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1. Kinematics of Machinery
(Formally Known as Theory of Machine)
SE Mechanical (2019 Course)
Unit - II
Kinematic Analysis of Mechanisms (Analytical Method)
By
Dr. Somnath Kolgiri
Associate Professor
Mechanical Engineering Department
P G Moze College of Engineering, Wagholi

2. 2
Kinematics of Machinery 2019
Course
Syllabus UNIT - II
Kinematic Analysis of Mechanisms : Analytical Methods
• Analytical method for displacement, velocity and acceleration analysis of slider crank
mechanism.
• Position analysis of links with vector and complex algebra methods, Loop closure
equation, Chace solution, Velocity and acceleration analysis of four bar and slider crank
mechanisms using vector and complex algebra methods.
• Hooke’s joint, Double Hooke’s joint.

3. Analytical Method for Slider Crank Mechanism
Consider Slider Crank Mechanism as below
Line
of Crank with
of Connecting
OP
Rod with Line of Stroke, OP
of Stroke,
of Connecting Rod ,CP
of Crank,OC
n  Obliquity Ratio 
l
r
l  Length
r  Radius
  Angle
  Angle
3

4. Analytical Method for Piston P in Slider Crank Mechanism
P’ is extreme Left position of PISTON P
(i.e. when Crank is at 0° from I.D.C.)
x is Displacement of P at given instant
(i.e. when OC is at θ from I.D.C.)
Hence, VELOCITY of Piston P,
ACCELERATION of Piston P,
dt

dx
vP
dt

dvP
aP
4

5. Analytical Method for Piston P in Slider Crank Mechanism
Displacement, x  PP' OP'OP  (OC'C' P')  (OQ  QP)
x  (r  l)  (r.cos  l.cos)
x  r[(1 cos)  n(1 cos)]
n
sin 
sin
From Fig
CQ  l.sin  r.sin
2
1
sin 
2
n2
We know
cos  1 sin  
2n2
sin2

2n2
Expanding by binomial theorem we get
sin2

 .....
cos  1 1 cos 
Displacement, 


 sin2
 
x  r(1 cos) 
2n 5

6. 

sin 2 
2 n2
sin2
 
Analytical Method for Piston P in Slider Crank Mechanism
Velocity of P wrt O,
vPO
d
dx dx d dx
 vP 
dt

d
.
dt

d
.crank
P
v 
dx
.
Velocity of P wrt O,
6


 2n
P

vP  .rsin 
sin 2 
v  .r in 
s

7. Analytical Method for Piston P in Slider Crank Mechanism
d

dv

dv
.
d 
dv
.
dt d dt d
P
a 
dv
.
crank
P
PO
Acceleration of P wrt O,
a  a
 
n
P
 2
.r os 
cos 2 
c
Acceleration of P wrt O, a
7

8. Analytical Method for Piston P in Slider Crank Mechanism
Acceleration of P wrt O,
d
dv dv d dv
 aP 
dt

d
.
dt

d
.crank
aPO
P
a 
dv
.






n
P
2n
sin 2 
  r.sin 
cos 2 
a  2
.rcos 
8

d  

d  
Acceleration of P wrt O,
In case Crank has non uniform Angular
velocity dt

dω
αOC


sin 2 
 2n
P
v  .r in 
s
Using

9. 
dt n cos
CP

d


.
cos
Analytical Method for Connecting Rod PC in Slider Crank Mechanism
As we know,
CQ  l.sin  r.sin
cos.
d

cos
.
d
dt n dt
d dt d

n
.
dt
d
sin.
d

d  sin  d
n
sin 
sin
 

dt dt

n
 
d
sin 
d  sin 
Angular Velocity of CP,
2
n2
sin2

cos  1 sin   1
And we also know,
n
CP
Angular Velocity of CP, 
n2
sin2


.cos 
.cos
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10. Analytical Method for Connecting Rod PC in Slider Crank Mechanism
Angular Acceleration of CP,
d
dt
CP
 
dCP
.
CP
CP
 
d


(n2
sin2
)3/ 2

2
.sin(n2
1)
CP
Angular Acceleration of CP,
- ve sign indicates sense of
acceleration of CP such that it tends
to reduce angle φ
10

11. In case Crank has non uniform Angular
Velocity
Analytical Method for Connecting Rod PC in Slider Crank Mechanism
Angular Acceleration of CP,
wcos
dt
CP
d dt d n
 
dCP
.
d
 w.
d
CP
CP
 
d

n
n
.cos

2
.sin
Angular Acceleration of CP, CP 
dt

dω
αOC
d  

d  
Using
11
CP
 
.cos
n

12. Thank You
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