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Optimizing 4-bar crank-rocker mechanism
Mechanism and Machine Theory,Vol, 14, pp. 319-325 © Pergamon Press Ltd., 1979. Printed in Great Britain
0(~J4-114X/79/1001-0319/$02.00/0
Optimizing 4-Bar Crank-Rocker Mechanism A. K. Kharet
and R. K. Davet
Received 20 September 1978 Abstract
Closed form equations are developed for the synthesis of the 4-bar crank-rocker mechanism in which the angle between dead-centre positions of the rocker and the corresponding angle turned by the crank are prescribed. All the four different situations are considered and the minimum transmission angle, for the complete rotation of the crank, is obtained for each case. The upper and lower bounds on the crank angle, corresponding to a dead centre position of the rocker, are obtained by using Grashof's criterion. The design is optimized by maximizing the minimum transmission angle. Each case is illustrated by providing a numerical example. Introduction TIlE 4-bar crank-rocker mechanism is employed to transform rotary motion into oscillating motion. In such cases, a large force is usually transmitted. Designers have recognised the necessity of optimizing the transmission angle of moderate and large force transmitting linkages. For best results, the transmission angle/z should be as close to 90° as possible during the entire rotation of the crank. In a crank-rocker mechanism angular displacement of the rocker and the corresponding displacement of the crank are usually given. These angles are measured from the dead-centre positions of the rocker where it reverses its direction of motion. Thus ~'o, the angle between the two dead-centre positions of the rocker and 00, the corresponding angle turned by the crank, are known. There is a family of linkages satisfying the above requirement. For the sake of convenience, the transmission angle is defined as the acute angle made by the rocker with the coupler. The transmission angle in a crank-rocker mechanism is a minimum when the crank is in line with the fixed link[l]. The problem undertaken is to develop an analytical procedure to select the linkage, from the linkage family, by maximising the minimum transmission angle. Theory
Figures 1 and 2 represent the graphical method of synthesizing the crank-rocker mechanism for different values of 00[2]. a, b, c and d are the link lengths. Usually d, the length of the fixed link, is taken equal to unity. Then a, b and c will represent the link length ratios. Refering to Fig. 1 LAoRBo = LARB
= Oo _ ~o 2 2 7)"
Z_R O A o = / - R A A o = ~ .
tDepartmentof MechanicalEngineering,GovernmentEngineeringCollege,REWA-486002(MP) India.
319
320
R
i
[
i Ao~8o/2
A,
d:lB ~ : ~
Figure 1.
L
B
Figure 2.
Therefore
Ao0 = sin
4'o 2
R---B = s i n
AoR =
-cos
AR AoR =
sin
AR RB = From
-
(_~
) +/3
(~ $o) cos
similar triangles Ao0
+
-
.
AoR0 and
~1)
RAB
b
AoR RB"
[2)
321 Simplifying eqns (1) and (2) sin a, cos/3, a =
sin ~2
b = sin al sin/31
(3)
COS or2
Where
~'o
O/1= ~ -
0o a2- 2
~bo 2
80
/3, = ~ +/3.
(4)
Applying the cosine law to the triangle AoBBo c 2 -- (a
+
b) 2 + 1 - 2(a + b) cos/3.
(5)
Substitution for a and b in eqn (5) and simplification results in the following expression c 2 = KI + K2 cos (a3 + 2/3) +
cos (2a3 + 2/3).
(6)
Where KI = 1 + K2 cos as + K2--~2 2 2 sin a l K2 - sin 2a2
a3 = 00
~o 2"
(7)
Case I. 0o > 180°
Grashof's criterion for the crank-rocker mechanisms
is
(1 + s) ~<(p + q). Where 1 is the length of the longest link, s is the length of the shortest link--the crank, p and q represent the lengths of the remaining two links. This condition is satisfied if l_LAoBo >1[3 >10 or
(8) Equation (8) gives the upper and lower bounds on/3. MMT VoL 14, No. 5--C
322
If Hois greater than 180”, the transmission the fixed link as shown in Fig. I. b’+
cos lm;”
(.-(I
=
angle is a minimum when the crank superimpose\
_a)?
