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Synthesis of an involute based dwell mechanism
Mechanism and Machine Theory, 1975,Vol.10, pp. 33-40.
PergamonPress. Printedin GreatBritain
ii
i
Synthesis of an Involute Based Dwell Mechanism R. Samuel* and G. Ramaiyan~Received on 12 October 1973
Abstract A dwell mechanism has been developed based on a coupler curve approximating an involute and utilising a rack and pinion arrangement. The proportions of the basic linkage have been synthesised and the dwell characteristics of the output link have been computed and presented. 1. I n t r o d u c t i o n ONE OF the important classes of m e c h a n i s m s e m p l o y e d in automatic machinery is the dwell mechanism. A dwell m e c h a n i s m essentially keeps a point or a body, sensibly at rest o v e r an interval of time during a cycle of operation. Dwell m e c h a n i s m s can be derived f r o m different approaches. One method, often illustrated in the literature on this subject, is to utilise a m e c h a n i s m with a coupler c u r v e having segments approximating circular arcs and to add a dyad at the coupler point. In this p a p e r a different method is suggested; a coupler c u r v e having a segment approximating an involute is utilised as the driving path and a rack and pinion combination has been e m p l o y e d to realise the dwell.
2. Principle If, in a rack (Tr) and pinion (w) a r r a n g e m e n t shown in Fig. 1, the pinion is held stationary and the rack is rolled o v e r the pinion f r o m position 1 to 2, a point K on the rack will trace an involute path K,K'~ (suffixes 1 and 2 will refer to initial and final states for a finite interval of motion). Conversely, if K is guided along an involute KIK~, the
K;
~Involute ~2 ~ ~ Arbitrarily ~ assumedcurve I
~
,
~
~
"KI
Figure 1. Basic Principle. *Professor of Mechanical Engineering, College of Engineering, Guindy, Madras-600025, India. tLecturer in Mechanical Engineering, College of Engineering, Guindy, Madras-600025, India.
33
M.M.T. Vol. 10 No. I ~
34
pinion remains stationary. If, however, K is guided along a path different from the involute, say K1K2, the pinion is constrained to rotate corresponding to the deviation K~K2 of the driving path K,K2 from the involute K~K;. Thus the pionion is at rest as long as the driving path coincides with the involute and starts moving if this path deviates from the involute. Rack and pinion arrangements have been suggested as driving combinations[l] in general periodic mechanisms, even though such a combination has not been suggested for generating a basic path for deriving a dwell. 2.1 Angular displacement of the pinion The displacement of the pinion when the point K is guided along any arbitrary path from K, to K~, may be calculated as follows: If P is the point of tangency on the pinion and if K , K ~ is the involute passing through K,, the displacement of the pinion will correspond to an arcuate length equal to K~K2 (see Fig. 2). Let K , P , = t, and K2P2 = t2. Then K ; P 2 = t , + P ] P 2 = t, + R ( ~
- 4,,). Involute Actual driving path
Figure 2. Calculation of angular displacement of the pinion. Therefore, the arcuate displacement of the pinion is {t, + R (~b2 - ~b,)} - t2
and the angular displacement of the pinion is 1
k- [{t, + R ( ~ - 4,,)}- t:] t2
=-~+ t+2- O,) R
(1)
3.0 Synthesis of a Four Bar Pin Jointed Linkage to Generate an Approximate Involute A mechanism will now be synthesised to generate a coupler curve serving as the driving path which will approximate an involute over a part of the cycle, during which a dwell of the pinion can be obtained, and which will deviate from the involute during the remaining part of the cycle, when the pinion is made to move. For arriving at this, the fixed polode p~ and the moving polode p,. (for the coupler
35
motion) of the mechanism must coincide with the pinion and rack considered already (see Fig. 3). T o enable the proportions of the linkage to be expressed as dimensionless quantities we shall assume the radius of the pinion (base circle of the involute) as unity. Pairs of points on the cubic of stationary curvature may be used as moving joints. For such pairs of points the application of the E u l e r - S a v a r y equation will yield the corresponding pairs of fixed joints, and hence the four bar mechanisms whose coupler points coincident with pm describe paths which approximate involutes over an interval. Referring to Fig. 3, C, represents the cubic of stationary curvature containing the moving joints, such as A and B. Its diameter is three times the radius of the base circle [2, 3]. C: represents the corresponding circle containing the fixed joints, such as Ao and B0 (lying on AP and BP extended). (23 represents the inflection circle. The point K on the coupler A B of the four bar linkage AoABBo will generate the desired driving path for the rack and pinion arrangement.
