displacement and velocity analysis - school of …moreno/kinematics/me 5150 kinematics spring... ·...

1

Displacement and Velocity Analysis

• 4 bar configurations – range of motion

• Grashof criteria

• Kinematic inversions

• Geometric inversions

• Dead center

• Circuits

• Graphical and Analytical Methods

http://www.uconn.edu/index.php

2

Grashof, F. (1883). Theoretische Maschinenlehre. Vol. 2. Voss: Hamburg.

Kinematics Introduction

Grashof condition addresses mobility

s=shortest link

l=longest link

p=one remaining link

q=other remaining ling

If s+l ≤ p+q one link capable of full rotation (Class I kinematic chain)

If s+l> p+q no link capable of complete revolution (Class II kinematic chain)


3

Crank-rocker

Double Rocker Rocker-Crank

Double Crank

4 variations

Position of s



4


Kinematic Inversions


5


Geometric Inversions -

α α

Dead Center position Rocker-Crank Rocker-Crank


6


Crank-rocker – driving with short crank

- no dead center positions

Inversion 1,2 on 2 circuits

Circuit 1 Circuit 2


7


Crank-rocker

2 circuits

Circuit 1

Circuit 2


8

Kinematics Introduction

Rocker- crank – driving with longest crank

- 2 dead center positions

2 Inversions on 2 circuits


9


Displacement Analysis – Graphical Method

Input (driver) and

Output links

Arcs


10


• Displacement Analysis

• E&S 3.3

• Complex Polar Vector Notation

• Displacement Analysis

• S&R

• Velocity Analysis

• E&S 3.5

• S&R

• Acceleration

• S&R

• Homework


11


Displacement Analysis Analytical – E&S 3.3 (ex B)

52

355

4123

744

74

2

3

2

7

2

4

74

2

4

2

7

2

3

127

)arg()arg(

)arg()arg(

2arccos

cos2

cos

rrrr

rr

rrrr

rr

rrrrr

rrrrr

ineslaw

rrrdia

oac

c

4

5

21

11

inversionccw

inversioncw

knownrrr

knownrrr

c

oa

,,,

,,

543

21


12


Complex Polar Vector Notation

)(

)2(

)(

)(

)sincos()(

)sincos()sin(cos

)(

)sin(cos

ii

ii

ii

i

iii

yx

i

reirei

reirirei

reire

iriirire

reree

irrirrer

x

iy

r

x

iy

r

ri

)( rii


13


x

iy

r

r

iii

iii

iererredt

d

iererredt

d

)(

)()(

ii

iiii

iii

ierrerr

iierierrierer

iererdt

dredt

d

)2()(

)()()(

)()(

2

2

2

velocity

acceleration


14


Link Velocity

ground

0A

x

iy A

Ar

AV

2

2

Ai

AAA

i

AA

iA

AA

i

AA

AA

riorierV

eirerdt

rdV

err

dtd

A

AA

A

0

)cossin(2 AAAA irV

Cartesian form


15


AV

B

A BV

2

222

2

2

iririr

VVV

irV

irV

BAAB

ABBA

BB

AA

AV

BVBAV

Relative Velocity

BA

BA

BA

BA

rV

riV

2

2


16


2

3

y

x0A

A

B

2r

1r

3r

3322321

3

223

32

323211

321

coscos

)sin

arcsin(

)()(

rrrrr

rr

rr

rrrrrr

rrr

xx

yy

232 ,,: rrknow

Crank-slider

)(2 Arr )(3 BArr

1rposition


17


Velocity of B

BA

BAx

AxBABA

BAyBAxAxAyBx

BAx

Ax

By

BAAB

BAAi

B

BAAB

rr

ririV

rrrrV

rr

Vimaginaryreal

iririV

ririeV

VVV

)(

)(

)(

)0(

)sincos()sincos())0sin()0(cos(

23

22

23

333222

32

BAV

BV

AV

22 ,


18


22 ,

sec/1

"13

"6

22

3

2

rad

rr

rr

BA

A


19


Homework 4 : Problem 3.10 E&S

Find

Given geometry:

