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BNG 202 – Biomechanics II Lecture 14 – Rigid Body Kinematics Instructor: Sudhir Khetan, Ph.D. Wednesday, May 1, 2013

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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)

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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

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Kinematics of translation Kinematics – Position – Velocity – Acceleration True for all points in R.B. (follows particle kinematics) B A x y rBrB rArA fixed in the body Simplified case of our relative motion of particles discussion – this situation same as cars driving side-by-side at same speed example

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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/dt Counterclockwise is positive! r

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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!

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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

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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!

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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.

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Example problem 2 The disk is originally rotating at ω 0 = 8 rad/s. If it is subjected to a constant angular acceleration of α = 6 rad/s 2, 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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