Velocity analysis

1. Velocity Analysis
Using Instantaneous Center

2. Instantaneous Center (IC)
of Velocity
It is the point about which
a rigid body rotates at a
particular instant of time.

3. Finding IC
O
VA
VB
A
B
C
VC

4. Relative and Absolute IC
Definition
When a rigid body is moving with respect to the other, we
may find a pair of coincident points having
zero relative velocity
at a particular time instant. This point is called the
instantaneous center of velocity.
•Relative IC: When two links 1 and 2 are moving, we define
relative IC as P12.
•It is the point about which link-2 is rotating instantaneously
with respect to link-1.
•If one link is fixed, the IC is called the absolute IC.

5. Positions of the ICs for two links
directly connected together

6. Things to remember
Relative motion
between two
links
Location of Relative IC
Pure rotation
about a hinge
Hinge location
Pure sliding Infinity
Rolling without
slip
Point of contact
Sliding on
curved surface
Center of curvature
Combined
rolling and
sliding
On the common normal to surfaces of these
links passing through the point of contact. The
exact determination of location requires some
other criteria including geometry of motion.

7. Number of ICs
ICs
of
Number
N
links
of
Number
2
)
1
(




n
n
n
N

8. Aronhold-Kennedy Theorem of
Three centers
If three bodies are in relative motion with respect to
one another, the three relative ICs of velocity are
collinear.

9. Aronhold-Kennedy Theorem of
Three centers
Proof
Say three links 1, 2 and 3 are having
relative motions of which 1 is fixed and
2 and 3 are rotating about O2 and O3,
respectively. Thus, P12 and P13 are the
two instantaneous centers. Suppose the
other one, P23 is not on the line joining
P12P13. If P23 is on the link-2, then its
velocity is along the perpendicular to
P12P23. Similarly, if P23 is on the link-3,
then its velocity is along the
perpendicular to P13P23. However, the
point P23 must have the same velocity (to
result zero relative velocity) irrespective
of whether it is on the link-2 or 3. This is
only possible if P23 is on the line joining
P12P13. Hence, the proof.
2
3
1
P12
P13
P23
V23
(2)
V23
(3)
w2
w3

10. Animation

11. Locating ICs of
Four-Bar Mechanism
P13 and P24 are obtained
using Aronhold-Kennedy
Theorem
P12 P13 P14
P21 P23 P24
P31 P32 P34

12. IC Animation
The locus of an IC is called centrode or polode

13. Circle Diagram
1 2
3
4
5
6
P12
P13
P23
P34
1
2
3
4
5 6
P15
P16 at inf.
P56
P45
P14
P16 at inf.
P46
P24
P26
12x 13 14x 15 16x
23x 24 25 26
34x 35 36
45x 46
56x

14. Classroom Assignment

15. Classroom Assignment (solution)
12× 13 14 ×
23 × 24
34 ×

16. Angular-Velocity-Ratio Theorem
Link-i (say fixed)
Link-j
Link-k
wj/i
wk/i
Pij
Pik
Pjk
Vjk
( ) ( )
/ /
/
/
jk ij jk ik
jk ij
jk ij
j k
jk j i P P jk k i P P
P P
k i
j i P P
V R V R
R
R
w w
w
w
    
 

17. Example #1;w2=20 rad/s find w4 (Using Angular Velocity-ratio theorem)
2)
-
link
as
direction
rad/s(same
81
.
7
20
93
.
60
95
95
P
O
P
O
0
V
P
O
V
:
4
-
link
on
also
is
P
As
P
O
V
:
2
-
link
on
is
P
As
2
24
C
24
B
4
)
4
(
24
)
2
(
24
24
24
C
4
(4)
24
24
24
B
2
(2)
24
24













w
w
w
w
V
V

18. w2=40rad/s
Example #2:
Find the slider Velocity

19. Finding ICs of Quick-Return mechanism

20. Example #3 Quick-Return mechanism w2=50 rad/s
As P24 is on link-2
V24 =w2.O2P24
As P24 is also on link-4
V24 =w4.O4P24
s
rad
P
O
P
O
Thus
/
66
.
22
50
07
.
42
07
.
19
,
2
24
4
24
2
4



 w
w

21. Freudenstein's Theorem
• Context: 4-bar linkage
• Collineation axis: the line
connecting P13 and P24
• The Theorem:
At an extreme of the output to
input angular velocity ratio of a
four-bar linkage, the collineation
axis is perpendicular to the
coupler link
P12 P14
P34
P23
P13
P24
Collineation axis
2
3
4

22. Analytical Method

25. Singularity

26. 3R1P Chain

29. Singularity