Analytical Linkage Synthesis
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Transcript of Analytical Linkage Synthesis
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Analytical Linkage
SynthesisChapter 5
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Types of Kinematic Synthesis Types
Function Generation
Correlation of an input function with an output function in a mechanism
Path Generation
Control of a point in the plane such that it follows some prescribed path
Motion Generation
Control of a line in the plane such that it assumes some sequential set of prescribed
positions
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Chapter 3 - Example 1 (revisited) Rocker Output- Two Position with
Angular Displacement (Function)
Design a four bar Grashof crank-rocker speed motor input to give 45 of rocker
motion with equal time forward and back,
from a constant speed motor input.
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Two-Position Synthesis for Rocker Output
Generic annotation
Link 4 is the output link to be driven by a dyad consisting
of link 2 and 3.
To be determine: links 1, 2, 3, and O2.
Defined: link 4, O4, 4 and
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Procedure Choose a location on link 4 to
attached link 3, B1 and B2 in its
extreme locations. This defined R4.
Two-Position Synthesis for Rocker Output
4441 cos ROB xx 4441 sin ROB yy
4442 cosROB xx 4442 sinROB yy
12 BBRRM
uMuRuL B 1
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Procedure Place the crank pivot O2 suitable far
from B1 along line L
The length of the crank must be half the length of M
Two-Position Synthesis for Rocker Output
3 Class ,
2 Class ,
1 Class ,
21
21
21
MOB
MOB
MOB
32 12
KKMRR BO
2/sin5.0 42 RMR
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Procedure Link 3 can be found by
subtracting R2 from the
magnitude of RB1-RO2
Link 1 is found by subtracting RO2 from RO4
Grashof crank-rocker /no quick return
Two-Position Synthesis for Rocker Output
23 21RRRR OB
241 OORRR
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Two-Position Motion Generation AL
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Two-Position Motion Generation AL Problem Statement
Design a fourbar linkage which will move a line on its coupler link such that
a point P on that line will be first at P1and later at P2 and will also rotate the
line through an angle 2 between those two positions. Find the lengths and
angles of the four links and the coupler
link dimension A1P1 and B1P1.
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Two-Position Motion Generation AL
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Two-Position Motion Generation AL Procedure
Define the two desired precision positions
The dyad W1Z1 (red)defines the left half of the linkages.
U1S1 (black)the right.
The pin-to-pin length and angle of link 3 is define in
terms of vector Z1and S1.
1221 RRP
111 SZV
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Two-Position Motion Generation AL Procedure
The ground links is define in terms of the two dyads
First solve for the left side of the linkage
To solve for W1 and Z1write the vector loop
equation
1111 UVWG
0112122 WZPZW
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Two-Position Motion Generation AL0112122 WZPZW
0
0
222
222
21
21
jjjjjjj
jjjjj
wezeepezeewe
wezeepzewe
222 2111 jjjjj epezeewe
22122
22
cossinsin1coscos
sinsin1coscos
pzz
ww
Real part:
Imaginary part:
22122
22
sinsincos1cossin
sincos1cossin
pzz
ww
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Two-Position Motion Generation AL
22122 and,,,,,,,
:ableseight vari are There
pzw
2212 and,,:statement
problem in the defined are Three
p
,, :assumed are Three 2
equations two with thesolve are Two
g,simplifyin
221
22
22
cos
sinsin1coscos
sinsin1coscos
pC
B
A
Strategy 1
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Two-Position Motion Generation AL
g,simplifyin
221
22
22
sin
sincos1cossin
sincos1cossin
pF
