# Chapter 12 : Toothed Gearing¶

## Example 12.1 Page No : 393¶

In [3]:
import math

# Variables:
P = 120.*1000 			#W
d = 250./1000
r = d/2 			#m
N = 650. 			#rpm
phi = 20. 			#degrees

#Solution:
#Calculating the angular speed of the gear
#Calculating the torque transmitted
T = P/omega 			#N-m
#Calculating the math.tangential load on the pinion
FT = T/r 			#N
#Calculating the total load due to power transmitted

#Results:
print " Total load due to power transmitted, F  =  %.2f kN."%(F)

# rounding off error

 Total load due to power transmitted, F  =  15.01 kN.


## Example 12.2 Page No : 397¶

In [2]:
import math

# Variables:
T = 40.
t = T
phi = 20. 			#degrees
m = 6.   			#mm

#Solution:
#Calculating the circular pitch
pc = math.pi*m 			#mm
#Calculating the length of arc of contact
Lac = 1.75*pc 			#Length of arc of contact mm
#Calculating the length of path of contact
Lpc = Lac*math.cos(phi) 			#Length of path of contact mm
#Calculating the pitch circle radii of each wheel
R = m*T/2 			#mm
r = R 			#mm
RA = math.sqrt(R**2*(math.cos(phi))**2+(Lpc/2+R*math.sin(phi))**2) 			#mm
#Calculating the addendum of the wheel

#Results:

 Addendum of the wheel  =  6.17 mm.


## Example 12.3 Page No : 398¶

In [10]:
import math

# Variables:
t = 30.
T = 80.
phi = 20. 			#degrees
m = 12. 			#mm

#Solution:
#Length of path of contact:
#Calculating the pitch circle radius of pinion
r = m*t/2 			#mm
#Calculating the pitch circle radius of gear
R = m*T/2 			#mm
#Calculating the length of path of approach
#Refer Fig. 12.11
#Calculating the length of path of recess
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Calculating the length of arc of contact
Lac = KL/math.cos(math.radians(phi)) 			#Length of arc of contact mm
#Contact ratio:
#Calculating the circular pitch
Pc = math.pi*m 			#mm
#Calculating the contact ratio
CR = Lac/Pc 			#Contact ratio

#Results:
print " Length of path of contact, KL  =  %.1f mm."%(KL)
print " Length of arc of contact  =  %.2f mm."%(Lac)
print " Contact ratio  =  %.1f."%(CR)

 Length of path of contact, KL  =  52.3 mm.
Length of arc of contact  =  55.61 mm.
Contact ratio  =  1.5.


## Example 12.4 Page No : 399¶

In [11]:
import math

# Variables:
phi = 20. 			#degrees
t = 20.
G = 2.
m = 5. 			#mm
v = 1.2 			#m/s

#Solution:
#Angle turned through by pinion when one pair of teeth is in mesh:
#Calculating the pitch circle radius of pinion
r = m*t/2 			#mm
#Calculating the pitch circle radius of wheel
R = m*G*t/2 			#mm
#Calculating the length of path of approach
#Calculating the length of path of recess
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Calculating the length of arc of contact
#Calculating the angle turned by the pinion
angle = Lac*360/(2*math.pi*r) 			#Angle turned by the pinion degrees
#Maximum velocity of sliding:
#Calculating the angular speed of pinion
#Calculating the angular speed of wheel
#Calculating the maximum velocity of sliding
vS = (omega1+omega2)*KP 			#mm/s

#Results:
print " Angle turned through by pinion when one pair of teeth is in mesh  =  %.2f degrees."%(angle)
print " Maximum velocity of sliding, vS  =  %.1f mm/s."%(vS)

 Angle turned through by pinion when one pair of teeth is in mesh  =  29.43 degrees.
Maximum velocity of sliding, vS  =  455.3 mm/s.


