# Chapter 7 : Velocity in Mechanisms (Relative Velocity Method)¶

## Example 7.1 Page No : 148¶

In [1]:
import math

# Variables:
NBA = 120. 			#rpm
AB = 40./1000
CD = 80./1000 			#m

#Solution:
#Refer Fig. 7.7
#Calculating the angular velocity of the crank AB
#Calculating the velocity of B with respect to A
vBA = omegaBA*AB 			#m/s
vB = vBA
#By measurement from the velocity diagram Fig. 7.7(b)
vCD = 0.385 			#m/s
vC = vCD
#Calculating the angular velocity of link CD

#Results:

 The angular velocity of link CD omegaCD  =  4.8 rad/s clockwise about D.


## Example 7.2 Page No : 148¶

In [2]:
import math

# Variables:
NBO = 180. 			#rpm
OB = 0.5
PB = 2.
dO = 50./1000
dB = 60./1000
dC = 30./1000 			#m

#Solution:
#Refer Fig. 7.8
#Calculating the angular velocity of the crank BO
#Calculating the linear velocity of B with respect to O
vBO = omegaBO*OB 			#m/s
vB = vBO
#By measurement from the velocity diagram Fig. 7.8(b)
vP = 8.15
vPB = 6.8
vE = 8.5
bg = 5.
bp = vPB
vG = 8. 			#m/s
#Calculating the angular velocity of the connecting rod PB
#Calculating the velocity of rubbing at the pin of crank-shaft
vCS = dO/2*omegaBO 			#Velocity of rubbing at the pin of crank-shaft m/s
#Calculating the velocity of rubbing at the pin of crank
vC = dB/2*(omegaBO+omegaPB) 			#Velocity of rubbing at the pin of crank m/s
#Calculating the velocity of rubbing at the pin of cross-head
vPCH = dC/2*omegaPB 			#Velocity of rubbing at the pin of cross-head m/s
#Calculating the position of point G on the connecting rod
BG = bg/bp*PB 			#m

#Results:
print " The velocity of piston P vP  =  %.2f m/s."%(vP)
print " The angular velocity of connecting rod omegaPB  =  %.1f rad/s anticlockwise."%(omegaPB)
print " The velocity of point E on the connecting rod vE  =  %.1f m/s."%(vE)
print " The velocity of rubbing at the pin of crank-shaft is %.2f m/s."%(vCS)
print " The velocity of rubbing at the pin of crank is %.4f m/s."%(vC)
print " The velocity of rubbing at the pin of cross-head is %.3f m/s."%(vPCH)
print " The position of point G on the connecting rod BG  =  %.2f m."%(BG)
print " The linear velocity of point G vG  =  %d m/s."%(vG)

 The velocity of piston P vP  =  8.15 m/s.
The angular velocity of connecting rod omegaPB  =  3.4 rad/s anticlockwise.
The velocity of point E on the connecting rod vE  =  8.5 m/s.
The velocity of rubbing at the pin of crank-shaft is 0.47 m/s.
The velocity of rubbing at the pin of crank is 0.6675 m/s.
The velocity of rubbing at the pin of cross-head is 0.051 m/s.
The position of point G on the connecting rod BG  =  1.47 m.
The linear velocity of point G vG  =  8 m/s.


## Example 7.3 Page No : 150¶

In [3]:
import math

# Variables:
NAO = 600. 			#rpm
OA = 28./1000       #m
BD = 46./1000 		#m

#Solution:
#Refer Fig. 7.10
#Calculating the angular velocity of crank AO
#Calculating the velocity of A with respect to O
vAO = omegaAO*OA 			#m/s
vA = vAO
#By measurement from the velocity diagram Fig. 7.10(b)
vD = 1.6
vDB = 1.7 			#m/s
#Calculating the angular velocity of D with respect to B

#Results:
print " The velocity of the slider D vD  =  %.1f m/s."%(vD)
print " The angular velocity of the link BD omegaBD  =  %.2f rad/s clockwise sbout B."%(omegaBD)

 The velocity of the slider D vD  =  1.6 m/s.
The angular velocity of the link BD omegaBD  =  36.96 rad/s clockwise sbout B.


