Synthesis of adjustable four-bar mechanisms generating circular arcs with specified tangential velocities

Mechanism and Machine Theory 36 (2001) 387±395

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Synthesis of adjustable four-bar mechanisms generating circular arcs with speci®ed tangential velocities Chi-Feng Chang * Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 80759, Taiwan, ROC Received 17 February 1999; accepted 20 July 2000

Abstract This work proposes synthesis methods to design the mechanism that is adjustable for tracing variable circular arcs with prescribed velocities. The constraint equations and useful properties of the desired mechanism are derived by using the concept of cross-ratio. The desired mechanism can be a crank-rocker or a slider-crank mechanism. The coupler curve of the mechanism, if necessary, can be symmetrical to itself. The practical applications of the mechanism are also discussed. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Four-bar mechanisms can function as shearing machines, as shown in Fig. 1, to cut a continuously moving strip of material into sheets [1,2], or to punch a series of holes on the moving strip [3]. In the former application, a blade is attached to coupler link AB. During the cutting portion of the motion cycle, the tip of the blade traces an approximate circular arc; meanwhile, its horizontal velocity nearly matches the strip velocity Vs . The period of the motion cycle is 2p=x, where x is the angular velocity of driving link A0 A. The length, L, of sheets after cutting can then be expressed as L ˆ Vs …2p=x†

or …L=2p† ˆ …Vs =x†:

…1†

As the mechanism reaches the position in Fig. 1, it has a maximum transmission angle. Links A0 A and A0 B0 are collinear, and pole P coincides with ®xed pivot B0 . Let V denote the velocity of *

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0094-114X/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 0 ) 0 0 0 4 9 - 5

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C.-F. Chang / Mechanism and Machine Theory 36 (2001) 387±395

Fig. 1. The four-bar mechanism functioning as shearing machine.

coupler point C in this position. The value of V can be selected to be kVs . The k is a constant whose value is very near to 1. Fig. 1 also shows that VA =V ˆ B0 A=B0 C, where VA ˆ …A0 A†x is the velocity of crankpin A. Combining these relations and using Eq. (1) yield …A0 A† ˆ e…B0 A=B0 C†;

…2†

where e ˆ …V =x† ˆ k…L=2p†: Eq. (2) reveals that both V =x and L are proportional to link length A0 A if the value of …B0 A=B0 C† remains constant. Thus, an adjustable four-bar mechanism, as shown in Fig. 2, can be applied to cut the moving strip into various sizes of sheet by adjusting ®xed pivot A0 and link length A0 A along line B0 A. For this purpose, the desired adjustable four-bar mechanism should satisfy the following two requirements:

Fig. 2. The mechanism adjustable for variable circular arcs.

