Chapter6-SIMPLE MACHINES

Example 6.1

In [73]:
import math
W = 10000.0  #Load
P = 500.0    #Effort
D = 20.0   #Distance moved by the effort 
d = 0.8     #Distance moved by the load 
MA=W/P         #Mechanical advantage
VR=D/d      #Velocity Ratio
Efficiency=MA/VR
Pi =W/VR        #Ideal effort
Wi = P*VR        #ideal load
efl=P-Pi            #Effort lost in friction
Fr=Wi-W            #frictional resistance
print "Mechanical advantage--",MA
print "Velocity Ratio",VR
print "Efficiency",Efficiency
print "Ideal Load",Wi
print "Ideal Effort",Pi
print "Effort lost in friction",efl
print "frictional resistance",Fr

Mechanical advantage-- 20.0
Velocity Ratio 25.0
Efficiency 0.8
Ideal Load 12500.0
Ideal Effort 400.0
Effort lost in friction 100.0
frictional resistance 2500.0

Example 6.2

In [74]:
import math
W1 = 2400.0  #Load 1
P1= 150.0    #Effort1

W2 = 3000.0  #Load 2
P2= 180.0    #Effort2
P3= 200.0    #Effort3
#law of machine is given by P=mW+C
m=(P2-P1)/(W2-W1)
C=P2-m*W2
print "Law of machine is P=",m,"W","+",C
W3=(P3-C)/m                   #Load 2
print "Load is ",W3,"N"
MA=W3/P3         #Mechanical advantage
print "Mechanical advantage--",MA
VR=30.0      #Velocity Ratio
Efficiency=MA/VR*100
Pi =W3/VR        #Ideal effort
print "Ideal effort is",Pi,"N"

efl=P3-Pi            #Effort lost in friction

print "Effort lost in friction",efl
print "Efficiency",Efficiency

Law of machine is P= 0.05 W + 30.0
Load is  3400.0 N
Mechanical advantage-- 17.0
Ideal effort is 113.333333333 N
Effort lost in friction 86.6666666667
Efficiency 56.6666666667

Example 6.3

In [75]:
import math
W1 = 7700.0  #Load 1
P1= 150.0    #Effort1
MA=W1/P1         #Mechanical advantage
Efficiency=0.6
VR=MA/Efficiency        #Velocity Ratio
print "Mechanical advantage--",MA
print "Velocity Ratio",VR
W2 = 13200.0  #Load 2
P2= 250.0    #Effort2
MA=W2/P2
Efficiency=MA/VR*100
print "Efficiency",Efficiency
#law of machine is given by P=mW+C
m=(P2-P1)/(W2-W1)


MMA=1/m         #Maximum Mechanical advantage
print "Maximum Mechanical advantage--",MMA

MaxEfficiency=MMA/VR*100

print "Maximum Efficiency",MaxEfficiency

Mechanical advantage-- 51.3333333333
Velocity Ratio 85.5555555556
Efficiency 61.7142857143
Maximum Mechanical advantage-- 55.0
Maximum Efficiency 64.2857142857

Example 6.4

In [76]:
%matplotlib notebook
import matplotlib
import numpy as np
import matplotlib.pyplot as plt2
W=[100.0,200.0,300.0,400.0,500.0,600.0]           #loads  
P=[16.0,22.5,28.0,34.0,40.5,46.5]    #Efforts
VR=25.0             #velocity ratio
E=[0,0,0,0,0,0]  #Efficiency
#calculating average slope
m=(P[4]-P[1])/(W[4]-W[1])
C=P[4]-m*W[4]
print "Law of machine is P=",m,"W","+",C
for i in range(0,6):
    
    E[i]=W[i]/(25*P[i])*100                     #E=W/(P*VR)
   
plt2.plot(W,E)
plt2.ylabel("Efficiency")
plt2.xlabel("Load")
plt2.show()    

    
MaxEfficiency=1/VR*100*1/m

print "Maximum Efficiency",MaxEfficiency

       
    

