FROM PARTICLE TO RIGID BODY.

FROM PARTICLE TO

RIGID BODY

From Particle to Rigid Body

This lecture concentrates on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the kinematics of a rigid body in 3-D space

After this lecture, the student should be able to:•Appreciate the concept of rigid bodies and extended rigid bodies•Define the mathematical condition for the rigidity of a body•Define the rotational matrix between two frames of reference

Particle vs. Rigid Body

Particle (size/shape

not important)

Combination of many particles (size/shape has to be considered)

XA B

The size/shape has to be considered, for example, because the length between point “A” and point “B” may affect the kinematics solution.

For a particle, size/shape is not important as it is treated as a point.

Rigid implies that body cannot be deformed.

Extended Rigid Body

The 3 rigid bodies are linked: Body 1 and body 2 joined together by body 3 (pin). As body 2 rotates about axis <a>, any point on the axis (e.g. point “P”) remains fixed relative to all 3 bodies. We can consider this axis as an extension of any of the 3 rigid bodies. In general, we can associate a space outside as an extension of a rigid body if no point of that space undergoes motion relative to the body.

Body 1

Body 2

Body 3

Axis <a>

Point “P”

Spatial Configuration

Consider the following rigid body and the inertia reference frame {X,Y,Z} with origin located at point “O”:

X-axis

Y-axis

Z-axis

XA B

To study the movement of the rigid body w.r.t. the inertia frame, we use another set of orthonormal frame attached to the body. For example, frame {e1, e2, e3} is attached to the body with origin fixed at “A”

1e2e

3e“O”

Note that points”O” and “A” are at the same point at time t=0. As the body moves, the points are not the same

Spatial Configuration

Consider the body has moved to a new location after time t=1 sec. To define the new position of the body w.r.t. frame {a}, we can use vector (from “O” to “A”)

X

AB

1e

2e3e

X-axis

Y-axis

Z-axis

“O”

OAA

Obviously, position is not enough to describe the motion as the orientation of the body has changed (orientation is not a problem in particle kinematics)

XA B

1e2e

3e

Position and Translation

•Translation can be defined by the displacement between the origins of frame {b} =(e1, e2, e3) and frame {a}=(X,Y,Z). Translation will not change the orientation between the frames:

X-axis

Y-axis

Z-axis

“O”X

A B1e

2e

3e

XA B

1e2e

3e

OrientationTo study the effects of the orientation, we note that at time t=0:

X-axis

Y-axis

Z-axis

“O”

XA B

1e2e

3e

is in the positive X-axis:1e Te ]001[1

is in the positive Y-axis:2e Te ]010[ˆ2

is in the positive Z-axis:3e Te ]100[ˆ3

T

T

T

e

e

e

]100[ˆ

]001[ˆ

]010[ˆ

3

2

1

At time t=1 sec.:

X

AB

1e

2e3e

We can use to define the orientation of the body

321 ˆˆˆ eee

XA

B

X-axis

Y-axis

Z-axis

“O”X

A B1e

2e

3e

X-axis

Y-axis

Z-axis

“O” 1e

2e3e

Orientation and Rotation

Orientation: The changes in the orientation of the can be viewed as a result of rotations: e.g. 90° rotation about Z-axis

}ˆ,ˆ,ˆ{ 321 eee

General Motion of a Rigid Body

The analysis so far revealed that the motion of a rigid body can be defined w.r.t. the inertia reference frame {X,Y,Z} using another orthonormal frame attached to the body.The rigid body motion has two parts:•Translation: defined by the vector from the origin of frame {X,Y,Z} to the origin of frame•Orientation: The changes in orientation of frame can be defined by rotations w.r.t. frame {X,Y,Z}

}ˆ,ˆ,ˆ{ 321 eee

}ˆ,ˆ,ˆ{ 321 eee}ˆ,ˆ,ˆ{ 321 eee

General Motion += Translation Rotation

Translation of a Rigid Body

Consider the following translation of a rigid body:

X-axis

Y-axis

Z-axis

“O”X

A B1e

2e

3e

XA B

1e2e

3e

Notice that all the points in the rigid body must have the same velocity and the same acceleration at any time instance

BBAA

BA

avdtdv

dtda

vv

Rotation of a Rigid Body

Unlike translation, rotation is defined using an angular coordinate like . The angular velocity and angular acceleration are respectively defined as

2

2

dtd

dd

dtd

dtd

Rotation, angular velocity and angular acceleration are vectors: they have both magnitudes and directions (the direction is defined using the right-hand rule)

Vector direction

Direction of rotation

Rigidity Condition

The example on the motion of a rigid body also revealed the rigidity conditions:

X-axis

Y-axis

Z-axis

“O”

XA B

1e2e

3e

CX

AB

1e

2e3eC

The distance between points “A” and “B” on the body remains unchanged after the body has moved. Similarly the CAB also remains unchanged after the motion. The preservation of these properties are called rigidity conditions.

)()(

)()(

10

10

tCABtCAB

tABtAB

Rotational Tensor

The changes in orientation can be defined by the rotational tensor. Let us review the case of pure rotation:

X-axis

Y-axis

Z-axis

“O”

XA B

1e2e

3e

is in the positive X-axis:1e TT rrre ][]001[ˆ 3121111

is in the positive Y-axis:2e TT rrre ][]010[ˆ 3222122

is in the positive Z-axis:3e TT rrre ][]100[ˆ 3323133

The rotational tensor is defined as

100010001

333231

232221

131211

rrrrrrrrr

R

There is no rotation!

