# Ch 16 -Planar (2D) Kinematics of Rigid Bodies (P301)

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```					Ch 16 – Planar (2D) Kinematics of Rigid Bodies (P301)

• Kinematics of rigid bodies: relations between time and the positions,
velocities, and accelerations of the particles forming a rigid body.

• Classification of 2D Rigid Body Motion:

Rectilinear translation
Translation

Curvilinear translation
2D
Rigid
Body      Rotation about a fixed axis
Motion

General planar motion = translation + rotation

Spring 2008                  Prof. Winncy Du
Important Points

1. Rotation about a point is a 3D motion, not a 2D
(planar) motion.

2. Kinematics of a RB is actually to find the position,
velocity or acceleration of a particle in the RB.

3. A RB is formed by many particles. Since the relative
distances between these particles are fixed, their
velocities and accelerations are related. Knowing the
motion of one particle in the RB , any other particle’s
motion can be founded – this is the main idea of
solving RB Kinematics problems.

Spring 2008             Prof. Winncy Du
Translation Motion (16.2)– Important Points
• When a RB does translation: its direction/orientation is constant
• For any two particles in the body,
The relative position                                  All particles in
All particles in
between particles in the                               the RB have
the RB have the
RB is never changed.                                   the same
same velocity.
acceleration.
r r r 0                            r0                        r 0
rB = rA + rB A          r     r              r
VB = V A + VB / A = V A
r    r              r
aB = a A + aB / A = a A

Therefore, if a RB does translation motion, its motion can be
treated as a particle motion.
Spring 2008                Prof. Winncy Du
Angular Motion: Rotation about a Fixed Axis (16.3)
Consider rotation of rigid body about a fixed axis AA’:
Angular Position of a point at the RB: θ
Angular Velocity of any point at the RB:
dθ &
ω=      =θ
dt
Angular Acceleration of any point at RB:
dω d 2θ &&
α=    = 2 =θ
dt  dt
If α is constant,
ω = ω0 + α c t                     When a RB rotates about a fixed
axis, the angular velocity and
1                   angular acceleration are same
θ = θ 0 + ω0t + α c t 2
2                  every where in the RB. OR, the
ω 2 = ω0 + 2α c (θ − θ 0 )
2                           entire RB has only one
angular velocity and angular
αdθ = ωdω                          acceleration.
Spring 2008                      Prof. Winncy Du

• A rigid body rotates about a fixed axis. The position, velocity,
and acceleration of a point P on the rigid body are described by:

r
r
r
r dr r r
v=    =ω×r
dt

r2 r
r d r dv r r r r r
a= 2 =    = α × r + ω × (ω × r )
dt   dt

Spring 2008               Prof. Winncy Du
Examples of Rotation about a Fixed Axis

Spring 2008           Prof. Winncy Du
General Plane Motion (16.5)

• General plane motion can be considered as the
sum of a translation and rotation.

• Displacement of particles A and B to A2 and B2
can be divided into two parts:
′
- translation to A2 and B1
′
- rotation of B1 about A2 to B2

Spring 2008                   Prof. Winncy Du
Position, Velocity, and Acceleration of a Point on the RB
• Any plane motion can be replaced by a translation of an arbitrary reference
point A and a simultaneous rotation about A.

Three Important formula for solving a general planar motion problems

r r r
Position        rB = rA + rB / A
r     r r           r      r r
Velocity        vB = v A + vB / A = v A + ω × rB A
r     r r            r      r r r r r
Acceleration    a B = a A + aB / A = a A + [α × r + ω × (ω × r )]

Note: Point A and B must be on the same rigid body

Spring 2008                     Prof. Winncy Du
Examples of General Plane Motion

Spring 2008          Prof. Winncy Du
Examples of General Plane Motion

Spring 2008          Prof. Winncy Du

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