Law of machine is P= 0.06 W + 10.5

Maximum Efficiency 66.6666666667

Example 6.5

In [77]:
W = 5000.0  #Load
P = 360.0    #Effort

MA=W/P         #Mechanical advantage
VR=30.0      #Velocity Ratio
Efficiency=MA/VR*100.0
var="reversible machine"
if Efficiency < 50.0:
    var="self-locking machine"



Wi = P*VR        #ideal load

Fr=Wi-W            #frictional resistance
print "Mechanical advantage--",MA
print "Velocity Ratio",VR
print "Efficiency",Efficiency
print var
print "Ideal Load",Wi

print "frictional resistance",Fr

Mechanical advantage-- 13.8888888889
Velocity Ratio 30.0
Efficiency 46.2962962963
self-locking machine
Ideal Load 10800.0
frictional resistance 5800.0

Example 6.6

In [78]:
import math
W = 6000.0  #Load
N=3.0              #number of pulleys
VR=2**N              #Velocity Ratio
L=0.05                     #Efficiency loss  in each pulley
Efficiency=0.8
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"
#second case
P=520.0
n=0,
for i in range(3,20):
    if((P*(0.8-(i-3)*0.05)*(2**i)))>6000:
        n=i
        break
        
        
print "number of pulley is ",n

Effort is 937.5 N
number of pulley is  4

Exmple 6.7

In [79]:
import math
W = 12000.0  #Load
N=3.0              #number of movable pulleys
VR=2*N              #Velocity Ratio
L=0.05                     #Efficiency loss  in each pulley
Efficiency=0.85
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Effort is 2352.94117647 N

Example 6.8

In [80]:
import math
W = 12000.0  #Load
N1=2.0              #number of movable pulleys in  system 1
N2=2.0              #number of movable  puleys in system 2
VR=2*N1+2*N2              #Velocity Ratio
L=0.05                     #Efficiency loss  in each pulley
Efficiency=0.78
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Effort is 1923.07692308 N

Example 6.9

In [81]:
import math
W = 1000.0  #Load
N=3.0              #number of pulleys
VR=2**N-1              #Velocity Ratio
P = 180.0    #Effort
MA=W/P        #Mechanical advantage
Efficiency=MA/VR*100
print "Efficiency",Efficiency
Pi =W/VR        #Ideal effort

efl=P-Pi            #Effort lost in friction
print "Effort lost in friction",efl

Efficiency 79.3650793651
Effort lost in friction 37.1428571429

Example 6.10

In [82]:
import math
W = 2500.0  #Load
N1=2.0              #number of movable pulleys in  system 1 in figure B
N2=2.0              #number of movable  puleys in system 2 in figure C
VR=2**N1-1+2**N2-1              #Velocity Ratio
Efficiency=0.70
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Effort is 595.238095238 N

Example 6.11

In [83]:
D=500.0            #diameter of the wheel
d=200.0            #diameter of axle
tcw=6.0             #thickness of the cord on the wheel
tca=20.0           #thickness of the cord on the axle
W=1200              #effort
ED=D+tcw            #Effective diameter of the wheel
Ed=d+tca            #Effectivediameter of axle
VR=ED/Ed           #Velocity Ratio
print "Velocity ratio is ",VR
Efficiency=0.7
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Velocity ratio is  2.3
Effort is 745.341614907 N

Example 6.12

In [84]:
D=800.0            #diameter of the wheel
d1=250.0            #diameter of axle 1
d2=300.0            #diameter of axle 2

W=20000.0              #effort

VR=(2*D)/(d2-d1)           #Velocity Ratio
print "Velocity ratio is ",VR
Efficiency=0.55
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Velocity ratio is  32.0
Effort is 1136.36363636 N

Example 6.13

In [85]:
D=500.0            #diameter of the wheel
d=200.0            #diameter of axle 

