١

Dr. Hitham Tlilan ١

Theory of Theory of MachinesMachines

042341

Instructor: Instructor: Dr. Hitham TlilanDr. Hitham Tlilan

Office: E3118 (Engineering Building) Tel.: 4463

Dr. Hitham Tlilan ٢

Chapter 4Acceleration Analysis of Mechanisms

33..1 1 Basic ConceptsBasic Concepts

dtRdV

Ppoint of Velocity

P R

x, i

y, j

z, k

i

j

k

k

kRjRiRR zyx

kvjviv

kRjRiRkRjRiRdtdV

zyx

zyxzyx

)(

٢

Dr. Hitham Tlilan ٣

dtd

tt

0lim

that such time; with positionangular in change The

velocity Angular

2

2Ppoint of onAccelerati

dtRd

dtVda

kRjRiR

kvjvivkvjvivdtdV

zyx

zyxzyx

)(

Dr. Hitham Tlilan ٤

2

2

that such time; with velocityangular in change The

ncceleratio Angular A

dtd

dtd

٣

Dr. Hitham Tlilan ٥

Motion of a Rigid Body a bout a Fixed axisMotion of a Rigid Body a bout a Fixed axis(without Translation)(without Translation)

If a rigid body rotates a bout a fixed axis in a stationary coordinate system

R

RV

RrV

sin

v

R

r

sin Rr

Then, the magnitude of the velocity of P is

sin Rrv R.H.R using and of plane the is of Direction

R

Dr. Hitham Tlilan ٦

)(Ppoint of onAccelerati

Rdtd

dtVda

)(

)(

RRaRV

VRdtRdR

dtdR

dtda

٤

Dr. Hitham Tlilan ٧

)(

RRaB

Plane MotionPlane Motion

B R

x, i

y, j

i

j

sinRVRV BB

BV

k

k

ntBB aa

B of path the toTeangent

Ra tB

Dr. Hitham Tlilan ٨

RaRRa

RRRR

Ra

nRn

R

n

BB

B

2) and between(90

) and between(90

sin)(

sin

) (

B

R

x, i

y, j

i

j

rotation of origin the towards

of Direction

tnBB aa

tB

anB

a

RRaB2

٥

Dr. Hitham Tlilan ٩

33..2 2 Moving Coordinate System and Relative VelocityMoving Coordinate System and Relative Velocity

X, I

Y, J

Z, K

J

I

K

y, j

z, k

x, i

i

j

k

r

P

vectorsunit ingCorrespondsystem Coordinate Fixed

),,(),,(

KJIZYX

vectorsunit ingCorrespondsystem Coordinate Moving

),,(),,(

kjizyx

O

o

oR

sCoordinate moving the of origin torepect with

P ofvector Position :

origin fixed the to repect with sCoordinate

moving the of origin the ofvector Position :

r

Ro

Dr. Hitham Tlilan ١٠

X, I

Y, J

Z, K

J

I

K

y, j

z, k

x, i

i

j

k

r

P

O

o

oR

R

The total position vector of P, which denotes a point on a linkage is:

rRR o

r

rr

oRdt

kdrdt

jdrdt

idrkrjrirKRJRIRR zyxzyxoZoYoX

٦

Dr. Hitham Tlilan ١١

(1)

rrRdtd

dtRd

dtVda ro

PP

(2)

ooo aR

dtRd

k

dtkdr

j

dtjdr

i

dtidr

rr

krjrir

krjrirdtdr

dtrd

rzryrxrzryrx

rzryrxrr

Dr. Hitham Tlilan ١٢

krjrirr

krjrirrdtrd

rzryrxr

rzryrxrr

(3)

rrr rr

dtrd

٧

Dr. Hitham Tlilan ١٣

r

dtkdr

dtjdr

dtidr

rr

krjrir

r

krjrirdt

d

krjrirdtdr

dtd

zyx

zyxzyx

zyx

)(

)()(

)(

(4) )()(

rrrrdtd

r

Dr. Hitham Tlilan ١٤

is Ppoint of naccelertio absolute The

Eq.(1) into .(2,3,4)Eq Sub. s

)(

rr

r

rr

a

r

a

RRa rr

r

r

o

oP

)()2()(

rvraaa rroP

٨

Dr. Hitham Tlilan ١٥

33..7 7 ComplexComplex--Numbers Methods Applied to Acceleration AnalysisNumbers Methods Applied to Acceleration Analysis

SliderSlider--Contact MechanismContact Mechanism

Dr. Hitham Tlilan ١٦

1R

2R

oR

12 RRR o

Real axisReal axis

Im. axisIm. axis

11111

22222

0

12

1

2

0 00

sincos

sincos

sincos

jReRR

jReRR

RjReRR

RRR

j

j

ooj

o

o

٩

Dr. Hitham Tlilan ١٧

1212

jo

j eRReR

21

12212

2

11

22

11

112

22

12

0

BB

jjjjo

j

vdt

dRdt

d dt

d

dt

dRlength constant has R thatNote

eRdt

djdt

dReeRdt

djeRReRdtd

,

(1) 221

212211

jBB

jj eveRjeRj

increasing if positive is , B

torespect with B of (sliding) velocity relative The :