## Example 12.5 Page No : 400¶

In [1]:
import math

# Variables:
T = 40.
t = 20.
N1 = 2000. 			#rpm
phi = 20. 			#degrees
m = 5. 			#mm

#Solution:
#Calculating the angular velocity of the smaller gear
#Calculating the angular velocity of the larger gear
#Calculating the pitch circle radius of the smaller gear
r = m*t/2 			#mm
#Calculating the pitch circle radius of the larger gear
R = m*T/2 			#mm
#Calculating the length of path of approach
#Calculating the length of path of recess
#Calculating the velocity of sliding at the point of engagement
vSK = (omega1+omega2)*KP 			#mm/s
#Calculating the velocity of sliding at the point of disengagement
vSL = (omega1+omega2)*PL 			#mm/s
#Angle through which the pinion turns:
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Calculating the length of arc of contact
Lac = KL/math.cos(math.radians(phi)) 			#Length of arc of contact mm
#Calculating the circumference of pinion
C = 2*math.pi*r 			#Circumference of pinion mm
#Calculating the angle through which the pinion turns
angle = Lac*360/C 			#Angle through which the pinion turns degrees

#Results:
print " Velocity of sliding at the point of engagement, vSK  =  %.f mm/s."%(vSK)
print " Velocity of sliding at the point of disengagement, vsL  =  %.f mm/s."%(vSL)
print " Angle through which the pinion turns  =  %.2f degrees."%(angle)

# answers differ due to rounding off error

 Velocity of sliding at the point of engagement, vSK  =  3973 mm/s.
Velocity of sliding at the point of disengagement, vsL  =  3610 mm/s.
Angle through which the pinion turns  =  29.43 degrees.


## Example 12.6 Page No : 401¶

In [3]:
import math

# Variables:
phi = 20. 			#degrees
m = 6.
t = 17.
T = 49.

#Solution:
#Number of pairs of teeth in contact:
#Calculating the pitch circle radius of pinion
r = m*t/2 			#mm
#Calculating the pitch circle radius of gear
R = m*T/2 			#mm
#Calculating the length of path of approach
#Refer Fig. 12.11
#Calculating the length of path of recess
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Calculating the length of arc of contact
Lac = KL/math.cos(math.radians(phi)) 			#Length of arc of contact mm
#Calculating the circular pitch
pc = math.pi*m 			#mm
#Calculating the number of pairs of teeth in contact
n = Lac/pc 			#Number of pairs of teeth in contact
#Angle turned by the pinion and gear wheel when one pair of teeth is in contact:
#Calculating the angle turned through by the pinion
anglep = Lac*360/(2*math.pi*r) 			#Angle turned through by the pinion degrees
#Calculating the angle turned through by the wheel
angleg = Lac*360/(2*math.pi*R) 			#Angle turned through by the gear wheel degrees
#Ratio of sliding to rolling motion:
#At the instant when the tip of a tooth on the larger wheel is just making contact with its mating teeth
r1 = ((1+t/T)*KP)/r 			#Ratio of sliding velocity to rolling velocity
#At the instant when the tip of a tooth on a larger wheel is just leaving contact with its mating teeth
r2 = ((1+t/T)*PL)/r 			#Ratio of sliding velocity to rolling velocity

#Results:
print " Number of pairs of teeth in contact  =  %.f."%(n)
print " Angle turned through by the pinion  =  %.1f degrees."%(anglep)
print " Angle turned through by the gear wheel  =  %.f degrees."%(angleg)
print " At the instant when the tip of a tooth on the larger wheel is just\
making contact with its mating teeth, ratio of sliding \nvelocity to rolling velocity  =  %.2f."%(r1)
print " At the instant when the tip of a tooth on a larger wheel is just leaving contact\
with its mating teeth, ratio of sliding velocity to rolling velocity  =  %.3f."%(r2)

 Number of pairs of teeth in contact  =  2.
Angle turned through by the pinion  =  34.6 degrees.
Angle turned through by the gear wheel  =  12 degrees.
At the instant when the tip of a tooth on the larger wheel is just making contact with its mating teeth, ratio of sliding
velocity to rolling velocity  =  0.41.
At the instant when the tip of a tooth on a larger wheel is just leaving contact with its mating teeth, ratio of sliding velocity to rolling velocity  =  0.354.


