The document discusses kinematics of rigid bodies, including definitions of translation, rotation about a fixed axis, and general plane motion. It provides equations relating position, velocity, and acceleration for particles undergoing translation and rotation. Examples are presented of determining velocities and accelerations of points on rigid bodies in translation, rotation, and rolling contact motion. Key concepts covered include absolute and relative velocity diagrams.

1. 1
All figures taken from Vector Mechanics for Engineers: Dynamics, Beer
and Johnston, 2004
ENGR 214
Chapter 15
Kinematics of Rigid Bodies

2. 2
Introduction
• Kinematics of rigid bodies: relations
between time and the positions, velocities,
and accelerations of the particles forming
a rigid body.
• Classification of rigid body motions:
- general motion
- motion about a fixed point
- general plane motion
- rotation about a fixed axis
• curvilinear translation
• rectilinear translation
- translation:

3. 3
Motion of the plate: is it translation or rotation?
Curvilinear translation Rotation

4. 4
Translation
• Consider rigid body in translation:
- direction of any straight line inside the
body is constant,
- all particles forming the body move in
parallel lines.
• For any two particles in the body,
A
B
A
B r
r
r





• Differentiating with respect to time,
A
B
A
A
B
A
B
v
v
r
r
r
r














All particles have the same velocity.
A
B
A
A
B
A
B
a
a
r
r
r
r


















• Differentiating with respect to time again,
All particles have the same acceleration.

5. 5
Rotation About a Fixed Axis
The change in angular position, d, is called the
angular displacement, with units of either
radians or revolutions. They are related by
1 revolution = 2 radians
When a body rotates about a fixed axis, any
point P in the body travels along a circular path.
The angular position of P is defined by .
Angular velocity, , is obtained by taking the
time derivative of angular displacement:
 = d/dt (rad/s) +
Similarly, angular acceleration is
 = d2/dt2 = d/dt or  = (d/d) + rad/s2

6. 6
Rotation About a Fixed Axis: Velocity
• Consider rotation of rigid body about a
fixed axis AA’
• Velocity vector of the particle P is
tangent to the path with magnitude
dt
r
d
v



dt
ds
v 
   
  






sin
sin
lim
sin
0

r
t
r
dt
ds
v
r
BP
s
t












locity
angular ve
k
k
r
dt
r
d
v


















• The same result is obtained from

7. 7
Rotation About a Fixed Axis: Acceleration
• Differentiating to determine the acceleration,
 
v
r
dt
d
dt
r
d
r
dt
d
r
dt
d
dt
v
d
a




























•
k
k
k
celeration
angular ac
dt
d


















component
on
accelerati
radial
component
on
accelerati
l
tangentia










r
r
r
r
a

















• Acceleration of P is combination of two
vectors,

8. 8
Rotation About a Fixed Axis: Representative Slab
• Consider the motion of a representative slab in
a plane perpendicular to the axis of rotation.
• Velocity of any point P of the slab,



r
v
r
k
r
v










• Acceleration of any point P of the slab,
r
r
k
r
r
a









2













• Resolving the acceleration into tangential and
normal components,
2
2




r
a
r
a
r
a
r
k
a
n
n
t
t












9. 9
Examples

10. 10
Equations Defining the Rotation of a Rigid Body
About a Fixed Axis
• Motion of a rigid body rotating around a fixed axis is
often specified by the type of angular acceleration.










d
d
dt
d
dt
d
d
dt
dt
d





2
2
or
• Recall
• Uniform Rotation,  = 0:
t


 
 0
• Uniformly Accelerated Rotation,  = constant:
 
0
2
0
2
2
2
1
0
0
0
2 



















t
t
t

11. 11
Sample Problem 5.1
Cable C has a constant acceleration of 9 in/s2 and an initial
velocity of 12 in/s, both directed to the right.
Determine (a) the number of revolutions of the pulley in 2 s,
(b) the velocity and change in position of the load B after 2 s,
and (c) the acceleration of the point D on the rim of the inner
pulley at t = 0.

