Intersecting inflection circles in adjustable mechanisms

Mech. Mach. Theory Vol. 22, No. 2, pp. 107-114, 1987 Printed in Great Britain. All fights reserved

0094-114X/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd

INTERSECTING INFLECTION CIRCLES IN ADJUSTABLE MECHANISMS S. K R I S H N A M O O R T H Y a n d A. J. K O T H A D I Y A Victoria Jubilee Technical Institute, Bombay 400 019, India

Abstract--The conditions for intersection between two inflection circles are discussed. It is then shown how to synthesize a four-bar mechanism with adjustable fixed pivot for obtaining two intersecting approximate straight lines.

INTRODUCTION The problem of synthesizing adjustable mechanisms for obtaining two intersecting approximate straight lines, has been dealt with by Tao and Amos [1], Tesar and Watts [2] and Krishnamoorthy [3]. Some restrictions have been placed on the choice of the inflection points, poles, etc., in the synthesis procedure. Intersecting straight line generation requires that the inflection circles intersect at the coupler point. In general, there is no guarantee that the two inflection circles corresponding to before and after adjustment of a four-bar mechanism, will intersect. The conditions to be satisfied by the mechanism for obtaining intersecting inflection circles is related to the locations of the poles and inflection points for the pole rays corresponding to the two cranks. In this paper for various types of adjustments, the conditions to be satisfied for intersecting inflection circles have been analysed. The conditions so obtained have been then applied to develop a synthesis procedure for adjustable mechanism.

If certain conditions are satisfied then the intersection will be assured. With reference to Fig. 2, these conditions are: (1) One of the three points P~ or Aw I or Bw I should lie on or inside the inflection circle defined by P, Aw and Bw. (2) Out of the remaining two points, at least one should lie on or outside the inflection circle defined by P, Aw and Bw. If the adjustable mechanism satisfies these two conditions, it can be ensured that the two inflection circles will intersect and the points of intersection can be used as a coupler point for generating two approximate intersecting straight lines.

circle

TWO INTERSECTING STRAIGHT LINES Two inflection circles corresponding to the two mechanisms before and after adjustment must intersect. In general for a four-bar mechanism, the inflection circle can be determined if the two cranks are known. This is so because three points define the inflection circle, one of which is the pole (instant center) and the other two are the inflection points for the two pole rays coinciding with the cranks as shown in Fig. 1. It is seen that after any type of adjustment, these three points are likely to change positions relative to the earlier locations. If P, Aw, Bw are the three points corresponding to first inflection circle and P~, AWl, Bw~ the points corresponding to second inflection circle, whether the two inflection circles will intersect or not will depend upon the locations of P~, Awt, Bw~ relative to P, Aw, Bw. 107

Fig. 1. Inflection circle defined by P, Aw and Bw.

P

A

Aw Bw w

Fig. 2. Conditions for intersection between two inflection

circles.

108

S. KRISHNAMOORTHYand A. J. KOTHADIYA ADJUSTMENTS OF FOUR-BAR MECHANISM

Figure 3 shows all possible ways in which a fourbar mechanism can be adjusted. It is seen that for all the cases, one pole ray has a fixed orientation and hence the two poles and the two inflection points associated with the two mechanisms (given mechanism and adjusted mechanism) relative to this pole ray will lie on the same pole ray. For example in Fig. 15, the pole ray passing through AA 0 is fixed for the adjustment in B0. The two poles P and Pl and inflection points Aw and Aw I lie on the pole ray passing through A A 0. Hence, it is sufficient to consider the interrelationship between these four points with regard to the above conditions for intersection between two inflection circles. It is also seen that any adjustment associated with pivots A, A0, or B, B0 result in one pole ray being fixed. The interrelationship between the four points P, Aw, P~ and Aw~ will be similar to the inter-

relationship between the four points P, Bw, P~ and Bw I . Hence the results obtained by considering adjustments involving pivots B and B0 can be applied to the cases of adjustment involving pivots A and A0. Hence further attention is restricted only to the adjustments involving B and B0. CONDITION FOR INTERSECTION BETWEEN TWO INFLECTION CIRCLES As mentioned earlier, for all cases of adjustments involving B and B0, points P, PI, Aw and Aw~ lie on one pole ray passing through A A 0. Consider the inflection circle I, passing through P and Aw. Inflection point Bw corresponding to pole ray passing through B and B0 (unadjusted state) lies on this circle. After adjustment, the position of B or B0 shifts (for the types of adjustment under consideration). Consequently, the pole ray through B and B0 changes orientation which may result in shifting of pole along AA 0 to a new location PI. Depending on the orien-

