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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Lecture 14 – Rigid Body Kinematics

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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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