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Intersecting inflection circles in four bar mechanisms adjustable for velocities
Mech. Math. Theory Vol. 24. No. 6, pp. 527-540. 1989 Printed in Great Britain
0094-114X/89 $3.00 + 0.00 Pergamon Press plc
INTERSECTING INFLECTION CIRCLES IN FOUR BAR MECHANISMS ADJUSTABLE FOR VELOCITIES A. J. KOTHADIA and S. KRISHNAMOORTHY Victoria Jubilee Technical Institute, Bombay-400 019, India
Abstract--The conditions for intersection of inflection circles have been investigated by the authors and reported in earlier papers. Using these conditions, the synthesis pi'ocedures were developed for adjustable mechanisms for generating intersecting approximate straight line paths. In this paper, the velocity characteristics of intersection points of inflection circles have been investigated. Based on the properties thus obtained, this paper provides various synthesis procedures for four bar mechanisms adjustable for velocities of coupler points that are located at the intersection of inflection circles.
1. I N T R O D U C T I O N
Although applications of adjustable mechanisms have been numerous, less effort has been put in the area of adjustable mechanisms for prescribed velocities, accelerations etc. Tao and Bonnell[l] have developed graphical method for varying angular velocity ratio between input-output cranks of four bar mechanism. Sandor and McGroven[2] have used complex number method for synthesis of adjustable four bar mechanism for prescribed angular velocities and angular acceleration. The theory related to the intersection of inflection circles was developed and reported in the paper[3]. Based on this theory, paper [4] provides analytical synthesis procedure for four bar mechanism adjustable for generating intersecting straight line paths. In the present paper, velocity characteristics of intersection points of inflection circles have been investigated. Using these properties, various synthesis procedures have been developed for four bar mechanism adjustable for prescribed velocities. 2. VELOCITY CHARACTERISTICS OF INTERSECTION POINTS OF INFLECTION CIRCLES In Fig. I, the coupler point C is located on the inflection circle and so has an approximate straight line path. Vc is its instant velocity. Shifting either Ao along the pole ray PA or Bo along the pole ray PB, the position of pole P remains unchanged. In Fig. 2, shifting Ao along PA the inflection circles pass through the inflection point B,.. Selecting coupler point C at Bw, its velocity magnitude changes, direction of velocity remains unchanged,
A
~
Bw
Fig. 1. Four bar mechanism with coupler point on inflection circle. 527
528
A . J . KOTHADIA and S. KRISHNAMOORTHY
'°3 ,~ InfLection
cJrcLe TZ
nfLection circle I
A
Fig. 2. V, magnitude changes with shift of A. along PA.
Shifting Bo along P B , the inflection circles pass through the inflection point A,.. Selecting coupler point C at A,, its velocity remains unchanged as the orientation of P C and the angular velocity of coupler remain unchanged as shown in Fig. 3. Change in orientation of either input crank A A o or output crank B B o (keeping the orientations of other links unchanged) changes the position of pole P. The orientation of P C changes and so also the direction of V, as shown in Fig. 4. Thus with a particular type of adjustment, it is possible to change the velocity of coupler point with the change in either magnitude or direction or both magnitude and direction.
InfLection
circle IT
'_/
InfLection
circle
T
V¢~
Bw,/ Fig. 3. V,. remains unchanged with shift of B. along PB.
Intersecting inflectioncircles
529
InfLection circ~,e I Infkection circle I I
Lb
Bo
AoI
La
IVc,l
IVczl Fig. 4. Adjustable mechanismfor V,.:= 1.5 V,.~.in the same sense. 3. A D J U S T M E N T TO CHANGE VELOCITY M A G N I T U D E In Section 2, we have seen that to change the magnitude of V,, direction remaining unchanged, following conditions are to be satisfied, (A) Selection of coupler point C at the inflection point B, of the unadjusted crank BBo. (B) Varying the length of input crank AAo by shifting Ao along the pole ray PA. 3.1. Kinematic relation Vcl, Vc:--desired velocities before and after adjustment respectively. to3, a);--angular velocities of coupler before and after adjustment. a~2Mangular velocity of input crank AAo (eJ2 is assumed constant) r--velocity magnitude ratio = +_V,.,/V,, Positive sign is for same sense of velocities and negative sign is for opposite sense. All directed distances are considered to be positive along the pole rays through P. The angular velocities are considered positive in counterclockwise direction. Referring Fig. 3; I V¢:l = l + r " Vc, I
(1)
Substituting IV,:[ = [o~;. PCI and I V,.,I = la)3.PCI in equation (1) and simplifying results in
__a~;= _+r (.O3
(2)
530
A.J. KO'mADtAand S. KRISHNAMOORTHY
also
[ VA I =1-o95" AAo, I = lo93" PA[
(3)
I v~,l = I-og2"AAo,. I = 1co'3" PAI
(4)
and
Dividing (4) by (3) and equating to (2),
AAo2 = - +_ r • AAo,
(5)
The above kinematic relation provides the amount of shift of Ao along the pole ray PA for desired change in magnitude of I/",..
3.2. Synthesis procedure Based on the above analysis, a graphical synthesis procedure for specified value of + r, is as follows: (1) Draw the line la arbitrarily and locate P, A and Ao, on it arbitrarily. (2) Use kinematic relation to determine AAo: and locate A o2 on the line l~. (3) Draw the line lb through P arbitrarily and select suitable locations of B and Bo on it, then BBo be the output link. (4) Determine AA,.~, AA,., and BB,. using Euler-Savary equation
AA.~, = PA"/AAo~
AA. 2 = PAS/AAo.