C~
B
A
,
\
\Base circle P~ (Coincides with i'he pinion)
Figure 3. Synthesis of the four bar linkage to be coupled with rack and pinion arrangement. 3.1 Proportions and parameters of the basic four bar linkage for generating the involute As any two points on the cubic of stationary curvature can be taken as A and B, an infinite number of combinations of A and B are possible. However, not all of these combinations lend themselves to practical application. If it is decided to employ symmetrical mechanisms, i.e. with equal crank and follower lengths, it is convenient to introduce a parameter ~b, being the angle which the follower BoB makes with the pole normal. Once ~ is fixed, the proportions of all four links of the four bar linkage AoABBo are fixed automatically. ~ can very from 0 ° to 90 °. In terms of ~, and for an assumed base circle radius of unity, the following proportions can be derived from the geometry of Fig. 3. AoA = 4"5 cos
36
A B = 3"0 sin 2to BoB = 4-5 cos to AoBo = 1"5 sin 24,. AoBo is b e l o w the centre of the base circle b y a distance (1.5 c o s 2 t o - 1) when to < 33.8 °, is at the same level as the centre of the base circle w h e n to = 33.8 °, and is a b o v e the centre b y (1 - 1.5 cos2to) w h e n to > 33.8 °. T h e next stage of synthesis will be to fix the actual c o u p l e r point meant to g e n e r a t e the involute path. All such points lie on the line p,,. I n t r o d u c i n g a n o t h e r p a r a m e t e r L (equal to the distance P K ) , the following p r o p o r t i o n s f o r the coupler triangle B A K are easily derived f r o m the g e o m e t r y . L is c o u n t e d positive to the right of the pole normal. BAK
arctan
3 cos2to 1.5 sin2to L
A K = ~/-(1-5 sin 2to - L)~-+ (3 cos~to) :. A F o r t r a n p r o g r a m was written to c o m p u t e the coupler c u r v e c o o r d i n a t e s and the final angular displacement of the pinion. Based on these c o m p u t a t i o n s the following o b s e r v a t i o n s and c o m m e n t s are made. 3.2 Effect of changing to with constant L For an a s s u m e d value of L = 3, th was varied f r o m 10° to 50 ° and the resulting coupler c u r v e s have b e e n ploted in Fig. 4 along with the true involute. In the region rnn there is very close c o i n c i d e n c e of the coupler c u r v e s with the involute. Figure 5 indicates the output displacement of the pinion vs. the input c r a n k (A,,A) displacement. T h e range of dwell obtained increases with d e c r e a s i n g to. Permitting a tolerance of 1° m o v e m e n t of the pinion during the dwell period, a to value of l0 ° yields a dwell c o r r e s p o n d i n g to 85 ° of crank motion, t0 values of 30 ° and 50 ° similarly yield dwell True involute
Figure 4. Coupler curves traced by the point K in Fig. 3 shown with reference to the pinion.
37 500
L
40C
/
q~: a0*
f
30(:
,optI ~ "r-, ID
20(:
0
40
\ \ ~ - " ,~ ~ " C I 1 1 80
/
120
160
30 °
IO0 Dwel I
8
I
0
40
I
~
I-]
80
t-
120
~
v"7
160
J
J
200
I n p u t cronk (AoA)/ongte,
240
J
I
280
JJl 320
360
degrees
F i g u r e 5. Displacement curve showing the dwell, the inset being the enlarged
initial portion of the curve. periods of 76 ° and 68 ° respectively. For higher values of ~ the transmission angles were found to be poor. Thus a low value of + is preferable while, however, too low a value will result in a very small and impractical frame length AoBo. H e n c e 2 0 ° < + < 30 ° is suggested for purposes of the dwell mechanism. Sieker [2] has given an example with a value tk = 30 ° (approximately) for developing a four bar linkage to generate an approximate involute (His example, however, does not employ this four bar linkage for the synthesis of a dwell mechanism.) 3.3 Effect of changing L with constant Changing L would mean that the point K under consideration is shifted on the rack, either toward or away from the pole. This merely means that the corresponding involutes are the same but with a 'phase shift'. L can theoretically vary from - o c to + 0o and a value of L = 0 corresponds to K coinciding with pole. Increasing L from the already assumed value of 3, will serve only to increase the space requirements of the mechanisms; the range of dwell is unaffected since this depends only on the basic four bar linkage which, in turn, is determined solely by ~. It may be seen from the coupler curve (Fig. 4) that a portion of these curves is very near the base circle. The distance between the coupler point and the base circle in this region will b e c o m e too small and impractical if we decrease L below 3; ultimately, with L = 0, a cusp is formed at P. Hence L = 3 (times the base circle radius) may be taken as an optimum value.
