# Lecture 14 - Planar Rigid Body Kinematics

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14 Νοε 2013 (πριν από 4 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

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

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

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
/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