four bar force analysis_20150224
TRANSCRIPT
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Force analysis of planar four bar AND kinematics of SPHERICAL four bar MECHANISM
Hassen N.
2015.02.24
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Content
• Statics of four bar
• Dynamic of four bar
• Sp erical mec anism
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Static force analysislink %
5
1θ
1T
x o ,1
y o ,1
1 g
1W
1O
x A
y A
A∑ =0x F )1(0,1 =+ x x o A
∑ =0 y F )2(01,1 =−+ w o A y y ∑ =0 g M ( ) ( ) )3(0sinsincossin 111,111,1111111 =+−+−+−− T g o g o g Ag A y x y x θ θ θ θ
111
11
g g o
Ao
=
=
014,13,121 =+−++− T ! o ! o ! A! A y x y x
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Static force analysislink -
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• .n t e free bo#y #ia'ram of link - t e #irection of forceat A as to be opposite of t e #irection assume# on link%&
∑ =0x F )4(O " A # # =+−
∑ =0 y F )5(2 O w " A y y =−+−
2θ
x A
y A
2 g
2W
A
x "
y "
"
32
2
g Ag
A"
==
( ) ( ) )$(0cossincossin
0
2222222222 =+−−+−−
=∑θ θ θ θ g Ag Ag " g "
M
y # y #
g
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Static force analysislink /
• .n t e free bo#y #ia'ram of link / (e must set t e #irection of force at 0 in t e opposite #irection assume# on link- &
7
x o ,3
y o ,3
3 g
3W
1O
x "
y "
"
c P
%
∑ =0x F
)&(3 O P " O # # =+−
∑ =0 y F
)'(43 O w " O y y =−−−
( ) ( ) )(0cossincossin
0
334334333333 =−−+−+−−
=∑)% g O g O g " g "
M
y x y #
g
θ θ θ θ
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Static force analysis.,erall e"uation
• No( (e a,e nine e"uations (it nine unkno(ns an# assemblin' all e"uations in matri$ form 'i,es t e follo(in'&
8
)10(
0
0
0
0
00000
010001000
001000100
00000
000001010
000000101
10000
000100010
000010001
3
2
1
1
3
3
1
1
1211109
6587
4321
−
=
−−−
−−−
−−
−−
pd
w
p
w
w
T O
O
O
O
B
B
A
A
cccc
cccc
cccc
y
x
y
x
y
x
y
x
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Dynamics of four bar
9
*e+oci y e- a ion
)11(0
0
coscoscos
sinsinsin
3
2
1
332211
332211 =−−
−
θ
θ
θ
θ θ θ
θ θ θ
d d d
d d d
)12(cos
sin
coscos
sinsin
11
11
2
2
3322
3322
−=−
−θ
θ
θ
θ
θ θ
θ θ
d
d
d d
d d
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Dynamics of four bar
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Use LaGrange formulation for the dynamics. It canbe de ned as the total kinetic energy minus totalpotential energy. The total kinetic energy of thesystem can be de ned as
( )( )
( )( )
( )( )
)13(sinsin
,sinsin
,,,
,,
323
1213
122
13121
3211
3211
3
2
θ θ
θ θ θ
θ θ
θ θ θ θ
θ θ θ
θ θ θ
θ
θ
−−=−
−==d d
d d
E
E
( ) ( ) ( ) (14) 21
21
21 2
332
332
222
222
112
11 θ θ θ I vm I vm I vmT
ccc +++++=
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Dynamics of four bar• ! en #etermine t e total potential ener'y&
• In t e abo,e e"uation (e can eliminate t e ,elocity terms usin' E"&1%/2&
11
)15(3,32,21,1 ccc gym gym gymV ++=
)16(V T L −=( )
( ) ( ) )17(,,,
,,,,,
321212111
233
222
211321321
θ θ θ θ θ θ θ
θ θ θ θ θ θ θ θ θ
GC J
K K K L
+
+++=
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Dynamics of four bar
12
( )
( )
( )
( ) ( )( ) ( )
( )18
sinsinsin,,
cos,
212121
33322211211321
21211
2221
32
333
22
222
2221
2111
−−−−=−=
=+=
+=
++=
θ θ θ θ θ θ
θ θ θ θ
ccc
c
c
c
c
gd m gd m gd m gd mG
C
d d m J
I d m K
I d m K
d m I d m K
( ) ( ) ( ) ( ) ( )[ ]( ) )19(,,
,,,,,,,,,,,,
321
21321
212111321
223321
2121321321
θ θ θ
θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ
G
E C J E K E K K L
++++=
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Dynamics of four bar• ! e e"uation of motion of t e entire system can
(ritten
• ! en
13
( )2011
τ
θ θ
=∂
∂
∂
∂ L Ldt
d
( ) ( ) ( ) ( )[ ] ( )21,,,,,,,2 3211211132122332121211
θ θ θ θ θ θ θ θ θ θ θ
θ E C J E K E K K
L +++=∂
∂
( )222111
θ θ θ
B A L
dt d +=
∂∂