W=5000.0              #effort

VR=(2*D)/(D-d)           #Velocity Ratio
print "Velocity ratio is ",VR
Efficiency=0.6
MA=Efficiency*VR         #Mechanical advantage
P = W/MA    #Effort
print "Effort is",P,"N"

Velocity ratio is  3.33333333333
Effort is 2500.0 N

Example 6.14

In [86]:
D=40.0  #Screw diameter
l=20.0   #Screw lwngth
p=l/3.0  #Lead of the screw
W=40000.0              #effort
R = 400  #Lever length
u = 0.12    #coefficient of friction between screw and nut
P = (D/(2*R))*W*((u+(p/(3.14*D)))/(1-u*(p/(3.14*D))))   #Effort
print "Effort is",P,"N"

Effort is 348.376068376 N

Example 6.15

In [87]:
import math
d=50.0  #mean diameter of screw
p=10.0  #pitch of screw
u=0.05 #coefficient of friction at the screw thread
R=300.0 ##Lever length
W=6000.0 #Load
o1=math.atan(p/(3.14*d))
o2=math.atan(0.05)
P=d/(2*R)*(W*math.tan(o1+o2))  #effort
print "Effort is",P,"N"
VR=2*3.14*R/p       #Velocity Ratio
MA=W/P         #Mechanical advantage
Efficiency=MA/VR*100.0
print "Efficiency",Efficiency,"%"
var="reversible machine"
if Efficiency < 50.0:
    var="self-locking machine"
print var
T =d/2.0*W*math.tan(o1-o2) #The torque required to keep the load from descending
print "The torque required to keep the load from descending",T,"Nm"

Effort is 57.0287539936 N
Efficiency 55.8439936484 %
reversible machine
The torque required to keep the load from descending 2047.61904762 Nm

Example 6.16

In [88]:
import math
p1=5.0   #Pitch of smaller screw
p2=10.0    #Pitch of larger screw
R=500.0     #Lever arm length from centre of screw
W=15000.0  #Load
P=185.0    #Effort
VR=2*3.14*R/(p2-p1)       #Velocity Ratio
MA=W/P         #Mechanical advantage
Efficiency=MA/VR*100.0

print "Efficiency",Efficiency,"%"

Efficiency 12.9110001721 %

Example 6.17

In [89]:
d=200.0  #Diameter of the load drum 
R = 1200.0   # Length of lever arm  
T1 = 10.0  #Number of teeth on pinion, 
T2 = 100.0   #Number of teeth on spur wheel
VR=R*T2/(d*T1)*2.0       #Velocity Ratio
print "Velocity Ratio is ",VR
W1 = 3000.0  #Load 1
P1= 100.0    #Effort1

W2 = 9000.0  #Load 2
P2= 160.0    #Effort2

#law of machine is given by P=mW+C
m=(P2-P1)/(W2-W1)
C=P2-m*W2
print "Law of machine is P=",m,"W","+",C
MA=W1/P1         #Mechanical advantage
Efficiency=MA/VR*100.0

print "Efficiency for first case",Efficiency,"%"
MA=W2/P2         #Mechanical advantage
Efficiency=MA/VR*100.0

print "Efficiency for second case",Efficiency,"%"

Velocity Ratio is  120.0
Law of machine is P= 0.01 W + 70.0
Efficiency for first case 25.0 %
Efficiency for second case 46.875 %

Example 6.18

In [90]:
d=150.0  #Diameter of the load drum 
R = 400.0   # Length of lever arm  
T1 = 15.0  #Number of teeth on pinion, 
T3 = 20.0  #Number of teeth on pinion, 
T2 = 45.0   #Number of teeth on spur wheel
T4 = 40.0   #Number of teeth on spur wheel
P= 250.0    #Effort
Efficiency=0.4
VR=R*T2/(d*T1)*2.0*T4/T3       #Velocity Ratio
print "Velocity Ratio is ",VR

W=VR*Efficiency*P       #Load 

print "LOad",W,"N"

Velocity Ratio is  32.0
LOad 3200.0 N