## Example 12.7 Page No : 403¶

In [16]:
import math

# Variables:
t = 18.
T = 72.
phi = 20. 			#degrees
m = 4. 			    #mm

#SOlution:
#Refer Fig. 12.12
#Calculating the pitch circle radius of the pinion
r = m*t/2 			#mm
#Calculating the pitch circle radius of the gear
R = m*T/2 			#mm
#Calculating the radius of the base circle of the pinion
#Calculating the radius of the base circle of the gear
#Calculating the length of path of approach
#Calculating the length of path of recess
#Calculating the length of the path of contact
KL = KP+PL 			#mm

#Results:
print " Length of the path of contact, KL  =  %.2f mm."%(KL)

 Length of the path of contact, KL  =  28.04 mm.


## Example 12.8 Page No : 406¶

In [18]:
import math

# Variables:
t = 20.
T = 40.
m = 10. 			#mm
phi = 20. 			#degrees

#Solution:
#Addendum height for each gear wheel:
#Calculating the pitch circle radius of the smaller gear wheel
r = m*t/2 			#mm
#Calculating the pitch circle radius of the larger wheel
R = m*T/2 			#mm
#Calculating the addendum height for larger gear wheel
#Calculating the addendum height for smaller gear wheel
#Calculating the length of the path of contact
Lpc = (r+R)*math.sin(math.radians(phi))/2 			#Length of the path of contact mm
#Calculating the length of the arc of contact
Lac = Lpc/math.cos(math.radians(phi)) 			#Length of the arc of contact mm
#Contact ratio:
#Calculating the circular pitch
pc = math.pi*m 			#mm
#Calculating the contact ratio
CR = Lpc/pc 			#Contact ratio

#Results:
print " Length of the path of contact  =  %.1f mm."%(Lpc)
print " Length of the arc of contact  =  %.1f mm."%(Lac)
print " Contact ratio  =  %d."%(CR+1)

# book answer is wrong for 2nd

 Addendum height for larger gear wheel  =  6.5 mm.
Addendum height for smaller gear wheel  =  16.2 mm.
Length of the path of contact  =  51.3 mm.
Length of the arc of contact  =  54.6 mm.
Contact ratio  =  2.


## Example 12.9 Page No : 410¶

In [20]:
import math

# Variables:
G = 3.
phi = 20. 			#degrees
Aw = 1. 			#module

#Solution:
#Calculating the minimum number of teeth for a gear ratio of 3:1
#Calculating the minimum number of teeth for equal wheel

#Results:
print " Minimum number of teeth for a gear ratio of 3:1, t  =  %.f."%(t1+1)
print " Minimum number of teeth for equal wheel, t  =  %d."%(t2+1)

 Minimum number of teeth for a gear ratio of 3:1, t  =  16.
Minimum number of teeth for equal wheel, t  =  13.


## Example 12.10 Page No : 410¶

In [26]:
import math
import numpy

# Variables:
G = 4.
phi = 14.5 			#degrees

#Solution:
#Least number of teeth on each wheel:
#Calculating the least number of teeth on the pinion
#Calculating the least number of teeth on the gear
T = G*t

#Results:
print " Least number of teeth on the pinion, t  =  %.1f."%(t)
print " Least number of teeth on the gear, T  =  %.f."%(round(T,-1))

 Least number of teeth on the pinion, t  =  24.3.
Least number of teeth on the gear, T  =  100.


## Example 12.11 Page No : 411¶

In [28]:
import math

# Variables:
phi = 16. 			#degrees
m = 6. 			    #mm
t = 16.
G = 1.75
T = G*t
N1 = 240. 			#rpm

#Solution:
#Calculating the angular speed of the pinion
#Addenda on pinion and gear wheel:
#Length of path of contact:
#Calculating the pitch circle radius of wheel
R = m*T/2 			#mm
#Calculating the pitch circle radius of pinion
r = m*t/2 			#mm
#Calculating the length of path of approach
#Calculating the length of path of recess
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Maximum velocity of sliding of teeth on either side of pitch point:
#Calculating the angular speed of gear wheel
#Calculating the maximum velocity of sliding of teeth on the left side of pitch point
vmaxl = (omega1+omega2)*KP 			#Maximum velocity of sliding of teeth on the left side of pitch point mm/s
#Calculating the maximum velocity of sliding of teeth on the right side of pitch point
vmaxr = (omega1+omega2)*PL 			#Maximum velocity of sliding of teeth on the right side of pitch point mm/s

