EM-Module 2.2-ICR method_MAP (1).pdf

11/12/2022 1
K J SOMAIYA COLLEGE OF ENGINEERING, MUMBAI-77
(CONSTITUENT COLLEGE OF SOMAIYA VIDYAVIHAR UNIVERSITY)
Presented by:
Prof. M. A. Palsodkar

11/12/2022 2
Introduction
15 - 2
• 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:

11/12/2022 3
Types of rigid body motion
• Kinematically speaking…
o Translation
−Orientation of AB
constant
o Rotation
−All particles rotate
about fixed axis
o General Plane Motion (both)
−Combination of both
types of motion
B
A
B
A
B
A
B
A

11/12/2022 4
Translation
15 - 4
• 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.

11/12/2022 5
Rotation About a Fixed Axis. Velocity
15 - 5
• 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

11/12/2022 6
Rotation About a Fixed Axis. Acceleration
15 - 6
• 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,

11/12/2022 7
Rotation About a Fixed Axis. Representative Slab
15 - 7
• 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












11/12/2022 8
Equations Defining the Rotation of a Rigid Body About a Fixed Axis
15 - 8
• 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

11/12/2022 9
General Plane Motion
15 - 9
• General plane motion is neither a translation nor a rotation.
• General plane motion can be considered as the sum of a
translation and rotation.
• Displacement of particles A and B to A2 and B2 can be
divided into two parts:
- translation to A2 and
- rotation of about A2 to B2
1
B
1
B

11/12/2022 10
Absolute and Relative Velocity in Plane Motion
15 - 10
• Any plane motion can be replaced by a translation of an arbitrary reference
point A and a simultaneous rotation about A.
A
B
A
B v
v
v






 r
v
r
k
v A
B
A
B
A
B 





A
B
A
B r
k
v
v






 

11/12/2022 11
Absolute and Relative Velocity in Plane
Motion
15 - 11
• Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the
angular velocity  in terms of vA, l, and .
• The direction of vB and vB/A are known. Complete the velocity diagram.


tan
tan
A
B
A
B
v
v
v
v






cos
cos
l
v
l
v
v
v
A
A
A
B
A




11/12/2022 12
Absolute and Relative Velocity in Plane
Motion
15 - 12
• 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.

11/12/2022 13
Instantaneous Center of Rotation in Plane Motion
15 - 13
• Plane motion of all particles in a slab can always be replaced by the
translation of an arbitrary point A and a rotation about A with an angular
velocity that is independent of the choice of A.
• The same translational and rotational velocities at A are obtained by allowing
the slab to rotate with the same angular velocity about the point C on a
perpendicular to the velocity at A.
• The velocity of all other particles in the slab are the same as originally
defined since the angular velocity and translational velocity at A are
equivalent.
• As far as the velocities are concerned, the slab seems to rotate about the
instantaneous center of rotation C.

11/12/2022 14
15 - 14
• If the velocity at two points A and B are known, the instantaneous center of
rotation 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, the
instantaneous center of rotation 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 parallel, the instantaneous center of rotation is at
infinity and the angular velocity is zero.
• If the velocity magnitudes are equal, the instantaneous center of rotation is
at infinity and the angular velocity is zero.

11/12/2022 15
15 - 15
• The instantaneous center of rotation 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 velocities of all particles on the rod are as if they were rotated about C.
• The particle at the center of rotation has zero velocity.
• The particle coinciding with the center of rotation changes with time and the
acceleration of the particle at the instantaneous center of rotation is not zero.
• The acceleration of the particles in the slab cannot be determined as if the slab
were simply rotating about C.
• The trace of the locus of the center of rotation on the body is the body centrode
and in space is the space centrode.

11/12/2022 16
• A rod AB 26 m long leans against a vertical wall. The end A on the floor
is drawn away from the wall at a rate of 24 m/s. When the end A of the
rod is 10 m from the wall, determine the velocity of B sliding down
vertically and the angular velocity of the rod.

11/12/2022 18
• At the instant shown in figure, the rod AB is rotating clockwise at 2.5
rad/sec. If the end C of the rod BC is free to move on horizontal surface,
find the angular velocity of the point C.

11/12/2022 20
• A wheel of radius o.75 m rolls without slipping on a horizontal surface to
right. Determine the velocities of the points P and Q shown in figure
when the velocity of the wheel is 10 m/s towards right.
P
Q

11/12/2022 21
• Block D shown in figure moves with a speed of 3 m/s. Determine the
angular velocities of link BD and AB and the velocity of point B at the
instant shown.

11/12/2022 23
• A slider crank mechanism is shown in the figure. The crank OA rotates
anticlockwise at 100 rad/sec. Find the angular velocity of the rod AB and
the velocity of the slider B.

EM-Module 2.2-ICR method_MAP (1).pdf

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