4. P r a c t i c a l A r r a n g e m e n t Theoretically it is seen from the displacement curve (Fig. 5) that, as the crank AoA makes one revolution, the output pinion also makes one full rotation, but with a dwell. However, the practical assembly of the entire mechanism, as indicated in Fig. 3, does not permit complete rotation of the link AoA. Still, this mechanism is possible in practice, if we are contented with an oscillating input through AoA. There are many applications where an oscillatory output motion with dwell is required. For such
38 instances it is suggested that the link AoA be made to oscillate suitably by means of an auxiliary crank-rocker AoCoCE, as shown in Fig. 6. For example, when $ = 30° and L = 3, if AoA is made to oscillate between 24 ° and 200 ° (measured from a reference line coinciding with AoBo) an output pinion oscillation of 60 ° with end dwell is obtained. As another example, if AoA oscillates between - 26 ° and 185 °, dwell will be achieved in the middle of both the forward and backward motions of the output pinion. B
A
K
Pinion
Ao
E~
\\
\
//
Figure 6. A practical a r r a n g e m e n t with an oscillatory input. 5. Synthesis of a Crank Slider Linkage as an Alternate for Generating an Approximate Involute Referring to Fig. 3 again, the pole normal is also a part of the cubic of stationary curvature. H e n c e one moving joint can also be chosen on the pole normal. If the point B is chosen at infinity on this line we have a particular case resulting in a slider as shown in Fig. 7. In this case also the parameter ~ can be varied from 0 ° to 90 °. However, this
K
i Figure 7. Crank-slider linkage as an alternate.
39
..cc,L
.¢_
bile region
2C o ->, .~
0
_
20
~,~6o
=o -~c ~m ~
--6C --6C
-IOC
Figure 8.
Immobile regions in crank-slider linkage.
~, 350 ~.
/
~ 3°0 f 25O rSO
5o o
, N o t mobile
DwelI 4O
I 80
. 120
I
160
'
~
j
200
Input crank (AoA) angle,
Figure 9.
240
I
IL:31
280
320
360
degrees
Displacement curve for crank-slider linkage.
mechanism behaves like a double rocker mechanism since, for each value of 4', there is an immobile region. The immobile regions are shown in Fig. 8. Therefore, as in the previous case, a first stage crank lever mechanism would be necessary. The output displacement of the pinion was computed (see Fig. 9) and it was found that the deviation of the locus of K from an involute path is faster than in the previous corresponding symmetrical drag link mechanism (Fig. 3). Hence, for the assumed tolerance of 1° movement, the dwell periods will be less in the case of this slider linkage arrangement. The dwell periods are compared in Table 1. When this mechanism is used for obtaining the dwell, the slider can be made integral
Table 1.
Dwell periods measured as the angle of rotation of AoA for a tolerance of 1° (with constant L = 3) @=10 o @=30 ° @=50 ° Four bar pin jointed mechanism Mechanism with slider
85°
76°
68°
66°
64°
61°
40 with t h e r a c k , T h e p h y s i c a l a r r a n g e m e n t will b e s i m p l e r t h a n t h e p r e v i o u s f o u r b a r d r a g link m e c h a n i s m .
6. Cognate Linkages For the mechanisms discussed above, the Robert-Chebyshev theorem may be applied for the coupler point K and cognate linkages obtained for a wider choice of physical arrangements.
References [1] MEYER ZUR CAPELLEN W., Der einfache Zahnstangen-Kubeltrieb und das entsprechende Bandgetriebe. Weskstatt und Betrieb. 89 Jahrgang Heft 2 (1956). [2] SIEKER K. H., Einfache Getriebe, C. F. Winter'sche Verlagshandlung, Fussan (pp. 154) (1956). [3] HALL A. S., Kinematics and Linkage Design, p. 103. Prentice Hall (1964).