!9)
2hc
The desired linkage is one for which (90-p,,& is a minimum or cos pmin has the smallest positive value. This can be achieved by minimising cos pmin with respect to p subject to the condition given by eqn (8). This problem can be attempted either graphically or by using a calculator. Its solution will yield the desired value of /?. The link length ratios a, h and c may, then, be obtained by using eqns (3) and (6) and the mechanism can be synthesised by suitably assuming the length of the fixed link. Case II. 0,< 180 The Grashof’s condition
for crank-rocker
mechanism
is satisfied if
(10) When B0is less than 180”, the transmission angle, during the complete rotation of the crank, is minimum when the crank is in line with the fixed link and away from it. Then b2 + c2- (1 + a)’
COS /.hmin=
2bc
’
The desired value of p may be obtained by minimising condition given by eqn (10).
(11) cos pmin with respect to p subject to
Case III. 0, = 180 + & This situation deserves special attention, because eqn (3) does not hold good in this case. Referring to Fig. 2, of all the possible linkages, the link length ratios in the mechanism, in which the transmission angle A2B2Bo is 90”, are such that the minimum transmission angle AlBIB is nearest to 90”. In this case
p=.R-!b
(12)
2 a = cos p = -cos
$.
(13)
From similar triangles A2B2Bo and AIBIAo b ;=(I
+a)’
Therefore b = g[a(
1+
a)].
(l+a)*=
b2+c2
(14)
Also
or c=V(l+a)
(15)
323
and 2V'a cos/zrai,, - (1 + a)"
Case IV. 00 = 180° The centric crank-rocker mechanism, defined by 0o = 180°, has unit time ratio[3]. It also has equal transmission angles/Zmi, at both the positions when the crank is in line with the fixed link. Therefore (1 - a) 2 = b2+ c2-2bc cos/zrai. and (1 + a) 2 = b2+ c2+2bc cos grain. Elimination of cos P-mi. results in c 2 = 1 + a 2 - b 2.
(16)
Since 0o = 180 °, eqn (3) gives a = tan -~ sin/3 Z
and b = cos/3.
(17)
Simplifying (16) using (17) c = s e c - ~ sin/3.
(18)
Substituting for a, b and c in eqns (9) or (11) and simplifying a _ sin (~o/2) COS ~/,min bc cos/3 "
(19)
--
Equation (19) gives a direct relation between/Zmi, and/3. Any value of angle/3 (subject to the condition given by eqns (8) or (10)) corresponds to a mechanism for which /3,min may be calculated from eqn (19). Also, for any desired tzra~,, the angle /3 may be obtained. The link length ratios a, b and c may, then, be obtained by using eqns (17) and (18).
Example Case I Given Oo= 200 ° 4~o= 50 °. The upper and lower bounds on/3 may be obtained by using eqn (8). 65°~/3 ~>0.
324 T h e f o l l o w i n g t a b l e is p r e p a r e d by u s i n g e q n s (3), (6) a n d (9).
60° 5if" 48° 46° 4(1°
0.411 0.379 0.371 0.363 0.335
0.865 0.816 0.865 0.913 1.05
I. 162 0.945 0.934 0.925 0.892
2"~ 40.5~ 41 ° 40.5c' 39°
F o r t h e g i v e n v a l u e s o f 0o a n d q~o, w h e n / 3 is e q u a l to 48 °, ]Zmin h a s its m a x i m u m v a l u e 41 °.
Example Case II Given 0o = 160 ° $o = 40 °. U s i n g e q n (10) 70 ° / > / 3 / > 20 °. U s i n g e q n s (3), (6) a n d (! 1) t h e f o l l o w i n g t a b l e is p r e p a r e d .
j~
d
b
c
/Zmin
70° 60° 52° 500 48° 40°
0.342 0.302 0.264 0.254 0.24--4 0.198
0.337 0.44 0.508 0.524 0.537 0.584
1.0 0.899 0.803 0.778 0.752 0.643
25.5° 28.7° 31.5° 32° 300 28.5°
F o r t h e g i v e n v a l u e s o f 0o a n d $o,
Example Case III Given Oo = 220 ° ~o = 40 °. Therefore /3 = 70 ° a = 0.342 b = 0,677 c = 1.158 P, mi, = 29.4 °.
~mm
h a s its m a x i m u m v a l u e 32 ° a t / 3 = 50 °.
325
Example Case 1V Given 0o = 180° + o = 9 0 °. Using eqn (10) 45°~>/3 ~ 0 . Using eqns (17)-(19) the following table is prepared.
40° 30° 25o 20° 15°
a
b
c
,u.rnln
0.643 0.5 0.423 0.342 0.259
0.766 0.866 0.906 0.939 0.966
0.909 0.707 0.598 0.483 0.366
22.50 35.3° 38.7° 410 42.9°
Referring to the above table, it can be concluded that, with the decrease in/3, ~mi. increases. However, this is associated with decrease in a (the length of the crank). As/3 approaches zero, /z,,~, approaches its m a x i m u m value and a approaches zero. This is not practically feasible. Practical considerations will limit the m i n i m u m length of the crank. Therefore, a compromise is to be made b e t w e e n the reduction in crank length and ircreases in /~m~,.
