Lecture 14 - Planar Rigid Body Kinematics

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14 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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.