2

1

221

21

Rv

v

BB

BB

Dr. Hitham Tlilan ١٨

221

212211]Eq.(1)[ j

BBjj eveRjeRj

dtd

dtd

2

2

22

22

11

12

21

221

21

221

2

2

2

22

1

1

1

11

2

a

edt

dv

a

evj

a

eRdt

dj

a

eR

a

eRdt

dj

a

eR

tBB

cBB

tBn

B

tBn

B

jBBjBB

jjjj

aaaaaa tBB

cBB

tB

nB

tB

nB 21212211

١٠

Dr. Hitham Tlilan ١٩

increases) if ( ,2

,

,

22

2222

1111

22

12

21212121

21

2211

dtdR

dtdv

v

RRdt

dRRdt

d

RR

BBtBBBB

cBB

tB

tB

nB

nB

aa

aa

aa

Eq.(Eq.(22) can be solved as follows) can be solved as follows

2analysis Position From

1111 given are and , Since )1( θ,ω,θ,RRo

determined are and )2( B1B12analysis Velocity From vω

follows as determined be can tB1B2 and )3( B1B1

2 adtdv

Dr. Hitham Tlilan ٢٠

22212

2122

211

1

2

2

222

22

11

12 2

Eq.(2)

unknowns. the seperate willthat by Eq.(2) gMultiplyin 1)-(3

jjBBjBB

jjjj

j

j

eedt

dvevjeR

dtdjeReR

dtdjeR

e

e

2)sin()cos(

Eq.(3)

forr rectangula in formpolar the Expressing )23(

t2222

2212111

2

formr rectangula In

212121 BBavjRjRjRj BB

211

2121

212111212

22

22

12112112t

2)cos()sin(1 Part .Im

)sin()cos( Part Real

parts imaginary the andpart real the Seperate )13(

BBvRRR

RRRaBB

(3) 2 t2222

2)(11

221212

211 BB

avjRjReRj BBj

١١

Dr. Hitham Tlilan ٢١

ExampleExample

B

A

1R

2R

3R

mm 60R mm, 40R ),60 when ( mm 80R Given,

?g and Bpoint of onaccelerati the find shown, mechanism crank-slider theFor

3221

3

gg33

Dr. Hitham Tlilan ٢٢

SolutionSolution

(1)

321 RRR

(4)

(3)

(2) 00

33333

22222

1111

3

2

1

sinjcosReRR

sinjcosReRR

RsinjcosReRR

j

j

j

B

A

1R

2R

3R

gg33

١٢

Dr. Hitham Tlilan ٢٣

(5) 32321

jj eReRR

length) fixed are 3 and 2 link (

,

032

33

22

11

33

221

3213232

dtdR

dtdR

dtd

dtd

Vdt

dRlength constant has R thatNote

eRdt

djeRdt

djdt

dReReRRdtd

B

jjjj

(6) )sin(cos)sin(cos 33332222

332232

jRjjRjeRjeRjV jj

B

Dr. Hitham Tlilan ٢٤

74.32426.35sinsin (5) Eq. 23

213

analysis Position From

RR

)O (towards mm/s 634.975 mm/s 634.975 (CW) rad/s 8.165 Eq.(6)

2

3analysis velocity From

BV

3

32

2

32

332

222

3322

jjB

jjBB

eRjeRja

eRjeRjdtd

dtVda

3333

2

22222

sincos

sincos

3

2

jRj

jRjaB

١٣

Dr. Hitham Tlilan ٢٥

3

233332

2222

part Im.

333233222

22

part Real

sincossincos0

sincossincos

2

2

RR

RRaB

3

323

33

2222

23 cos

sincos

cossin2

R

R

A

2R

dR

Re.Re.

Im.Im.

3gRgg33

3gagg

32

3

d2

d2g

jj eReR

RRR

To Find the Acceleration of Point gTo Find the Acceleration of Point g33

Dr. Hitham Tlilan ٢٦

323 d322g

jj eRjeRjV

33

22

33 d

232

22

gg

jj eRjeRjdtVd

a

0constant Since 22

3

2d33d2

22

32

d33d22

2g

sincossin

cossincos

32

323

RRRj

RRRa

32

d33d22

2

32

d33d22

2

g

sincossin

cossincos where;

,

32

32

3

RRR

RRR

ja

١٤

Dr. Hitham Tlilan ٢٧

1g

22g tan ,

33a

11

2244

55

3

BB22, B, B44

CC55

Inverted SliderInverted Slider--Crank MechanismCrank Mechanism

Dr. Hitham Tlilan ٢٨

To fined the velocity of point B

Point (B) pinned on link Point (B) pinned on link 2 2 and slides over link and slides over link 4411

2244

55

3

BB22, B, B44

CC55

B

2R

4R

Re.Re.

Im.Im.

1R