Transcript Eğitim
ME 302 DYNAMICS OF
MACHINERY
Dynamic Force Analysis IV
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
1
Preliminary
Coordinate Transformation
– Reference coordinate frame
OXYZ
– Body-attached frame O’uvw
z
P
Point represented in OXYZ:
Pxyz [ px , py , pz ]
Pxyz px i x p y jy pz k z
T
Point represented in O’uvw:
Puvw pu i u pv jv pwk w
Two frames coincide ==>
y
w
v
x
O, O’
u
pu px pv py pw pz
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Preliminary
Properties: Dot Product
x
3
Let x and y be arbitrary vectors in R and be
the angle from x to y , then
x y x y cos
y
Properties of orthonormal coordinate frame
Mutually perpendicular
i j 0
i k 0
k j 0
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Unit vectors
| i | 1
| j | 1
| k | 1
3
Preliminary
Coordinate Transformation
z
– Rotation only
P
Pxyz px i x p y jy pz k z
Puvw pu i u pv jv pwk w
y
w
v
Pxyz RPuvw
u
x
How to relate the coordinate in these two frames?
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4
Preliminary
Basic
–
Rotation
px , p y , and pz
z
pz
represent the w
projections of P onto OX, OY,
py
OZ axes, respectively
– Since
P pu i u pv jv pwk w
P
y
v
u
x
px
px i x P i x i u pu i x jv pv i x k w pw
py jy P jy i u pu jy jv pv jy k w pw
pz k z P k z i u pu k z jv pv k z k w pw
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Preliminary
Basic
Rotation Matrix
px i x i u
p j i
y y u
p z k z i u
i x jv
j y jv
k z jv
i x k w pu
j y k w pv
k z k w pw
– Rotation about x-axis with
1 0
Rot( x , ) 0 C
0 S
0
S
C
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z
w
P
v
u
x
y
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Preliminary
Is it True?
– Rotation about x axis with
0
p x 1
p 0 cos
y
p z 0 sin
pu
sin pv
cos pw
0
z
w
P
p x pu
p y pv cos pw sin
p z pv sin pw cos
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v
y
u
x
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Basic Rotation Matrices
– Rotation about x-axis with
1 0
Rot( x , ) 0 C
0 S
– Rotation about y-axis with
C
Rot( y , ) 0
S
– Rotation about z-axis with
Pxyz RPuvw
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0
S
C
S
1 0
0 C
0
C
Rot( z , ) S
0
S
0
0
1
C
0
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Preliminary
Basic Rotation Matrix
i x i u i x jv i x k w
Pxyz RPuvw
R jy i u jy jv jy k w
k z i u k z jv k z k w
– Obtain the coordinate of Puvw from the coordinate of Pxyz
Dot products are commutative!
pu i u i x
p j i
v v x
pw k w i x
i u jy
jv j y
k w jy
i u k z px
jv k z p y
k w k z p z
QR RT R R1R I3
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Puvw QPxyz
Q R1 RT
<== 3X3 identity matrix
9
Example 2
A point auvw (4,3,2) is attached to a rotating frame, the
frame rotates 60 degree about the OZ axis of the
reference frame. Find the coordinates of the point
relative to the reference frame after the rotation.
a xyz Rot( z ,60)auvw
0.5 0.866 0 4 0.598
0.866
0.5
0 3 4.964
0
0
1 2 2
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Example 3
A point axyz (4,3,2) is the coordinate w.r.t. the
reference coordinate system, find the
corresponding point auvw w.r.t. the rotated OU-VW coordinate system if it has been rotated 60
degree about OZ axis.
auvw Rot( z ,60)T a xyz
0.866 0 4 4.598
0.5
0.866 0.5 0 3 1.964
0
0
1 2 2
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Coordinate Transformations
• position vector of P
in {B} is transformed
to position vector of P
in {A}
• description of {B} as
seen from an observer
in {A}
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}
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Coordinate Transformations
Two Special Cases
A P A
B P A o'
r RB r r
1. Translation only
– Axes of {B} and {A} are
parallel
A
RB 1
2. Rotation only
– Origins of {B} and {A}
are coincident
r 0
A o'
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Homogeneous Representation
• Coordinate transformation from {B} to {A}
A P A
r RB B r P Ar o'
A
A o'
r
RB
r B r P
1 1
1 013
• Homogeneous transformation matrix
A P
RB
TB
013
A
A
r R33
1 0
A o'
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P31
1
Scaling
Rotation
matrix
Position
vector
14
Homogeneous Transformation
Special cases
1. Translation
I 33
A
TB
013
r
1
A o'
2. Rotation
A
RB
A
TB
013
031
1
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Example 5
Translation along Z-axis with h:
1
0
Trans( z , h)
0
0
0
1
0
0
0
0
1
0
x 1
y 0
z 0
1 0
0
0
h
1
0
1
0
0
0
0
1
0
0 pu pu
0 pv pv
h pw pw h
1 1 1
z
z
P
P
y
w
v
v
x
O, O’
y
w
h
u
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O, O’
u
x
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Example 6
Rotation about the X-axis by
1 0
0 C
Rot( x, )
0 S
0 0
0
S
C
0
0
0
0
1
x 1 0
y 0 C
z 0 S
1 0 0
w
0
S
C
0
0 pu
0 pv
0 p w
1 1
z
P
u
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x University
v
y
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Homogeneous Transformation
Composite Homogeneous Transformation
Matrix
Rules:
– Transformation (rotation/translation) w.r.t
(X,Y,Z) (OLD FRAME), using premultiplication
– Transformation (rotation/translation) w.r.t
(U,V,W) (NEW FRAME), using postmultiplication
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Example 7
Find the homogeneous transformation matrix (T)
for the following operation:
Rotation about OX axis
T ranslation of a alongOX axis
T ranslation of d alongOZ axis
Rotationof about OZ axis
Answer :
T Tz , Tz ,d Tx,aTx, I 44
C S 0 0 1 0 0
S C 0 0 0 1 0
0
0
1 0 0 0 1
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0
0
0
1
0 0 0
0 1
0 0
d 0
1 0
0
1
0
0
0
0
1
0
a 1 0
0 0 C
0 0 S
1 0 0
0
S
C
0
0
0
0
119
Homogeneous Representation
A frame in space (Geometric
Interpretation)
z
R33 P31
F
1
0
nx
n
F y
nz
0
sx
sy
sz
0
ax
ay
az
0
P( px , py , pz )
a
s
n
px
p y
pz
1
y
x
Principal axis n w.r.t. the reference coordinate system
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Homogeneous Transformation
Translation
a
s
1
0
Fnew
0
0
nx
n
y
nz
0
0 0 d x nx s x
1 0 d y n y s y
0 1 d z nz s z
0 0 1 0 0
sx ax px d x
s y a y p y d y
sz az pz d z
0 0
1
ax
ay
az
0
px
p y
pz
1
z
a
n
s
n
y
Fnew Trans(d x , d y , d z ) Fold
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