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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 - PowerPoint PPT Presentation
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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
AABAB rrr /
AB vv
AB aa
x
y
rBrA
fixed in the bodySimplified 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, dθ
• Angular velocity ω=dθ/dt
• Angular Acceleration– α=dω/dtCounterclockwise 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, dθ
• Angular velocity ω=dθ/dt
• Angular Acceleration– α=dω/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 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.