BB., = PBZ/BBo
(5) Select coupler point C at B. and check for velocities. Illustrative example: Synthesize an adjustable mechanism to increase velocity magnitude by 50% when (A) sense of velocities remaining same (B) sense of velocities in opposite sense,
Solution: (A) In Fig. 4 the lines l~ and lb are drawn through P. Let PA = 60 ram,
PAo, = 90 ram,
PB = 50 mm,
PB o = 80 mm
As velocities have same sense r = +1.5 Using kinematic relation
AAo:= +1.5 x 30 = + 4 5 m m Ao: is marked on l., along the positive sense of PA. AA., =60:/30=12Omm;
AA..:=602/45 = 8 0 m m
and
BB..= 502/30=83.33mm
Coupler point C is selected at B.,. The synthesized mechanism is shown in Fig. 4. Solution: (B) In Fig. 5, PA = l l0 mm, PAo~ = 150mm, PB = 50mm, PBo --- 80 ram. As r = - 1.5, AAo2 = - 1.5 x 40 = - 6 0 mm, Ao: is located accordingly
AA.~ = 1102/40 --- 302.5 mm
A.:A = 1102/60 = 83.33 mm
BB.. -- 50:/30 = 83.33 mm
Coupler point C is selected at B.,. 4. A D J U S T M E N T TO CHANGE V E L O C I T Y D I R E C T I O N The change in velocity direction of V, is effected using following conditions. (A) Selection of coupler point C at the intersection of inflection circles. (B) Change of position of pole P with the adjustment. Position of pole P changes with the change in orientation of AAo or B8o. In this analysis orientation of BBo is changed.
Intersecting inflection circles
531
4.1. Kinematic relation
All directed distances are considered positive along the pole rays through P. The angular velocities are considered positive in counterclockwise direction. Let r be the velocity magnitude ratio. 1 Vc., ] = I r" Vc, I
(6)
substituting I Vc~I = l o3Pl C[ and i V. [= [c0~P2C[ in (I) and simplifying eJ'~/co 3 = r . Pz C/P2 C
(7)
V~ = I -e~2AAo, I = I o h P ~ A )
(8)
V'~ = 1 - o h A A o ,
(9)
Also
and ] = )co'3P2A l
I n f L e c t i o n circle I I
Vc I
l, a Infl.ection
circt, e I
w3
V~ 2
IVc~ I ~2
Fig. 5. AdJustable mechanism for V~. = 1.5 V,.,, in the opposite sense.
532
A . J . KOTHADIA and S. KRISHNAMOORTHY InfLecLion cfrcLe Z /%
Vcl
/ Inflection circle
,
lrI
~' wt
B
Fig. 6. Line L oriented through in CCL direction relative to line l I if A lies outsi/
PI P2.
The input crank is invariant, AA,,, = AAo,. Dividing (9) by (8) and equating with (7) results in
P] A/P:A = rPi C/P,.C = K
(10)
4.2. Location of poles PI and P, The kinematic relation (10) gives the desired velocity magnitudes. The orientation of V,. with respect to V~ depends on the locations of P2 and A relative to the location of P~. Let a be the angle made by V,., with respect to V,.~ measured in counterclockwise direction. The pole Pt is located on the line l~ drawn perpendicular to Vc, at C. The orientation of the line /2 on which P, is to be located, is decided as follows: (1) For A being lying outside P~P,, the line lz should be oriented through angle ~ in counterclockwise direction relative to the line l~ (Fig. 6) (2) For A being lying within P~P,, the line l~ should be oriented through the angle ( 1 8 0 - z~) in clockwise direction with respect to the location of line It (Fig. 7). Additional conditions are imposed on the location of A for the intersection of inflection circles.
8ol
Q •
... . ...°'.'."
\o,roL., ,
L1
;.'...'i
/
InfLecti circle I
Fig. 7. Line 1: oriented through ( 1 8 0 - ~ t ) in CL direction relative to line l) if A lies within PIP:.
Intersecting inflectioncircles
533
Table 1. Conditionsfor intersection of inflectioncircles Location of A Conditions for intersection K ~-r(PEC/P:C) A outside (I) A outside PIP2, on the side of P, ! > K > 2/3* Pi P: such that Pt/)2 < P, .4/2 (2) A.~ between P,A such that (1) .4 outside PIP2. on the side of Pi (2) A.., outside P~/'2 on the side P.,
between P) 1'2 A
(I) (2) (I) (2)
A between PIP..,PtA >PIP2~2 A..~ between P2A A between PIP~, PjA
K < I K>I
k< I
4.3. Intersection of inflection circles Since it is desired that the same coupler point should have velocities in different direction, then it should be located at the point of intersection between two inflection circles corresponding to the two phases of mechanism before and after adjustment. Intersection between two inflection circles depends on the location of A, Ao, Aw, and A,.: relative to the locations of PI and P:. The conditions to be satisfied for ensuring intersection were investigated and reported in paper [3]. After predetermined location of A, either to be outside PI P2 or within P~ P2 the line/2 is oriented (Section 4.2). Then for specified velocity magnitude ratio r, the distances P~ C and P2 C are selected such that the conditions of intersection between inflection circles given in Table l is ensured. Thus to change Vc, the conditions specified in Table l must be satisfied.
4.4. Synthesis: graphical approach For specified values of r and ct, based on above analysis, a graphical synthesis procedure is as follows: (1) Draw a line l~ perpendicular to V,~ at coupler point CP~ is to be located on this line.
Fig. 8, Adjustablemechanism to change direction of V~through 20°.
534
A.J. KOTHA~t^ and S. KRISH.~^MOORTH~
v~
A
P~ Fig. 9. Velocity analysis.
(2) Since A has to be located on the line passing through Pz and P2, so decide in advance the location of A either to be outside P, P2 or within P, P2. Then orient the line 1. on which P2 is to be located as explained in Section 4.2. (3) For the predetermined location of A, refer the Table 1 and (a) select the lengths P~ C and P , C to provide appropriate value of K. (b) Locate A on the line through Pt P2 using the kinematic equation. (c) Impose the condition on the location of A,,, and choose length AA,, accordingly on the line through P~ P2. (4) Determine unadjusted input crank length AAo using Euler-Savary equation
AAo = PIA2/AA,,t (5) Determine AA,. 2 and locate Aw:
AA,., = P2A2/AAo
(input crank AAo is invarient)
(6) Draw the inflection circle I, through the points P~, A,, and C and the inflection circle II through P2, A,., and C. (7) Select suitable location of moving pivot B. Join B and P, and if necessary extend it to intersect inflection circle I at B,.,. Similarly join B and P., and extend to intersect inflection circle II at B,,. (8) Measure P~ B, P2 B, BB,., and BB,,.. Determine the adjusted crank length using Euler-Savary equation.
