# Kinematics of Rigid Bodies

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```					       Kinematics of Rigid Bodies
By the end of this lesson, you should be able to:
• Calculate the velocity and acceleration
• of any point on a rigid body
• relative to any reference frame

1

Outline
•   Overview
•   Velocity
•   Acceleration
•   Sliding Contacts
•   Relative Reference Frames

2

Rigid Bodies
Rigid bodies are idealized models, that do not deform or change
shape
Rigid                               Not Rigid

3

1
Rigid Body vs. COM

4

Kinematics vs. Kinetics
• Kinematics only describes motion
(pos, velocity, acceleration)
• Kinetics considers the forces and couples
causing kinematics (next week)

5

Types of Motion
• Translation
• Planar Motion

6

2
Translation
•   Translation: the object does not rotate
•   Its trajectory may be curved
•   Every point has the same velocity and acceleration
•   Therefore, Motion is described by the motion of a single point
•   No need to discuss it more; we already know everything

7

• Each point of the rigid body on the axis is
stationary
• Each point not on the axis moves in a circular

8

Planar Motion
Combines translation
and rotation in a plane

9

3
• Easy, based on previous knowledge
dθ              dω d 2θ
ω=               α=      = 2
dt              dt  dt

v = rω          at = rα
v2
an = = rω 2
r

10

Example
Gear A turns gear B, raising hook H. If the gear A starts
from rest at t=0 and its clockwise angular acceleration is
αA=0.2t rad/s2, what vertical distance has the hook H
risen and what is its velocity at t = 10s?

11

General Motion: Velocity
• We must develop equations that relate relative motions
of points of a rigid body to its angular motion
• To do this, we use equations of relative motion:
rA = rB + rA/ B
v A = vB + v A/ B
Because rA/B is constant, velocity of A relative to B is
tangent to the circular path and equal to the product of rA/B
and the angular velocity ω of the rigid body
v A/ B = ω × rA/ B
v A = vB + ω × rA/ B

12

4
General
Motion
Velocity

13

Example
• What is the velocity of A? of B?

A             B

14

Example
Bar AB rotates with a clockwise angular velocity of 10
rad/s. Determine the angular velocity of bar BC and the
velocity of point C

15

5
Example
Bar AB rotates with a clockwise angular velocity of 10
rad/s. What is the vertical velocity vR of the rack?

16

Instantaneous Center
Instantaneous Center: The point
where velocity is 0 at a given instant

Useful concept, because all velocities
relative to this point are only angular
velocities

This point can change with time

17

Calculating Instantaneous Center
Instantaneous center C is the intersection
between 2 lines perpendicular to the motion of
2 points

C can be outside of the rigid body

18

6
Example
Bar AB rotates with a counterclockwise angular velocity of 10
rad/s. What are the angular velocities of bars BC and CD?
(Hint: locate instantaneous center of bar BC)

19

General Motion: Acceleration
v A = v B + ω × r A/ B
a A = a B + α × r A/ B + ω × (ω × r A/ B )

For planar motion, this simplifies to:
a A = a B + α × r A/ B − ω 2 r A/ B

20

Example
What is the acceleration of point A on the rolling disk?

21

7
Example
Bar AB has a CCW angular velocity of 10 rad/s and a CW
angular acceleration of 300 rad/s2. What are the angular
accelerations of bars BC and CD?

22

Sliding Contacts
We must rederive equations, without assuming that point A is
part of the rigid body

Movie illustration

23

Sliding contacts
v A = v B + v Arel + ω × r A/ B                   dx ˆ dy ˆ dz ˆ
v Arel =      i+    j+ k
dt    dt   dt
vA/B

a A = a B + a Arel + 2ω × v Arel + α × r A/ B + ω × (ω × r A/ B )

aA/B

d 2x ˆ d 2 y ˆ d 2z ˆ
a Arel   =      i+      j+     k
dt2     dt 2    dt2
In the case of planar motion:
a A = a B + a Arel + 2ω × v Arel + α × r A/ B − ω 2 r A/ B

24

8
Example
Bar AB has a CCW angular velocity of 2 rad/s and a
CCW angular acceleration of 10 rad/s2.
a) Determine the angular velocity of bar AC and the
velocity of the pin A relative to the slot in bar AB.
b) Determine the angular acceleration of bar AC and the
acceleration of the pin A relative to the slot in bar AB.

25

Example
The collar at B slides along the circular bar, causing the pin B
to move at constant speed v0 in a circular path of radius R.
Bar BC slides in the collar at A. At the instant shown,
determine the angular velocity and angular acceleration of bar
BC.

26

Example
Bar AB rotates with a constant CCW angular velocity of 1
rad/s. The block B slides in a circular slot in the curved bar
BC. At the instant shown, the center of the circular slot is at
D. Determine the angular velocity and angular acceleration
of bar BC.

27

9
Reference Frames
v A = v B + v Arel + ω × r A/ B
a A = a B + a Arel + 2ω × v Arel + α × r A/ B + ω × (ω × r A/ B )
Same equations as for sliding bodies
vA and aA , and vB, aB, are relative to the primary reference frame
vArel and aArel are relative to the secondary reference frame

28

Example
The merry-go-round rotates with constant angular
velocity ω. Suppose that you are in the center at B, and
observe the motion of a second person A, using a
coordinate system that rotates with the merry-go-round.
1) Person A is standing next to the merry-go-round. What
are his velocity and acceleration relative to your
coordinate system?
2) Person A is on the edge of the merry-go-round. What
are his velocity and acceleration relative to the earth?

29

Example
At the instant shown, the ship is moving north at a
constant speed of 15 m/s j and is turning toward the west
at a constant rate of 5° per second. Relative to the ship’s
coordinate system, its radar indicates that the motion of
the helicopter are:
ˆ               ˆ
r A/ B = 420i + 236 ˆ + 212k (m)
j
ˆ         ˆ
v A rel = −53.5i + 2 ˆ + 6k (m / s )
j
ˆ             ˆ
a A/ B = .4i − .2 ˆ − 13k ( m / s 2 )
j

What are the helicopter’s
velocity and acceleration
relative to the earth?

30

10

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