#Results:
print " Length of path of contact, KL  =  %.2f mm."%(KL)
print " Maximum velocity of sliding of teeth on the left side of pitch point  =  %d mm/s."%(vmaxl)
print " Maximum velocity of sliding of teeth on the right side of pitch point  =  %d mm/s."%(vmaxr)

# rounding error

 Addendum on pinion  =  10.76 mm.
Addendum on wheel  =  4.56 mm.
Length of path of contact, KL  =  38.39 mm.
Maximum velocity of sliding of teeth on the left side of pitch point  =  1044 mm/s.
Maximum velocity of sliding of teeth on the right side of pitch point  =  471 mm/s.


## Example 12.12 Page No : 412¶

In [30]:
import math

# Variables:
phi = 20. 			#degrees
t = 30.
T = 50.
m = 4.
N1 = 1000. 			#rpm

#Solution:
#Calculating the angular speed of thr pinion
#Sliding velocities at engagement and at disengagement of a pair of teeth:
#Calculating the addendum of the smaller gear
#Calculating the addendum of the larger gear
#Calculating the pitch circle radius of the smaller gear
r = m*t/2 			#mm
#Calculating the pitch circle radius of the larer gear
R = m*T/2 			#mm
#Calculating the path of approach
#Calculating the path of recess
#Calculating the angular speed of the larger gear
#Calculating the sliding velocity at engagement of a pair of teeth
v1 = (omega1+omega2)*KP 			#Sliding velocity at engagement of a pair of teeth mm/s
#Calculating the sliding velocity at disengagement of a pair of teeth
v2 = (omega1+omega2)*PL 			#Sliding velocity at disengagement of a pair of teeth mm/s
#Contact ratio:
#Calculating the length of the arc of contact
#Calculating the circular pitch
pc = math.pi*m 			#Circular pitch mm
#Calculating the contact ratio
CR = Lac/pc 			#Contact ratio

#Results:
print " Sliding velocity at engagement of a pair of teeth  =  %.3f m/s."%(v1/1000)
print " Sliding velocity at disengagement of a pair of teeth  =  %.3f m/s."%(v2/1000)
print " Contact ratio  =  %d."%(CR+1)

# rounding off error

 Sliding velocity at engagement of a pair of teeth  =  3.438 m/s.
Sliding velocity at disengagement of a pair of teeth  =  5.731 m/s.
Contact ratio  =  5.


## Example 12.13 Page No : 414¶

In [34]:
import math

# Variables:
G = 3.
m = 6.
AP = 1*m
AW = AP 			#mm
phi = 20. 			#degrees
N1 = 90. 			#rpm

#Solution:
#Calculating the angular speed of the pinion
#Calculating the number of teeth on the pinion to avoid interference on it
#Calculating the corresponding number of teeth on the wheel
T = G*t
#Length of path and arc of contact:
#Calculating the pitch circle radius of pinion
r = m*t/2 			#mm
rA = r+AP 			#mm
#Calculating the pitch circle radius of wheel
R = m*T/2 			#mm
RA = R+AW 			#mm
#Calculating the path of approach
#Calculating the path of recess
#Calculating the length of path of contact
KL = KP+PL 			#mm
#Calculating the length of arc of contact
Lac = KL/math.cos(math.radians(phi)) 			#Length of arc of contact mm
#Number of pairs of teeth in contact:
#Calculating the circular pitch
pc = math.pi*m 			#mm
#Calculating the number of pairs of teeth in contact
n = Lac/pc 			#Number of pairs of teeth in contact
#Maximum velocity of sliding:
#Calculating the angular speed of wheel
#Calculating the maximum velocity of sliding
vs = (omega1+omega2)*KP 			#mm/s

