Ch04 position analysis

POSITION ANALYSIS

Chapter 4

Introduction

Dynamics

Kinematics

Position Velocity Acceleration

Kinetics

Newton’s Second Law

Work & Energy

Impulse & Momentum

Stresses

Design

Graphical

Analytical

Coordinate Systems

Global or Absolute

Attached to Earth

Local

Attached to a link at some point of interest

LNCS (local nonrotatingcoordinate system)

LRCS (local rotating coordinate system)

Position and Displacement

22

XYA RRR

X

Y

R

R1tan

Coordinate

Transformation sincos yxX RRR

cossin yxY RRR

Displacement

The straight-line distance between the initial and final

position of a point which has moved in the reference

frameABBA RRR

Translation, Rotation and Complex

Motion

Translation

All points on the body have the

same displacement

BBAA RR ''


Motion

Rotation

Different points in the body

undergo different

displacements and thus there is

a displacement difference

between any two points chosen

Euler’s theorem

BAABBB RRR ''


Motion

Complex Motion

Is the sum of the translation and

rotation

Chasles’Theorem

'"'" BBBBBB RRR

'"'" ABAAAB RRR

Graphical Position Analysis

For any one-DOF, such a fourbar, only one

parameter is needed to completely define the

positions of all the links. The parameter usually

chosen is the angle of the input link.


Construction of the graphical solution

The a, b, c, d and the angle θ2 of the input link are

given.

1. The ground link (1) and the input link (2) are

drawn to a convenient scale such that they

intersect at the origin O2 of the global XY

coordinate system with link 2 placed at the input

angle θ2.

2. Link 1 is drawn along the X axis for

convenience.





3. The compass is set to the scaled length of link

3, and an arc of that radius swung about the end

of link 2 (point A).



4. Set the compass to the scaled length of link 4, and a second arc swung about the end of link 1 (point O4). These two arcs will have two intersections at B and B’ that define the two solution to the position problem for a fourbarlinkage which can be assembled in two configurations, called circuits, labeled open and crossed.

5. The angles of links 3 and 4 can be measured with a protractor.



Algebraic Position Analysis

Algebraic Algorithm

Coordinates of point A

Coordinates of point B

2cosaAx

2sinaAy

222

yyxx ABABb

222

yx BdBc


Algebraic Algorithm

Coordinates of point B

dA

BAS

dA

BA

dA

dcbaB

x

yy

x

yy

x

x

2

2

2

2

2

2222

02

2

2

c

dA

BASB

x

yy

y

P

PRQQBy

2

42

1

2

2

dA

AP

x

y

dA

SdAQ

x

y

2

22cSdR

dA

dcbaS

x

2

2222


Algebraic Algorithm

Link angles for the

given position

xx

yy

AB

AB1

3 tan

dB

B

x

y1

4 tan


Vector Loop

Links are represented as position vectors


Complex Numbers as Vector

Unit vectors



Complex number notation




Euler identity

sincos je j

jj

jed

de


Vector Loop Equation

for a Fourbar Linkage

Position vector

Complex number

notation

01432 RRRR

01432 jjjj

decebeae

0Independent

variableTo be

determine


Vector Loop Equation for a

Fourbar Linkage

Euler equivalents and separate

into two scalar equations

24

23

,,,,

,,,,

dcbaf

dcbaf

0sincossincossincossincos 11443322 jdjcjbja

Real part: 0coscoscos 432 dcba

0sinsinsin 432 cbaImaginary part:


Solve simultaneously

Square and add

dcab 423 coscoscos

423 sinsinsin cab

242

2

423

2

3

22 coscossinsincossin dcacab

242

2

42

2 coscossinsin dcacab

424242

2222 coscossinsin2cos2cos2 accdaddcab


To simplify, constants are define in terms of the

constant link length,

Substituting the identity,

Freudenstein’s equation

a

dK 1

424232241 sinsincoscoscoscos KKK

c

dK 2

ac

dcbaK

2

2222

3

424242 sinsincoscoscos

4232241 coscoscos KKK


Using half angle identities,

Simplified form

2tan1

2tan2

sin42

4

4

2tan1

2tan1

cos42

42

4

02

tan2

tan 442

CBA

3221

2

32212

cos1

sin2

coscos

KKKC

B

KKKA


The solution,

If the solution is complex conjugate, the link lengths

chosen are not capable of connection

The solution will usually be real and unequal

Crossed (+)

Open (-)

A

ACBB

2

4

2tan

2

4

A

ACBB

2

4arctan2

2

4 2,1


Solution for θ3

Square and add

dbac 324 coscoscos

324 sinsinsin bac

323252431 sinsincoscoscoscos KKK

b

dK 4

ab

badcK

2

2222

5

02

tan2

tan 332

FED

5241

2

52412

cos1

sin2

coscos

KKKF

E

KKKD


The solution,

If the solution is complex conjugate, the link lengths

chosen are not capable of connection

The solution will usually be real and unequal

Crossed (+)

Open (-)

D

DFEE

2

4arctan2

2

3 2,1


Fourbar Slider-Crank

Linkage

Position vector

Complex number

notation

01432 RRRR

01432 jjjj

decebeae

0Independent

variableTo be

determine


Vector Loop Equation for a

Fourbar Linkage

Euler equivalents and separate

into two scalar equations

0sincossincossincossincos 11443322 jdjcjbja

Real part: 0coscoscos 432 dcba

0sinsinsin 432 cbaImaginary part:


The solution,

32

23

coscos

sinarcsin

1

bad

b

ca

b

ca 23

sinarcsin

2


Inverted Slider-Crank (p193-p194)


Geared Fivebar Linkage

Position vector


015432 RRRRR

015432 jjjjj

fedecebeae



Using the relationship

between the two geared

links;


25

012432

jjjjjfedecebeae

ratiogear ,

angle phase ,



Solution (pag. 196)

D

DFEE

2

4arctan2

2

4 2,1

22

22

2

22222

22

22

sinsin2

coscos2

cos2

sinsin2

coscos2

ad

fad

affdcbaC

adcB

fadcA

CAF

BE

ACD

2



Solution (pag. 196)

L

LNMM

2

4arctan2

2

3 2,1

22

22

2

22222

22

22

sinsin2

coscos2

cos2

sinsin2

coscos2

ad

fad

affdcbaK

dabH

fdabG

KGN

HM

GKL

2


Sixbar Linkage


Sixbar Linkage

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Ch04 position analysis

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