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Dynamics of four bar• 3 ere
14
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( )
( ) ( ) ( )
∂
∂∂+∂
∂
+
∂
∂
∂+∂
∂
∂+∂
∂
+
∂
∂∂+∂∂
∂+∂∂
+
∂∂
+∂∂
+∂∂
=
+++=
1
2
2
211
1
2113211
1
3
3
3211
1
2
2
3211
1
32112111
1
3
3
3212
1
2
2
3212
1
321232123
3
3212
2
3212
1
321232111
321121113212
333212
221
,,,,
,,,,,,,
,,,,,,,,2
,,,,,,,,2
2
,,,,,,,2
θ
θ
θ
θ θ
θ
θ θ θ θ θ
θ
θ
θ
θ θ θ
θ
θ
θ
θ θ θ
θ
θ θ θ θ θ
θ
θ
θ
θ θ θ
θ
θ
θ
θ θ θ
θ
θ θ θ θ θ θ
θ
θ θ θ
θ
θ θ θ
θ
θ θ θ θ θ θ
θ θ θ θ θ θ θ θ θ θ θ
C C E
E E E C J
E E E E K
E E E E K
B
E C J E K E K K A
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Dynamics of four bar
15
( )
Where
232
11 C H
L
+=∂∂ θ θ
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( )
( ) ( ) ( )
∂∂
∂∂+∂
∂
+
∂∂
∂∂+∂
∂∂
∂+∂∂
+
∂∂
∂∂
+∂∂
∂∂
+∂∂
+
∂∂
∂∂+∂
∂
=
1
2
2
211
1
2113211
1
3
3
3211
1
2
2
3211
1
32112111
1
3
3
3212
1
2
2
3212
1
3212
32123
3
3211
2
3211
1
321132121
,,,,
,,,,,,,J
,,,,,,,,2
,,,,,,,,2
θ
θ
θ
θ θ
θ
θ θ θ θ θ
θ
θ
θ
θ θ θ
θ
θ
θ
θ θ θ
θ
θ θ θ θ θ
θ
θ
θ
θ θ θ
θ
θ
θ
θ θ θ
θ
θ θ θ θ θ θ
θ
θ θ θ
θ
θ θ θ
θ
θ θ θ θ θ θ
C C E
E E E C
E E E E K
E E E E K
H
( ) ( ) ( )1
3
3
321
1
2
2
321
1
321 ,,,,,,θ
θ
θ
θ θ θ
θ
θ
θ
θ θ θ
θ
θ θ θ
∂∂
∂∂+∂
∂∂
∂+∂∂= GGGC
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Dynamics of four bar• Substitute E"&1-%+-- 4 -/2 into E"&1-52 yiel#s
?
16
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Sp erical mec anism• A sp erical mec anism is linka'es ( ose )oints
a,e a$es t at intersect at a common point&
17
−−
=−
1000
01
iii
iiiiiii
iiiiiii
ii
d c s
sac scc sca s s scc
Aα α
θ θ α θ α θ
θ θ α θ α θ
−−
=−
ii
iiiii
iiiii
ii
c s
c scc s
s s scc
Rα α
θ α θ α θ
θ α θ α θ
0
1
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Sp erical mec anism• 3 ere
• 6se loop closure e"uation
18
axes. jointaof normainci!ento "et#een t#an$ eJointaxes jointa!jacento "et#een t#an$ e%#ist
axes jointaof normainci!ento "et#een t#!istancena%rans atio!
axes jointa!jacento "et#een t#!istance&ffset
i
i
i
==
==
θ
α
ia
I R R R R =14
43
32
21
14
43
32
21 R R R R =
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Sp erical mec anism• 3 ere
'i,es t e follo(in'
19
−
−
−
−=
−−
−−
11
11111
11111
44
44444
44444
33
33333
22333
22
22222
22222
00
00
α α
θ α θ α θ
θ α θ α θ
α α
θ α θ α θ
θ α θ α θ
α α
θ α θ α θ
θ α θ α θ
α α
θ α θ α θ
θ α θ α θ
c s
c scc s
s s scc
c s
c scc s
s s scc
c s
c scc s
s s scc
c s
c scc s
s s scc
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Sp erical mec anism
20
6sin' t is matri$ (e can #etermine t e #isplacement of t e output )oint& Also t e interme#iate an'les&
+−+−−−+−+
++−−−=
+−+−−−+−+
++−−−
141141411414
1441144114144114411414414
144144144114411414414
323323233232
3223322332322332233232232
322332322332233232232
α α θ α α α θ θ α α θ α
α θ α θ α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α α θ α α α θ θ α α θ α
α θ α θ α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
ccc s s sccc s s s
cc sc scc s s s sc scccc sc s sccc s
s sc s sc s sccc sc scc s sccc
ccc s s sccc s s s
cc sc scc s s s sc scccc sc s sccc s
s sc s sc s sccc sc scc s sccc
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Sp erical mec anism
21
6sin' t is matri$ (e can #etermine t e #isplacement of t e output )oint& Also t e interme#iate an'les&
+−+−−−+−+
++−−−=
+−+−−−+−+
++−−−
141141411414
1441144114144114411414414
144144144114411414414
323323233232
3223322332322332233232232
322332322332233232232
α α θ α α α θ θ α α θ α
α θ α θ α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α α θ α α α θ θ α α θ α
α θ α θ α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
α θ α θ α θ α θ α θ α θ α θ α θ θ θ α θ θ
ccc s s sccc s s s
cc sc scc s s s sc scccc sc s sccc s
s sc s sc s sccc sc scc s sccc
ccc s s sccc s s s
cc sc scc s s s sc scccc sc s sccc s
s sc s sc s sccc sc scc s sccc