## Example 7.4 Page No : 151¶

In [4]:
import math

# Variables:
NBA = 120. 			#rpm
AB = 150./1000      #mm
DC = 450./1000      #mm
BC = 450./1000      #mm
dC = 50./1000
rC = dC/2 			#m

#Solution:
#Refer Fig. 7.12
#Calculating the angular velocity of the crank AB
#Calculating the linear velocity of B with respect to A
vBA = omegaBA*AB 			#m/s
vB = vBA
#By measurement from the velocity diagram Fig. 7.12(b)
vF = 0.7
vCD = 2.25
vCB = 2.25 			#m/s
#Calculating the angular velocity of DC
#Calculating the angular velocity of BC
#Calculating the rubbing speed at the pin C
vr = (omegaCB-omegaDC)*rC 			#The rubbing speed at the pin C m/s

#Results:
print " The velocity of block F vF  =  %.1f m/s."%(vF)
print " The rubbing speed at the pin C is %d m/s."%(vr)

 The velocity of block F vF  =  0.7 m/s.
The rubbing speed at the pin C is 0 m/s.


## Example 7.5 Page No : 153¶

In [5]:
import math

# Variables:
NAO = 120. 			#rpm
OA = 100./1000
CE = 350./1000 			#m

#Solution:
#Refer Fig. 7.13
#Calculating the angular speed of the crank OA
#Calculating the velocity of A with respect to O
vAO = omegaAO*OA 			#m/s
vA = vAO
#By measurement from the velocity diagram Fig. 7.14(b)
vF = 0.53
od = 1.08
vCE = 0.44 			#m/s
#Calculating the angular velocity of CE

#Results:
print " The velocity of F  vF  =  %.2f m/s."%(vF)
print " The velocity of sliding of CE in the trunnion is %.2f m/s."%(od)
print " The angular velocity of CE  omegaCE  =  %.2f rad/s  clockwise about E."%(omegaCE)

 The velocity of F  vF  =  0.53 m/s.
The velocity of sliding of CE in the trunnion is 1.08 m/s.
The angular velocity of CE  omegaCE  =  1.26 rad/s  clockwise about E.


## Example 7.6 Page No : 155¶

In [7]:
import math

# Variables:
NCO = 120. 			#rpm
OC = 125./1000 			#m

#Solution:
#Refer Fig. 7.15
#Calculating the angular velocity of the crank CO
#Calculating the linear velocity of C with respect to O
vCO = omegaCO*OC 			#m/s
vC = vCO
#By measurement from the velocity diagram  Fig. 7.16(b)

vCO = 1.57
vE = 0.7 			#m/s

#Results:
print " The absolute velocity of point E of the lever  vE  =  %.1f m/s."%(vE)

 The absolute velocity of point E of the lever  vE  =  0.7 m/s.


## Example 7.7 Page No : 156¶

In [8]:
import math

# Variables:
NBO1 = 40. 			#rpm
O1O2 = 800./1000
O1B = 300./1000
O2D = 1300./1000
DR = 400./1000 			#m

#Solution:
#Refer Fig. 7.18
#Calculating the angular speed of the crank BO
#Calculating the velocity of B with respect to O1
vBO1 = omegaBO1*O1B 			#m/s
vB = vBO1
#By measurement from the velocity diagram  Fig. 7.18(b)
vR = 1.44
vDO2 = 1.32 			#m/s
vD = vDO2
#Calculating the angular velocity of the link O2D

#Results:
print " The velocity of the ram R  vR  =  %.2f m/s."%(vR)

 The velocity of the ram R  vR  =  1.44 m/s.


