position analysis

8/18/2019 Position Analysis

1/21

Position synthesis1

Analytic Approach to

Mechanism Design

ME 324

Fall 2000

http://www.engr.colostate.edu/me/program/courses/ME324/notes/PositionAnalysis.ppt


2/21

Position synthesis2

Chapter 4 -

Analytic Position Analysis

A !ector can "e

represented "y a

comple# num"er $eal part is #%a#is

&maginary part is y%

a#is

'se(ul when we

"egin to ta)e

deri!ati!es

RealAxis

ImaginaryAxis

Point A

RA

θ

R cos θ

jR sin θ


3/21


4/21

Position synthesis4

a!eling o" in#s an$

in# engths

in) la"eling starts

with ground lin)

a"eling o( lin)lengths starts with lin)

ad+acent to ground

lin)

Ma)es no sense % +ust

go with it

Ground Link

Coupler

Link 1, length d

Pivot 02 Pivot 04

A

B

Link 2,

length a

Link 3, length b

Link 4,

length


5/21

Position synthesis!

Angle Meas%rement Convention

All angles measured

(rom angle o( the

ground lin) 1e(ine θ 0-

ne 1F5 so can

descri"e all angles interms o( one input5

usually θ2

1

A

B

2

3

4

θ2

θ3θ4

θ1 = 0-


6/21

Position synthesis"

More on Complex &otation

Polar (orm:

re j θ

artesian (orm:r cosθ 6 j r sinθ

Euler identity:

7e j θ

cosθ 7 j sin

θ

1i((erentiation:

de jθ

d θ

= je jθ


7/21Position synthesis#

'he Vector oop 'echni(%e

8ector loop e9uation:

$2 6 $3 % $4 % $1 0

Alternati!e notation: $A%2 6 $BA % $B%4 % $%4%2 0

nomenclature % tip then tail

omple# notation:

ae +θ2 6 "e +θ3 % ce +θ4 % de +θ 0

u"stitute Euler e9uation:

a ;cos θ26+ sinθ2< 6 " ;cos θ36+ sinθ3<

% c ;cos θ46+ sinθ4< % d ;cos θ6+ sinθ< 0

d

A

B

a

b

θ2

θ3θ4

R1

R2

R3 R4

O2

O4


8/21

Position synthesis&

Vector oop 'echni(%e - contin%e$

eparate into real and imaginary parts:

$eal:

a cos θ2 6 " cos θ3 % c cos θ4 % d cos θ 0a cos θ2 6 " cos θ3 % c cos θ4 % d 05since θ 05 cos θ

&maginary: +a sin θ2 6 +" sin θ3 % +c sin θ4 % +d sin θ 0

a sin θ2 6 " sin θ3 % c sin θ4 05 since θ 05 sin θ 0


9/21

Position synthesis'

Vector oop 'echni(%e -

contin%e$

a cos θ2 6 " cos θ3 % c cos θ4 % d 0

a sin θ2 6 " sin θ3 % c sin θ4 0

a5"5c5d are )nown

ne o( the three angles is gi!en

2 un)nown angles remain

2 e9uations gi!en a"o!e

ol!e simultaneously (or remaining angles


10/21

Position synthesis10

Vector oop -

)%mmary

1raw and la"el !ector loop (or mechanism

rite !ector e9uations

u"stitute Euler identity

eparate into real and imaginary

2 e9uations5 2 un)nown angles

ol!e (or 2 un)nown angles

=ote: there will "e two solutions since

mechanism can "e open or crossed


11/21


*xample+


&nput position θ2 gi!en

ol!e (or θ3 > θ4

θ2?.3-

a.@

"2.4

c2.0@

d3.?

θ3-

θ4-


12/21


*xample+

Vector oop *(%ation

$2 6 $3 % $4 % $1 0

ae +θ2 6 "e +θ3 % ce +θ4 % de +θ 0

.@e +?.3B 6 2.4e +θ3 % 2.0@e +θ4 % 3.?e ,0- 0

θ2?.3-

a.@

"2.4

c2.0@

d3.?

θ3-

θ4-

$2

$3

$4

$1


13/21


*xample+


ae +θ2 6 "e +θ3 % ce +θ4 % de +θ 0

a;cosθ26+sinθ2< 6 ";cosθ36+sinθ3< % c;cosθ46+sinθ4< % d;cosθ6+sinθ


14/21


)ol%tion+ pen in#age

2 e9uations (rom real > imaginary e9uations

.@ cos 51.3 6 2.4 cos θ3 % 2.0@ cos θ4 % 3.? 0

.@ sin 51.3 6 2.4 sin θ3 % 2.0@ sin θ4 02 un)nowns: θ3 > θ4ol!e simultaneously to yield

2 solutions.%pen solution(

3 ) 21*, 4 ) 104+ θ2?.3A

a.@

"2.4

c2.0@

d3.?

C32A

C404A


15/21

Position synthesis1!

Revie. - a. o" Cosines

A

B

C

θ

cosθ = A

2 + B2 − C 2

2 AB

θ =arccos A

2 + B2 − C 2

2 AB


16/21

Position synthesis1"

'ransmission Angles

*ransmission angle is the angle

"etween the output angle and the

coupler A"solute !alue o( the acute angle

Measure o( 9uality o( (orce

transmission

&deally5 as close to ,0-

as possi"le

µ1180°- µ1

acte

µ2180°°- µ2


17/21

Position synthesis1#

*xtreme 'ransmission Angles -

/rasho" Cran# Roc#er

For a Drasho( (our"ar5e#treme !alues occur

when cran) is collinearwith ground

For the e#tended position shown:µarccos ;"26;a6d< 2 % c2


18/21

Position synthesis1&


/rasho" Cran# Roc#er

For the o!erlapped case shown:

µHHHHHHHHHHHHHHHHHHHHHHHHHH

µ2HHHHHHHHHHHHHHHHHHHHHHHHHH a

!

c

"µ1

µ2


19/21

Position synthesis1'


/rasho" Do%!le Roc#er

$emem"er: coupler

ma)es a (ull re!olution

with respect toroc)ers

*ransmission angle

!aries (rom 0- to ,0-


20/21



&on-/rasho" in#age

*ransmission angle is Iero

degrees in toggle position:output roc)er > coupler

ther transmission angle gi!en

as:

µ 2HHHHHHHHHHHHHHHHHHHHHHHHHH

imilar analysis (or other toggle

position

a

!

c

"

µ1=0°°µ2


21/21


Calc%lation o" 'oggle Angles

*he input angle5 θ2 5 (or the (irsttoggle position gi!en as:

θ2HHHHHHHHHHHHHHHHHHHHHHHHHH

imilar analysis (or the other toggleposition

a

!

c

"θ2

position analysis

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