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Synthesis of a planar seven-link mechanism with variable topology for motion between two dead-center positions
Mechanism and Machine Theory 38 (2003) 1271–1287 www.elsevier.com/locate/mechmt
Synthesis of a planar seven-link mechanism with variable topology for motion between two dead-center positions Shrinivas S. Balli a b
a,*
, Satish Chand
b
Department of Mechanical Engineering, Basaveshwar Engineering College, Bagalkot 587 102, Karnataka, India Department of Mechanical Engineering, M.N. National Institute of Technology, Allahabad 211 004, UP, India Accepted 18 March 2003
Abstract An analytical method of synthesis of a planar seven-link mechanism with variable topology for motion between two dead-center positions is suggested. The method is useful to reduce the solution space. The proposed method is non-iterative. A complex number, dyadic approach of synthesis of mechanism for motion, path (with prescribed timing) and function generation is considered. An alternative choice for Phase-II is also suggested. 2003 Elsevier Ltd. All rights reserved. Keywords: Synthesis; Seven-link mechanism; Variable topology; Dead-center positions
1. Introduction A seven-link mechanism has two degrees of freedom. Various methods are proposed to synthesize it. A seven-bar linkage with variable topology operates in two phases. In each phase, a link adjacent to the permanently fixed link of seven-bar linkage is fixed temporarily and the resulting portion acts like a six-bar mechanism with single degree of freedom. In the beginning, an overview of the available literature on the variable topology mechanism has been given to form the basis of the method developed in the present work. Rose [1], Ting and Tsai [2] and Ting [3] with the help of graphical methods made indirect reference of five-bar variable topology mechanism. Rawat [4] established a synthesis technique for a five-bar variable
*
Corresponding author. Tel.: +91-8354-433025/426397; fax: +91-835-420504/425102. E-mail addresses: ssballi@rediffmail.com (S.S. Balli), satishchand@rediffmail.com (S. Chand).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00077-6
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topology mechanism operating in two phases. Joshi et al. [5] and Joshi [6] used the dyad synthesis of a five-bar variable topology mechanism for circuit breaker applications. Balli and Chand deal with various aspects like transmission angle control [7], motion between extreme positions [8,9], defects and solution rectification [10,15,17] of five-bar variable topology mechanism. Chand and Balli proposed a method of synthesis of a seven-link mechanism with variable topology [26]. In these methods, the arbitrarily prescribed or chosen values lead to an infinite number of solutions to a given problem resulting in a very large domain of solution space. The solutions thus obtained may be infeasible. One has to reiterate the procedure till s/he gets a practically feasible mechanism that functions satisfactorily requiring a large amount of time. The criterion of motion of mechanism between its dead-center positions may be used to reduce the solution space. Therefore, the objective of the problem undertaken is to develop a non-iterative analytical method of solution to determine the links of a seven-link mechanism with variable topology for motion between two dead-center positions of the mechanism. 2. Variable topology of seven-link mechanism Any mechanism with five or more links and with two or more degrees of freedom can be made to act as variable topology mechanism operating in two or more phases [4–6]. The planar sevenlink mechanism is shown in Fig. 1. Phase-I, Phase-II and Phase-II (alternative) shown in Figs. 2–4 respectively are discussed in the following paragraphs. 2.1. Phase-I In Phase-I, the link DOd is temporarily fixed and therefore, linkage becomes a six-bar mechanism with single degree of freedom. It is a combination of two four-bar linkages in series. Oa A1 is
Fig. 1. A planar seven-link mechanism.
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Fig. 2. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-I.
Fig. 3. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-II.
the input link. B is the possible path tracer point. Suffix 1 and 2 of alphabets in Fig. 2 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-I. It is to be noted that D is a temporarily fixed pivot. Oa and Oc are the permanently fixed pivots.
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Fig. 4. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-II (alternative).
