Lecture 14 – 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.

• 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.