Synthese eines Evolventenrastgetriebes R. Samuel und G. Ramaiyan
Kurzfassung--ln Verarbeitungsmachinen ben6tigt man u.a.periodische Getriebe mit einem zeitweiligen Stillstand (Rast) des Abtriebes. Rastgetriebe k6nnen auf verschiedenen Arten abgeleitet werden. Ublich ist es, an einen Koppelpunkt einen Zweischlag anzuschliessen und die Anschmiegung des Kr0mmungskreises an die Koppelkurve aus zu n0tzen. Es I&sst sich zeigen, dass ohne Benutzung kreisf6rmiger Bahnkurven Stillst&nde des Abtriebes erreicht werden k/Snnen, und zwar mit Hilfe einer Evolvente als "F0hrungskurve". Statt des Zweischlages w~ihlt man eine Zahnstange und ein Zahnrad. Rollt eine Zahnstange a0f einem Zahnrad ab, beschreibt jeder Punkt K der Zahnstange eine Evolvente. F0hrt man ein Punkt K der Zahnstange auf einer Evolvente, steht das Zahnrad im Gestell still. Ausgehend von den Kr0mmungsverh~ltnissen der Evolvente kann man Viergelenkgetriebe for die Erzeugung der Evolvente entwickeln. Diese Viergelenkgetriebe wird mit der dazugeh6rigen Zahnstange und dem Zahnrad verbunden. Fur die Wahl der Lage des Gelenkes gibt es unendlich viele M6glichkeiten. Es werden zwei Arten des Getriebes ber0cksichtigt. Die erste Art besteht aus den symmetrischen Viergelenkgetrieben, die eine Evolvente erzeugen. Die Gliedl~ingenverh<nisse sind mittels tO bestimmbar (Fig. 3). Bei dem Getriebe zweiter Art ist ein Glied des Grundgetriebes unendlich gross; es ist eine exzentrische Schubkurbel. Die von den beiden arte Gewonnene Rasten werden untersucht und optimale Verh~.ltnisse der Gliedl~ingen vorgeschlagen.
PergamonPress. Printedin GreatBritain
ii
i
Synthesis of an Involute Based Dwell Mechanism R. Samuel* and G. Ramaiyan~Received on 12 October 1973
Abstract A dwell mechanism has been developed based on a coupler curve approximating an involute and utilising a rack and pinion arrangement. The proportions of the basic linkage have been synthesised and the dwell characteristics of the output link have been computed and presented. 1. I n t r o d u c t i o n ONE OF the important classes of m e c h a n i s m s e m p l o y e d in automatic machinery is the dwell mechanism. A dwell m e c h a n i s m essentially keeps a point or a body, sensibly at rest o v e r an interval of time during a cycle of operation. Dwell m e c h a n i s m s can be derived f r o m different approaches. One method, often illustrated in the literature on this subject, is to utilise a m e c h a n i s m with a coupler c u r v e having segments approximating circular arcs and to add a dyad at the coupler point. In this p a p e r a different method is suggested; a coupler c u r v e having a segment approximating an involute is utilised as the driving path and a rack and pinion combination has been e m p l o y e d to realise the dwell.
2. Principle If, in a rack (Tr) and pinion (w) a r r a n g e m e n t shown in Fig. 1, the pinion is held stationary and the rack is rolled o v e r the pinion f r o m position 1 to 2, a point K on the rack will trace an involute path K,K'~ (suffixes 1 and 2 will refer to initial and final states for a finite interval of motion). Conversely, if K is guided along an involute KIK~, the
K;
~Involute ~2 ~ ~ Arbitrarily ~ assumedcurve I
~
,
~
~
"KI
Figure 1. Basic Principle. *Professor of Mechanical Engineering, College of Engineering, Guindy, Madras-600025, India. tLecturer in Mechanical Engineering, College of Engineering, Guindy, Madras-600025, India.