References 1. D. C. Tao, Applied Linkage Synthesis, p. 10. Addison-Wesley,New York (1964). 2. J. Volmer and P. W. Jensen, Four bar power linkage. Product Engng Nov. 12 (1%2). 3. R. J. Brodell and A. H. Soni, Design of crank-rocker mechanism with unit time ratio. J. Mech. 5, (1970). 4. J. Volmer, Zur Totlagenskonstruktion der zentrischer Kurbelschwinge. Maschinenbautechnik 3, 228 (1954). 5. H. Strauchmann, Ein Betrag zur Synthese der zentrischen Kurbelschwinge. Maschinenbautecknik 15, 587 (1%6).
OPTIMISATION A.K.
Khare
R~sumd dans
DU M E C A N I S M E
et R.K.
lesquelles
UNE M A N I V E L L E
ET UN B A L A N C I E R
des ~ q u a t i o n s
sont p r ~ c i s ~ s
l'angle
de clSture
entre
pour
la s y n t h ~ s e
du m ~ c a n i s m e
les points m o r t s du b a l a n c i e r
~ quatre
et l'angle
barres,
correspon-
par la m a n i v e l l e .
Nous
consid~rons
de t r a n s m i s s i o n
requis
On o b t i e n t l'angle
B A R R E S AVEC
Dave
- Nous d ~ v e l o p p o n s
dant d~crit
A QUATRE
les q u a t r e
possibilit~s
et o b t e n o n s
pour chaque
par
de la m a n i v e l l e
Une c o n c e p t i o n
la voie du c r i t ~ r e correspondant
de G r a s h o f
les limites
sup~rieure
illustrons
o p t i m u m est o b t e n u e
chaque
et i n f ~ r i e u r e
~ un point m o r t du b a l a n c i e r .
en p o r t a n t
au m a x i m u m
l'angle m i n i m u m
mission.
NOUS
cas l'angle m i n i m u m
pour un tour c o m p l e t de la m a n i v e l l e .
cas au m o y e n
d'un e x e m p l e
num~rique.
de t r a n s -
de
0(~J4-114X/79/1001-0319/$02.00/0
Optimizing 4-Bar Crank-Rocker Mechanism A. K. Kharet
and R. K. Davet
Received 20 September 1978 Abstract
Closed form equations are developed for the synthesis of the 4-bar crank-rocker mechanism in which the angle between dead-centre positions of the rocker and the corresponding angle turned by the crank are prescribed. All the four different situations are considered and the minimum transmission angle, for the complete rotation of the crank, is obtained for each case. The upper and lower bounds on the crank angle, corresponding to a dead centre position of the rocker, are obtained by using Grashof's criterion. The design is optimized by maximizing the minimum transmission angle. Each case is illustrated by providing a numerical example. Introduction TIlE 4-bar crank-rocker mechanism is employed to transform rotary motion into oscillating motion. In such cases, a large force is usually transmitted. Designers have recognised the necessity of optimizing the transmission angle of moderate and large force transmitting linkages. For best results, the transmission angle/z should be as close to 90° as possible during the entire rotation of the crank. In a crank-rocker mechanism angular displacement of the rocker and the corresponding displacement of the crank are usually given. These angles are measured from the dead-centre positions of the rocker where it reverses its direction of motion. Thus ~'o, the angle between the two dead-centre positions of the rocker and 00, the corresponding angle turned by the crank, are known. There is a family of linkages satisfying the above requirement. For the sake of convenience, the transmission angle is defined as the acute angle made by the rocker with the coupler. The transmission angle in a crank-rocker mechanism is a minimum when the crank is in line with the fixed link[l]. The problem undertaken is to develop an analytical procedure to select the linkage, from the linkage family, by maximising the minimum transmission angle. Theory
Figures 1 and 2 represent the graphical method of synthesizing the crank-rocker mechanism for different values of 00[2]. a, b, c and d are the link lengths. Usually d, the length of the fixed link, is taken equal to unity. Then a, b and c will represent the link length ratios. Refering to Fig. 1 LAoRBo = LARB
= Oo _ ~o 2 2 7)"
Z_R O A o = / - R A A o = ~ .
tDepartmentof MechanicalEngineering,GovernmentEngineeringCollege,REWA-486002(MP) India.