(a) Fixed pivot adjusted along fixed link

A

Ao

B Bo

Bo A° (o

~

)

(b) Fixed pivot adjusted perpendicular to fixed link

A

B

A

B

(c) Fixed pivot rotated with respective moving pivot

Ao (d) Fixed pivot rotated with respective other fixed pivot

Bo B° A o ~ B o

Ao~

A Ao

(e) Fixed pivot adjusted along corresponding pole ray

/, (f) Moving pivot adjusted along corresponding pole ray

(g) Moving pivot adjusted along a coupler

B

A#o~Bo

AocL

~ Bo

Ao~Bo

AoA~~Bo

(~ 4 ^o A I~'~ =

(h) Moving pivot adjusted in circular groove keeping

coupler length unchanged ~'o Adjustment either in A 0 or A

Fig. 3. Types of adjustment.

B° A o ~ B o Adjustment either in Bo or B

Intersecting inflection circles in adjustable mechanisms

2 - - 1

P,

Pl

)P~ )p

A

)A A

P

P Pl Ao

4

3

) A o A~

5 ipl pl( ,8.

A A

109

7

6 I

I

i

I - -

I

) P~

A I) )/"

P

P1

A o A(

)A o A

P

Pt

PI

P

A l

)P ) Ao Ao(

Ao

) Ao Ao~

) A o A° I

)P P~(Z) 3 P

P

) P,T

Fig. 4. Relative positions of P and P~ on the pole ray passing through AA0.

tation of the two cranks of the initial mechanism and the nature of adjustment, P and P~ can be located in three regions:

p

P

P1

PI A

(i) outside AoA, on the same side as A; (ii) outside AoA, on the same side as A0; (iii) between AoA.

Aw~ A

Figure 4 shows the possible combination of relative positions of P and P~ along AoA. Since P and P~ can be interchanged (being only a matter of notation) only seven cases with respect to the disposition of P and P~ on the pole ray have been considered for analysis. All directed distances are considered to be positive in the direction of A from P.

Ao Aw

Aw

-'~w ~

Ao

Co)

(b)

PA > AA o

PA < AA o

Case I

Case I

P and P~ located outside of AA0 and on the same side of A. The inflection point Aw corresponding to P, is located on the pole ray passing through AA 0. By Euler-Savary equation, l/PAw = 1 / P A - I/PA 0 with PAo = PA + AA0, the equation is, PAw = PA + pA2/AA0 .

(1)

Similarly the location of inflection point Aw~ corresponding to P~ is given by Pl Awl = Pl A + PI A2/AA0 •

(2)

If Pl is located between P and A, the inflection point Aw I lies between P and Aw. This implies PAw > PAw I . This can be proved as follows: PAwl = PPI + PiAwl • Using equation (2) and substituting P~A = PA - PPI PAw1 = PPI + (PA - PP~) +

(PA - PPI )2 AA0

(3)

Using equation (1), equation (3) can be written as PAWl = PAw + PP~/AA0 - 2PA'PP~/AAo.

(4)

From equation (4) it is seen that PAw~ < PAw if 2PA-PP~/AAo > PP2/AA0 i.e. 2PA > PP~. From Fig.

Fig. 5

5(a) and (b), it is seen that this condition is satisfied and hence Aw I lies between P and Aw. Now, applying the condition for intersection of two inflection circles, it is seen that the inflection circles passing through P, Aw and P~, Awj may not intersect. There is no assurance of intersection. Case H

P and Pj located between A and Ao (Fig. 6). Consider P between A and Ao. Applying EulerSavary equation PAw = PA - pA2/A0 A.