2.2. Phase-II Once the above six-bar portion of seven-bar mechanism, reaches the position 2, the link DOd is released to move and the link Oa A is fixed temporarily, thus switching on to the Phase-II. In Phase-II also, it is a single degree of freedom, six-bar mechanism. Link DOd is the input link. B is the possible path tracer point. Suffix 2 and 3 of alphabets in Fig. 3 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-II. It is to be noted that D is no more a fixed pivot whereas A2 is a temporarily fixed pivot. Oc and Od are the permanently fixed pivots.
2.3. Phase-II (alternative) Instead of link Oa A of seven-link mechanism, if link COc is fixed temporarily, then also it results in six-bar linkage mechanism (Fig. 4) with single degree of freedom. Link DOd is the input link. B is the possible path tracer point. Suffix 2 and 3 of alphabets in Fig. 4 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-II (alternative). Again it is to be noted that D is no more a fixed pivot whereas C2 is a temporarily fixed pivot. Oa and Od are the permanently fixed pivots. Phase-I, Phase-II and Phase-II (alternative) might be interchanged i.e. instead of Phase-I being synthesized first, Phase-II synthesis might be made first and Phase-I later. Likewise, Phase-II (alternative) synthesis might be made first and Phase-I later and so on. Accordingly the values are prescribed or chosen and the unknowns are determined.
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3. Description of the problem In dead-center positions, a mechanism becomes instantaneous structure [27]. However, the mobility can be retained if its input link is changed properly. It is required to move the mechanism in both the phases between two dead-center positions. The six-bar portion of seven-link variable topology mechanism could be in dead-center position in more than one way [13]. 3.1. Transmission angle in planar six-bar mechanism Hain [18] has identified the different cases of six-bar linkage such as: (i) Six-bar linkage consisting of two four-bar linkages in series having two transmission angles. In this case, the positions of the mechanism in which the extreme values of transmission angle occur are dependent only on the dimensions of the basic four-bar linkage and on the position of the fixed pivot. (ii) Six-bar linkage with coupler drive (a) with one transmission angle and (b) with two transmission angles 3.2. Transmission angle in dead-center position of mechanism The dead-center position of a planar four-bar mechanism is defined as the position when transmission angle becomes zero [18,27]. The dead-center position of six-bar linkage and transmission angle are discussed in [13]. For symmetric 2-DOF, seven-link mechanism in a position of symmetry the transmission angle is equal to 0 or 180. Then the mechanism will be in a locking position. If such a linkage is required, it is necessary to provide an auxiliary drive [22]. A second driving link is used in dead-center position and then there is generally another transmission angle at another point of linkage so that it is possible to overcome the dead lock. This driving link may be a link, which gives up some of its kinetic energy to overcome dead-center position. Spring, flywheel, lever may be used for the purpose [18]. In the phases discussed above, the six-bar portion of seven-link mechanism is considered to be a combination of two four-bar linkages in series. The two transmission angles resulting are merged to be a single transmission angle [11]. The value of transmission angle in each position of each phase is equal to either 0 or to 180 (Figs. 2–4). Thus, all the positions represent the dead-center positions of mechanism. 3.3. Solution steps The problem to be solved consists of the following steps: iii(i) To identify the link to be fixed temporarily and input link in each phase of operation. ii(ii) To recognize the type of mechanism in each phase (Stephenson, Watt or any other). i(iii) To write down the standard dyad equations for the motion between position 1 and position 2 of Phase-I and also between position 2 and position 3 of Phase-II. i(iv) To identify the values to be specified, values to be chosen freely and the unknowns based on the task to be performed.
