four bar linkage
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slides deal with mathematics of four bar linkageTRANSCRIPT
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MD/AY2011-2012 ME304 KDM.1
Indian Institute of Technology Gandhinagar
Dept. of Mechanical Engineering
L09: POSITION ANALYSIS OF
MECHANISMS
Dr. Murali Damodaran
Spring Semester of AY2011-2012
ME304: KINEMATICS AND DYNAMICS
OF MACHINES
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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/
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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
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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
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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
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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
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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’
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MD/AY2011-2012 ME304 KDM.8
POSITION ANALYSIS OF MECHANISMS
Coordinate Systems for Airplane Fight Dynamics
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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
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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
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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
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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
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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
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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
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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
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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
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MD/AY2011-2012 ME304 KDM.17
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage in Octave
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MD/AY2011-2012 ME304 KDM.18
KINEMATICS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage
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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
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MD/AY2011-2012 ME304 KDM.20
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage
FOURBAR Program
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MD/AY2011-2012 ME304 KDM.21
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage
FOURBAR Program
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MD/AY2011-2012 ME304 KDM.22
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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MD/AY2011-2012 ME304 KDM.33
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Inverted Crank-Slider Linkage
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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
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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
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MD/AY2011-2012 ME304 KDM.36
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Geared Fivebar Linkage
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MD/AY2011-2012 ME304 KDM.37
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Geared Fivebar Linkage
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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
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MD/AY2011-2012 ME304 KDM.39
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Watt’s Sixbar Linkage
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MD/AY2011-2012 ME304 KDM.40
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Watt’s Sixbar Linkage
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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
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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
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MD/AY2011-2012 ME304 KDM.43
POSITION ANALYSIS OF MECHANISMS
Algebraic Position Analysis of Fourbar Linkage
x’ y’
Rp
RA
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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
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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
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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
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MD/AY2011-2012 ME304 KDM.47