Transcript Lecture 14 - Planar Rigid Body Kinematics

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.
particle
• 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
Rigid-body (continuum of
particles)
Types of rigid body motion
• Kinematically
speaking…
B
– Translation
A
• Orientation of AB
constant
– Rotation
B
B
• All particles rotate
about fixed axis
– General Plane Motion
(both)
• Combination of both
types of motion
A
A
B
A
Kinematics of translation
• Kinematics
– Position
y
B



rB  rA  rB / A
– Velocity


vB  vA
A
rB
rA
– Acceleration
x


aB  aA
• True for all points in R.B.
(follows particle kinematics)
Simplified case of our relative motion of particles
discussion – this situation same as cars driving
side-by-side at same speed example
fixed in the body
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, dθ
• Angular velocity
ω=dθ/dt
• Angular Acceleration
– α=dω/dt
Counterclockwise is positive!
r
Angular velocity
angular velocity vector always
perpindicular to plane of rotation!
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?
Relating angular and linear velocity
http://lancet.mit.edu/motors/angvel.gif
http://www.thunderbolts.info
• 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
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, dθ
• Angular velocity
ω=dθ/dt
• Angular Acceleration
r
– α=dω/dt
• Angular motion kinematics
– Can handle the same way as rectilinear
kinematics!
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 )
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 ω0 = 8 rad/s. If it is subjected to
a constant angular acceleration of α = 6 rad/s2, determine the
magnitudes of the velocity and the n and t components of
acceleration of point A at the instant t = 0.5 s.