT
print'%s %.1f %s %.1f %s %.1f %s%.1f %s%.1f %s'%('Number of teeth on wheel A is ',TA,''  'Number of teeth on wheel B is ',TB,''  '  Exact centre distance is ',Ce,' cm '  'If centre distance is ',C,' m' 'then  Velocity ratio is',VR,'')

##================================END OF PROGRAM=============================================

Number of teeth on wheel A is  819.0 Number of teeth on wheel B is  234.0   Exact centre distance is  119.7  cm If centre distance is 1.2  mthen  Velocity ratio is3.5


## Ex5-pg136¶

In [7]:
##Chapter-4, Illustration 5, Page 136
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
C=600;##Distance between shafts in mm
Cp=30;##Circular pitch in mm
NA=200;##Speed of wheel A in rpm
NB=600;##Speed of wheel B in rpm
F=18;##Tangential pressure in kN
pi=3.141

##CALCULATIONS
a=Cp/(pi*10.);##Ratio of pitch diameter of wheel A to teeth of wheel A in cm
b=Cp/(pi*10.);##Ratio of pitch diameter of wheel B to teeth of wheel B in cm
T=(2*C)/(a*10.);##Sum of teeth of wheels
r=NB/NA;##Ratio of teeth of wheels
TB=T/(r+1);##Teeth of wheel B
TB1=math.ceil(TB);##Teeth of wheel B
TA=TB1*r;##Teeth of wheel A
DA=a*TA;##Pitch diameter of wheel A in cm
DB=b*TB1;##Pitch diameter of wheel B in cm
CPA=(pi*DA)/TA;##Circular pitch of gear A in cm
CPB=(pi*DB)/TB1;##Circular pitch of gear B in cm
C1=(DA+DB)*10/2.;##Exact centre distance in mm
P=(F*1000.*pi*DA*NA)/(60.*1000.*100.);##Power transmitted in kW

##OUTPUT
print'%s %.1f %s %.1f %s %.1f %s %.1f %s %.1f %s %.1f %s %.1f %s %.1f %s '%('Number of teeth on wheel A is ',TA,' '' Number of teeth on wheel B is ',TB1,' '' Pitch diameter of wheel A is ',DA,' cm'' Pitch diameter of wheel B is ',DB,' cm'' Circular pitch of wheel A is',CPA,'cm ' 'Circular pitch of wheel B is ',CPB,' cm '' Exact centre distance between shafts is ',C1,' mm'' Power transmitted is',P,' kW')
##================================END OF PROGRAM=============================================

Number of teeth on wheel A is  96.0   Number of teeth on wheel B is  32.0   Pitch diameter of wheel A is  91.7  cm Pitch diameter of wheel B is  30.6  cm Circular pitch of wheel A is 3.0 cm Circular pitch of wheel B is  3.0  cm  Exact centre distance between shafts is  611.3  mm Power transmitted is 172.8  kW


## Ex6-pg137¶

In [8]:
##Chapter-4, Illustration 6, Page 137
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
r=16.;##Speed ratio
mA=4.;##Module of gear A in mm
mB=mA;##Module of gear B in mm
mC=2.5;##Mosule of gear C in mm
mD=mC;##Module of gear D in mm
C=150.;##Distance between shafts in mm

##CALCULATIONS
t=math.sqrt(r);##Ratio of teeth
T1=(C*2.)/mA;##Sum of teeth of wheels A and B
T2=(C*2.)/mC;##Sum of teeth of wheels C and D
TA=T1/(t+1.);##Teeth of gear A
TB=T1-TA;##Teeth of gear B
TC=T2/(t+1.);##Teeth of gear C
TD=T2-TC;##Teeth of gear D

##OUTPUT
print'%s %.1f %s %.1f %s %.1f %s %.1f %s '%('Number of teeth on gear A is ',TA,' '' Number of teeth on gear B is ',TB,'' 'Number of teeth on gear C is ',TC,'' ' Number of teeth on gear D is ',TD,'')

##================================END OF PROGRAM=============================================

Number of teeth on gear A is  15.0   Number of teeth on gear B is  60.0 Number of teeth on gear C is  24.0  Number of teeth on gear D is  96.0


## Ex7-pg138¶

In [9]:
##Chapter-4, Illustration 7, Page 138
##Title: Gears and Gear Drivers
##=============================================================================
import math

##INPUT DATA
N=4.5;##No. of turns

##CALCULATIONS
Vh=N/2.;##Velocity ratio of main spring spindle to hour hand spindle
Vm=12.;##Velocity ratio of minute hand spindle to hour hand spindle
T1=8.##     assumed no of teeth on gear 1
T2=32.##    assumed no of teeth on gear 2
T3=(T1+T2)/4.##       no of teeth on gear 3
T4=(T1+T2)-T3## no of teeth on gear 4
print'%s %.1f %s %.1f %s %.1f %s %.1f %s '%('no of teeth on gear 1=',T1,'' 'no of teeth on gear 2=',T2,' ''no of teeth on gear 3=',T3,' ''no of teeth on gear 4=',T4,'')

no of teeth on gear 1= 8.0 no of teeth on gear 2= 32.0  no of teeth on gear 3= 10.0  no of teeth on gear 4= 30.0


