Post on 17-Jan-2018
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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
is in the positive Z-axis:3e TT rrre ][]100[ˆ 3323133
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 positive Z-axis:2e TT rrre ][]100[ˆ 3222122
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
TzyxT ffftf 22202 ]001[)(ˆ
TzyxT ffftf 33303 ]010[)(ˆ
Let
001100
010)(
321
321
321
0
zzz
yyy
xxx
fffffffff
tF
Rotation between two configurations
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
Rotation between two configurations
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
Rotation between two configurations
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
Rotation between two configurations
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