Lecture 14 – Rigid Body Kinematics

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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

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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

B

B

B

A

A

A

A

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

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!

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?

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

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 )

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.