BBo, = Pz B2/BB,.~ and
BBo: = P,. B2/BB..
Illustrative example I: Synthesize an adjustable mechanism to change velocity direction through 20 ~. The coupler point generates an approximate straight line path. Solution: Let the moving pivot A be located outside P, P:. The line l~ is drawn perpendicular to Vc, at C. As explained in Section 4.2, the line l, is drawn at 20 ~ in CCL direction with respect to the line It. Referring Table 1, consider the case where A,, is located outside PIP: and on the side of P,. The required value of K < I. Therefore let Pj C = 45 mm and P : C = 60 mm so that for r = 1, K -- 0.75 PIA/P2A = P1A/(PjP2 + PtA) = 0.75 Putting Pj P , - 25 mm (measured value) in above equation and solving PtA = 75 mm
and
P2A = 100 mm
Intersecting inflectioncircles
535
A,, is selected on the line through Pt P2 such that the length Pi A,., = 37.5 mm (This ensures intersection of inflection circles.) The unadjusted crank length
AAo = 755/(75 + 37.5) = 50 mm Also
AA,, = P:AS/AAo = 1005/50 = 200 mm The steps (6) and (7) of synthesis procedure explained in Section 4.4, are followed. The inflection points B,, and B,, are located on inflection circles I and II respectively. On measuring
BB,,=154mm,
P~B=81mm,
PsB=lOOmm,
BB,,=291mm
and
The adjusted crank lengths
BBo, = 812/154 = 42.6 mm
BBo2= 1002/291 = 34.36 mm
The orientation as well as length of adjusted crank varies. Figure 8 shows the synthesized adjustable mechanism; and it's velocity analysis in Fig. 9. Illustrative example 2: Synthesize the mechanism to meet the following specifications of velocity = 150 °, r = 0.5. The coupler point has approximate straight line path. Solution: Let A be located within Pj Ps- The line lj is drawn perpendicular to V,.. The line l., is oriented through ( 1 8 0 - 150)= 30 ° in clockwise direction. Referring Table 1, consider the case in which A lies between P~P2 such that P~A < PzA. So K < I. Let P~C= 150mm, P:C= 100mm so for r =0.5, K =0.75 P, P2 = 82 mm (measured) ?, ~ / P : A = P, A/(P, ~'2 + e, A) = 0.75 Solving
P~A = 35.1mm
and
PsA = 4 6 . 9 m m
Let P~ A,.j = 100 mm (for intersection of inflection circles). The unadjusted crank AoA
AoA = PtA2/AA,s = 35.15/64.9 = 18.98 mm AA~.2 = PsA:/AAo = 46.95/18.98 = 115.89 mm Moving pwot B is selected arbitrary. Inflection circles are drawn and B,., and B,.~ are located (following steps 6 & 7). Then
PjB = 5 6 . 5 m m ,
P,.B = 9 2 m m ,
BB,., = 157mm
and
BBw,.= 197.5mm
The output crank lengths are
BBos = 56.52/157 = 20 mm
before adjustment
BBo~= 922/197.5 = 42.8 mm after adjustment Figures 10 and 11 show the resulting mechanism and its velocity analysis.
4.5. Synthesis: analytical approach The synthesis procedure outlined above does not gaurantee the invarience of the length of link rotated for adjusting the mechanism. A trial and error procedure will be needed to achieve constant link length. However it is possible to keep the adjusted link length constant by using analytical technique. The appendix contains the analysis of the case in which A is located outside PI P: and on the side of PI and A,, outside P~P: and on the side of P:. Knowing ~, :q, ~2, 2 and fij, the Appendix equation (A14) is solved for the angle 0, the angle through which the output crank is to be rotated. M M T 24 (>-'-F
536
A. J. KOTHADIA and S. KRISHNAMOORTHY
V¢,
Bwz V¢z
Infkection
circle II
Aw~
Bw,
l2
Inflection circle I
Bo 1
-8
Fig. 10. Adjustable mechanism for [ It.. [ = 10.5 V,., I. Finally the adjusted crank length BoB is determined using equations either (A l), (A4) and (AI0) or (A2), (A5) and (AI3). The input crank length is determined as follows: by the kinematic relation
P,A/P~A = K
(ll)
Putting P2A = Pj P2 + P~ A in above equation and solving
P, A = P, e,_. K/(I - K)
(12)
Also for intersection A.tA > AP2, therefore A..,A is selected accordingly,
AoA = p~A2/A.,A
(13)
A,.P2 = (P2 A 2/A oA ) -- P2 A
(14)
and
Illustrative example: Design on adjustable mechanism to change direction of velocity through 45 ° and has magnitude 50% more than that of before. Solution: The line l~ is drawn perpendicular to V,.t. As A lies outside P, P,, the line 12 is drawn at 45: in CCL direction with respect to the line 1,. P~ C and P2C are selected such that K < 1. So let P,C=50mm
P2C=150mm.
Intersecting inflection circles
537
v¢~ /
150"
V¢=
Ire, I
Lz IV~l Boz
fl*l
B
Fig. 11. Velocity analysis.
Therefore
P I A / P : A = r • Pi C/P2 C = 1.5 (50/150) = 0. U s i n g e q u a t i o n (12), Pz A = 120 m m . Let P~ A,., = 182 mm. T h e u n a d j u s t e d c r a n k
AAo = 120/(120 + 182) = 47.68 m m O n m e a s u r i n g 2 = 17.13 °, ctt = 50 °. By e q u a t i o n (14)
A,.~P: = 240:/47.68 - 240 = 968 m m
and
ct2 = 60 ° ( m e a s u r e d )
Let 61 = 45 °. K n o w i n g cq, ~t2, a n d &l the v a r i o u s p a r a m e t e r s o f the t r a n s c e n d e n t a l e q u a t i o n s are o b t a i n e d . T h e r e f o r e k~ = 1.975, k 2 = 33.64 a n d k3 = 60 °. S u b s t i t u t i n g in ( A I 4 ) a n d solved for 0, the angle t h r o u g h which BBo is r o t a t e d . T h e r e f o r e 0 = 30 ° a n d so 6_, = 15 °. Pj B = 120
sin 15/sin 30 = 62.11 m m
120 sin(17.13) • sin(45 + 50)
B'tPJ --" sin 4 5 . sin(45 + 17.13 - 50) = 237 m m and
BoB -- 62.112/(237 + 62.11) = 12.89 m m The synthesized m e c h a n i s m is shown in Fig. 12 t h a t satisfies the desired velocity relationship.