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ii(v) To solve the equations of motion in each phase separately for the link lengths. i(vi) To retain the link lengths of Phase-I while solving the equations in Phase-II. (vii) To find out the total number of solutions possible in all, by the method. 3.4. Assumptions made In the present discussion, when the coupler and driving link are inline (i.e., when the mechanism is at dead-center positions), the angle between the coupler and the follower link (transmission angle) at these extreme positions is also either maximum or minimum [16] (at which the specified extreme values of transmission angles also occur). In the foregoing derivation, the max. transmission angle is to be 180 and the min. transmission angle is 0. When the transmission angle is 0 or 180 i.e. too much deviated from the ideal one (90), the mechanism exhibit poor transmission characteristics. If the transmission angle l reaches 180, the mechanism will be self-locked and the further motion is seized. It is desirable for some of the applications like toggling mechanism, circuit breakers, etc. The limits of transmission angle l are to be fixed depending upon the application. 4. Types of six-bar linkage and dead-center positions From Figs. 2–4 one can understand that the designer may get different types of six-bar linkage mechanism in different phases depending upon the link fixed temporarily. For Phase-II, an alternative choice (Fig. 4) of fixing the middle link temporarily is also possible if the original sevenlink mechanism has three fixed pivots [26]. Yan and Wu [13] discussed the different configurations of six-bar mechanisms in their dead-center positions. According to them [13] Watt-I six-bar linkage has seven dead-center positions, Watt-II six-bar linkage has seven dead-center positions, Stephenson-I six-bar linkage has seven dead-center positions, Stephenson-II six-bar linkage has five dead-center positions and Stephenson-III six-bar linkage has seven dead-center positions. If the six-bar portion of seven-link variable topology mechanism is other than these Watt or Stephenson types, the different dead-center positions are to be identified. Symmetric seven-bar linkage [22], Type-I and Type-II plane seven-link mechanisms, special configurations like co-axial and the cyclically symmetric mechanisms [23–25] may also be considered for the variable topology synthesis. These may yield different types of six-bar linkages in two phases after fixing temporarily a link adjacent to permanently fixed link. 5. Synthesis It is required to synthesize a planar seven-link mechanism (shown in Fig. 1) with variable topology. One can have three options as follows: ii(i) one end link is fixed temporarily, i(ii) other end link is fixed temporarily, (iii) middle link is fixed temporarily.
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Table 1 Convention followed to denote the angles in Phase-I and Phase-II Link Phase-I Vector representation and From 1st position to 2nd angle between two different position (Suffix-1, Fig. 2) position of link
Phase-II From 2nd position to 3rd position (Suffix-2, Fig. 3)
Phase-II Alternative From 2nd position to 3rd position (Suffix-2, Fig. 4)
Oa Oc –Z1 Oa A–Z2 –/ AB–Z3 –a BC–Z4 –u COc –Z5 –b BD–Z6 –m DOd –Z7 –k Oc D–Z8 Oc A2 –Z9 Oc Od –Z10 Oa C2 –Z11 Od C2 –Z12 Displacement vector
Fixed link Temporarily fixed a2 u2 b2 m2 – – Temporarily fixed Fixed link – – B2 B3 ¼ d2
Fixed link /2 a2 u2 Temporarily fixed m2 k2 – – Fixed link Temporarily fixed Temporarily fixed B2 B3 ¼ d2a
Fixed link /1 a1 u1 b1 m1 Temporarily fixed Temporarily fixed – Fixed link – – B1 B2 ¼ d1
Sign convention
CCW motion +ve
CW motion )ve
Any two of the above three can be used for two phases of a seven-link mechanism with variable topology and the remaining one can be used as an alternative. For the effective transmission of motion/force, these transmission angles are to be controlled so that they vary within some reasonable range [19]. Instead of landing into unnecessary problems while controlling the two transmission angles simultaneously, the method of Hain [18] or Wu and Yan [11] of reducing the two transmission angles into one and controlling this transmission angle within some feasible region is followed. For the dead-center positions of six-bar portion of mechanism, this combined transmission angle becomes 0 or 180. It is assumed that the mechanism moves from dead-center position 1 to the dead-center position 2 in Phase-I and from dead-center position 2 to the deadcenter position 3 in Phase-II. In the present case, as soon as the mechanism moves from one deadcenter position to the other, it stops and then switches on to the Phase-II. So there is no question of overcoming the dead lock and hence, no auxiliary drive is needed. The input motion in Phase-I is /1 . 1 The displacement vector B1 B2 is given by d1 . Writing the dyad equations [14] for Phase-I (refer Fig. 2), Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ d1 Z5 ðe
ib1
iu1
1Þ þ Z4 ðe
1Þ ¼ d1
Z6 ðeim1 1Þ ¼ d1
1
Conventions followed to denote link angles are given in Table 1.
