Chapter 11: Gear Trains

Example 1, Page 369

In [2]:
#Variable declaration
Ns=26#rpm of spindle
N1=4#rpm of lead screw
#the only wheel in the set of which 13 is a factor is that with 65 teeth
T1=65
T2=25#to satisfy the Ns/n1 ratio and to select from given set
T3=75#to satisfy the Ns/n1 ratio and to select from given set

#Calculations
T4=T1*T3*N1/(Ns*T2)
#solution b
Ns1=35
N1=4
Tb1=105#to satisfy the Ns/n1 ratio and to select from given set
Tb2=30#to satisfy the Ns/n1 ratio and to select from given set
Tb3=100#to satisfy the Ns/n1 ratio and to select from given set
Tb4=Tb1*Tb3*N1/(Ns1*Tb2)

#Results
print "a)The change wheel used will have %.f, %.f, %.f and %.f teeths\nb) The change wheel used will have %.f, %.f, %.f "\
      "and %.f teeths"%(T1,T2,T3,T4,Tb1,Tb2,Tb3,Tb4)

a)The change wheel used will have 65, 25, 75 and 30 teeths
b) The change wheel used will have 105, 30, 100 and 40 teeths

Example 2, Page 371

In [3]:
import math

#Variable declaration
v=15#ft/min
d=2#ft
N=450#rpm

#Calculations
N1=d*v/(2*math.pi)#rpm of barrel
s=N/N1#total reduction speed required
#With a minimum number of teeth = 20
T=20
T1=T*(s)**(1./3)
R=(T1/T)**3

#Results
print "If the minimum number of teeth is fixed at 20, the might be as follow ( %.f / 20 )^3 = %.1f"\
      "\nThis is sufficiently close to the required ratio"%(T1,R)

If the minimum number of teeth is fixed at 20, the might be as follow ( 91 / 20 )^3 = 94.2
This is sufficiently close to the required ratio

Example 3, Page 374

In [6]:
import math

#Variable declaration
d=7.#in; central distance
k1=2.*7*7#T1+t1/(2*7)=7
k2=2.*7*5#T2+t2/(2*5)=7
G=9./1

#Calculations
t1=(-(k1+k2)+((k1+k2)**2+4*(G-1)*(k1*k2))**(1/2))/(2*(G-1))
a=math.ceil(t1)
b=math.floor(t1)
T1=k1-a
T2=k2-a
T3=k2-b
G1=T1*T2/(a*a)
G2=T1*T3/(a*b)
dp=a/d
#case b)
tb1=23#let t1 = 23
Tb1=k1-tb1
Gb1=Tb1/tb1
Gb2=G/Gb1
tb2=k2/(Gb2+1)
p=math.ceil(tb2)
Tb2=k2-p
l=Tb1-1
m=tb1+1
n=Tb2+1
o=p-1
Gb2=l*n/(m*o)

#Results
print "a) No of teeth = %.f, %.f, %.f, %.f\nG = %.2f\n\nb) No of teeth = %.f, %.f, %.f, %.f\nG = %.2f\n\n"%(T1,T2,a,b,G2,l,m,n,o,Gb2)

a) No of teeth = 108, 80, -10, -11
G = 79.53

b) No of teeth = 74, 24, 52, 18
G = 8.91

Example 5, Page 388

In [9]:
#Variable declaration
Tb=27.
Tc=30
Td=24
Te=21

#Calculations
k=Te*Tb/(Tc*Td)#k=Nd/Ne
#by applying componendo and dividendo, using Ne=0 and reducing we get
a=(1-k)#where a = Nd/Na
b=1./a

#Results
print "The ratio of the speed of driving shaft to the speed of driven shaft, Na/Nd = %.2f"%b

The ratio of the speed of driving shaft to the speed of driven shaft, Na/Nd = 4.71

Example 6, Page 391

In [10]:
#Variable declaration
Tb=75.
Tc=18
Td=17
Te=71
N1=500#rpm

#Calculations
k=Tb*Td/(Tc*Te)#k=Ne/Nb
#case a)
#using componendo and dividendo , Nb=0 and reducing we get
a=1-k#a=Ne/Na
Na=N1
Ne=Na*a
#case b)
Na1=500#given
Nb1=100#given
Ne1=k*(Nb1-Na1)+Na1

#Results
print "case a) Ne= %.3f rpm\ncase b) Ne= %.1f rpm"%(Ne,Ne1)

case a) Ne= 1.174 rpm
case b) Ne= 100.9 rpm

Example 8, Page 398

In [12]:
import math

#Variable declaration
Td=23.
Ta=19
Tb=20
Tc=22

#Calculations
k=Td*Ta/(Tb*Tc)
#using componendo and dividendo, Nc=0 and reducing we get
a=1./k-1#a=Nd/Ne
b=1./a#- denotes opposite direction
d=5280*12/(math.pi*5*b)
p=math.ceil(d)

#Results
print "The diameter must be = %.1f in\nThe numbers of teeths are therefore suitable for a cyclometer for bicycle with %.f "\
      "inches wheels"%(d,p)

The diameter must be = 27.7 in
The numbers of teeths are therefore suitable for a cyclometer for bicycle with 28 inches wheels

Example 10, Page 403

In [13]:
#Variable declaration
s1=26
s2=24
s3=23
sr=31
i1=70
i2=72
i3=61
ir=71
t=1500#lb in 

#Calculations
k1=-i3/s3#Ns3-Ni2/(Ni3-Ni2)=k
#S3 is fixed thus 
k2=1-(1./k1)#k2=Ni3/Ni2
k3=-i2/s2#k3=Ns2-Ni3/(Ni2-Ni3)
k4=(1./k2-1)*k3+1#k4=Ns2/Ni3  ; reducing using k2 and k3
k5=-i1/s1#Ns1-Nf/(Ni1-Nf)
k6=(1-k5)/(1-k5/k4)#k6=Ns1/Nf

#Result
print "Ns1/Nf = %.2f"%k6

Ns1/Nf = 1.47