33
M.M.T. Vol. 10 No. I ~
34
pinion remains stationary. If, however, K is guided along a path different from the involute, say K1K2, the pinion is constrained to rotate corresponding to the deviation K~K2 of the driving path K,K2 from the involute K~K;. Thus the pionion is at rest as long as the driving path coincides with the involute and starts moving if this path deviates from the involute. Rack and pinion arrangements have been suggested as driving combinations[l] in general periodic mechanisms, even though such a combination has not been suggested for generating a basic path for deriving a dwell. 2.1 Angular displacement of the pinion The displacement of the pinion when the point K is guided along any arbitrary path from K, to K~, may be calculated as follows: If P is the point of tangency on the pinion and if K , K ~ is the involute passing through K,, the displacement of the pinion will correspond to an arcuate length equal to K~K2 (see Fig. 2). Let K , P , = t, and K2P2 = t2. Then K ; P 2 = t , + P ] P 2 = t, + R ( ~
- 4,,). Involute Actual driving path
Figure 2. Calculation of angular displacement of the pinion. Therefore, the arcuate displacement of the pinion is {t, + R (~b2 - ~b,)} - t2
and the angular displacement of the pinion is 1
k- [{t, + R ( ~ - 4,,)}- t:] t2
=-~+ t+2- O,) R
(1)
3.0 Synthesis of a Four Bar Pin Jointed Linkage to Generate an Approximate Involute A mechanism will now be synthesised to generate a coupler curve serving as the driving path which will approximate an involute over a part of the cycle, during which a dwell of the pinion can be obtained, and which will deviate from the involute during the remaining part of the cycle, when the pinion is made to move. For arriving at this, the fixed polode p~ and the moving polode p,. (for the coupler
35
motion) of the mechanism must coincide with the pinion and rack considered already (see Fig. 3). T o enable the proportions of the linkage to be expressed as dimensionless quantities we shall assume the radius of the pinion (base circle of the involute) as unity. Pairs of points on the cubic of stationary curvature may be used as moving joints. For such pairs of points the application of the E u l e r - S a v a r y equation will yield the corresponding pairs of fixed joints, and hence the four bar mechanisms whose coupler points coincident with pm describe paths which approximate involutes over an interval. Referring to Fig. 3, C, represents the cubic of stationary curvature containing the moving joints, such as A and B. Its diameter is three times the radius of the base circle [2, 3]. C: represents the corresponding circle containing the fixed joints, such as Ao and B0 (lying on AP and BP extended). (23 represents the inflection circle. The point K on the coupler A B of the four bar linkage AoABBo will generate the desired driving path for the rack and pinion arrangement.
C~
B
A
,
\
\Base circle P~ (Coincides with i'he pinion)
Figure 3. Synthesis of the four bar linkage to be coupled with rack and pinion arrangement. 3.1 Proportions and parameters of the basic four bar linkage for generating the involute As any two points on the cubic of stationary curvature can be taken as A and B, an infinite number of combinations of A and B are possible. However, not all of these combinations lend themselves to practical application. If it is decided to employ symmetrical mechanisms, i.e. with equal crank and follower lengths, it is convenient to introduce a parameter ~b, being the angle which the follower BoB makes with the pole normal. Once ~ is fixed, the proportions of all four links of the four bar linkage AoABBo are fixed automatically. ~ can very from 0 ° to 90 °. In terms of ~, and for an assumed base circle radius of unity, the following proportions can be derived from the geometry of Fig. 3. AoA = 4"5 cos
36
A B = 3"0 sin 2to BoB = 4-5 cos to AoBo = 1"5 sin 24,. AoBo is b e l o w the centre of the base circle b y a distance (1.5 c o s 2 t o - 1) when to < 33.8 °, is at the same level as the centre of the base circle w h e n to = 33.8 °, and is a b o v e the centre b y (1 - 1.5 cos2to) w h e n to > 33.8 °. T h e next stage of synthesis will be to fix the actual c o u p l e r point meant to g e n e r a t e the involute path. All such points lie on the line p,,. I n t r o d u c i n g a n o t h e r p a r a m e t e r L (equal to the distance P K ) , the following p r o p o r t i o n s f o r the coupler triangle B A K are easily derived f r o m the g e o m e t r y . L is c o u n t e d positive to the right of the pole normal. BAK
arctan
3 cos2to 1.5 sin2to L
A K = ~/-(1-5 sin 2to - L)~-+ (3 cos~to) :. A F o r t r a n p r o g r a m was written to c o m p u t e the coupler c u r v e c o o r d i n a t e s and the final angular displacement of the pinion. Based on these c o m p u t a t i o n s the following o b s e r v a t i o n s and c o m m e n t s are made. 3.2 Effect of changing to with constant L For an a s s u m e d value of L = 3, th was varied f r o m 10° to 50 ° and the resulting coupler c u r v e s have b e e n ploted in Fig. 4 along with the true involute. In the region rnn there is very close c o i n c i d e n c e of the coupler c u r v e s with the involute. Figure 5 indicates the output displacement of the pinion vs. the input c r a n k (A,,A) displacement. T h e range of dwell obtained increases with d e c r e a s i n g to. Permitting a tolerance of 1° m o v e m e n t of the pinion during the dwell period, a to value of l0 ° yields a dwell c o r r e s p o n d i n g to 85 ° of crank motion, t0 values of 30 ° and 50 ° similarly yield dwell True involute
Figure 4. Coupler curves traced by the point K in Fig. 3 shown with reference to the pinion.