12. 12
Sample Problem 5.1
• The tangential velocity and acceleration of D are equal to the
velocity and acceleration of C.
   
 
 
s
rad
4
3
12
s
in.
12
0
0
0
0
0
0







r
v
r
v
v
v
D
D
C
D



  
 
  2
s
rad
3
3
9
s
in.
9







r
a
r
a
a
a
t
D
t
D
C
t
D




• Apply the relations for uniformly accelerated rotation to
determine velocity and angular position of pulley after 2 s.
   s
rad
10
s
2
s
rad
3
s
rad
4 2
0 



 t



     
rad
14
s
2
s
rad
3
s
2
s
rad
4 2
2
2
1
2
2
1
0




 t
t 


  revs
of
number
rad
2
rev
1
rad
14 








N rev
23
.
2

N
  
  
rad
14
in.
5
s
rad
10
in.
5







r
y
r
v
B
B
in.
70
s
in.
50




B
B
y
v


13. 13
Sample Problem 5.1
• Evaluate the initial tangential and normal acceleration
components of D.
  

 s
in.
9
C
t
D a
a


     2
2
2
0 s
in
48
s
rad
4
in.
3 

 
D
n
D r
a
    


 2
2
s
in.
48
s
in.
9 n
D
t
D a
a


Magnitude and direction of the total acceleration,
   
2
2
2
2
48
9 


 n
D
t
D
D a
a
a
2
s
in.
8
.
48

D
a
 
 
9
48
tan


t
D
n
D
a
a


 4
.
79


14. 14
General Plane Motion
A combination of translation & rotation

15. 15
General Plane Motion
Pure translation, followed by rotation about A2 (to move B'1 to B' 2)
Motion of B w.r.t. A is pure rotation, i.e. B draws a circle centered at A
Any plane motion can be represented as a translation of an
arbitrary reference point A and a rotation about A.

16. 16
Absolute and Relative Velocity
A
B
A
B r
k
v
v






 
B A B A
v v v
 
For any two points lying on the same rigid body:
Note: B A
v r

 r = distance from A to B
/
B A B A
v k r

 
Equation can be represented graphically by a velocity diagram

17. 17
Absolute and Relative Velocity
Assuming that the velocity vA of end A is known, determine
the velocity vB of end B and the angular velocity .
The direction of vB and vB/A are known. Complete the velocity diagram.
tan tan
B
B A
A
v
v v
v
 
  
B A B A
v v v
 
Locus for vB
vA
vB/A

vB
Locus
for vB/A
cos
cos cos
B A B A A
B A
v v v
v l
l l l

 
 
 
    
 
 

18. 18
Absolute and Relative Velocity in Plane Motion
• Selecting point B as the reference point and solving for the velocity vA of end A
and the angular velocity  leads to an equivalent velocity triangle.
• vA/B has the same magnitude but opposite sense of vB/A. The sense of the
relative velocity is dependent on the choice of reference point.
• Angular velocity  of the rod in its rotation about B is the same as its rotation
about A. Angular velocity is not dependent on the choice of reference point.

19. 19
Rolling Motion
Consider a circular disc that rolls without slipping on a flat surface

r
s
A1
A2
O1 O2
s r

From geometry:
s = displacement of center
v r r
 
 

20. 20
Sample Problem 15.2
The double gear rolls on the stationary lower rack; the velocity
of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear, and (b) the
velocities of the upper rack R and point D of the gear.

21. 21
Sample Problem 15.2
A A A
v r


1.2
8 /
0.15
A
A
A
v
rad s
r
   
A
P
A
A
P
A
P r
k
v
v
v
v










 
For any point P on the gear:
 
1.2 8 0.1 2 /
B A B A
v v v m s
      
For point B:
 
1.2 8 0.15 1.2 1.2
D A D A
v v v i j i j
      
For point D:
2 /
R B
v v m s
  
vB/A
vD/A

22. 22
Sample Problem 15.3
The crank AB has a constant clockwise angular velocity of
2000 rpm. For the crank position indicated, determine (a) the
angular velocity of the connecting rod BD, and (b) the velocity
of the piston P.

