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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’

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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

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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?

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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

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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

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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

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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

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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!

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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

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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

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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}

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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

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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

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Homogeneous TransformationHomogeneous Transformation Special cases

1. Translation

2. Rotation

10

0

31

13BA

BA RT

10 31

'33

oA

BA rIT

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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

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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

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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

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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

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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

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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 ),,(