acceleration analysis

8/8/2019 Acceleration Analysis

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Chapter 7Acceleration Analysis


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Definition

Rate of change of velocity with respect totime Angular

Linear

Position Vector

Velocity

dt

d

dt

dVA

j

PA peR

jjPApa ep

dt

dpje

dt

RV


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Definition

dt

jepd

dt

d jPAPA

A

A

sincoscossin 2 jpjpPA A

"Absolute"PPA AA

dt

dje

dt

dejp

jj

PA

A

jj

PA epjep2A

PAn

PAt

PA AAA


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Definition

Acceleration (difference)

Relative Acceleration

"bodysameon the"

PAAP AAA

APPA AAA

nPAtPAnAtAnPtP AAAAAA


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Graphical Analysis

Graphical Acceleration Analysis

Solve for

PAAP AAA

rAtt

A

CBA ,,on;acceleratilinear

,;onacceleratiangular 43

2rA

nn A


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Graphical Analysis

Example 7-1 Given 2, 3, 4, 2 ,3, 4,2

find 3 , 4 , AA, AB and AC

Velocity analysis alreadyperformed

1. Start at the end of thelinkage about which you havethe most information.Calculate the magnitude ofthe acceleration of point A,

222 AOAn

A 22 AOAt

A


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Graphical Analysis

Example 7-1 2. Draw the acceleration AA 3. Move next to a point which

you have some information,point B. Draw theconstruction line pp throughB perpendicular to BO4

4. Write the acceleration

difference equation for pointB vs. A

BAAB AAA

n

BA

t

BA

n

A

t

A

n

B

t

B AAAAAA 2

44 BOAn

B

n

BA


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Graphical Analysis

5. Draw construction line qqthrough point B andperpendicular to BA torepresent the direction ofABA

6. The vector equation can besolve graphically by drawing

the following vector diagram

23BAAn

BA

nBAtBAnAtAnBtB AAAAAA BAAB AAA

n

B

n

BA

n

A

t

A AAAA ?& t

B

t

BA

AA


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Graphical Analysis

7. The angular velocities of link3 and 4 can be calculated,

8. Solve for VC

4

4BO

At

B

BA

At

BA3

CAAC AAA

3cAtCA

2

3cAn

CA


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Graphical Analysis


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Analytical Analysis


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Fourbar Pin-Joint Linkage Vector Loop

Position

Velocity

Acceleration

Analytical Solution

01432 RRRR

01432

jjjj

decebeae

0432 432

jjj

ejcejbeja

0443322 244233222 jjjjjj ecjecebjebeajea


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Fourbar Pin-Joint Linkage

Analytical Solution

04433

22

2

44

2

33

2

22

jjn

B

t

BB

jjn

BA

t

BABA

jjn

A

t

AA

ecjec

ebjeb

eajea

AAA

AAA

AAA

0 BBAA AAA

identityEuler

partreal

partimaginary


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Analytical Solution

4

2

43

2

32

2

222

3

4

coscoscossin

sin

sin

cbaaC

bB

cA

BDAE

AFCD

3

BDAE

BFCE

4

4

2

43

2

32

2

222

3

4

sinsinsincos

cos

cos

cbaaF

bE

cD


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Analytical Solution

442

4444

33

2

3333

22

2

2222

sincoscossin

sincoscossin

sincoscossin

jcjc

jbjb

jaja

B

BA

A

A

A

A

0 BBAA AAA


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Analytical Solution

Slider-Crank

d

ebjeb

eajea

t

BB

jjn

BA

t

BABA

jjn

A

t

AA

AA

AAA

AAA

33

22

2

33

2

22

0 BBAA AAA

BABA

AA

BAAB AAA


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Analytical Solution

Slider-Crank

3

2

3332

2

222

3

3

2

32

2

2223

cossincossincos

sinsincos

bbaadb

baa

identityEuler

partreal

partimaginary


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Coriolis Acceleration Is an additional component

of acceleration present whena sliding joint is on a rotating

link Thy position vector is both

rotating and changing lengthas the system moves

Analytical Solution

2j

P peR


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Coriolis Acceleration The velocity expression is

The acceleration,

Analytical Solution

22

2

jj

P epep V

sliptrans PPP VVV

2222

2

22

22

2

jjjj

P

epjepejpjep A

slipcoriolisnormaltangetial PPPPPAAAAA


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Coriolis Acceleration

The acceleration,

Analytical Solution

2222

222

22 2 jjjj

P epjepejpjep A

slipcoriolisnormaltangetial PPPPPAAAAA


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Acceleration of Any Point

Once the angular acceleration of all the links arefound it is easy to define and calculate theacceleration of any point on any link for any input

position of the linkage


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To find the acceleration of points S

22220

sincos222

jssej

SSRR

222222 cossin22

jsjsej

SV

22222

2

22222

sincos

cossin

js

jsS

A


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To find the acceleration of points U

44440

sincos444

juuej

UR

444444

cossin44

jujsej

UV

444424

44444

sincos

cossin

ju

juUA


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To find the acceleration of point P

PAAP

j

PA jppe

RRR

R

3333sincos33

PAAP

jPA jpjpe

VVV

V

333333cossin33

3333

2

3

33333

sincos

cossin

jp

jpPA

PAAP

A

AAA


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Jerk page 367-369 Human Tolerance to Acceleration page 364-366 Fourbar Inverted Slider-Crank page 354-360

Acceleration Analysis of the Geared Fivebar- page361 -362

Important topics to review

acceleration analysis

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