(5)

Since PA < AoA, PAw is greater than zero, hence is directed in the sence of PA. For P~ between P and A0, Aw~ lies between P and Aw and is given by PI Awl = Pl A - Pl A2/A0A-

(6)

Since this satisfies the conditions for intersection, the two inflection circles intersect. This implies that PAw should be greater than PAWl for intersection. This can be shown as follows. PAwl = Pl Awl -- Pl P-

110

S. KRISHNAMOORTHYand A. J. KOTHADIYA intersection. PAwj = PP~ + PIAw~ &w

Substituting for PIAWl PAwt = PPI + Pi A - PIA:/Ao A = - PIP + P1A - Pt A2/Ao A.

(7)

Now using PIA = PA + Pj P, PAw~ can be written as, ( P P ~ - 2PA" PPI'~ PAw, = PA - PA2/AoA - ~. "A"~ / Ao

= PAw + 2PA" PPI/Ao A - P t p2/Ao A Case TT

= PAw -

Fig. 6

2PA'PIP

p~p2

A0A

AoA"

Hence PAw~ > PAw and the intersection of inflection Substitution, for P1Aw, using equation (6) and: circles is ensured. with P~A = PIP + PA, PAw~ becomes PAw~ = PA - PA:/Ao A _ (p2 P/Ao A + 2 P A ' P IP/Ao A). By equation (5) PAw1 = PAw - (PI P2/AoA + 2PA.P~ P/AoA).

Case I V

P located outside A0A and on the same side as A and Pl between A and ,% (Fig. 8). Using Euler-Savary equation, location of inflection point Aw is given by PAw = PA + PA2/AAo.

Since P~P:/AoA + 2PA-PIP/AoA > 0 it can be concluded that PAw > PAWl.

Refering to Fig. 8(a), If PA > AAo, Aw lies beyond Ao. IfPt lies between A and Ao, Aw~ also lies between A and P, and is given by AwIP1 = API - AP~/AAo.

Case IlI

P and P~ located outside AA o and on the same side as Ao (Fig. 7). Using Euler-Savary equation, PAw can be written as PAw = PA - PA2/AoA. As PA > AoA, PAw is negative and hence Aw is located as shown in Fig. 7. With P~ between P and Aw, Pt Awl is given by PiAwl = P ~ A - P I A 2 / A o A and is negative. The intersection between two inflection circles is assured if Aw~ lies outside P and Aw. This implies PAw~ should be greater than PAw, for the

Hence, the two inflection circles may or may not intersect. Referring to Fig. 8(b), if PA < AA o, Aw lies between A and A o. If Pt lies between A and Ao, corresponding Aw~ will lie between A and Pj. In this case also, the two inflection circles may or may not intersect. The intersection is assured if Awj corresponding to P,, lies between A and Aw [Fig. 8(c)]. For this to be true PAw > PAwj

A C A Ao

Aw

Aw~(

Aw1

Aw (

P~ I

Pt

Ao

Ao

P~ P

Ao

PI Aw

~w (a)

Avq,

PA ~ AAo

Case ITr

Fig. 7

(8)

(0)

(c)

PA
Intersecting inflection circles in adjustable mechanisms Therefore

111

Hence PAw, will be greater than PAw if PAWl = PPI - Awl Pl.

Substituting for Awl Pl = AP] - AP~/AA0 putting AP] = PP] - PA, PAw] becomes

PP~/AAo - 2PPI" PA/AA0 > 0. and

PAw I = PAw + PP~/AA 0 - 2PPI - P A / A A 0. Hence for PAw > PAw] PP~ - 2PPI' PA/AAo < 0. This will be true if 2PA > PPI.

This implies, PPI > 2PA. If this condition is satisfied, the two inflection circles will intersect. Case VI

P located between A and A o. P] located outside A A 0 and on the same side of Ao (Fig. 10). By EulerSavary equation, the inflection point Aw correspond to P is given by, PAw = PA - pA2/AoA.