iAB

mmiBA

mmiDE

mmiCE

mmiCB

mmiBB

mmiDA

mmiAA

5.981

1080104

395.2232

5.251322

645631

32285.15

505.4619

205.185.7

00

0

0

0

0

65 ,, EV


20


r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

d

d

d3

x

iy S&R ref 2

))(())((

1

22

2

4132

2121

21

4132

iiii

ii

ii

iiiiB

errerrdede

ee

errde

ererererr

dd

d

Complex conjugate

Loop 1

2 unknowns

Complex polar vector notation


21


221

2

2

2

1

221

2

2

2

121

2

2

2

1

2

cos2

cos2)( 22

rrrrd

rrrreerrrrdii

)cos(sintan

sin0sinsin:

coscoscos:

)sin(cos)sin(cos

22122

22

122

1222

212

rrr

dri

rdrR

ridir

errde

d

dd

dd

dd

ii d

Loop 1

identity


22


r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

d

d

d3

x

iy

3

2

4

22

33

2

4

22

333

)()(

3

)(

3

)(

3

22

3

2

4

3344

2)cos(

)cos(2)(

))(())((

33

33

3344

drrdr

rdrdreedr

edredrdrr

deerdeererer

d

d

ii

ii

iiiiii

dd

dd

dd

Loop 2

diiideerer

34

34


23


r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

d

d

d3

x

iy

)(:2

2)cos(

3

3

2

4

22

33

d

d

solutions

drrdr

r4 r3

r2

r1

Bo Ao

A

B

2

2

2

4

d

d

d

3

x

iy

Inversion 1

Inversion 2


24


4

334

4433

4433

43

34

)sinsin(sin

sinsinsin:

coscoscos:

43

34

rdr

rdri

rdrR

erdeer

deerer

d

d

d

iii

iii

d

d

),,(

),,(

),,(

),,(

434

433

221

221

drrf

drrf

rrf

rrfd

d

Loop 2


25


)sincos(

)sincos()sincos(

)(

4444

33332222

443322

4132

432

4132

ir

irir

ierierier

ererererdt

d

iii

iiii

velocity


26


Real and imaginary parts

4433

4433

44222

44223

222

222

4

3

4433

4433

coscos

sinsin

coscos

sin2sin

cos

sin

coscos

sinsin

rr

rr

rr

rr

r

r

rr

rr

Cramer’s Rule


27


),,,,,,(,

)sin(

)sin()(

)sin(

)sin()(

432432243

43

23

4

224

43

24

3

223

rrrf

rr

rr

Again using Cramer’s rule

Using identities


28

Kinematic Analysis using Cartesian Vector Notation

Scalar Product – Dot Product

...0

1

cos

etcji

kkjjii

bababaabba zzyyxx

Vector Product – Cross Product

...,

sin

etcikxjkjxi

bbb

aaa

kji

bxa

abbxa

zyx

zyx


29


Differentiation of Vectors

First Derivative- velocity

rrr

rrr

tolarperpendicubemustrdt

dr

ttdtd

dtdrrr

rr

tt

)(

limlim00

r

y

x


30


Differentiation of Vectors

Second Derivative -acceleration

)(2)(

)()()(

)(

)(

)(

rrrrr

rrrrrdt

d

rrdt

dr

rdt

drdt

drrr

rrdt

dr

rrr


31


Find B,4

Vector Loop Equation

21

12

rrd

rdr

r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

4 Bar Linkage

d

d

d3

Closed vector loop

Head to tail chain

Or

Two branch paths

+


32


21 rrd

xxyy

d rrrr

rrrrd

rrrrd

rrrrdd

2121

221

2

2

2

1

221

2

2

2

1

2

2121

tan

cos2

cos2

)()(


33


drr

rdr

34

43

r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

d

d

d3

2nd loop equation

d

drrdr

drdrr

3

3

2

4

22

3

3

22

3

2

4

2cos

cos2

xxyy

d rrrr

2121tan

3r

4r

+ inversion 1

- inversion 2

3


34


)(

2cos

cos2

54

4

2

3

22

45

54

22

4

2

3

d

drrdr

drdrr

r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

4 Bar Linkage

d

d

d3

5


35


Velocity Analysis

r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

4 Bar Linkage

d

d

d3

Two paths to common point B

443322

1

444111333222

4132

0,0

)(

rrr

axisfixedrbut

rrrrrrrrdt

d

rr

rrrr


36


443322

443322

444433332222

444333222

443322

)(

)(

)()()(

xxx

yyy

yxyxyx

yxyxyx

rrrj

rrri

irjrirjrirjr

jrirkjrirkjrirk

vectorsunit

rrr

yxyx

yxyx

yxyx

xyyx

rrrr

rrrr

rrrr

rrrr

4334

233224

4334

424223

Same as complex polar

r2y=r2sinθ2 , etc….


37


Acceleration

r4

r3

r2

r1

Bo Ao

A

B

2

2

2 4

4 Bar Linkage

d

d

d3

......

..........)))((()(

))(())(())((

)(

2

2

22

2

22222

2222222

444443333322222

443322

jririrjr

jrirkkjrirk

rrrrrr

rdt

rd

rrrdt

d

yxyx

yxyx

Expand in vector form

Take cross products


38


Rearranging

xx

yy

x

y

xx

yy

x

y

xx

yy

yyyxxx

xxxyyy

rr

rr

Br

Ar

rr

rr

rB

rA

B

A

rr

rr

Brrrrrrj

Arrrrrri

43

43

3

3

4

43

43

4

4

3

4

3

43

43

4

2

43

2

32

2

2224433

2

2

43

2

32

2

2224433

)

)


39


)( 54 d

yxyx

yxyx

yxyx

xyyx

rrrr

rrrr

rrrr

rrrr

4334

233224

4334

424223

xx

yy

x

y

xx

yy

x

y

rr

rr

Br

Ar

rr

rr

rB

rA

43

43

3

3

4

43

43

4

4

3

d3

5

r4

r3

r2

r1

Bo Ao

A

B

2

2

2

4

d

d

d3

xxyy

d rrrr

rrrrd

2121

221

2

2

2

1

tan

cos2

drrdr

3

2

4

22

3

2cos

drrdr

4

2

3

22

45 2

cos

All the equations

known

knownrrrr

222

4321

,,

,,,


40


Oscillating Slider

)(

1

4

4

1342

1

oo abrrfr

DOF

f

n

Position equation

33

20012

,

),(,,

runknowns

abrfrknowns

2r

3r

f

2

d

4

3

24

3

d

0a

a

0b

d


41


222

03

2

3

22

22

0

2

00

000000

21

)()(

)()(

)()(

)()()()(

fababr

rfdalso

ababddd

jabiabd

jaaiaajabiabd

rrddiagonal

yyoxx

yyoxx

yyxx

yyxxyyxx

d

xox

yoyd

xox

yoyd

rf

rf

ab

ab

ab

ab

3

3

1

3

1

)(tantan

tantan

b

ad


42

0

1

2

3

4

5

6

7

0 90 180 270 360

R3

R3

2

2

3

Oscillating Slider

r2=a=2.20”

f=2.15”

r1=4.51”

0

10

20

30

40

50

60

70

80

90

100

0 90 180 270 360

Th 3 vs Th2

Th 3 vs Th2


43


Velocity

Homework #2

Derive expressions for Velocity and acceleration


displacement and velocity analysis - school of …moreno/kinematics/me 5150 kinematics spring... ·...

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