## Ex8-pg139¶

In [10]:
##Chapter-4, Illustration 8, Page 139
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
Tb=70.;##Teeth of wheel B
Tc=25.;##Teeth of wheel C
Td=80.;##Teeth of wheel D
Na=-100.;##Speed of arm A in clockwise in rpm
y=-100.##Arm A rotates at 100 rpm clockwise

##Calculations
Te=(Tc+Td-Tb);##Teeth of wheel E
x=(y/0.5)
Nc=(y-(Td*x)/Tc);##Speed of wheel C in rpm

##Output
print'%s %.1f %s'%('Speed of wheel C is ',Nc,' rpm ''Direction of wheel C is anti-clockwise')

##================================END OF PROGRAM=============================================

Speed of wheel C is  540.0  rpm Direction of wheel C is anti-clockwise


## Ex9-pg140¶

In [12]:
##Chapter-4, Illustration 9, Page 140
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
Tb=25.;##Teeth of wheel B
Tc=40.;##Teeth of wheel C
Td=10.;##Teeth of wheel D
Te=25.;##Teeth of wheel E
Tf=30.;##Teeth of wheel F
y=-120.;##Speed of arm A in clockwise in rpm

##Calculations
x=(-y/4.)
Nb=x+y;##Speed of wheel B in rpm
Nf=(-10/3.)*x+y;##Speed of wheel F in rpm

##Output
print'%s %.1f %s %.1f %s'%('Speed of wheel B is',Nb,' rpm  Direction of wheel B is clockwise' ' Speed of wheel F is ',Nf,' rpm Direction of wheel F is clockwise')

##================================END OF PROGRAM=============================================

Speed of wheel B is -90.0  rpm  Direction of wheel B is clockwise Speed of wheel F is  -220.0  rpm Direction of wheel F is clockwise


## Ex10-pg141¶

In [13]:
##Chapter-4, Illustration 10, Page 141
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
Ta=96.;##Teeth of wheel A
Tc=48.;##Teeth of wheel C
y=-20.;##Speed of arm C in rpm in clockwise

##Calculations
x=(y*Ta)/Tc
Tb=(Ta-Tc)/2.;##Teeth of wheel B
Nb=(-Tc/Tb)*x+y;##Speed of wheel B in rpm
Nc=x+y;##Speed of wheel C in rpm

##Output
print'%s %.1f %s %.1f %s'%('Speed of wheel B is ',Nb,' rpm' 'Speed of wheel C is ',Nc,' rpm')

##================================END OF PROGRAM=============================================

Speed of wheel B is  60.0  rpmSpeed of wheel C is  -60.0  rpm


## Ex11-pg142¶

In [14]:
##Chapter-4, Illustration 11, Page 142
##Title: Gears and Gear Drivers
##=============================================================================
import math
import numpy
from numpy.linalg import inv
##Input data
Ta=40.##       no of teeth on gear A
Td=90.##       no of teeth on gear D

##Calculations
Tb=(Td-Ta)/2.##      no of teeth on gear B
Tc=Tb##           no of teeth on gear C
##
##x+y=-1
##-40x+90y=45

A=([[1, 1],[-Ta, Td]])##Coefficient matrix

B=([[-1],[Td/2]])##Constant matrix

X=numpy.dot(inv(A) ,B)##Variable matrix
##
##x+y=-1
##-40x+90y=0
A1=([[1, 1],[-Ta, Td]])##Coefficient matrix
B1=([[-1],[0]])##Constant matrix
X1=numpy.dot(inv(A1) ,B1)##Variable matrix
b=X1[1]
print(X[1])
print'%s %.4f %s'%('speed of the arm =',b,' revolution clockwise')

[ 0.03846154]
speed of the arm = -0.3077  revolution clockwise


## Ex12-pg144¶

In [15]:
##Chapter-4, Illustration 12, Page 144
##Title: Gears and Gear Drivers
##=============================================================================

##Input data
Te=30.;##Teeth of wheel E
Tb=24.;##Teeth of wheel B
Tc=22.;##Teeth of wheel C
Td=70.;##Teeth of wheel D
Th=15.;##Teeth of wheel H
Nv=100.;##Speed of shaft V in rpm
Nx=300.;##Speed of spindle X in rpm

##Calculations
Nh=Nv;##Speed of wheel H in rpm
Ne=(-Th/Te)*Nv;##Speed of wheel E in rpm
Ta=(Tc+Td-Tb);##Teeth of wheel A
##x+y=-50
##y=300
x=(Ne-Nx)
Nz=(187/210.)*x+Nx;##;##Speed of wheel Z in rpm

##Output
print'%s %.1f %s'%('Speed of wheel Z is ',Nz,' rpm  Direction of wheel Z is opposite to that of X')

Speed of wheel Z is  -11.7  rpm  Direction of wheel Z is opposite to that of X


## Ex13-pg145¶

In [16]:
##Chapter-4, Illustration 13, Page 145
##Title: Gears and Gear Drivers
##=============================================================================

