kinematics of planar rigid body motion - may 11 lecture

8/3/2019 Kinematics of Planar Rigid Body Motion - May 11 Lecture

1/19

Kinematics of planar rigid bodymotion analysis using

rans a ng axesHibbeler chapter 16

May 11, 2011 lecture


2/19

Objective of kinematic analysis

using translating axes

Establish:

a) relationships between the

motion (velocity &

acceleration) ofpoints on

the same rigid body;

b) relationships between the

motion (velocity &acceleration) ofpoints on

rigid bodies connected with

non-slipping joints.


3/19

Last lecture Established relationships

between the motion (velocity &

acceleration) of points on the

same rigid body:

ABA rrv /rrr

+=

Black vectors absolute

motion of A & B (seen from

the fixed frame).

Green vectors relative

motion of B w.r.t. the axes

translating with A (motion of

B seen from the frame thattranslates with A).

ABAABnABtA

ABABA

ABAB

AB

ABAABA

aaaaa

rra

rrva

vvrv

//,/,

//

//

//

)(rrrrr

rrr

rr

r

rr

rrr

r

rrrrr

+=++=

++=

++=

+=+=


4/19

Today Use velocity & acceleration

equations from last lecture to:

a) Illustrate problem solving

procedure.

c) Establish relationships betweenthe motion (velocity &acceleration) ofpoints on rigid

bodies connected with non-slipping joints.

na yze part cu ar r g o ymotions:

Translation.

Rotation about a fixed axis.


5/19

Problem solving procedure1. Identify body connections:

they provide kinematicconstraints (relationships thatvelocities & accelerations mustobey).

2. Write velocity &acceleration equations:

connect point with known

direction of velocity (locallyknown path) to point withknown velocity & acceleration.

3. Solve for the unknowns.

Example

1. Joints: sliders at A & B. Write

corresponding kinematic constraints.

2. Point with known motion: A.3. Point with known direction of

velocity: B.

4. Write velocity & acceleration of B

w.r.t. A.5. Solve.


6/19

Slider (joint) B imposes that:

A.

B.

OBa

OBv

Bn

B

,

r

Oxa

Oxv

A

A

r

r

C.

D.

ABa

ABv

Bn

B

,

r

r

Oxa

Oxv

OBa

OBv

A

A

Bn

B

r

r

r

r

,


7/19

Translation Definition: any line of the body remains parallel to

itself throughout the motion.

r

0

=

AB

AB

def

aa

vvrr

rr

rr

0


8/19

Translation

important notes.

Rectilinear translation

All points on a translating

rigid body move on parallel

paths, with the same velocityand the same acceleration.

The paths of points on a

Curvilinear translation( )trr AArr

=

trans at ng r g o y are

determined by the bodys

connections (kinematic

constraints).

A rigid body in planartranslation has 2DOF

(translation is fully described

by the displacement of one

point ).


9/19

Translation


10/19

Rotation about a fixed axis Definition: a line/axis of the

body is at rest throughout themotion.

==

kk r

Choose O on axis of rotation:

0

0

O

O

a

vr

r

====

kkk rr


11/19

Rotation about a fixed axis

velocity equation

rr

In the plane of motion (perpendicular on the fixed axis of

rotation).

Perpendicular on the distance from P to the axis of rotation.

=

OP

OPrif

P

OP

OPOPP

rvvrv

/

/

/

//

,

rr

rrrr


12/19


acceleration equation

( )

PnPtOP

OPOPP

aaa

rra

,,/

//rrr

rrr

rr

r

+

+

( )OP

OPrif

OPPn

OP

OPrif

Pt

OP

OPPt

rra

ra

r

ra

/

2/

/,

/

/

,

/

/,

,

rrrr

r

rr

rr

rr

==

=

=

In the plane of motion.

Perpendicular on the

distance from P to the axisof rotation.

In the plane of motion.

From P to the axis ofrotation.


13/19


important notes All points on a rigid body rotating

about a fixed axis move on

circular paths centered on the

axis of rotation.

fixed axis of rotation move fasterthan points closer to it.

A rigid body in rotation about a

fixed axis has 1DOF (rotationabout a fixed axis is fully

described by the angular

displacement about the axis of

rotation ).( )t=


14/19

Examples: bodies pinned to the ground.Rotation about a fixed axis


15/19

Examples (2): bodies pinned to the ground.Rotation about a fixed axis


16/19

Examples (3): bodies pinned to the ground.Rotation about a fixed axis


17/19

A. B rotates about A.

B. A translates with B and

rotates about B.

C. A does not move.

D. All of the above.

.

A does not move.


18/19

Rotation abouta fixed axis


19/19

Rotation abouta fixed axis

kinematics of planar rigid body motion - may 11 lecture

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