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1. It is adjustable to generate circular arcs k1 and k2 tangential at precision point C with prescribed radii of curvature C01 C and C02 C, respectively (Fig. 2). 2. At the precision point, it traces the two arcs with prescribed velocities. The two velocities de®ne the two limits of sheet size, and the two curvatures control the shapes of the coupler curves in the cutting portion within available range. The synthesis method for satisfying the ®rst requirement has been proposed in literatures [4,5]. However, this work presents a further study, and proposes straightforward methods to synthesize the mechanisms that satisfy both the requirements. 2. Synthesis of four-bar mechanism with two pairs of curvature and velocity prescribed Providing ®xed pivot A0 in Fig. 2 is adjusted from A01 to A02 , the radius of curvature is changed from C01 C to C02 C. Meanwhile, the velocity of coupler point C is changed from V1 to V2 , and the length of sheets after cutting is changed from L1 to L2 . It can be seen from the Aronhold's construction for path curvature [6] that the relative pole QAC in Fig. 2 is a point on line AC and satis®es \CB0 B ˆ \QAC B0 A. The position of QAC is thus kept still in the adjustment. Accordingly, with CC01 ; CC02 ; V1 =x and V2 =x prescribed, a desired mechanism must ful®ll the following six independent conditions (refer to Fig. 2 and Eq. (2)): Condition (1): Pivots A; A01 , and A02 are on a ray through B0 . Condition (2): Points C; C01 and C02 are on another ray through B0 . Condition (3): Lines A01 C01 ; A02 C02 and AC are concurrent in point QAC . Condition (4): A01 A ˆ e1 …B0 A=B0 C†, where e1 ˆ V1 =x ˆ kL1 =2p. Condition (5): A02 A ˆ e2 …B0 A=B0 C†, where e2 ˆ V2 =x ˆ kL2 =2p. Condition (6): \CB0 B ˆ \QAC B0 A. According to conditions (1)±(3), point sets …A; A01 ; A02 ; B0 † and …C; C01 ; C02 ; B0 † are the projection of each other through point QAC . These two point sets are therefore having the same cross-ratio [7,8], as shown below on the two sides of Eq. (3).   A02 A B0 A C02 C B0 C ˆ : …3† A02 A01 B0 A01 C02 C01 B0 C01 Note that the segments in Eq. (3) are sensed line segments. The segments on rays B0 A and B0 C are de®ned to be positive if their senses are the same as B0 A and B0 C, respectively. Besides, referring to Fig. 2 and conditions (4)±(5), the ratios of the segments in Eq. (3) can be expressed as: B0 C=B0 C01 ˆ B0 C=…B0 C ‡ CC01 †; A02 A=A02 A01 ˆ A02 A=…A02 A ÿ A01 A† ˆ e2 =…e2 ÿ e1 †; B0 A=B0 A01 ˆ B0 A=…B0 A ÿ A01 A† ˆ B0 C=…B0 C ÿ e1 †: Substituting these ratios of segments in Eq. (3) yields B0 C ˆ

n…m ÿ 1†e1 ‡ m…n ÿ 1†CC01 ; mÿn

…4†

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C.-F. Chang / Mechanism and Machine Theory 36 (2001) 387±395

where m ˆ CC02 =CC01 ; n ˆ e2 =e1 ˆ V2 =V1 ˆ L2 =L1 ˆ A02 A=A01 A: Since B0 C in Eq. (4) is in terms of the prescribed data: CC01 ; CC02 ; e1 and e2 ; the position of ®xed pivot B0 can be directly determined from Eq. (4). After locating B0 , the rest of the design work is to locate the other pivots such that conditions (4)±(6) are met. The general procedures for synthesizing the desired mechanisms are as follows (see Fig. 2): 1. Specify e1 and e2 , and locate points C; C01 ; C02 on a line such that CC01 and CC02 are the prescribed radii of curvature. 2. Locate ®xed pivot B0 on CC01 according to the B0 C obtained from Eq. (4). 3. Locate pivot A at a proper position. 4. Locate A01 and A02 on line B0 A to meet conditions (4) and (5), respectively. 5. Draw line AC to intersect line A01 C01 in QAC , and determine the direction of line B0 B according to condition (6). 6. Locate pivot B at a proper point on line B0 B to complete the design. In this synthesis method, there are three free choices: the value of B0 B and the coordinates of pivot A. Notably, one can specify …V1 ; V2 ; x† or …k; L1 ; L2 † instead of …e1 ; e2 † since ej ˆ Vj =x ˆ kLj =2p …j ˆ 1; 2†. 3. Explicit design equations for locating the pivots of desired mechanisms Let precision point C in Fig. 3 be the origin of the coordinate system. The pivot positions of a desired mechanism can be expressed in explicit equations with the aid of the following properties: 1. The line drawn through QAC and parallel to B0 A always cuts line C01 C02 at a ®xed point T. The location of T is given by:

Fig. 3. Useful properties of the desired mechanisms.