319
320
R
i
[
i Ao~8o/2
A,
d:lB ~ : ~
Figure 1.
L
B
Figure 2.
Therefore
Ao0 = sin
4'o 2
R---B = s i n
AoR =
-cos
AR AoR =
sin
AR RB = From
-
(_~
) +/3
(~ $o) cos
similar triangles Ao0
+
-
.
AoR0 and
~1)
RAB
b
AoR RB"
[2)
321 Simplifying eqns (1) and (2) sin a, cos/3, a =
sin ~2
b = sin al sin/31
(3)
COS or2
Where
~'o
O/1= ~ -
0o a2- 2
~bo 2
80
/3, = ~ +/3.
(4)
Applying the cosine law to the triangle AoBBo c 2 -- (a
+
b) 2 + 1 - 2(a + b) cos/3.
(5)
Substitution for a and b in eqn (5) and simplification results in the following expression c 2 = KI + K2 cos (a3 + 2/3) +
cos (2a3 + 2/3).
(6)
Where KI = 1 + K2 cos as + K2--~2 2 2 sin a l K2 - sin 2a2
a3 = 00
~o 2"
(7)
Case I. 0o > 180°
Grashof's criterion for the crank-rocker mechanisms
is
(1 + s) ~<(p + q). Where 1 is the length of the longest link, s is the length of the shortest link--the crank, p and q represent the lengths of the remaining two links. This condition is satisfied if l_LAoBo >1[3 >10 or
(8) Equation (8) gives the upper and lower bounds on/3. MMT VoL 14, No. 5--C
322
If Hois greater than 180”, the transmission the fixed link as shown in Fig. I. b’+
cos lm;”
(.-(I
=
angle is a minimum when the crank superimpose\
_a)?
!9)
2hc
The desired linkage is one for which (90-p,,& is a minimum or cos pmin has the smallest positive value. This can be achieved by minimising cos pmin with respect to p subject to the condition given by eqn (8). This problem can be attempted either graphically or by using a calculator. Its solution will yield the desired value of /?. The link length ratios a, h and c may, then, be obtained by using eqns (3) and (6) and the mechanism can be synthesised by suitably assuming the length of the fixed link. Case II. 0,< 180 The Grashof’s condition
for crank-rocker
mechanism
is satisfied if
(10) When B0is less than 180”, the transmission angle, during the complete rotation of the crank, is minimum when the crank is in line with the fixed link and away from it. Then b2 + c2- (1 + a)’
COS /.hmin=
2bc
’
The desired value of p may be obtained by minimising condition given by eqn (10).
(11) cos pmin with respect to p subject to
Case III. 0, = 180 + & This situation deserves special attention, because eqn (3) does not hold good in this case. Referring to Fig. 2, of all the possible linkages, the link length ratios in the mechanism, in which the transmission angle A2B2Bo is 90”, are such that the minimum transmission angle AlBIB is nearest to 90”. In this case
p=.R-!b
(12)
2 a = cos p = -cos
$.
(13)
From similar triangles A2B2Bo and AIBIAo b ;=(I
+a)’
Therefore b = g[a(
1+
a)].
(l+a)*=
b2+c2
(14)
Also
or c=V(l+a)
(15)
323
and 2V'a cos/zrai,, - (1 + a)"
Case IV. 00 = 180° The centric crank-rocker mechanism, defined by 0o = 180°, has unit time ratio[3]. It also has equal transmission angles/Zmi, at both the positions when the crank is in line with the fixed link. Therefore (1 - a) 2 = b2+ c2-2bc cos/zrai. and (1 + a) 2 = b2+ c2+2bc cos grain. Elimination of cos P-mi. results in c 2 = 1 + a 2 - b 2.
(16)
Since 0o = 180 °, eqn (3) gives a = tan -~ sin/3 Z
and b = cos/3.
(17)
Simplifying (16) using (17) c = s e c - ~ sin/3.
(18)
Substituting for a, b and c in eqns (9) or (11) and simplifying a _ sin (~o/2) COS ~/,min bc cos/3 "
(19)
--
Equation (19) gives a direct relation between/Zmi, and/3. Any value of angle/3 (subject to the condition given by eqns (8) or (10)) corresponds to a mechanism for which /3,min may be calculated from eqn (19). Also, for any desired tzra~,, the angle /3 may be obtained. The link length ratios a, b and c may, then, be obtained by using eqns (17) and (18).
Example Case I Given Oo= 200 ° 4~o= 50 °. The upper and lower bounds on/3 may be obtained by using eqn (8). 65°~/3 ~>0.