Case V

P located outside A0A and on the same side as A. P~ located outside A0A and on the same side as Ao (Fig. 9). The inflection point Aw corresponding to P, is located by Euler-Savary equation PAw = PA 4- PA2/AA0 . Similarly, Aw I corresponding to P] is located by the equation. I / P i A w I = -- I/AP l 4- I/AoPl. Substituting AoPl = API - AA0, Pl Awl = API - AP~/AAo. F r o m Fig. 9(b), for PA > AAo, Aw is located outside A A 0 and on the same side of Ao. Since API > AA0, Aw I lies further to P] on the same side of A o. Hence the two inflection circles may or may not intersect. The two inflection circles will intersect provided Aw lies between Pl and AWl. This requires that PAw I > PAw. Consider PAwl

Since P A < A o A , PAw is positive. The inflection point Aw I for Pl is given by P IA2 Ao A '

PIAw]=PIA

Since PIA > AoA, PiAw~ is negative. Awl will be located beyond P] as shown in Fig. 10. This case does not satisfy the conditions for intersection between two inflection circles. Hence the two inflection circles may or may not intersect. Case VII

P and Pl coincide (Fig. l l). For adjustment involving change in position of B or 130 along BB0 the

~w P

PAwl = PPI 4- Pl Awl. Substituting for Pl Awl by equation (8)

Ao PI

PAwl = PPI - API 4- AP21/AA0. With API = PPI - PA,

Aw

PAwl = PPI - (PPI - PA) 4- (PPI -- pA)2/AA0 Case

PAwl = PAw 4- pp2/AA0 - 2PPI" PA/AA0. P

(~

A

()

P

()

P

(

F i g . 10

P

P1 P~ &

Aw () A

()

& q A

Ao ()

Ao (

Ao (

p~

Aw (

P1 (

F,I (

AW ,

AwI ( )

Awl( )

Awll

(a)

(b)

PA < A~ o

PA > AA o

()

C a s e 2Z

Fig. 9

(c) P A > AA o

Aw

Aw,1

Ao

/~w

~,w,~

A0

(o)

(b)

PA > AA o

P~ < AA o

C a s e xrrr

F i g . 11

112

S. KRISHNAMOORTHYand A. J. KOTHADIYA PwIPT at 0

0

~ V U

R

B

Pu ~PR S

T

Locus of Bo

Ps ~Pv /7 PW IPT

Ot CO

Fig. 12. Change in position of pole with adjustment of B0.

pole ray passing through BB0, does not change orientation and hence PI coincides with P. By EulerSavary equation for Awl, PAwl = PA + pA2/AA0 and for Awl, P~ Awl = PI A + PI A2/AA0 • Since P and PI coincide, P A = P I A and A0A is constant, the inflection points Aw and Aw~ coincide. Two out of three points defining inflection circle remain fixed. This is a special case wherein the points of intersection between two inflection circles are P and Aw [the type of adjustment shown in Fig. 3(e) and (f), falls under this case]. ILLUSTRATIVE EXAMPLE For a given four-bar mechanism, applying any one type of adjustment which in Fig. 3 leads to one of the seven cases which we have ~liscussed. Hence one can

easily make out whether the two inflection circles will intersect or not. Consider the adjustment [Fig. 3(c)] where the pivot B0 is rotated with respect to B. With this adjustment only AoB0 is varied. The pivot B0 can occupy any position in the circle with B as center and radius equal to BB0 as shown in Fig. 12. Depending on given position of B0 and B0, in various ranges (B0,--after adjustment) the corresponding poles P and PI will lie along AA 0. For the position of B0 and B0, either between the range R and S or U and V, the poles P and Pl will lie between A and Ao. This case is similar to Case II. Hence adjustment in these ranges, assures the intersection between the two inflection circles. For the position of B0 and B0, either between the range S and T or V and W lead to Case III where, two poles P and Pl lie outside AA o and on the same side of Ao. Hence adjustment in these ranges assures the intersection. If B0 is located either in the range T and U or W and R and B0, is adjusted either in the range R and S or U and V, the intersection is assured provided the adjustment satisfies the condition given in Case IV [Fig. 8(c)]. If B0 is located either in the range T and U or W and R and B0, adjusted either in the range S and T or V and W, leads to Case V. The intersection is assured if the condition given in Case V [Fig. 9(c)] is satisfied. Tao and Amos [1] have used a similar type of adjustment for the mechanisms shown in Fig. 14, to get two approximate intersecting straight lines. In these mechanisms, the pivot A0 is adjusted and hence the pole P and P~ will move along the pole ray through B and B0. For the mechanism of Fig. 14(a), P and P, lie

nfeconcrceI p/ o,,.c,,oocrce A

8o ;

Fig. 13. Intersecting inflection circles with adjustment.