## Example 7.8 Page No : 158¶

In [9]:
import math

# Variables:
NAO1 = 60.         			#rpm
O1A = 85.
rQ = 50. 			#mm

#Solution:
#Refer Fig. 7.20 and Fig. 7.21
#Calculating the angular velocity of AO1
#Calculating the velocity of A with respect to O1
vAO1 = omegaAO1*O1A 			#mm/s
vA = vAO1
#By measurement from the velocity diagram Fig. 7.20(b)
vDO2 = 410. 			#mm/s
O2D = 264.   			#mm
#To vary the Scilab function 'beta'
alpha = 120
beta = 240 			#degrees
#Calculating the angular velocity of the quadant Q
#Calculating the linear speed of the rack
vR = omegaQ*rQ 			#mm/s
#Calculating the ratio of times of lowering and raimath.sing the rack
r = beta/alpha
#Calculating the length of stroke of the rack
L = rQ*angleB1O2B2 			#mm

#Results:
print " The linear speed of the rack vR  =  %.1f mm/s."%(vR)
print " The ratio of times of lowering and raising the rack is %d."%(r)
print " The length of the stroke of the rack is %.2f mm."%(L)

 The linear speed of the rack vR  =  77.7 mm/s.
The ratio of times of lowering and raising the rack is 2.
The length of the stroke of the rack is 52.36 mm.


## Example 7.9 Page No : 160¶

In [2]:
import math

# Variables:
NPO = 120. 			#rpm
OQ = 100./1000      #mm
OP = 200./1000      #mm
RQ = 150./1000      #mm
RS = 500./1000 		#m

#Solution:
#Refer Fig. 7.22
#Calculating the angular speed of the crank PO
#Calculating the velocity of P with respect to O
vPO = omegaPO*OP 			#m/s
vP = vPO
#By measurement from the velocity diagram Fig. 7.23(b)
vS = 0.8
vSR = 0.96
vTP = 0.85 			#m/s
#Calculating the angular velocity of link RS

#Results:
print " The velocity of the slider S cutting tool vS  =  %.1f m/s."%(vS)
print " The velocity of the sliding block T on the slotted lever QT vTP  =  %.2f m/s."%(vTP)

# note :  answer in book is wrong

 The velocity of the slider S cutting tool vS  =  0.8 m/s.
The velocity of the sliding block T on the slotted lever QT vTP  =  0.85 m/s.


## Example 7.10 Page No : 162¶

In [11]:
import math

# Variables:
TA = 50. 			#N-m
DA = 300./1000
CB = 360./1000
AB = CB
DC = 600./1000 			#m
eta = 70./100 			#%

#Solution:
#Refer Fig. 7.25
#Calculating the angular velocity of the crank AD
#Calculating the velocity of A with respect to D
#By measurement from the velocity diagram Fig. 7.25(b)
vBC = 2.25 			#m/s
vB = vBC
#Calculating the angular velocity of the driven link CB
omegaB = omegaBC
MAactual = eta*omegaA/omegaB
#Calculating the resisting torque
TB = eta*TA*omegaA/omegaB 			#N-m

#Results:
print " The velocity of the point B. vB  =  %.2f m/s."%(vB)
print " The angular velocity of the driven link CB. omegaBC  =  %.2f rad/s."%(omegaBC)
print " The actual mechanical advantage. M.A.actual)  =  %.2f."%(MAactual)
print " The resisting torque. TB  =  %.1f N-m."%(TB)

 The velocity of the point B. vB  =  2.25 m/s.
The angular velocity of the driven link CB. omegaBC  =  6.25 rad/s.
The actual mechanical advantage. M.A.actual)  =  1.17.
The resisting torque. TB  =  58.6 N-m.