324 T h e f o l l o w i n g t a b l e is p r e p a r e d by u s i n g e q n s (3), (6) a n d (9).
60° 5if" 48° 46° 4(1°
0.411 0.379 0.371 0.363 0.335
0.865 0.816 0.865 0.913 1.05
I. 162 0.945 0.934 0.925 0.892
2"~ 40.5~ 41 ° 40.5c' 39°
F o r t h e g i v e n v a l u e s o f 0o a n d q~o, w h e n / 3 is e q u a l to 48 °, ]Zmin h a s its m a x i m u m v a l u e 41 °.
Example Case II Given 0o = 160 ° $o = 40 °. U s i n g e q n (10) 70 ° / > / 3 / > 20 °. U s i n g e q n s (3), (6) a n d (! 1) t h e f o l l o w i n g t a b l e is p r e p a r e d .
j~
d
b
c
/Zmin
70° 60° 52° 500 48° 40°
0.342 0.302 0.264 0.254 0.24--4 0.198
0.337 0.44 0.508 0.524 0.537 0.584
1.0 0.899 0.803 0.778 0.752 0.643
25.5° 28.7° 31.5° 32° 300 28.5°
F o r t h e g i v e n v a l u e s o f 0o a n d $o,
Example Case III Given Oo = 220 ° ~o = 40 °. Therefore /3 = 70 ° a = 0.342 b = 0,677 c = 1.158 P, mi, = 29.4 °.
~mm
h a s its m a x i m u m v a l u e 32 ° a t / 3 = 50 °.
325
Example Case 1V Given 0o = 180° + o = 9 0 °. Using eqn (10) 45°~>/3 ~ 0 . Using eqns (17)-(19) the following table is prepared.
40° 30° 25o 20° 15°
a
b
c
,u.rnln
0.643 0.5 0.423 0.342 0.259
0.766 0.866 0.906 0.939 0.966
0.909 0.707 0.598 0.483 0.366
22.50 35.3° 38.7° 410 42.9°
Referring to the above table, it can be concluded that, with the decrease in/3, ~mi. increases. However, this is associated with decrease in a (the length of the crank). As/3 approaches zero, /z,,~, approaches its m a x i m u m value and a approaches zero. This is not practically feasible. Practical considerations will limit the m i n i m u m length of the crank. Therefore, a compromise is to be made b e t w e e n the reduction in crank length and ircreases in /~m~,.
References 1. D. C. Tao, Applied Linkage Synthesis, p. 10. Addison-Wesley,New York (1964). 2. J. Volmer and P. W. Jensen, Four bar power linkage. Product Engng Nov. 12 (1%2). 3. R. J. Brodell and A. H. Soni, Design of crank-rocker mechanism with unit time ratio. J. Mech. 5, (1970). 4. J. Volmer, Zur Totlagenskonstruktion der zentrischer Kurbelschwinge. Maschinenbautechnik 3, 228 (1954). 5. H. Strauchmann, Ein Betrag zur Synthese der zentrischen Kurbelschwinge. Maschinenbautecknik 15, 587 (1%6).
OPTIMISATION A.K.
Khare
R~sumd dans
DU M E C A N I S M E
et R.K.
lesquelles
UNE M A N I V E L L E
ET UN B A L A N C I E R
des ~ q u a t i o n s
sont p r ~ c i s ~ s
l'angle
de clSture
entre
pour
la s y n t h ~ s e
du m ~ c a n i s m e
les points m o r t s du b a l a n c i e r
~ quatre
et l'angle
barres,
correspon-
par la m a n i v e l l e .
Nous
consid~rons
de t r a n s m i s s i o n
requis
On o b t i e n t l'angle
B A R R E S AVEC
Dave
- Nous d ~ v e l o p p o n s
dant d~crit
A QUATRE
les q u a t r e
possibilit~s
et o b t e n o n s
pour chaque
par
de la m a n i v e l l e
Une c o n c e p t i o n
la voie du c r i t ~ r e correspondant
de G r a s h o f
les limites
sup~rieure
illustrons
o p t i m u m est o b t e n u e
chaque
et i n f ~ r i e u r e
~ un point m o r t du b a l a n c i e r .
en p o r t a n t
au m a x i m u m
l'angle m i n i m u m
mission.
NOUS
cas l'angle m i n i m u m
pour un tour c o m p l e t de la m a n i v e l l e .
cas au m o y e n
d'un e x e m p l e
num~rique.
de t r a n s -
de