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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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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
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
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

kinematics of translation
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
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

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
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 motion1
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
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
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.

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