E
D
then,
FEzDw
CBzAw
usly,simultaneo solving
BDAE
BFCEw
BDAE
CDAFz
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Two-Position Motion Generation AL
Strategy Second
2212 and,,:statement
problem in the defined are Three
p
2
1
2link
,, Zvector :assumed are Three
z
1W vector for the solveThen
22122 and,,,,,,,
:ableseight vari are There
pzw
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Two-Position Motion Generation AL
11 Z and W vectorsof componentsy and x The
2212121
2121
cossin1cos
sin1cos
pZZ
WW
yx
yx
Real part:
Imaginary part:
sin sin
cos cos
11
11
zZwW
zZwW
yy
xx
2212121
2121
sinsin1cos
sin1cos
pZZ
WW
xy
xy
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Two-Position Motion Generation ALg,simplifyin
2212212
222
sin cos sin
1cos sin 1cos
pFpED
CBA
substituting:
FDZCZBWAW
EDZCZBWAW
xyxy
yxyx
1111
1111
the solution is;
A
FDZCZBEDZCZAW
xyyx
x 2
1111
1
A
EDZCZBFDZCZAW
yxxy
y 2
1111
1
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Two-Position Motion Generation AL
0112122 USPSU
222 2111 jjjjj epeseeue
22122
22
cossinsin1coscos
sinsin1coscos
pss
uu
For the right hand dyad, U1S1
Imaginary part:
22122
22
sinsincos1cossin
sincos1cossin
pss
uu
Real part:
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Two-Position Motion Generation AL
22122 and,,,,,,,
:ableseight vari are There
psu
2212 and,,:statement
problem in the defined are Three
p
,, :assumed are Three 2
equations two with thesolve are Two
g,simplifyin
221
22
22
cos
sinsin1coscos
sinsin1coscos
pC
B
A
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Two-Position Motion Generation AL
g,simplifyin
221
22
22
sin
sincos1cossin
sincos1cossin
pF
E
D
then,
FEsDu
CBsAu
usly,simultaneo solving
BDAE
BFCEu
BDAE
CDAFs
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Two-Position Motion Generation AL
Strategy Second
2212 and,,:statement
problem in the defined are Three
p
2
1
3link
,, Svector :assumed are Three
s
1U vector for the solveThen
22122 and,,,,,,,
:ableseight vari are There
psu
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Two-Position Motion Generation AL
11 Z and W vectorsof componentsy and x The
2212121
2121
cossin1cos
sin1cos
pSS
UU
yx
yx
Real part:
Imaginary part:
sin sin
cos cos
11
11
sSuU
sSuU
yy
xx
2212121
2121
sinsin1cos
sin1cos
pSS
UU
xy
xy
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Two-Position Motion Generation ALg,simplifyin
2212212
222
sin cos sin
1cos sin 1cos
pFpED
CBA
substituting:
FDSCSBUAU
EDSCSBUAU
xyxy
yxyx
1111
1111
the solution is;
A
FDSCSBEDSCSAU
xyyx
x 2
1111
1
A
EDSCSBFDSCSAU
yxxy
y 2
1111
1
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Comparison Graphical/Analytical
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Example 5-1 Design a fourbar linkage to
move the link APB shown
from position A1P1B1 to
A2P2B2.
Solution
1. Draw the link APB in its two desired positions to
scale
2. Measure or calculate the values of the magnitude and
angle of vector P21
2.165
416.2
2
21
p
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Example 5-1 3. Measure or calculate the
values of the change in
angle, 2, of vector Z from position 1 to 2,
4. Assume three additional free choice. Using strategy 2,
3.432
4.38
5.26
298.1
z
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Example 5-1 5. Substitute these values in
the corresponding equation
A-F, W1x .. and obtain,
6. Compare to graphical,
This vector W1 is link 2
6.71
467.2
w
71
48.2
w
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Example 5-1 7. Repeat the procedure for
the link-4 side, the free
choices,
8. Substitute these values along with the original
values,
9. Compare the graphical solution
6.85
1.104
035.1
2
s
3.432.165416.2 2221 p
4.15
486.1
u
1453.1 u Vector U1 is link 4
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Example 5-1 10. Line A1B1 is link 3,
Line O2O4 is link 1,
11. Check the grashofcondition
12. Construct a model in CAD. Check for limiting
conditions
13. Check transmission angles
111 SZV
1111 UVWG