1 me 302 dynamics of machinery dynamic force analysis iv dr. sadettin kapucu © 2007 sadettin kapucu
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1
ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY
Dynamic Force Analysis IV
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
2Gaziantep University
PreliminaryPreliminary Coordinate Transformation
– Reference coordinate frame OXYZ
– Body-attached frame O’uvw
wvu kji wvuuvw pppP
zyx kji zyxxyz pppP
x
y
z
P
u
v
w
O,
Point represented in OXYZ:
zwyvxu pppppp
Tzyxxyz pppP ],,[
Point represented in O’uvw:
Two frames coincide ==>
O’
3Gaziantep University
PreliminaryPreliminary
Mutually perpendicular Unit vectors
Properties of orthonormal coordinate frame
0
0
0
jk
ki
ji
1||
1||
1||
k
j
i
Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
3R
cosyxyx
x yx y
x
y
4Gaziantep University
PreliminaryPreliminary Coordinate Transformation
– Rotation only
wvu kji wvuuvw pppP
x
y
zP
zyx kji zyxxyz pppP
uvwxyz RPP u
vw
How to relate the coordinate in these two frames?
5Gaziantep University
yp
xp
PreliminaryPreliminary
Basic Rotation– , , and represent the
projections of onto OX, OY, OZ axes, respectively
– Since
xpP
yp zp
wvux pppPp wxvxuxx kijiiii
wvuy pppPp wyvyuyy kjjjijj
wvuz pppPp wzvzuzz kkjkikk
wvu kji wvu pppP
x
y
zP
u
vw
zp
6Gaziantep University
PreliminaryPreliminary Basic Rotation Matrix
– Rotation about x-axis with
w
v
u
z
y
x
p
p
p
p
p
p
wzvzuz
wyvyuy
wxvxux
kkjkik
kjjjij
kijiii
x
z
y
v
wP
u
CS
SC),x(Rot
0
0
001
7Gaziantep University
PreliminaryPreliminary Is it True?
– Rotation about x axis with
cospsinpp
sinpcospp
pp
p
p
p
cossin
sincos
p
p
p
wvz
wvy
ux
w
v
u
z
y
x
0
0
001
x
z
y
v
wP
u
8Gaziantep University
Basic Rotation MatricesBasic Rotation Matrices– Rotation about x-axis with
– Rotation about y-axis with
– Rotation about z-axis with
uvwxyz RPP
CS
SC),x(Rot
0
0
001
CS
SC
),y(Rot
0
010
0
100
0
0
),(
CS
SC
zRot
9Gaziantep University
PreliminaryPreliminary Basic Rotation Matrix
– Obtain the coordinate of from the coordinate of
uvwxyz RPP
wzvzuz
wyvyuy
wxvxux
kkjkik
kjjjij
kijiii
R
xyzuvw QPP
TRRQ 1
31 IRRRRQR T
uvwP xyzP
<== 3X3 identity matrix
z
y
x
w
v
u
p
p
p
p
p
p
zwywxw
zvyvxv
zuyuxu
kkjkik
kjjjij
kijiii
Dot products are commutative!
10Gaziantep University
Example 2Example 2
A point 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.
)2,3,4(uvwa
2
964.4
598.0
2
3
4
100
05.0866.0
0866.05.0
)60,( uvwxyz azRota
11Gaziantep University
Example 3Example 3 A point is the coordinate w.r.t. the
reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.
)2,3,4(xyza
uvwa
2
964.1
598.4
2
3
4
100
05.0866.0
0866.05.0
)60,( xyzT
uvw azRota
12Gaziantep University
Coordinate TransformationsCoordinate 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}
13Gaziantep University
Coordinate TransformationsCoordinate Transformations Two Special Cases
1. Translation only– Axes of {B} and {A} are
parallel
2. Rotation only– Origins of {B} and {A}
are coincident
1BAR
'oAPBB
APA rrRr
0' oAr
14Gaziantep University
Homogeneous RepresentationHomogeneous Representation• Coordinate transformation from {B} to {A}
• Homogeneous transformation matrix
'oAPBB
APA rrRr
1101 31
' PBoAB
APA rrRr
10101333
31
' PRrRT
oAB
A
BA
Position vector
Rotation matrix
Scaling
15Gaziantep University
Homogeneous TransformationHomogeneous Transformation Special cases
1. Translation
2. Rotation
10
0
31
13BA
BA RT
10 31
'33
oA
BA rIT
16Gaziantep University
x
y
z
P
u
v
w
O, O’
Example 5Example 5 Translation along Z-axis with h:
1000
100
0010
0001
),(h
hzTrans
111000
100
0010
0001
1
hp
p
p
p
p
p
hz
y
x
w
v
u
w
v
u
x
y
z P
u
vw
O, O’h
17Gaziantep University
Example 6Example 6 Rotation about the X-axis by
1000
00
00
0001
),(
CS
SCxRot
x
z
y
v
w
P
u
11000
00
00
0001
1w
v
u
p
p
p
CS
SC
z
y
x
18Gaziantep University
Homogeneous TransformationHomogeneous Transformation Composite Homogeneous Transformation
Matrix Rules:
– Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplication
– Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication
19Gaziantep University
Example 7Example 7 Find the homogeneous transformation matrix (T)
for the following operation:
:
axis OZabout ofRotation
axis OZ along d ofn Translatio
axis OX along a ofn Translatio
axis OXabout Rotation
Answer
44,,,, ITTTTT xaxdzz
1000
00
00
0001
1000
0100
0010
001
1000
100
0010
0001
1000
0100
00
00
CS
SC
a
d
CS
SC
20Gaziantep University
Homogeneous RepresentationHomogeneous Representation A frame in space (Geometric
Interpretation)
x
y
z),,( zyx pppP
1000zzzz
yyyy
xxxx
pasn
pasn
pasn
F
n
sa
101333 PR
F
Principal axis n w.r.t. the reference coordinate system
21Gaziantep University
Homogeneous TransformationHomogeneous Transformation
Translation
y
z
n
sa n
sa
1000
10001000
100
010
001
zzzzz
yyyyy
xxxxx
zzzz
yyyy
xxxx
z
y
x
new
dpasn
dpasn
dpasn
pasn
pasn
pasn
d
d
d
F
oldzyxnew FdddTransF ),,(