538
A.J. KOTHADIAand S.
KRISHNAMOORTHY
8., I
vA
Inflect)on
circle I
V¢ I
Iv=
Bw2
LZ
Bo=
Fig. 12
5. C O N C L U S I O N S Velocity characteristics o f the intersection points o f inflection circles have been investigated. Varying input crank length A A o (orientation remaining unchanged) velocity magnitude of coupler point changes if it is located at the inflection point B, o f unadjusted crank B B o. C h a n g i n g orientation o f one o f the cranks results in change in direction o f velocity o f coupler point, Thus the desired velocity characteristics o f coupler point can be obtained using a particular type o f adjustment. The work o f investigation o f velocity characteristics o f points other than inflection points is being done by the authors.
REFERENCES 1. D. C. Tao and R. D. Bonell, Four bar linkage adjusted for constant angular velocity ratio, A S M E Poper. 68, Mech-39 (1968). 2. J. McGrovern and G. Sandor, Tram A S M E 417-429 (1973). 3. S. Krishnamoorthy and A. J. Kothadia, Mech. Mach. Theory 22(2), 107-114 (1987). 4. A. J. Kothadia and S. Krishnamoorthy, Synthesis of adjustable mechanism for intersecting straight lines, Proceedings, VHth Worm Congress on Theory of Machines & Mechanisms (1987).
APPENDIX Here the analysis is done for the case in which A is located outside P~P, and on the side of P~ and A., outside P~P: on the side of P:. The output link BB, is rotated about B. In Fig. AI, the position of moving pivot B is fixed by drawing lines at angles 3t and 3, with the conditions c~)# t/: ~ 0 and 6t # 3,. The angles are taken positive measured in counterclockwise direction. By Euler-Savary equation Bo, B = pi B2/(B~, pi + pi B )
(AI)
Bo:B = P,.B"/(B.:P2 + P2B)
(A2)
Intersecting inflection circles
539
Bwa
InfLection circte 1I
Inft.ec~.ioncircle! -~
Bw,
V¢t
Aw 1
Awz
I. 2
Fig. AI
It is desired that the adjusted crank should be invarient. Equating (AI) and (A2)
P~ B2/'(B..~ Pt + Pt B) = P,.B:/(B..:P 2 + P2 B)
(A3)
Various parameters of above equations are determined referring Fig. A I Pi B = Pt P: sin 6,-/sin 0
From AP~ PzB
(A4)
P,B = P~ P,. sin h I/sin 0
From AP t P,-B
(A5)
From APIA.,Bw,
(A6)
From Ap, P..C
(A7)
From AP, P2C
(A8)
B..,P 1 = A,,,P l • sin ~/sin~ t P, C = P, P:- sin ;./sin ~
P,.C = PIP,.sin(~, + 2)/sin" A., P, = P, C sin :q/sin ~
From ~A.., P~ C
Putting Pt C from (A7) and E = ;. - ( ' , ~ - - , ) in above equation A., Pi
P4 P:" sin 2 • sin :q sin a • sin(~, + ;. -- "t )
(A9)
Substituting A,., Pt from (A9) and ¢ = 1 8 0 - (6 t + 'Yl) in (A6) and simplifying results in
B.,P I =
P t P 2 • sin ~. • sin(di t + "~) sin ,, • sin(~, + 2 - "~ )
(AI0)
A.~. P,- " sin
(All)
B.~P,- is obtained as follows. From AP,.A,:B.~ B,: P z =
Q/sin(a,-
--
',)
A,: P, = P., C • sin(': - ,~)sin ~ from AA ,.: P: C Substituting P:C from (A8) and ff = ). - ( : q -
") in the above equation
A,,:P,- =
P= P,-" sin(a, - ",). sin(', + 2) sin " • sin(" + / . - :e,-)
(AI2)
540
A . J . KOTHADIA and S. KRISHNAMOORTHY
Putting
,4,,:P, from (AI2) and f2 = 1 8 0 - (6: + : t : - :t) in (AI I) and simplifying B, zP:=
P~ P:. sin(;( - 2)sin(6, + :t: - =) sinzc -sin(= +,;. - : h )
(AI3)
Substituting (A4), (AS), (AI0) and (A 13) in (A3) results in a transcendental equation sin:(61 - 0)
sin" 6 t
Kz.sinO+sin(dt-O)
Kz.sinOsin(K~-O)+sin6~
(AI4)
where, Kt
sin 2 - sin(6t + ~t) sin :t • sin(:t + ). - :q) K, =
SICH
(AI5)
sin(', + ).) sin ~ . sin(:c + £ - :t_,)
(AI6)
K 3 = 6~ + :t, - -
(AI7)
SCHNEIDENDE WENDEKREISE EINES VIERGELENKGETRIEBES ERSTELLBAR F~R GESCHWENDIGKEITEN
Zusammenfassung--Fr,'iher wurden die Bedingungen f/Jr sich schneidende Wendekreise eines Viergelenkgetriebes abgeleitetl'. Diese Bedingungen waren ffir die Synthese eines Viergelenkgetriebes mit verstellbaren Gestellpunkten zur angennherten Geradfuehrung eines Koppelpunktes l~ings zweier gegebener sich schneidender Geraden. In diesem Artikel werden die Geschwindigkeitseharakterstiken der Schnittpunkten der Wendekreise untersucht. Daraus ergeben sich verschiedene Methoden der Synthee eines Viergelenkgetriebes verstellbar ffir Geschwindigkeiten yon Koppelpunkten in den Schnittpunkten der Wenderkreise.
tSich schneidende Wendekreise eines verstellbaren Viergelenketriebes.
