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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

- What is the difference between particle and rigid body mechanics?
- Rigid body – can be of any shape
- Block
- Disc/wheel
- Bar/member
- Etc.

- Rigid body – can be of any shape
- Still planar
- All particles of the rigid body
move along paths equidistant

from a fixed plane

- All particles of the rigid body
- Can determine motion of
any single particle (pt)

in the body

particle

Rigid-body (continuum of particles)

- Kinematically
speaking…

- Translation
- Orientation of AB
constant

- Orientation of AB
- Rotation
- All particles rotate
about fixed axis

- All particles rotate
- General Plane Motion
(both)

- Combination of both
types of motion

- Combination of both

- Translation

B

B

B

B

A

A

A

A

- 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

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!

- α=dω/dt

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?

- 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

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 )

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.

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.