## Example 7.11 Page No : 163¶

In [12]:
import math

# Variables:
WC = 2.5*1000
WD = 4.*1000 			#N
OA = 175./1000          #mm
AB = 180./1000          #mm
BC = 325./1000 			#m

#Solution:
#Refer Fig. 7.26
#Assuming the speed of crank OA to be 'N'
#Calculating the angular velocity of crank OA
#Assume the vector oa (i.e. velocity of A) as 20 m/s
N = 20/(2*math.pi/60*OA) 			#mm
#By measurement from the velocity diagram Fig. 7.27(b)
vC = 35.
vD = 21. 			#mm
#Calculating the velocity ratio between C and the ram D
r = vC/vD 			#The velocity ratio between C and the ram D
#Calculating the efficiency of the machine
eta = (WD*vD)/(WC*vC)*100 			#%

#Results:
print " The velocity ratio between C and the ram D is %.2f."%(r)
print " The efficiency of the machine eta  = n %d %%."%(eta)

 The velocity ratio between C and the ram D is 1.67.
The efficiency of the machine eta  = n 96 %.


## Example 7.12 Page No : 165¶

In [15]:
import math

# Variables:
NAO = 180. 			#rpm
OA = 180./1000      #mm
CB = 240./1000      #mm
AB = 360./1000      #mm
BD = 540./1000 		#mm
FD = 2.*1000 			#N
DA = 30./1000
DD = DA
rA = DA/2
rD = DD/2 			#m

#Solution:
#Refer Fig. 7.28
#Calculating the angular velocity of the crank OA
#Calculating the velocity of A with respect to O
vAO = omegaAO*OA
vA = vAO
#By measurement fro the velocity diagram  Fig. 7.29(b)
vD = 2.05
vBA = 0.9
vBC = 2.8
vDB = 2.4 			#m/s
#Calculating the angular velocity of the link AB
#Calculating the angular velocity of the link CB
#Calculating the angular velocity of the link BD
#Calculating the relative angular velocity at A
rvA = omegaCB-omegaAB+omegaBD 			#The relative angular velocity at A  rad/s
#Calculating the relative angular velocity at D
rvD = omegaBD 			#The relative angular velocity at D  rad/s
#Calculating the velocity of rubbing on the pin A
vrA = rvA*rA*1000 			#The velocity of rubbing on the pin A  mm/s
#Calculating the velocity of rubbing on the pin D
vrD = rvD*rD*1000 			#The velocity of rubbing on the pin D mm/s
#Calculating the torque applied to crank OA
TA = FD*vD/omegaAO 			#N-m

#Results:
print " The velocity of slider D vD  =  %.2f m/s."%(vD)
print " The velocity of rubbing on the pin A is %d mm/s."%(vrA)
print " The velocity of rubbing on the pin D is %d mm/s."%(vrD)
print " The torque applied to the crank OA, TA  =  %.1f N-m."%(TA)

 The velocity of slider D vD  =  2.05 m/s.
The velocity of rubbing on the pin A is 204 mm/s.
The velocity of rubbing on the pin D is 66 mm/s.
The torque applied to the crank OA, TA  =  217.5 N-m.


## Example 7.13 Page No : 167¶

In [16]:
import math

# Variables:
NBA = 180 			#rpm
AB = 0.45           #m
BD = 1.5            #m
BC = 0.9            #m
CE = BC 			#m
FD = 500.
FE = 750. 			#N

#Solution:
#Refer Fig. 7.31
#Calculating the angular velocity of the crank AB
#Calculating the velocity of B with respect to A
vBA = omegaBA*AB 			#m/s
vB = vBA
#By measurement from the velocity diagram Fig. 7.31(b)
vD = 9.5
vE = 1.7 			#m/s
#Calculating the power input
Pi = FD*vD-FE*vE 			#N-m/s
#Calculating the turning moment at A
TA = Pi/omegaBA 			#N-m

#Results:
print " The velocity of slider D, vD  =  %.1f m/s."%(vD)
print " The velocity of slider E, vE  =  %.1f m/s."%(vE)
print " The turning moment at A, TA  =  %.1f N-m."%(TA)

 The velocity of slider D, vD  =  9.5 m/s.
The velocity of slider E, vE  =  1.7 m/s.
The turning moment at A, TA  =  184.4 N-m.