V
0094-114X/89 $3.00 + 0.00 Pergamon Press plc
INTERSECTING INFLECTION CIRCLES IN FOUR BAR MECHANISMS ADJUSTABLE FOR VELOCITIES A. J. KOTHADIA and S. KRISHNAMOORTHY Victoria Jubilee Technical Institute, Bombay-400 019, India
Abstract--The conditions for intersection of inflection circles have been investigated by the authors and reported in earlier papers. Using these conditions, the synthesis pi'ocedures were developed for adjustable mechanisms for generating intersecting approximate straight line paths. In this paper, the velocity characteristics of intersection points of inflection circles have been investigated. Based on the properties thus obtained, this paper provides various synthesis procedures for four bar mechanisms adjustable for velocities of coupler points that are located at the intersection of inflection circles.
1. I N T R O D U C T I O N
Although applications of adjustable mechanisms have been numerous, less effort has been put in the area of adjustable mechanisms for prescribed velocities, accelerations etc. Tao and Bonnell[l] have developed graphical method for varying angular velocity ratio between input-output cranks of four bar mechanism. Sandor and McGroven[2] have used complex number method for synthesis of adjustable four bar mechanism for prescribed angular velocities and angular acceleration. The theory related to the intersection of inflection circles was developed and reported in the paper[3]. Based on this theory, paper [4] provides analytical synthesis procedure for four bar mechanism adjustable for generating intersecting straight line paths. In the present paper, velocity characteristics of intersection points of inflection circles have been investigated. Using these properties, various synthesis procedures have been developed for four bar mechanism adjustable for prescribed velocities. 2. VELOCITY CHARACTERISTICS OF INTERSECTION POINTS OF INFLECTION CIRCLES In Fig. I, the coupler point C is located on the inflection circle and so has an approximate straight line path. Vc is its instant velocity. Shifting either Ao along the pole ray PA or Bo along the pole ray PB, the position of pole P remains unchanged. In Fig. 2, shifting Ao along PA the inflection circles pass through the inflection point B,.. Selecting coupler point C at Bw, its velocity magnitude changes, direction of velocity remains unchanged,
A
~
Bw
Fig. 1. Four bar mechanism with coupler point on inflection circle. 527
528
A . J . KOTHADIA and S. KRISHNAMOORTHY
'°3 ,~ InfLection
cJrcLe TZ
nfLection circle I
A
Fig. 2. V, magnitude changes with shift of A. along PA.
Shifting Bo along P B , the inflection circles pass through the inflection point A,.. Selecting coupler point C at A,, its velocity remains unchanged as the orientation of P C and the angular velocity of coupler remain unchanged as shown in Fig. 3. Change in orientation of either input crank A A o or output crank B B o (keeping the orientations of other links unchanged) changes the position of pole P. The orientation of P C changes and so also the direction of V, as shown in Fig. 4. Thus with a particular type of adjustment, it is possible to change the velocity of coupler point with the change in either magnitude or direction or both magnitude and direction.
InfLection
circle IT
'_/
InfLection
circle
T
V¢~
Bw,/ Fig. 3. V,. remains unchanged with shift of B. along PB.
Intersecting inflectioncircles
529
InfLection circ~,e I Infkection circle I I
Lb
Bo
AoI
La
IVc,l
IVczl Fig. 4. Adjustable mechanismfor V,.:= 1.5 V,.~.in the same sense. 3. A D J U S T M E N T TO CHANGE VELOCITY M A G N I T U D E In Section 2, we have seen that to change the magnitude of V,, direction remaining unchanged, following conditions are to be satisfied, (A) Selection of coupler point C at the inflection point B, of the unadjusted crank BBo. (B) Varying the length of input crank AAo by shifting Ao along the pole ray PA. 3.1. Kinematic relation Vcl, Vc:--desired velocities before and after adjustment respectively. to3, a);--angular velocities of coupler before and after adjustment. a~2Mangular velocity of input crank AAo (eJ2 is assumed constant) r--velocity magnitude ratio = +_V,.,/V,, Positive sign is for same sense of velocities and negative sign is for opposite sense. All directed distances are considered to be positive along the pole rays through P. The angular velocities are considered positive in counterclockwise direction. Referring Fig. 3; I V¢:l = l + r " Vc, I
(1)
Substituting IV,:[ = [o~;. PCI and I V,.,I = la)3.PCI in equation (1) and simplifying results in
__a~;= _+r (.O3
(2)
530
A.J. KO'mADtAand S. KRISHNAMOORTHY
also
[ VA I =1-o95" AAo, I = lo93" PA[
(3)
I v~,l = I-og2"AAo,. I = 1co'3" PAI
(4)
and
Dividing (4) by (3) and equating to (2),
AAo2 = - +_ r • AAo,
(5)
The above kinematic relation provides the amount of shift of Ao along the pole ray PA for desired change in magnitude of I/",..
3.2. Synthesis procedure Based on the above analysis, a graphical synthesis procedure for specified value of + r, is as follows: (1) Draw the line la arbitrarily and locate P, A and Ao, on it arbitrarily. (2) Use kinematic relation to determine AAo: and locate A o2 on the line l~. (3) Draw the line lb through P arbitrarily and select suitable locations of B and Bo on it, then BBo be the output link. (4) Determine AA,.~, AA,., and BB,. using Euler-Savary equation
AA.~, = PA"/AAo~
AA. 2 = PAS/AAo.
