Lecture 14 – Rigid Body Kinematics

1. BNG 202 – Biomechanics II Lecture 14 – Rigid Body Kinematics Instructor: Sudhir Khetan, Ph.D. Wednesday, May 1, 2013

2. 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)

3. 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 B B B A A A A

4. Kinematics of translation • Kinematics • Position • Velocity • Acceleration • True for all points in R.B. (follows particle kinematics) y rB rA x B A 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

5. Rotation about a fixed axis – Angular Motion r • 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!

6. 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 & αto motion of a point on the body?

7. Relating angular and linear velocity • v = ωx 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 http://lancet.mit.edu/motors/angvel.gif

8. Rotation about a fixed axis – Angular Motion r • 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! 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 )

9. 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.

10. 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.