23. 23
Sample Problem 15.3
B
D
B
D v
v
v





• The velocity is obtained from the crank rotation data.
B
v

    
s
rad
4
.
209
in.
3
s
rad
4
.
209
rev
rad
2
s
60
min
min
rev
2000
























AB
B
AB
AB
v 


• The direction of the absolute velocity is horizontal.
The direction of the relative velocity is
perpendicular to BD.
D
v

B
D
v

Locus for vD
Locus for vD/B

24. 24
Sample Problem 15.3
• Determine the velocity magnitudes
from the vector triangle drawn to scale.
B
D
D v
v and
B
D
B
D v
v
v





s
in.
9
.
495
s
ft
6
.
43
s
in.
4
.
523



B
D
D
v
v
s
rad
0
.
62
in.
8
s
in.
9
.
495




l
v
l
v
B
D
BD
BD
B
D



25. 25
Instantaneous Center of Rotation
For any body undergoing planar motion, there always exists a
point in the plane of motion at which the velocity is
instantaneously zero (if it were rigidly connected to the body).
This point is called the instantaneous center of
rotation, or C.
It may or may not lie on the body!
If the location of this point can be determined, the velocity
analysis can be simplified because the body appears to rotate
about this point at that instant.

26. 26
Instantaneous Center of Rotation
To locate the C, we use the fact that the velocity of a point on a
body is always perpendicular to the position vector from C to that
point.
If the velocity at two points A and B are
known, C lies at the intersection of the
perpendiculars to the velocity vectors
through A and B .
If the velocity vectors at A and B
are perpendicular to the line AB,
C lies at the intersection of the
line AB with the line joining the
extremities of the velocity
vectors at A and B.
If the velocity vectors are equal & parallel, C is at infinity and the angular
velocity is zero (pure translation)

27. 27
If the velocity vA of a point A on the body and
the angular velocity  of the body are
known, C is located along the line drawn
perpendicular to vA at A, at a distance
r = vA/ from A. Note that the C lies up and
to the right of A since vA must cause a
clockwise angular velocity  about C.
Instantaneous Center of Rotation

28. 28
The velocity of any point on a body undergoing general plane
motion can be determined easily if the instantaneous center
is located.
Since the body seems to rotate about
the IC at any instant, the magnitude of
velocity of any arbitrary point is v =  r,
where r is the radial distance from the IC
to that point. The velocity’s line of action
is perpendicular to its associated radial
line. Note the velocity has a direction
which tends to move the point in a
manner consistent with the angular
rotation direction.
Velocity Analysis using Instantaneous Center

29. 29
Velocity Analysis using Instantaneous Center

30. 30
Instantaneous Center of Rotation
C lies at the intersection of the
perpendiculars to the velocity
vectors through A and B .


cos
l
v
AC
v A
A


   




tan
cos
sin
A
A
B
v
l
v
l
BC
v



The velocity of any point on the rod can be obtained.
Accelerations cannot be determined using C.

31. 31
Sample Problem 15.4, using instantaneous center
The double gear rolls on the stationary lower rack; the velocity
of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear, and (b) the
velocities of the upper rack R and point D of the gear.

32. 32
Sample Problem 15.4
Point C is in contact with the stationary lower rack
and, instantaneously, has zero velocity. It must be
the location of the instantaneous center of rotation.
1.2
8rad s
0.15
A
A A
A
v
v r
r
 
    
  
8 0.25
R B B
v v r

  
 i
vR


s
m
2

 
  
0.15 2 0.2121 m
8 0.2121
D
D D
r
v r

 
 
  
s
m
2
.
1
2
.
1
s
m
697
.
1
j
i
v
v
D
D







33. 33
Sample Problem 15.5 using instantaneous center
The crank AB has a constant clockwise angular velocity of
2000 rpm. For the crank position indicated, determine (a) the
angular velocity of the connecting rod BD, and (b) the velocity
of the piston P.
Crank-slider mechanism

34. 34
Sample Problem 15.5
C is at the intersection of the perpendiculars to the velocities through B and D.
   