37 500
L
40C
/
q~: a0*
f
30(:
,optI ~ "r-, ID
20(:
0
40
\ \ ~ - " ,~ ~ " C I 1 1 80
/
120
160
30 °
IO0 Dwel I
8
I
0
40
I
~
I-]
80
t-
120
~
v"7
160
J
J
200
I n p u t cronk (AoA)/ongte,
240
J
I
280
JJl 320
360
degrees
F i g u r e 5. Displacement curve showing the dwell, the inset being the enlarged
initial portion of the curve. periods of 76 ° and 68 ° respectively. For higher values of ~ the transmission angles were found to be poor. Thus a low value of + is preferable while, however, too low a value will result in a very small and impractical frame length AoBo. H e n c e 2 0 ° < + < 30 ° is suggested for purposes of the dwell mechanism. Sieker [2] has given an example with a value tk = 30 ° (approximately) for developing a four bar linkage to generate an approximate involute (His example, however, does not employ this four bar linkage for the synthesis of a dwell mechanism.) 3.3 Effect of changing L with constant Changing L would mean that the point K under consideration is shifted on the rack, either toward or away from the pole. This merely means that the corresponding involutes are the same but with a 'phase shift'. L can theoretically vary from - o c to + 0o and a value of L = 0 corresponds to K coinciding with pole. Increasing L from the already assumed value of 3, will serve only to increase the space requirements of the mechanisms; the range of dwell is unaffected since this depends only on the basic four bar linkage which, in turn, is determined solely by ~. It may be seen from the coupler curve (Fig. 4) that a portion of these curves is very near the base circle. The distance between the coupler point and the base circle in this region will b e c o m e too small and impractical if we decrease L below 3; ultimately, with L = 0, a cusp is formed at P. Hence L = 3 (times the base circle radius) may be taken as an optimum value.
4. P r a c t i c a l A r r a n g e m e n t Theoretically it is seen from the displacement curve (Fig. 5) that, as the crank AoA makes one revolution, the output pinion also makes one full rotation, but with a dwell. However, the practical assembly of the entire mechanism, as indicated in Fig. 3, does not permit complete rotation of the link AoA. Still, this mechanism is possible in practice, if we are contented with an oscillating input through AoA. There are many applications where an oscillatory output motion with dwell is required. For such
38 instances it is suggested that the link AoA be made to oscillate suitably by means of an auxiliary crank-rocker AoCoCE, as shown in Fig. 6. For example, when $ = 30° and L = 3, if AoA is made to oscillate between 24 ° and 200 ° (measured from a reference line coinciding with AoBo) an output pinion oscillation of 60 ° with end dwell is obtained. As another example, if AoA oscillates between - 26 ° and 185 °, dwell will be achieved in the middle of both the forward and backward motions of the output pinion. B
A
K
Pinion
Ao
E~
\\
\
//
Figure 6. A practical a r r a n g e m e n t with an oscillatory input. 5. Synthesis of a Crank Slider Linkage as an Alternate for Generating an Approximate Involute Referring to Fig. 3 again, the pole normal is also a part of the cubic of stationary curvature. H e n c e one moving joint can also be chosen on the pole normal. If the point B is chosen at infinity on this line we have a particular case resulting in a slider as shown in Fig. 7. In this case also the parameter ~ can be varied from 0 ° to 90 °. However, this
K
i Figure 7. Crank-slider linkage as an alternate.