C.-F. Chang / Mechanism and Machine Theory 36 (2001) 387±395

T ˆ ÿiTC ˆ ÿi‰m…n ÿ 1†CC01 Š=…m ÿ n†:

391

…5†

Proof. Let S be an in®nite point on line B0 A. Point sets …S; A; A01 ; A02 † and …T ; C; C01 ; C02 † are the projection of each other through QAC , and have the same cross-ratio. Thus,   CC02 TC02 AA02 SA02 ˆ : CC01 TC01 AA01 SA01 Since the value of SA02 =SA01 approaches to 1, the above equation can be rewritten as  TC02 CC02 AA02 ˆ ˆ m=n: TC01 CC01 AA01

…6†

Thus, point T is a ®xed point on line C01 C02 because points C01 and C02 as well as the values of m and n are the prescribed data. Further, Eq. (6) can be rewritten as TC ˆ ‰m…n ÿ 1†CC01 Š=…m ÿ n† because of TC02 =TC01 ˆ …TC ‡ CC02 †=…TC ‡ CC01 †, and this complete the proof. 2. TQAC =B0 A ˆ TC=B0 C ˆ constant. This property is revealed from the ®rst one. Denoting b as the direction of B0 A, the position of QAC can be expressed as QAC ˆ T ‡ TQAC eib ˆ T ‡ ‰…B0 A†…TC†=…B0 C†Š eib :

…7†

With the aid of Eqs. (5) and (7), the positions of the pivots of a desired mechanism can be directly determined by using the following equations after selecting a proper position for pivot A and a suitable value for B0 B. B0 ˆ ÿiB0 C;

…8†

A01 ˆ B 0 ‡ …B0 A ÿ A01 A† eib ˆ B0 ‡ B0 A‰1 ÿ …e1 =B0 C†Š eib ;

…9†

A02 ˆ B 0 ‡ …B0 A ÿ A02 A† eib ˆ B0 ‡ B0 A‰1 ÿ …ne1 =B0 C†Š eib ;

…10†

B ˆ B0 ‡ B0 B ei…a‡p=2† ;

…11†

where B0 C is determined from Eq. (4), B0 A ˆ jA ÿ B 0 j; b ˆ arg…A ÿ B 0 †; a ˆ b ÿ arg…QAC ÿ B 0 †, and QAC is determined from Eqs. (5) and (7). Example 1. Synthesize a mechanism to satisfy the following data: C…0; 0†; C01 …0; 8:8†, C02 …0; 20†, e1 ˆ 75=2p, and e2 ˆ 150=2p (see Fig. 3). Since m ˆ CC02 =CC01 ˆ 20=8:8 and n ˆ e2 =e1 ˆ 150=75, using Eqs. (4) and (8) yields B0 ˆ ÿi184:7418. Select A ˆ ÿ175 ÿ i23. From Eqs. (9) and (10), obtained here are A01 ˆ ÿ163:6928 ÿi33:45053 and A02 ˆ ÿ152:3857 ÿ i43:90108. Then T ˆ ÿi73:33334 and QAC ˆ ÿ69:46633 ÿi9:129862 are determined from Eqs. (5) and (7), respectively. Finally, obtain B ˆ ÿ95:99896 ‡i14:97403 from Eq. (11) by specifying \B0 BA ˆ p=2, that is, jB0 Bj ˆ jB0 A cos…b ÿ a ÿ p=2†j. Fig. 3 shows the synthesized mechanism. The coupler curves shown as k1 and k2 are generated by linkages A01 ABB0 and A02 ABB0 , respectively.

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4. Slider-crank mechanism with two pairs of curvature and velocity prescribed The preceding synthesis method is general because the prescribed radii of curvature can be positive or negative and can range from zero to in®nity. In addition, the synthesized mechanism can be a slider-crank mechanism. The two cases are as follows: Case 1: m ˆ n 6ˆ 1. In this case, the denominator of Eq. (4) is zero. Thus, B0 C is in®nite and a slider-crank mechanism as shown in Fig. 4(a) is synthesized. Since B0 …P † is at in®nity, the preceding conditions (4)±(6) should be modi®ed here as: Condition (40 ): A01 A ˆ e1 . Condition (50 ): A02 A ˆ e2 . Condition (60 ): the distance measured from line PC to the parallel line PB equals to that measured from PQAC to PA (Fig. 4). Accordingly, the synthesis method for this case is modi®ed as follows (see Fig. 4(a)): 1. Specify e1 and e2 , and locate points C; C01 ; C02 on a line such that CC01 and CC02 are the prescribed radii of curvature. 2. Locate pivot A at a proper position. 3. Locate A01 and A02 on a line through A and parallel to C01 C02 such that A01 A and A02 A satisfy conditions (40 ) and (50 ), respectively.

