four bar linkage

MD/AY2011-2012 ME304 KDM.1

Indian Institute of Technology Gandhinagar

Dept. of Mechanical Engineering

L09: POSITION ANALYSIS OF

MECHANISMS

Dr. Murali Damodaran

murali@iitgn.ac.in

Spring Semester of AY2011-2012

ME304: KINEMATICS AND DYNAMICS

OF MACHINES

MD/AY2011-2012 ME304 KDM.2

LECTURE 09

23 January 2012

ME304: Kinematics and Dynamics of Machines

Introduction to Mechanisms. ..continuing with Kinematic Fundamentals Position, Velocity and Acceleration Analysis. (Kinematics) Analytical Method for Position Synthesis (The basis for computer modeling and synthesis of linkages) Design of Cam Follower Mechanisms. Gear tooth profiles, Spur gears and Helical gears,Epicyclic Gear Trains Belt drives Dynamic Analysis of Mechanisms (Dynamics) Balancing Analysis and Applications of Discrete and Continuous System Vibration Course Website @ https://sites.google.com/a/iitgn.ac.in/me304/

MD/AY2011-2012 ME304 KDM.3

POSITION ANALYSIS OF MECHANISMS

Coordinate Systems for Analysis of Planar Mechanism

• Converting between the two

• Position Difference, Relative position

– Difference (one point, two times)

– relative (two points, same time)

RBA=RB-RA

X

Y

RB

RA

A

B RBA

2 2

arctan

A X Y

Y X

R R R

R R

cos

sin

X A

Y A

R R

R R

,Cartesian:

Polar: ( , )

X Y

A

R R

R

o

o

MD/AY2011-2012 ME304 KDM.4

POSITION ANALYSIS OF MECHANISMS

Position Analysis of Fourbar Linkage Mechanism

2 4

2

3 4

Given :

The lengths of links

, , and

position of the ground link O O

and the angle

Objective: Find and

a b c d

MD/AY2011-2012 ME304 KDM.5

POSITION ANALYSIS OF MECHANISMS

Graphical Analysis of Fourbar Linkage

• Draw an arc of radius b, centered at A

• Draw an arc of radius c, centered at O4

• The intersections are the two possible positions for the linkage, open and crossed

a

d 2

b

c 3

4

A

O2 O4

B1

B2

MD/AY2011-2012 ME304 KDM.6

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

2

2

Coordinates of Point

cos

sin

A:

x

y

A a

A a

222

22 2

Solve these equations to

Coordinates of Point B

See Page 17

fin

7-1

d

a

78 for solution

nd

x x y y

x y

x y

b B A B A

c B d B

B B

MD/AY2011-2012 ME304 KDM.7

POSITION ANALYSIS OF MECHANISMS

Coordinate Systems for Analysis of Planar Mechanism

Coordinate Systems:

GCS = Global Coordinate System, (X, Y)

LNCS = Local Non-Rotating Coordinate System , (x, y) LRCS = Local Rotating Coordinate System , (x’, y’)

x’

y’

MD/AY2011-2012 ME304 KDM.8

POSITION ANALYSIS OF MECHANISMS

Coordinate Systems for Airplane Fight Dynamics

MD/AY2011-2012 ME304 KDM.9

POSITION ANALYSIS OF MECHANISMS

Non-planar Linkages-3D Spherical Linkages

http://synthetica.eng.uci.edu/~mcca

rthy/Linkages.html

MD/AY2011-2012 ME304 KDM.10

POSITION ANALYSIS OF MECHANISMS

Representation of Position Vectors

• For planar motion complex numbers on the real-imaginary plane can be used to model position vectors

• Euler identity e±iθ=cos θ ± i sin θ (Note-Norton uses j instead of i in his text book i.e. e±jθ =cos θ ± j sin θ )

• Cartesian form: RAcos θ + i RAsin θ

• Polar form: RAei θ

MD/AY2011-2012 ME304 KDM.11

POSITION ANALYSIS OF MECHANISMS

Representation of Position Vectors

Multiplying a vector by ei corresponds to rotating the vector through

2i i i

A AR e e R e

2

For a rotation

through 90 degrees

cos sin2 2

i

e i i

cos sin

i

A

Re

R iR

R R

BR iR

2

CR i R R

3

DR i R iR

MD/AY2011-2012 ME304 KDM.12

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

Write the vector loop equation:

(Positive from tail to tip)

Substitute with complex vectors

Split the knowns on one side and the unknowns on the other.