##Input data
Tp=20.;##Teeth of wheel P
Tq=30.;##Teeth of wheel Q
Tr=10.;##Teeth of wheel R
Nx=50.;##Speed of shaft X in rpm
Na=100.;##Speed of arm A in rpm

##Calculations
##x+y=-50
##y=100
x=(-Nx-Na)
y=(-2.*x+Na);##Speed of Y in rpm

##Output
print'%s %.1f %s'%('Speed of driven shaft Y is ',y,' rpm  Direction of driven shaft Y is anti-clockwise')

Speed of driven shaft Y is  400.0  rpm  Direction of driven shaft Y is anti-clockwise


## Ex4-pg146¶

In [17]:
##Chapter-4, Illustration 14, Page 146
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
d=216.;##Ring diameter in mm
m=4.;##Module in mm

##Calculations
Td=(d/m);##Teeth of wheel D
Tb=Td/4.;##Teeth of wheel B
Tb1=math.ceil(Tb);##Teeth of wheel B
Td1=4.*Tb1;##Teeth of wheel D
Tc1=(Td1-Tb1)/2.;##Teeth of wheel C
d1=m*Td1;##Pitch circle diameter in mm

##Output
print'%s %.1f %s %.1f %s %.1f %s%.1f %s '%('Teeth of wheel B is ',Tb1,' ' 'Teeth of wheel C is ',Tc1,' ' 'Teeth of wheel D is ',Td1,' '' Exact pitch circle diameter is ',d1,' mm')

Teeth of wheel B is  14.0  Teeth of wheel C is  21.0  Teeth of wheel D is  56.0   Exact pitch circle diameter is 224.0  mm


## Ex15-pg147¶

In [18]:
##Chapter-4, Illustration 15, Page 147
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
Ta=100.##       no of teeth on gear A
Tc=101.##       no of teeth on gear C
Td=99.##        no of teeth on gear D
Tp=20.##        no of teeth on planet gear
y=1.##          from table 4.9(arm B makes one revolution)
x=-y##         as gear is fixed

##Calculations
Nc=(Ta*x)/Tc+y##            Revolution of gear C
Nd=(Ta*x)/Td+y##            Revolution of gear D

##Output
print'%s %.4f %s %.4f %s  '%('Revolution of gear C =',Nc,'' ' Revolution of gear D = ',Nd,'')

Revolution of gear C = 0.0099  Revolution of gear D =  -0.0101


## Ex16-pg148¶

In [19]:
##Chapter-4, Illustration 16, Page 148
##Title: Gears and Gear Drivers
##=============================================================================
import math

##Input data
Ta=12.##       no of teeth on gear A
Tb=60.##       no of teeth on gear B
N=1000.##      speed of propeller shaft in rpm
Nc=210.##      speed of gear C in rpm

##Calculations
Nb=(Ta*N)/Tb##     speed of gear B in rpm
x=(Nb-Nc)
Nd=Nb+x##          speed of road wheel driven by D

##Output
print'%s %.1f %s'%('speed of road wheel driven by D= ',Nd,' rpm')

speed of road wheel driven by D=  190.0  rpm


## Ex17-pg148¶

In [1]:
##Chapter-4, Illustration 17, Page 148
##Title: Gears and Gear Drivers
##=============================================================================
import math
import numpy
from numpy.linalg import inv
##Input data
Ta=20.##   no of teeth on pinion A
Tb=25.##   no of teeth on wheel B
Tc=50.##   no of teeth on gear C
Td=60.##   no of teeth on gear D
Te=60.##  no of teeth on gear E
Na=200.##   SPEED of the gear A
Nd=100.##   speed of the gear D

##calculations
##(i)
##(5/6)x+y=0
##(5/4)x+y=200
A1=([[Tc/Td, 1],[Tb/Ta, 1]])##Coefficient matrix
B1=([[0],[Na]]) ##Constant matrix
X1=numpy.dot(inv(A1),B1)##Variable matrix
Ne1=X1[1]-(Tc/Td)*X1[0]##
T1=(-Ne1/Na)##     ratio of torques when D is fixed
##(ii)
##(5/4)x+y=200
##(5/6)x+y=100
A2=([[Tc/Td, 1],[Tb/Ta, 1]])##Coefficient matrix
B2=([[Nd],[Na]])##Constant matrix
X2=numpy.dot(inv(A2),B2)##Variable matrix
Ne2=X2[1]-(Tc/Td)*X2[0]
T2=(-Ne2/Na)##      ratio of torques when D ratates at 100 rpm

##Output
print'%s %.2f %s %.2f %s %.2f %s %.2f %s'%('speed of E= ',Ne1,' rpm in clockwise direction' and  'speed of E in 2nd case(when D rotates at 100 rpm)= ',Ne2,' rpm in clockwise direction' and 'ratio of torques when D is fixed= ',T1,' ' 'ratio of torques when D ratates at 100 rpm= ',T2,'')

speed of E=  -800.00 speed of E in 2nd case(when D rotates at 100 rpm)=  -300.00 ratio of torques when D is fixed=  4.00  ratio of torques when D ratates at 100 rpm=  1.50