BB., = PBZ/BBo
(5) Select coupler point C at B. and check for velocities. Illustrative example: Synthesize an adjustable mechanism to increase velocity magnitude by 50% when (A) sense of velocities remaining same (B) sense of velocities in opposite sense,
Solution: (A) In Fig. 4 the lines l~ and lb are drawn through P. Let PA = 60 ram,
PAo, = 90 ram,
PB = 50 mm,
PB o = 80 mm
As velocities have same sense r = +1.5 Using kinematic relation
AAo:= +1.5 x 30 = + 4 5 m m Ao: is marked on l., along the positive sense of PA. AA., =60:/30=12Omm;
AA..:=602/45 = 8 0 m m
and
BB..= 502/30=83.33mm
Coupler point C is selected at B.,. The synthesized mechanism is shown in Fig. 4. Solution: (B) In Fig. 5, PA = l l0 mm, PAo~ = 150mm, PB = 50mm, PBo --- 80 ram. As r = - 1.5, AAo2 = - 1.5 x 40 = - 6 0 mm, Ao: is located accordingly
AA.~ = 1102/40 --- 302.5 mm
A.:A = 1102/60 = 83.33 mm
BB.. -- 50:/30 = 83.33 mm
Coupler point C is selected at B.,. 4. A D J U S T M E N T TO CHANGE V E L O C I T Y D I R E C T I O N The change in velocity direction of V, is effected using following conditions. (A) Selection of coupler point C at the intersection of inflection circles. (B) Change of position of pole P with the adjustment. Position of pole P changes with the change in orientation of AAo or B8o. In this analysis orientation of BBo is changed.
Intersecting inflection circles
531
4.1. Kinematic relation
All directed distances are considered positive along the pole rays through P. The angular velocities are considered positive in counterclockwise direction. Let r be the velocity magnitude ratio. 1 Vc., ] = I r" Vc, I
(6)
substituting I Vc~I = l o3Pl C[ and i V. [= [c0~P2C[ in (I) and simplifying eJ'~/co 3 = r . Pz C/P2 C
(7)
V~ = I -e~2AAo, I = I o h P ~ A )
(8)
V'~ = 1 - o h A A o ,
(9)
Also
and ] = )co'3P2A l
I n f L e c t i o n circle I I
Vc I
l, a Infl.ection
circt, e I
w3
V~ 2
IVc~ I ~2
Fig. 5. AdJustable mechanism for V~. = 1.5 V,.,, in the opposite sense.
532
A . J . KOTHADIA and S. KRISHNAMOORTHY InfLecLion cfrcLe Z /%
Vcl
/ Inflection circle
,
lrI
~' wt
B
Fig. 6. Line L oriented through in CCL direction relative to line l I if A lies outsi/
PI P2.
The input crank is invariant, AA,,, = AAo,. Dividing (9) by (8) and equating with (7) results in
P] A/P:A = rPi C/P,.C = K
(10)
4.2. Location of poles PI and P, The kinematic relation (10) gives the desired velocity magnitudes. The orientation of V,. with respect to V~ depends on the locations of P2 and A relative to the location of P~. Let a be the angle made by V,., with respect to V,.~ measured in counterclockwise direction. The pole Pt is located on the line l~ drawn perpendicular to Vc, at C. The orientation of the line /2 on which P, is to be located, is decided as follows: (1) For A being lying outside P~P,, the line lz should be oriented through angle ~ in counterclockwise direction relative to the line l~ (Fig. 6) (2) For A being lying within P~P,, the line l~ should be oriented through the angle ( 1 8 0 - z~) in clockwise direction with respect to the location of line It (Fig. 7). Additional conditions are imposed on the location of A for the intersection of inflection circles.
8ol
Q •
... . ...°'.'."
\o,roL., ,
L1
;.'...'i
/
InfLecti circle I
Fig. 7. Line 1: oriented through ( 1 8 0 - ~ t ) in CL direction relative to line l) if A lies within PIP:.
Intersecting inflectioncircles
533
Table 1. Conditionsfor intersection of inflectioncircles Location of A Conditions for intersection K ~-r(PEC/P:C) A outside (I) A outside PIP2, on the side of P, ! > K > 2/3* Pi P: such that Pt/)2 < P, .4/2 (2) A.~ between P,A such that (1) .4 outside PIP2. on the side of Pi (2) A.., outside P~/'2 on the side P.,
between P) 1'2 A
(I) (2) (I) (2)
A between PIP..,PtA >PIP2~2 A..~ between P2A A between PIP~, PjA
K < I K>I
k< I
4.3. Intersection of inflection circles Since it is desired that the same coupler point should have velocities in different direction, then it should be located at the point of intersection between two inflection circles corresponding to the two phases of mechanism before and after adjustment. Intersection between two inflection circles depends on the location of A, Ao, Aw, and A,.: relative to the locations of PI and P:. The conditions to be satisfied for ensuring intersection were investigated and reported in paper [3]. After predetermined location of A, either to be outside PI P2 or within P~ P2 the line/2 is oriented (Section 4.2). Then for specified velocity magnitude ratio r, the distances P~ C and P2 C are selected such that the conditions of intersection between inflection circles given in Table l is ensured. Thus to change Vc, the conditions specified in Table l must be satisfied.
4.4. Synthesis: graphical approach For specified values of r and ct, based on above analysis, a graphical synthesis procedure is as follows: (1) Draw a line l~ perpendicular to V,~ at coupler point CP~ is to be located on this line.
Fig. 8, Adjustablemechanism to change direction of V~through 20°.
534
A.J. KOTHA~t^ and S. KRISH.~^MOORTH~
v~
A
P~ Fig. 9. Velocity analysis.
(2) Since A has to be located on the line passing through Pz and P2, so decide in advance the location of A either to be outside P, P2 or within P, P2. Then orient the line 1. on which P2 is to be located as explained in Section 4.2. (3) For the predetermined location of A, refer the Table 1 and (a) select the lengths P~ C and P , C to provide appropriate value of K. (b) Locate A on the line through Pt P2 using the kinematic equation. (c) Impose the condition on the location of A,,, and choose length AA,, accordingly on the line through P~ P2. (4) Determine unadjusted input crank length AAo using Euler-Savary equation
AAo = PIA2/AA,,t (5) Determine AA,. 2 and locate Aw:
AA,., = P2A2/AAo
(input crank AAo is invarient)
(6) Draw the inflection circle I, through the points P~, A,, and C and the inflection circle II through P2, A,., and C. (7) Select suitable location of moving pivot B. Join B and P, and if necessary extend it to intersect inflection circle I at B,.,. Similarly join B and P., and extend to intersect inflection circle II at B,,. (8) Measure P~ B, P2 B, BB,., and BB,,.. Determine the adjusted crank length using Euler-Savary equation.