C

Rotational Tensor

What is the rotational tensor?

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

is in the positive Y-axis:1e TT rrre ][]010[ˆ 3121111

is in the negative X-axis:2e TT rrre ][]001[ˆ 3222122


100001010

333231

232221

131211

rrrrrrrrr

R

The rotational tensor is

X

A

B

C

Rotational Tensor Example

What is the rotational tensor for the following configuration?

X-axis

Y-axis

Z-axis

“O”

1e

2e

3e

is in the positive X-axis:1e TT rrre ][]001[ˆ 3121111


is in the negative Y-axis:3e TT rrre ][]010[ˆ 3323133

010100

001

333231

232221

131211

rrrrrrrrr

R

The rotational tensor is

Rotational Motion

Consider two points “A” to “B” on the rigid body (the distance “AB” is 2) and the rotational tensor:

TtAB ]020[)( 0

X-axis

Y-axis

Z-axis

“O”

XA B

1e2e

3e

At time t=0:TtAB ]002[)( 1

At time t=1:

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

100001010

)( 1tR

Rotational Motion

TtAB ]020[)( 0

TtAB ]002[)( 1

)(002

020

100001010

)]()[( 101 tABtABtR

100001010

)( 1tR

Notice that:

R(t1) times vector (AB) before rotation =

Vector (AB) after rotation

Rotational Motion

In general, given two arbitrary point “P” and “Q” on the rigid body and the rotational tensor at time “t”:

)]([)()()()( 0tPQtRtPQtPtQ

The equation describes how changes its orientation in space as a result of a rotation defined by R(t)

PQ

Note that if R(t) is the identity matrix, then )]([)( 0tPQtPQ

This means that either:•No rotation takes place•The rotation returns the body to its original position•The rotation is such that every line connecting 2 arbitrary points of the body remains parallel to itself

Example: Rotational Motion

Given before rotation is

Find the vector on the rigid body after rotation.

AP TtAP 0010

33333

3333333333

331R

3333

3

001

333333333333333

331

)]([)()( 0tAPtRtAP

Rotation between two configurations

X-axis

Y-axis

Z-axis

“O”

Let us again examine the rotation between the following two configurations using a new orthonormal frame:

At time t=t0:

XA B

1e2e

3e 1f

2f3f

TzyxT ffftf 11101 ]100[)(ˆ



Let

001100

010)(

321

321

321

0

zzz

yyy

xxx

fffffffff

tF


At time t=t1:

X

AB

1e

2e3e

X-axis

Y-axis

Z-axis

“O”

2f

1f3f

100001010

)( 1tR

TzyxT ffftf 11111 ''']100[)(ˆ

TzyxT ffftf 22212 ''']010[)(ˆ

TzyxT ffftf 33313 ''']001[)(ˆ

001010100

'''''''''

)(

321

321

321

1

zzz

yyy

xxx

fffffffff

tF


At time t=t1:

001100

010)( 0tF

100001010

)( 1tR

001010100

)( 1tF

Note that

)(001010100

001100

010

100001010

)()(

1

01

tF

tFtR

R(t1) times Frame (t0) before rotation =

Frame (t1) after rotation


At time t=t2:

010001100

)( 2tR

TzyxT ffftf 11121 """]001[)(ˆ

TzyxT ffftf 22222 """]010[)(ˆ

TzyxT ffftf 33323 """]100[)(ˆ

100010001

"""""""""

)(

321

321

321

2

zzz

yyy

xxx

fffffffff

tF

X

A

B

1e

2e

3eX-axis

Y-axis

Z-axis

“O”

2f1f

3f

Rotation between two configurationsAt time t=t2:

001100

010)( 0tF

Again, note that

)(100

010001

001100

010

010001100

)()(

2

02

tF

tFtR

010001100

)( 2tR

100010001

)( 2tF

R(t2) times Frame (t0) before rotation =

Frame (t2) after rotation

Rotation between two configurationsAt time t=t1 and t=t2:

100010001

)( 2tF

010001100

)( 2tR

100001010

)( 1tR

001010100

)( 1tF

100001010

)( 11 tR

)(100

010001

001010100

001010100

001010100

100001010

010001100

)()()(

2

111

2

tF

tFtRtR

Rotation between two configurationsAt time t=t1 and t=t2:

100010001

)( 2tF

010001100

)( 2tR

100001010

)( 1tR

001010100

)( 1tF

001010100

)( 11 tF

001010100

001010100

100010001

)()(

001010100

100001010

010001100

)()(

11

2

11

2

tFtF

tRtRR


In general, the concept can be extended to any arbitrary orthonormal set attached to the rigid body, i.e. using

)()()()()()(

)()()()()()(

111

2022

111

0011

tFtRtRtFtRtF

tFtRtFtFtRtF

The above shows that if two configurations F(t2) and F(t1) are known relative to F(t0), then we can find the configuration of F(t2) relative to F(t1) using R, where )()( 1

12 tFtFR

Therefore:

)()()()()()(

12

11

211

2

tRFtFtFtFtRtRR

Summary

This lecture concentrates on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3-D space.

The following were covered:•The concept of rigid bodies and extended rigid bodies•The mathematical condition for the rigidity of a body•The rotational matrix between two frames of reference

FROM PARTICLE TO RIGID BODY.

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