#Results:
print " Number of teeth on the pinion to avoid interference, t  =  %d."%(t+1)
print " Corresponding number of teeth on the wheel, T  =  %.F."%(T+1)
print " Length of path of contact, KL  =  %.2f mm."%(KL)
print " Length of arc of contact  =  %.2f mm."%(Lac)
print " Number of pairs of teeth in contact  =  %d."%(n+1)
print " Maximum velocity of sliding, vs  =  %.f mm/s."%(vs)

# ROUNDING ERROR

 Number of teeth on the pinion to avoid interference, t  =  19.
Corresponding number of teeth on the wheel, T  =  56.
Length of path of contact, KL  =  29.24 mm.
Length of arc of contact  =  31.12 mm.
Number of pairs of teeth in contact  =  2.
Maximum velocity of sliding, vs  =  197 mm/s.


## Example 12.14 Page No : 416¶

In [41]:
import math

# Variables:
T = 20.
d = 125.          #mm
r = d/2
OP = r
LH = 6.25 			#mm
#Calculating the least pressure angle to avoid interference
phi = math.sin(math.sqrt(LH/r))*180/math.pi 			#degrees
#Length of arc of contact:
#Calculating the length of path of contact
#Calculating the length of arc of contact
Lac = KL/math.cos(math.radians(phi)) 			#Length of arc of contact mm
#Minimum number of teeth:
#Calculating the circular pitch
pc = math.pi*d/T 			#mm
#Calculating the number of pairs of teeth in contact
n = Lac/pc 			#Number of pairs of teeth in contact
#Calculating the minimum number of teeth in contact
nmin = n 			#Mimimum number of teeth in contact

#Results:
print " Least pressure angle to avoid interference, phi  =  %.3f degrees."%(phi)
print " Length of arc of contact  =  %.2f mm."%(Lac)
print " Minimum number of teeth in contact  =  %d or %d pair."%(nmin+1,(nmin+1)/2)

# rounding error

 Least pressure angle to avoid interference, phi  =  17.818 degrees.
Length of arc of contact  =  36.17 mm.
Minimum number of teeth in contact  =  2 or 1 pair.


## Example 12.15 Page No : 421¶

In [45]:
from numpy import linalg
from scipy.optimize import fsolve
import math

# Variables:
L = 175./1000
d2 = 100./1000        #m
r2 = d2/2 			  #m
theta = 70. 			#degrees
G = 1.5
T2 = 80.
Tf = 75. 			#Torque on faster wheel N-m

#Solution:
#Spiral angles for each wheel:
#Calculating the number of teeth on slower wheel
T1 = T2*G
#Calculating the pitch circle diameter of the slower wheel
d1 = (L*2)-d2 			#m
#Calculating the spiral angles
#We have d2/d1  =  (T2*math.cos(alpha1))/(T1*math.cos(alpha2)) or T2*d1*math.cos(alpha1)-T1*d2*math.cos(alpha2)  =  0    .....(i)
#Also alpha1+alpha2  =  theta or alpha1+alpha2-theta  =  0                                           .....(ii)
def f(x):
alpha1 = x[0]
alpha2 = x[1]
y = [0,0]
y[0] = T2*d1*math.cos(alpha1)-T1*d2*math.cos(alpha2)
y[1] = alpha1+alpha2-theta*math.pi/180
return y

z = fsolve(f,[1,1])
alpha1 = z[0]*180/math.pi 			#Spiral angle for slower wheel degrees
alpha2 = z[1]*180/math.pi 			#Spiral angle for faster wheel degrees
#Axial thrust on each shaft:
#Calculating the math.tangential force at faster wheel
F2 = Tf/r2 			#N
#Calculating the normal reaction at the point of contact
#Calculating the axial thrust on the shaft of slower wheel
#Calculating the axial thrust on the shaft of faster wheel