BBo, = Pz B2/BB,.~ and
BBo: = P,. B2/BB..
Illustrative example I: Synthesize an adjustable mechanism to change velocity direction through 20 ~. The coupler point generates an approximate straight line path. Solution: Let the moving pivot A be located outside P, P:. The line l~ is drawn perpendicular to Vc, at C. As explained in Section 4.2, the line l, is drawn at 20 ~ in CCL direction with respect to the line It. Referring Table 1, consider the case where A,, is located outside PIP: and on the side of P,. The required value of K < I. Therefore let Pj C = 45 mm and P : C = 60 mm so that for r = 1, K -- 0.75 PIA/P2A = P1A/(PjP2 + PtA) = 0.75 Putting Pj P , - 25 mm (measured value) in above equation and solving PtA = 75 mm
and
P2A = 100 mm
Intersecting inflectioncircles
535
A,, is selected on the line through Pt P2 such that the length Pi A,., = 37.5 mm (This ensures intersection of inflection circles.) The unadjusted crank length
AAo = 755/(75 + 37.5) = 50 mm Also
AA,, = P:AS/AAo = 1005/50 = 200 mm The steps (6) and (7) of synthesis procedure explained in Section 4.4, are followed. The inflection points B,, and B,, are located on inflection circles I and II respectively. On measuring
BB,,=154mm,
P~B=81mm,
PsB=lOOmm,
BB,,=291mm
and
The adjusted crank lengths
BBo, = 812/154 = 42.6 mm
BBo2= 1002/291 = 34.36 mm
The orientation as well as length of adjusted crank varies. Figure 8 shows the synthesized adjustable mechanism; and it's velocity analysis in Fig. 9. Illustrative example 2: Synthesize the mechanism to meet the following specifications of velocity = 150 °, r = 0.5. The coupler point has approximate straight line path. Solution: Let A be located within Pj Ps- The line lj is drawn perpendicular to V,.. The line l., is oriented through ( 1 8 0 - 150)= 30 ° in clockwise direction. Referring Table 1, consider the case in which A lies between P~P2 such that P~A < PzA. So K < I. Let P~C= 150mm, P:C= 100mm so for r =0.5, K =0.75 P, P2 = 82 mm (measured) ?, ~ / P : A = P, A/(P, ~'2 + e, A) = 0.75 Solving
P~A = 35.1mm
and
PsA = 4 6 . 9 m m
Let P~ A,.j = 100 mm (for intersection of inflection circles). The unadjusted crank AoA
AoA = PtA2/AA,s = 35.15/64.9 = 18.98 mm AA~.2 = PsA:/AAo = 46.95/18.98 = 115.89 mm Moving pwot B is selected arbitrary. Inflection circles are drawn and B,., and B,.~ are located (following steps 6 & 7). Then
PjB = 5 6 . 5 m m ,
P,.B = 9 2 m m ,
BB,., = 157mm
and
BBw,.= 197.5mm
The output crank lengths are
BBos = 56.52/157 = 20 mm
before adjustment
BBo~= 922/197.5 = 42.8 mm after adjustment Figures 10 and 11 show the resulting mechanism and its velocity analysis.
4.5. Synthesis: analytical approach The synthesis procedure outlined above does not gaurantee the invarience of the length of link rotated for adjusting the mechanism. A trial and error procedure will be needed to achieve constant link length. However it is possible to keep the adjusted link length constant by using analytical technique. The appendix contains the analysis of the case in which A is located outside PI P: and on the side of PI and A,, outside P~P: and on the side of P:. Knowing ~, :q, ~2, 2 and fij, the Appendix equation (A14) is solved for the angle 0, the angle through which the output crank is to be rotated. M M T 24 (>-'-F
536
A. J. KOTHADIA and S. KRISHNAMOORTHY
V¢,
Bwz V¢z
Infkection
circle II
Aw~
Bw,
l2
Inflection circle I
Bo 1
-8
Fig. 10. Adjustable mechanism for [ It.. [ = 10.5 V,., I. Finally the adjusted crank length BoB is determined using equations either (A l), (A4) and (AI0) or (A2), (A5) and (AI3). The input crank length is determined as follows: by the kinematic relation
P,A/P~A = K
(ll)
Putting P2A = Pj P2 + P~ A in above equation and solving
P, A = P, e,_. K/(I - K)
(12)
Also for intersection A.tA > AP2, therefore A..,A is selected accordingly,
AoA = p~A2/A.,A
(13)
A,.P2 = (P2 A 2/A oA ) -- P2 A
(14)
and
Illustrative example: Design on adjustable mechanism to change direction of velocity through 45 ° and has magnitude 50% more than that of before. Solution: The line l~ is drawn perpendicular to V,.t. As A lies outside P, P,, the line 12 is drawn at 45: in CCL direction with respect to the line 1,. P~ C and P2C are selected such that K < 1. So let P,C=50mm
P2C=150mm.
Intersecting inflection circles
537
v¢~ /
150"
V¢=
Ire, I
Lz IV~l Boz
fl*l
B
Fig. 11. Velocity analysis.
Therefore
P I A / P : A = r • Pi C/P2 C = 1.5 (50/150) = 0. U s i n g e q u a t i o n (12), Pz A = 120 m m . Let P~ A,., = 182 mm. T h e u n a d j u s t e d c r a n k
AAo = 120/(120 + 182) = 47.68 m m O n m e a s u r i n g 2 = 17.13 °, ctt = 50 °. By e q u a t i o n (14)
A,.~P: = 240:/47.68 - 240 = 968 m m
and
ct2 = 60 ° ( m e a s u r e d )
Let 61 = 45 °. K n o w i n g cq, ~t2, a n d &l the v a r i o u s p a r a m e t e r s o f the t r a n s c e n d e n t a l e q u a t i o n s are o b t a i n e d . T h e r e f o r e k~ = 1.975, k 2 = 33.64 a n d k3 = 60 °. S u b s t i t u t i n g in ( A I 4 ) a n d solved for 0, the angle t h r o u g h which BBo is r o t a t e d . T h e r e f o r e 0 = 30 ° a n d so 6_, = 15 °. Pj B = 120
sin 15/sin 30 = 62.11 m m
120 sin(17.13) • sin(45 + 50)
B'tPJ --" sin 4 5 . sin(45 + 17.13 - 50) = 237 m m and
BoB -- 62.112/(237 + 62.11) = 12.89 m m The synthesized m e c h a n i s m is shown in Fig. 12 t h a t satisfies the desired velocity relationship.