Fig. 4. (a) The mechanism corresponding to m ˆ n. (b) The mechanism with in®nite CC01 and CC02 .

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4. Draw line AC to intersect line A01 C01 in QAC , and then draw line PB according to condition (60 ). 5. Locate pivot B at a proper point on line PB to complete the design. Case 2: CC01 and CC02 are in®nite. In this case, the value of m in Eq. (4) approaches to one. Hence, B0 C ˆ ÿCC01 and B0 …P † is at in®nity. Fig. 4(b) shows the mechanism of this case. Its synthesis method is similar to that described in case 1. The only di€erence is that pivot B should be located on line PC to meet condition (60 ). However, the coupler curve of this mechanism is not available for the shearing machine.

5. Adjustable four-bar mechanism with symmetrical coupler curves The couple curve of the four-bar mechanism synthesized by using the preceding method is generally not symmetrical. If it is desired to generate coupler curves symmetrical to line B0 C (Fig. 5), the following two additional conditions should be imposed on the synthesis procedures [4]. Condition (7): \B0 QAC C ˆ 90°. Condition (8): BC ˆ B0 B. Condition (7) means that QAC is on a circle f with B0 C as the diameter. It leads to \B0 AC ˆ p=2 ÿ a, where a ˆ \CB0 B ˆ \QAC B0 A. From condition (8), pivot B is on the midnormal of B0 C. Consequently, \B0 BC ˆ p ÿ 2a ˆ 2\B0 AC. This fact reveals that points B0 , A, and C are on a circle centered at B and AB ˆ BC ˆ B0 B. Therefore, the coupler curve of the mechanism must be symmetrical to B0 C if conditions (7) and (8) are met. Since the desired mechanism here should satisfy two additional conditions, there is only one free choice left for the synthesis. Selecting angle a as the free choice, the desired mechanisms can be synthesized by using the following procedures (see Fig. 5): 1. Specify e1 and e2 , and locate points C; C01 ; C02 on a line such that CC01 and CC02 are the prescribed radii of curvature. 2. Locate ®xed pivot B0 on CC01 according to Eqs. (4) and (8).

Fig. 5. The mechanism generating symmetrical coupler curves.

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3. Specify a value for a, the position of pivot B is determined from Eq. (12) below. B ˆ B 0 ‡ jB0 Bj ei…a‡p=2† ˆ B 0 ‡ jB0 C=2 cos aj ei…a‡p=2† :

…12†

4. Determine the position of T from Eq. (5). Let point G be the intersection of circle f and the extension of line TQAC . Notice that TQAC is parallel to B0 A, and points C; G; B0 , and QAC are on circle f. Then, \B0 CG ˆ B0 QAC G ˆ a, and the position vector of G can be written as G ˆ jCGj ei…a‡3p=2† ˆ jB0 C cos aj ei…a‡3p=2† :

…13†

5. From Fig. 5, it is observed that B0 QAC ˆ B0 A cos a and B0 QAC ˆ B0 C cos…b ÿ a ÿ p=2†. Then B0 A ˆ B0 C sin…b ÿ a†= cos a, and the position of pivot A can be expressed as A ˆ B 0 ‡ jB0 Aj eib ˆ B0 ‡ jB0 C sin…b ÿ a†= cos aj eib ;