#Results:
print " Spiral angle for slower wheel, alpha1  =  %.2f degrees."%(alpha1)
print " Spiral angle for faster wheel, alpha2  =  %.2f degrees."%(alpha2)
print " Axial thrust on the shaft of slower wheel, Fa1 =  %d N."%(Fa1+1)
print " Axial thrust on the shaft of faster wheel, Fa2  =  %d N."%(Fa2+1)

 Spiral angle for slower wheel, alpha1  =  54.65 degrees.
Spiral angle for faster wheel, alpha2  =  15.35 degrees.
Axial thrust on the shaft of slower wheel, Fa1 =  1269 N.
Axial thrust on the shaft of faster wheel, Fa2  =  412 N.


## Example 12.16 Page No : 422¶

In [51]:
import math

# Variables:
L = 400./1000 			#m
G = 3.
theta = 50.
phi = 6. 			#degrees
pN = 18. 			#mm

#Solution:
#Number of teeth on each wheel:
#Calculating the spiral angles of the driving and driven wheels
alpha1 = theta/2 			#degrees
alpha2 = alpha1 			#degrees
#Calculating the number of teeth on driver wheel
#Calculating the number of teeth on driven wheel
T2 = G*T1
#Calculating the exact centre distance
#Calculating the efficiency of the drive

#Results:
print " Number of teeth on driver wheel, T1  =  %d."%(T1+1)
print " Number of teeth on driven wheel, T2  =  %.f."%(T2+1)
print " Exact centre distance, L1  =  %.1f mm."%(L1)
print " Efficiency of the drive, eta  =  %.1f %%."%(eta)

# rounding off error

 Number of teeth on driver wheel, T1  =  32.
Number of teeth on driven wheel, T2  =  96.
Exact centre distance, L1  =  400.0 mm.
Efficiency of the drive, eta  =  90.7 %.


## Example 12.17 Page No : 423¶

In [57]:
from numpy import linalg
from scipy.optimize import fsolve
import math

# Variables:
pN = 12.5
L = 134. 			#mm
theta = 80.
phi = 6. 			#degrees
G = 1.25

#Solution:
#Spiral angle of each wheel:
#Calculating the spiral angles of wheels 1 and 2
#We have d2/d1  =  (T2*math.cos(alpha1))/(T1*math.cos(alpha2)) or math.cos(alpha1)-G*math.cos(alpha2)  =  0          .....(i)
#Also alpha1+alpha2  =  theta or alpha1+alpha2-theta  =  0                                           .....(ii)
def f(x):
alpha1 = x[0]
alpha2 = x[1]
y = [0,0]
y[0] = math.cos(alpha1)-G*math.cos(alpha2)
y[1] = alpha1+alpha2-theta*math.pi/180
return y

z = fsolve(f,[1,1])
alpha1 = z[0]*180/math.pi 			#Spiral angle for slower wheel degrees
alpha2 = z[1]*180/math.pi 			#Spiral angle for faster wheel degrees
#Number of teeth on each wheel:
#Calculating the diameters of the wheels
d1 = L
d2  =  d1 			#mm
#Calculating the number of teeth on wheel 1
#Calculating the number of teeth on wheel 2
T2 = T1/G
#Calculating the efficiency of the drive
#Calculating the maximum efficiency

#Results:
print " Spiral angle for slower wheel, alpha1  =  %.2f degrees."%(alpha1)
print " Spiral angle for faster wheel, alpha2  =  %.2f degrees."%(alpha2)
print " Number of teeth on wheel 1, T1  =  %.1f."%(T1)
print " Number of teeth on wheel 2, T2  =  %.f."%(T2+1)
print " Efficiency of the drive, eta  =  %d %%."%(eta+1)
print " Maximum efficiency, etamax  =  %.1f %%."%(etamax)

 Spiral angle for slower wheel, alpha1  =  32.46 degrees.
Spiral angle for faster wheel, alpha2  =  47.54 degrees.
Number of teeth on wheel 1, T1  =  28.4.
Number of teeth on wheel 2, T2  =  24.
Efficiency of the drive, eta  =  83 %.
Maximum efficiency, etamax  =  83.9 %.