\/B"

Intersecting inflection circles in adjustable mechanisms (a)

I I3

outside BB 0 and on the same side of B0. The case is similar to Case III, and the intersection between two inflection circles is assured as Bw] lies outside PBw. In the mechanism of Fig. 14(b) P and Pl lie between B and B0. This case is similar to Case II. The intersection between two inflection circles is assured as PBw~ > PBw. In both the mechanisms to simplify the synthesis procedure, P1 is taken at the limiting position to Bw.

BW1

APPLICATION TO SYNTHESIS A

The conditions developed for the intersection of two inflection circles, and be fruitfully applied to synthesize the adjustable mechanism for obtaining two approximately intersecting straight lines. The construction steps are as follows (Fig. 15):

8

tb)

(I) Draw intersecting straight lines with the specified angle ~t between them at C. (2) Draw an arbitrary line a and draw perpendiculars to the intersecting straight lines at C. Let these perpendiculars meet the line a at P and PI respectively. (3) Select a point A on the line a such that 2PA > PP~ (the condition to be satisfied stated in Case III). (4) Select an arbitrary point A 0 between P and A and locate Aw and Aw I using Euler-Savary equations. Fig. 14. (a) Adjustable four-bar mechanism to get two intersecting straight lines with the specified angle between them. (b) Adjustable four-bar mechanism to get two intersecting straight lines with specified angle between them and the crank angle.

1/PAw = 1/PA - 1PAo and i/PI Awl = 1/P1 A - 1/PI A0-

InflecTion circle l'T

Inflection circle I

Aw Bo

Ao 8w

8wi

Fig. 15. Synthesis for two approximate intersecting straight lines.

114

S. KRISHNAMOORTHY and A. J. KOTHADIYA

(5) Draw inflection circle I through P, Aw and C and inflection circle II, through P~, Awl and C. (6) Draw arbitrary lines I and Il through P and P] intersecting at B. (7) The line I intersects the inflection circle I at Bw, locate B0 on it by using equation. 1/PBw = 1 / P B - I/PB 0. Similarly the line Ii intersects the inflection circle II at BW~. Locate B0~ on it by using equation l/Pl B % = l / P t B - I / P l Bo,. The AoABBo is the four-bar mechanism and AoABBo~ is the adjusted mechanism. The final mechanism is such that the crank BBo changes in magnitude and orientation. CONCLUSION For various types of adjustment that can be done on a four-bar linkage mechanism, the conditions to be satisfied for two inflection circles to intersect have been obtained. The conditions can be applied to the

problem of synthesizing adjustable mechanism for generating two intersecting approximate straight lines. This has been illlustrated by means of an example. The resulting mechanism in this case requires one of the cranks to change in orientation as well as length. Since the procedure of synthesis leads to innumerable solutions, it appears that an additional constraint for keeping the length of adjusted crank constant can be imposed. It should also be feasible to obtain mechanisms with other types of adjustments enumerated in this paper. Further work in this area is being done by the authors of this paper.

REFERENCES

1. D. C. Tao and L. G. Amos, A four-bar linkage adjustable for variable straight line motions. Trans. A S M E J. Engng Ind. 87, 287-289 (1965). 2. D. Tesar and E. H. Watts, The analytical design of an adjustable four-bar linkage for variable straight line motions. ASME Paper 66-Mech-30 (1966). 3. S. Krishnamoorthy, Kinematic synthesis of adjustable four-bar linkage mechanisms. Ph.D. thesis, University of Kentucky (1974).

SICH SCHNEIDENDE WENDEKREISE EINES VERSTELLBAREN VIERGELENKCETRIEBES Zumunmeufs~tsung--Es werden die Bedingungen ffir sich schneidende Wendekreise eines Viergelenkgetriebes abgeleitet. Das dient der Synthese eines Viergelenkgetriebes mit verstellbaren Gestellpunkten zur angenfiherten Geradfiihrung eines Koppelpunktes 1/ings zweier gegebener sich schneidender Geraden.