Call the knowns Z

2 3 4 1 0R R R R

32 4 1 0ii i i

ae be ce de

2 1

3 4 2 1

known

i i i

i i

ibe ce ae de Z

Z ae de

MD/AY2011-2012 ME304 KDM.13

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

3 4 2 1i i i ibe ce ae de Z

3 4

3 4

Define and i i

i i

s e t e

be c bs c Ze t

3 41 1

Define = and i.e. conjugates

and a conjugate based on known link lengths

1 noting that

i i

b cZ

s

s e t es t

bs ct Z

Zt Z

MD/AY2011-2012 ME304 KDM.14

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

1 no ting that

bs ct Z

b cZ

s ZtZ

bs ct Z

b cZ

s t

2

2 2

2 2 2

cb ct Z Z

t

cb c ctZ Z ZZ

t

b t c t ct Z cZ tZZ

MD/AY2011-2012 ME304 KDM.15

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

2 2 2 2 0c t ct Z t c b ZZ cZ

2

2 2 2 2

2 2

2

2 Solve 0 for

4

2

c b ZZ

ct Z t c b ZZ cZ

c b ZZ c ZZ

t

tcZ

ct Zs

b

MD/AY2011-2012 ME304 KDM.16

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

3 4

3 4

3 4

and

conjugate of

1 noting that

i i

i i

i i

s e t e

Z be ce

Z be ce Z

ZZ

2

2 2 2 2 24

2

c b ZZ c b ZZ c ZZt

cZ

ct Zs

b

2 2 2Solve 0 ct Z t c b ZZ cZ

Solve this in Matlab/Octave

MD/AY2011-2012 ME304 KDM.17

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage in Octave

MD/AY2011-2012 ME304 KDM.18

KINEMATICS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

MD/AY2011-2012 ME304 KDM.19

POSITION ANALYSIS OF MECHANISMS

Software Systems for Analysis of Various Linkages

R L NORTON’s SUITE OF PROGRAMS ACCOMPANYING THE TEXTBOOK Kinematics and Dynamics of Machinery FOURBAR FIVEBAR SIXBAR SLIDER DYNACAM ENGINE All are student SI editions

COMMERCIAL SOFTWARES FOR LINKAGE MECHANISMS WORKING MODEL 2D SAM (Synthesis and Analysis of Mechanisms) MATLAB Simechanics Toolbox

Ch MECHANISMS Toolkit AutoDesk Inventor In-Motion Module

MD/AY2011-2012 ME304 KDM.20

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

FOURBAR Program

MD/AY2011-2012 ME304 KDM.21

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

FOURBAR Program

MD/AY2011-2012 ME304 KDM.22

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

MD/AY2011-2012 ME304 KDM.23

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis Crank-Slider Linkage

• Given: link lengths a, b and c, θ1, θ2 (the motor position)

• Find: the unknown angle θ3 and length d

2 3 4 1 0R R R R

32 4 1 0ii i i

ae be ce de

MD/AY2011-2012 ME304 KDM.24

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis Crank-Slider Linkage

32 4 1

2 3 4 1

2 3 4 1

1

Using Euler Equivalents, collecting

real and imaginary terms and setting

each to zero results in:

cos cos cos cos 0

sin sin sin sin 0

As 0, these eq a io

0

u t

ii i iae be ce

a b c d

de

a b c d

2 3 4

2 3 4

cos cos c

ns simplify to

os 0

si

:

n sin sin 0

a b c d

a b c

1 2 4

3

sin sinsin

a c

b

MD/AY2011-2012 ME304 KDM.25

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Linkages with more than 4 bars

1

1 2 43

1 23

4

2 3

Here is the offset.

Initial set up of coordinate system

for slider block such that

0 and 90

Hence one solution i

sin sinsin

sinsin

cos c

s:

os

a c

b

a c

c

b

d a b

1 2

3

2 3

The next valid solution taking into

account the multi-valued

sinsin

cos

ness of arcsin

i

c

s

os

a c

b

d a b

MD/AY2011-2012 ME304 KDM.26

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Linkages with more than 4 bars

1 23

2 3

Hence one solution i

sinsin

:

cos

s

cos

a c

b

d a b

MD/AY2011-2012 ME304 KDM.27

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Linkages with more than 4 bars

1 23

2 3

sinsin

cos cos

a c

b

d a b

MD/AY2011-2012 ME304 KDM.28

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

• Given: link lengths a, c and d,

1, 2 (the motor position), and g the angle between the slider and rod

• Find: the unknown angles 3

and 4 and length b

2

MD/AY2011-2012 ME304 KDM.29

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

Write the vector loop equation:

(Positive from tail to tip)