…14†

where b ˆ arg…T ÿ G†: 6. Using Eqs. (9) and (10) to locate pivots A01 and A02 , respectively, the desired mechanism is synthesized. Example 2. Synthesize an adjustable mechanism generating coupler curves symmetrical to B0 C and satisfying the prescribed data in Example 1 (see Fig. 5). Selecting a ˆ 40° and using the synthesis method described above yields the following position vectors: B0 ˆ ÿi184:7418; B ˆ ÿ77:50839 ÿ i92:3709; A ˆ ÿ196:9669 ÿ i108:7903, A01 ˆ ÿ184:2404 ÿ i113:6977; A02 ˆ ÿ171:5139 ÿ i118:6051. Fig. 5 shows the synthesized mechanism. The symmetrical coupler curves generated by linkages A01 ABB0 and A02 ABB0 are shown as k1 and k2 , respectively. For a shearing machine, the coupler curve is not necessary to be symmetrical. However, the design equations in this section are useful for those design problems in which the radii of curvature are the main consideration [4,5]. For those design problems, e1 and e2 become two free

Fig. 6. The mechanism of Example 3.

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choices. Since ej ˆ Vj =x ˆ A0j A…B0 C=B0 A†; …j ˆ 1; 2†, then we can specify …a; V1 ; V2 , and x† or …a; A01 A; A02 A, and the ratio of B0 C=B0 A† to synthesize the mechanism generating symmetrical coupler curve with CC01 and CC02 prescribed. An example is given below to illustrate the application. Example 3. Synthesize a desired mechanism that generates symmetrical coupler curve and satis®es the following prescribed data: C…0; 0†; C…0; ÿ43†, C02 …0; ÿ52† (see Fig. 6). Select a ˆ ÿ16°; V1 ˆ 120 mm=s; V2 ˆ 180 mm=s, and x ˆ 2p rad=s. Then, m ˆ CC02 =CC01 ˆ …ÿ52†=…ÿ43† ˆ 1:2093, n ˆ e2 =e1 ˆ V2 =V1 ˆ 180=120 ˆ 1:5. Substituting these data into the design equations gives the following position vectors of the desired mechanism: B0 ˆ ÿi68:81351; B ˆ 9:865976 ÿ i34:40676; A ˆ ÿ25:85606 ÿ i32:14886; A01 ˆ ÿ18:67994 ÿ i42:32482; A02 ˆ ÿ15:09188 ÿ i47:4128. The synthesized mechanism is shown in Fig. 6. 6. Summary Two synthesis methods are proposed for synthesizing four-bar mechanisms with two pairs of curvature and velocity prescribed. The ®rst synthesis method involves three free choices; however, the coupler curve of the mechanism is generally non-symmetrical. The second one is proposed for generating symmetrical coupler curves. This method has only one free choice because two additional constraints must be imposed on the synthesis. However, if the radii of curvature are the main consideration, the two velocities of the coupler point and the angular velocity of the driving link can be speci®ed arbitrarily. The synthesis methods are simple to use since explicit equations are presented for locating the pivots of the desired mechanisms. Besides, the special cases that the desired mechanism may become a slider-crank mechanism are also discussed, and the corresponding synthesis method is proposed. Acknowledgements The ®nancial support of the National Science Council of the Republic of China (NSC 86-2212E-151-005) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8]

A.S. Hall Jr., Kinematics and Linkage Design, Prentice-Hall, Englewood Cli€s, NJ, 1961. J. Volmer, Getriebetechnik-Lehrbuch, VEB Verlag Technik, Berlin, 1979, p. 19. J. Volmer, Getriebetechnik-Koppelgetriebe, VEB Verlag Technik, Berlin, 1979, p. 279. D.C. Tao, H.S. Yan, Technology transfer in the design of adjustable linkages, ASME J. Mech. Design 101 (1979) 495±498. L. Huston, S. Kramer, Complex number synthesis of four-bar path generating mechanisms adjustable for multiple tangential circular arcs, ASME J. Mech. Design 104 (1982) 185±191. K.H. Hunt, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, 1978. H. Eves, A Survey of Geometry, vols. 1 & 2, Allyn & Bacon, Boston, MA, 1963 and 1965. P.C. Gasson, Geometry of Spatial Forms, Ellis Horwood, Chichester, West Sussex, 1983.