B AB BD
B
BD
v AB BC
v
BC
 

 
  
D BD
v CD



35. 35
Absolute and Relative Acceleration
Absolute acceleration of point B: A
B
A
B a
a
a





Relative acceleration includes tangential and normal components:
A
B
a

    2
B A B A
t n
a r a r
 
 

36. 36
Absolute and Relative Acceleration
• Given determine
,
and A
A v
a


.
and


B
a
   t
A
B
n
A
B
A
A
B
A
B
a
a
a
a
a
a











• Vector result depends on sense of and the
relative magnitudes of  n
A
B
A a
a and
A
a

• Must also know angular velocity .

37. 37
Absolute and Relative Acceleration
Draw acceleration diagram to scale:

38. 38
Sample Problem 15.6
The center of the double gear has a velocity and acceleration
to the right of 1.2 m/s and 3 m/s2, respectively. The lower rack
is stationary.
Determine (a) the angular acceleration of the gear, and (b) the
acceleration of points B, C, and D.

39. 39
Sample Problem 15.6
1.2
8 /
0.15
A
A
v
v r rad s
r
 
    
2
3
20 /
0.15
A
A
a
a r rad s
r
 
    
2
2
3 ( )
3 (8) (0.1) (20)(0.1)
5 6.4
B A B A
a a a
i r j ri
i j i
i j
 
 
   
  
 
2
2
3 ( ) ( )
3 (8) (0.15) (20)(0.15)
9.6
C A C A
a a a
i r j r i
i j i
j
 
 
   
  

2
2
3 ( )
3 (8) (0.15) (20)(0.15)
12.6 3
D A D A
a a a
i r i rj
i i j
i j
 
 
  
  
 

40. 40
Sample Problem 15.7
Crank AG of the engine system has a constant clockwise
angular velocity of 2000 rpm.
For the crank position shown, determine the angular
acceleration of the connecting rod BD and the acceleration
of point D.

41. 41
Sample Problem 15.7
   n
B
D
t
B
D
B
B
D
B
D a
a
a
a
a
a











  
AB
2
2 2
3
12
2000 rpm 209.4rad s constant
0
ft 209.4rad s 10,962ft s
AB
B AB
a r



  

  
From Sample Problem 15.3, BD = 62.0 rad/s, b = 13.95o.

42. 42
Sample Problem 15.7
   n
B
D
t
B
D
B
B
D
B
D a
a
a
a
a
a











Draw acceleration diagram:
       2
2
12
8
2
s
ft
2563
s
rad
0
.
62
ft 

 BD
n
B
D BD
a 
      BD
BD
BD
t
B
D BD
a 

 667
.
0
ft
12
8 


Drawn to scale

43. 43
Sample Problem 15.8
In the position shown, crank AB has a constant angular velocity
1 = 20 rad/s counterclockwise. Determine the angular
velocities and angular accelerations of the connecting rod BD
and crank DE.
Four-bar mechanism

44. 44
Sample Problem 15.8
B
D
B
D v
v
v





   k
k DE
BD




s
rad
29
.
11
s
rad
33
.
29 

 

Velocities
1( )
B
v AB


vD
vB
vD/B vB
vD
vD/B
( )
D B BD
v BD


( )
D DE
v DE


Velocity diagram
Shown here not to scale

45. 45
Sample Problem 15.8
B
D
B
D a
a
a





   k
k DE
BD



 2
2
s
rad
809
s
rad
645 

 

Accelerations
aB
aD/B n
aD/B t aD/E t
aD/E n
Acceleration diagram
aB
  2
( )
D B BD
n
a BD


aD/B n
2
( )
B AB
a AB


aD/E n
aD/E t
  2
( )
D E DE
n
a DE


  ( )
D E DE
t
a DE


Shown here not to scale