BNG
202
–
Biomechanics II
Lecture 14
–
Rigid Body Kinematics
Instructor: Sudhir Khetan, Ph.D.
Wednesday, May 1,
2013
Particle vs. rigid body mechanics
•
What is the difference between particle and rigid body
mechanics?
–
Rigid body
–
can be of any shape
•
Block
•
Disc/wheel
•
Bar/member
•
Etc.
•
Still planar
–
All particles of the rigid body
move along paths equidistant
from a fixed plane
•
Can determine motion of
any single particle (pt)
in the body
particle
Rigid

body (continuum of
particles)
Types of rigid body motion
•
Kinematically
speaking…
–
Translation
•
Orientation of AB
constant
–
Rotation
•
All particles rotate
about fixed axis
–
General Plane Motion
(both)
•
Combination of both
types of motion
B
A
B
A
B
A
B
A
Kinematics of translation
•
Kinematics
–
Position
–
Velocity
–
Acceleration
•
True for all points in R.B.
(follows particle
kinematics)
B
A
A
B
A
B
r
r
r
/
A
B
v
v
A
B
a
a
x
y
r
B
r
A
f
ixed in the body
Simplified case of our relative motion of particles
discussion
–
this situation same as cars driving
side

by

side at same speed example
Rotation about a fixed axis
–
Angular Motion
•
In this slide we discuss the motion of a line or
body
since these have dimension, only they
and not
points
can undergo angular motion
•
Angular motion
–
Angular position,
θ
–
Angular displacement, d
θ
•
Angular velocity
ω
=d
θ
/
dt
•
Angular Acceleration
–
α
=d
ω
/dt
Counterclockwise is positive!
r
Angular velocity
http://www.dummies.com/how

to/content/how

to

determine

the

direction

of

angular

velocity.html
Magnitude of
ω
vector = angular speed
Direction of
ω
vector
1) axis of rotation
2) clockwise or counterclockwise rotation
How can we relate
ω
&
α
to motion of a
point
on the body?
angular velocity vector always
perpindicular to plane of rotation!
Relating angular and linear velocity
http://lancet.mit.edu/motors/angvel.gif
•
v
=
ω
x
r, which is the cross product
–
However, we don’t really need it because
θ
= 90
°
between our
ω
and r vectors
we determine direction intuitively
•
So, just use v = (
ω
)(r)
multiply magnitudes
http://
www.thunderbolts.info
Rotation about a fixed axis
–
Angular Motion
r
Axis of
rotation
In solving problems, once know
ω
&
α
, we can get velocity and
acceleration of any point on
body
!!!
(
Or can relate the two types of motion if
ω
&
α
unknown )
•
In this slide we discuss the motion of a line or
body
since these have dimension, only they
and not
points
can undergo angular
motion
•
Angular
motion
–
Angular position,
θ
–
Angular displacement, d
θ
•
Angular velocity
ω
=d
θ
/
dt
•
Angular Acceleration
–
α
=d
ω
/dt
•
Angular motion kinematics
–
Can handle the
same way
as rectilinear
kinematics!
Example problem 1
When the gear rotates 20 revolutions, it achieves an
angular velocity of
ω
= 30
rad
/s, starting from rest.
Determine its constant angular acceleration and the time
required
.
Example problem 2
The disk is originally rotating at
ω
0
= 8 rad/s. If it is subjected to
a constant angular acceleration of
α
= 6 rad/s
2
, determine the
magnitudes of the velocity and the n and t components of
acceleration of point A at the instant t = 0.5 s.
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