PPT: Kinematics of Particles & Rigid Bodies | Engineering Mechanics - Civil Engineering (CE) PDF Download

 Page 1


CHAPTER 15
Kinematics of Rigid Bodies
Page 2


CHAPTER 15
Kinematics of Rigid Bodies
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:
Page 3


CHAPTER 15
Kinematics of Rigid Bodies
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:
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.
Page 4


CHAPTER 15
Kinematics of Rigid Bodies
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:
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.
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
? ? ?
? ? ?
ˆ ˆ
? ? ?
?
?
?
? ?
?
?
• The same result is obtained from
Page 5


CHAPTER 15
Kinematics of Rigid Bodies
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:
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.
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
? ? ?
? ? ?
ˆ ˆ
? ? ?
?
?
?
? ?
?
?
• The same result is obtained from
•
k k k
celeration angular ac
dt
d
ˆ ˆ ˆ
 
? ? ?
?
?
? ?
?
?
?
? ? ?
? ?
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
? ? ?
?
?
? ?
?
? ?
?
?
? ? ? ?
? ? ? ?
? ? ?
?
?
?
?
?
component on accelerati radial ) r ? ( ?
component on accelerati tangential r a
) r ? ( ? r a a
= × ×
= ×
× × + × =
? ? ?
? ?
? ? ? ? ? ?
• Acceleration of P is combination of two 
vectors,