39
..cc,L
.¢_
bile region
2C o ->, .~
0
_
20
~,~6o
=o -~c ~m ~
--6C --6C
-IOC
Figure 8.
Immobile regions in crank-slider linkage.
~, 350 ~.
/
~ 3°0 f 25O rSO
5o o
, N o t mobile
DwelI 4O
I 80
. 120
I
160
'
~
j
200
Input crank (AoA) angle,
Figure 9.
240
I
IL:31
280
320
360
degrees
Displacement curve for crank-slider linkage.
mechanism behaves like a double rocker mechanism since, for each value of 4', there is an immobile region. The immobile regions are shown in Fig. 8. Therefore, as in the previous case, a first stage crank lever mechanism would be necessary. The output displacement of the pinion was computed (see Fig. 9) and it was found that the deviation of the locus of K from an involute path is faster than in the previous corresponding symmetrical drag link mechanism (Fig. 3). Hence, for the assumed tolerance of 1° movement, the dwell periods will be less in the case of this slider linkage arrangement. The dwell periods are compared in Table 1. When this mechanism is used for obtaining the dwell, the slider can be made integral
Table 1.
Dwell periods measured as the angle of rotation of AoA for a tolerance of 1° (with constant L = 3) @=10 o @=30 ° @=50 ° Four bar pin jointed mechanism Mechanism with slider
85°
76°
68°
66°
64°
61°
40 with t h e r a c k , T h e p h y s i c a l a r r a n g e m e n t will b e s i m p l e r t h a n t h e p r e v i o u s f o u r b a r d r a g link m e c h a n i s m .
6. Cognate Linkages For the mechanisms discussed above, the Robert-Chebyshev theorem may be applied for the coupler point K and cognate linkages obtained for a wider choice of physical arrangements.
References [1] MEYER ZUR CAPELLEN W., Der einfache Zahnstangen-Kubeltrieb und das entsprechende Bandgetriebe. Weskstatt und Betrieb. 89 Jahrgang Heft 2 (1956). [2] SIEKER K. H., Einfache Getriebe, C. F. Winter'sche Verlagshandlung, Fussan (pp. 154) (1956). [3] HALL A. S., Kinematics and Linkage Design, p. 103. Prentice Hall (1964).
Synthese eines Evolventenrastgetriebes R. Samuel und G. Ramaiyan
Kurzfassung--ln Verarbeitungsmachinen ben6tigt man u.a.periodische Getriebe mit einem zeitweiligen Stillstand (Rast) des Abtriebes. Rastgetriebe k6nnen auf verschiedenen Arten abgeleitet werden. Ublich ist es, an einen Koppelpunkt einen Zweischlag anzuschliessen und die Anschmiegung des Kr0mmungskreises an die Koppelkurve aus zu n0tzen. Es I&sst sich zeigen, dass ohne Benutzung kreisf6rmiger Bahnkurven Stillst&nde des Abtriebes erreicht werden k/Snnen, und zwar mit Hilfe einer Evolvente als "F0hrungskurve". Statt des Zweischlages w~ihlt man eine Zahnstange und ein Zahnrad. Rollt eine Zahnstange a0f einem Zahnrad ab, beschreibt jeder Punkt K der Zahnstange eine Evolvente. F0hrt man ein Punkt K der Zahnstange auf einer Evolvente, steht das Zahnrad im Gestell still. Ausgehend von den Kr0mmungsverh~ltnissen der Evolvente kann man Viergelenkgetriebe for die Erzeugung der Evolvente entwickeln. Diese Viergelenkgetriebe wird mit der dazugeh6rigen Zahnstange und dem Zahnrad verbunden. Fur die Wahl der Lage des Gelenkes gibt es unendlich viele M6glichkeiten. Es werden zwei Arten des Getriebes ber0cksichtigt. Die erste Art besteht aus den symmetrischen Viergelenkgetrieben, die eine Evolvente erzeugen. Die Gliedl~ingenverh<nisse sind mittels tO bestimmbar (Fig. 3). Bei dem Getriebe zweiter Art ist ein Glied des Grundgetriebes unendlich gross; es ist eine exzentrische Schubkurbel. Die von den beiden arte Gewonnene Rasten werden untersucht und optimale Verh~.ltnisse der Gliedl~ingen vorgeschlagen.