Substitute with complex vectors

2 3 4 1 0R R R R

32 4 1

42 4 1

3 4

0

Since

0

ii i i

ii i i

ae be ce de

ae be ce de

g

g

2

MD/AY2011-2012 ME304 KDM.30

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

2

• Grouping knowns and unknowns

• Denoting

• Taking the conjugate to get the second equation

42 4 1 0ii i i

ae be ce de g

4 4 2 1

2 1 known

i i i i

i i

be ce ae de Z

Z ae de

g

4 ,i is e t e g

4 4

4 4

i i

i ii

Z be ce

Z be e ce bst cs

g

g

1 bZ bst cs c

s t

MD/AY2011-2012 ME304 KDM.31

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

2 2 2

Multiply and to g

1

et:

1

Z Z

Z bst cs

bZ bst cs c

s t

ZZ b bc t ct

2 21Solve 0 for b c t b c ZZ b

t

2

2 21 14

2

c t c t c ZZt t

b

MD/AY2011-2012 ME304 KDM.32

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

2 21Solve 0 for b c t b c ZZ b

t

2

2 21 14

2

c t c t c ZZt t

b

2 From Z bst cs

Zs

bt c

MD/AY2011-2012 ME304 KDM.33

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Inverted Crank-Slider Linkage

MD/AY2011-2012 ME304 KDM.34

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Geared Fivebar Linkage

• Consider Geared fivebar linkage • Write vector loop equation

• Apply complex number representation

2 3 4 5 1 0R R R R R

3 52 4 1 0i ii i i

ae be ce de fe

5 2

5 2

Gear Ratio: will relate and v

via a phase angle as follows:

32 4 2 1( )0

ii i i iae be ce de fe

MD/AY2011-2012 ME304 KDM.35

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Geared Fivebar Linkage

32 4 2 1

3 4 2 2 1

3 4

( )

( )

Separate unknowns and know

Denot

n

e

s

0ii i i i

i i i i i

i iZ

ae be ce de fe

be ce ae de fe Z

be ce

3 4

3 4

Form Conjugate

Define and i i

i i

s be t

bs

c

ct Z

b cZ bs ct

e

be ce

s t

The remaining analysis follows exactly the same way as shown for the fourbar linkage

MD/AY2011-2012 ME304 KDM.36

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Geared Fivebar Linkage

MD/AY2011-2012 ME304 KDM.37

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Geared Fivebar Linkage

MD/AY2011-2012 ME304 KDM.38

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Watt’s Sixbar Linkage

• Watt’s sixbar can be solved as 2 fourbar linkages

• R1R2R3R4, then R5R6R7R8

• R4 and R5 have a constant angle between them

MD/AY2011-2012 ME304 KDM.39

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Watt’s Sixbar Linkage

MD/AY2011-2012 ME304 KDM.40

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Watt’s Sixbar Linkage

MD/AY2011-2012 ME304 KDM.41

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Stephenson’s Sixbar Linkage

• Stephenson’s sixbar can sometimes be solved as a fourbar and then a fivebar linkage

• R1R2R3R4, then R4R5R6R7R8

• R3 and R5 have a constant angle between them

• If motor is at O6 you have to solve eqns. simultaneously

MD/AY2011-2012 ME304 KDM.42

POSITION ANALYSIS OF MECHANISMS

Finding the position of any point on a linkage

• Once the unknown angles have been found it is easy to find any position on the linkage (relative to pivot O2)

• For point S

Rs=sei(2+d2)

• For point P

RP=aei 2 +pei (3+d3)

• For point U

RU=d +uei (4+d4)

a

b

d

c

MD/AY2011-2012 ME304 KDM.43

POSITION ANALYSIS OF MECHANISMS

Algebraic Position Analysis of Fourbar Linkage

x’ y’

Rp

RA

MD/AY2011-2012 ME304 KDM.44

POSITION ANALYSIS OF MECHANISMS

Analysis of Transmission Angles of Fourbar Linkage

• Extreme value of transmission angle when links 1 and 2 are aligned.

22 2

1 arccos2

b c d a

bc

22 2

2 arccos2

b c d a

bc

MD/AY2011-2012 ME304 KDM.45

POSITION ANALYSIS OF MECHANISMS

Analysis of Toggle Positions of Fourbar Linkage

• Caused by the collinearity of links 3 and 4.

2 2 2 21

2 2cos 02toggle toggle

a d b c bc

ad ad

2

2

2

3

4

4

3

2

Overlapped

Extended

MD/AY2011-2012 ME304 KDM.46

POSITION ANALYSIS OF MECHANISMS

Analysis of Toggle Positions of a Fourbar Linkage

• Caused by the collinearity of links 3 and 4.

• For a non-Grashof linkage, only one of the values of

will be between –1 and 1

2 2 2 21

2 2cos 02toggle toggle

a d b c bc

ad ad

2 2 2 2

2

a d b c bc

ad ad

MD/AY2011-2012 ME304 KDM.47