538
A.J. KOTHADIAand S.
KRISHNAMOORTHY
8., I
vA
Inflect)on
circle I
V¢ I
Iv=
Bw2
LZ
Bo=
Fig. 12
5. C O N C L U S I O N S Velocity characteristics o f the intersection points o f inflection circles have been investigated. Varying input crank length A A o (orientation remaining unchanged) velocity magnitude of coupler point changes if it is located at the inflection point B, o f unadjusted crank B B o. C h a n g i n g orientation o f one o f the cranks results in change in direction o f velocity o f coupler point, Thus the desired velocity characteristics o f coupler point can be obtained using a particular type o f adjustment. The work o f investigation o f velocity characteristics o f points other than inflection points is being done by the authors.
REFERENCES 1. D. C. Tao and R. D. Bonell, Four bar linkage adjusted for constant angular velocity ratio, A S M E Poper. 68, Mech-39 (1968). 2. J. McGrovern and G. Sandor, Tram A S M E 417-429 (1973). 3. S. Krishnamoorthy and A. J. Kothadia, Mech. Mach. Theory 22(2), 107-114 (1987). 4. A. J. Kothadia and S. Krishnamoorthy, Synthesis of adjustable mechanism for intersecting straight lines, Proceedings, VHth Worm Congress on Theory of Machines & Mechanisms (1987).
APPENDIX Here the analysis is done for the case in which A is located outside P~P, and on the side of P~ and A., outside P~P: on the side of P:. The output link BB, is rotated about B. In Fig. AI, the position of moving pivot B is fixed by drawing lines at angles 3t and 3, with the conditions c~)# t/: ~ 0 and 6t # 3,. The angles are taken positive measured in counterclockwise direction. By Euler-Savary equation Bo, B = pi B2/(B~, pi + pi B )
(AI)
Bo:B = P,.B"/(B.:P2 + P2B)
(A2)
Intersecting inflection circles
539
Bwa
InfLection circte 1I
Inft.ec~.ioncircle! -~
Bw,
V¢t
Aw 1
Awz
I. 2
Fig. AI
It is desired that the adjusted crank should be invarient. Equating (AI) and (A2)
P~ B2/'(B..~ Pt + Pt B) = P,.B:/(B..:P 2 + P2 B)
(A3)
Various parameters of above equations are determined referring Fig. A I Pi B = Pt P: sin 6,-/sin 0
From AP~ PzB
(A4)
P,B = P~ P,. sin h I/sin 0
From AP t P,-B
(A5)
From APIA.,Bw,
(A6)
From Ap, P..C
(A7)
From AP, P2C
(A8)
B..,P 1 = A,,,P l • sin ~/sin~ t P, C = P, P:- sin ;./sin ~
P,.C = PIP,.sin(~, + 2)/sin" A., P, = P, C sin :q/sin ~
From ~A.., P~ C
Putting Pt C from (A7) and E = ;. - ( ' , ~ - - , ) in above equation A., Pi
P4 P:" sin 2 • sin :q sin a • sin(~, + ;. -- "t )
(A9)
Substituting A,., Pt from (A9) and ¢ = 1 8 0 - (6 t + 'Yl) in (A6) and simplifying results in
B.,P I =
P t P 2 • sin ~. • sin(di t + "~) sin ,, • sin(~, + 2 - "~ )
(AI0)
A.~. P,- " sin
(All)
B.~P,- is obtained as follows. From AP,.A,:B.~ B,: P z =
Q/sin(a,-
--
',)
A,: P, = P., C • sin(': - ,~)sin ~ from AA ,.: P: C Substituting P:C from (A8) and ff = ). - ( : q -
") in the above equation
A,,:P,- =
P= P,-" sin(a, - ",). sin(', + 2) sin " • sin(" + / . - :e,-)
(AI2)
540
A . J . KOTHADIA and S. KRISHNAMOORTHY
Putting
,4,,:P, from (AI2) and f2 = 1 8 0 - (6: + : t : - :t) in (AI I) and simplifying B, zP:=
P~ P:. sin(;( - 2)sin(6, + :t: - =) sinzc -sin(= +,;. - : h )
(AI3)
Substituting (A4), (AS), (AI0) and (A 13) in (A3) results in a transcendental equation sin:(61 - 0)
sin" 6 t
Kz.sinO+sin(dt-O)
Kz.sinOsin(K~-O)+sin6~
(AI4)
where, Kt
sin 2 - sin(6t + ~t) sin :t • sin(:t + ). - :q) K, =
SICH
(AI5)
sin(', + ).) sin ~ . sin(:c + £ - :t_,)
(AI6)
K 3 = 6~ + :t, - -
(AI7)
SCHNEIDENDE WENDEKREISE EINES VIERGELENKGETRIEBES ERSTELLBAR F~R GESCHWENDIGKEITEN
Zusammenfassung--Fr,'iher wurden die Bedingungen f/Jr sich schneidende Wendekreise eines Viergelenkgetriebes abgeleitetl'. Diese Bedingungen waren ffir die Synthese eines Viergelenkgetriebes mit verstellbaren Gestellpunkten zur angennherten Geradfuehrung eines Koppelpunktes l~ings zweier gegebener sich schneidender Geraden. In diesem Artikel werden die Geschwindigkeitseharakterstiken der Schnittpunkten der Wendekreise untersucht. Daraus ergeben sich verschiedene Methoden der Synthee eines Viergelenkgetriebes verstellbar ffir Geschwindigkeiten yon Koppelpunkten in den Schnittpunkten der Wenderkreise.
tSich schneidende Wendekreise eines verstellbaren Viergelenketriebes.
V