ð1Þ ð2Þ ð3Þ
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5.1. Motion generation In motion generation mechanisms, the body to be guided usually is a part of a floating link. Hence, the location of tracer point on the coupler and the coupler orientation are the part of design specifications as the entire motion of the coupler link is to take place. It requires that an entire body be guided through a prescribed motion order. 5.1.1. Phase-I synthesis In the standard dyad Eqs. (1)–(3), in motion generation, the coupler point motions (a1 and u1 ) and displacement vector ðd1 Þ are prescribed. /1 , b1 , m1 , Z2 and Z5 are the free choices. Hence, there will be 17 numbers of solutions. Then the unknowns Z3 , Z4 , Z6 , Z1 and Z8 are determined as follows: Z3 ¼
d1 Z2 ðei/1 1Þ ðeia1 1Þ
ð4Þ
Z4 ¼
d1 Z5 ðeib1 1Þ ðeiu1 1Þ
ð5Þ
Z6 ¼
d1 ðeim1 1Þ
ð6Þ
Z1 ¼ Oa Oc ¼ Z2 þ Z3 Z4 Z5
ð7Þ
Z 8 ¼ Oc D ¼ Z 4 þ Z 5 Z 6
ð8Þ
5.1.2. Phase-II Synthesis The input motion in Phase-II is k2 . The displacement vector B2 B3 is given by d2 . Writing the dyad equations [14] for Phase-II (refer Fig. 3), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2 Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ d2 ia1
Z3 e ðe
ia2
1Þ ¼ d2
ð9Þ ð10Þ ð11Þ
All the link lengths except the link Z7 , Z9 and Z10 are known from Phase-I. Eq. (9) is solved to determine the link 7 and Z9 and Z10 are determined by loop closure equations. Here, d2 and m2 are prescribed and k2 is a free choice. Therefore, there will be 1 numbers of solutions. Instead of d2 , b2 and u2 or a2 may also be prescribed. Then the unknowns Z7 , Z9 and Z10 are determined as follows: when d2 and m2 are prescribed, Z7 ¼
d2 Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð12Þ
Or when b2 , u2 and m2 are prescribed, Z7 ¼
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
or when a2 and m2 are prescribed,
ð13Þ
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Z7 ¼
Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
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ð14Þ
from loop closure equations, Z9 ¼ Oc A2 ¼ Z2 ei/1 Z1
ð15Þ
Z10 ¼ Oc Od ¼ Z5 þ Z4 Z6 Z7
ð16Þ
5.1.3. Phase-II synthesis (alternative) Here, m2 , the displacement vector B2 B3 i.e. d2a or u2 are prescribed and the input motion k2 is chosen freely. Therefore, there will be 1 numbers of solutions. Then the unknowns Z7 , Z11 and Z12 are determined as follows. Writing the dyad equations [14] for Phase-II (refer Fig. 4), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2a i/1
i/2
1Þ þ Z3 e ðe
iu1
iu2
1Þ ¼ d2a
Z2 e ðe Z4 e ðe
ia1
ia2
1Þ ¼ d2a
ð17Þ ð18Þ ð19Þ
When d2a is prescribed, Z7 ¼
d2a Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð20Þ
Or when is /2 and a2 prescribed, Z7 ¼
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð21Þ
When u2 is prescribed, Z7 ¼
Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð22Þ
Z11 ¼ Oa C2 ¼ Z1 þ Z5 eib1
ð23Þ
Z12 ¼ Od C2 ¼ Z7 þ Z6 eim1 Z4 eiu1
ð24Þ
5.2. Path generation with prescribed timing In path generation a point on a floating link is to trace a path defined with respect to the fixed frame of reference. If the path points are to be correlated with either time or input link positions, the task is called path generation with prescribed timings. For this purpose, location of a tracer point on the coupler and the position of input link are the part of the design specifications. The position of the input link must correspond to each position of the tracer point on the coupler. Path of tracer point (motion of coupler point) and rotation of one crank are to be prescribed. The values to be prescribed and the values to be free chosen in case of motion generation mechanism are to be interchanged if the mechanism is required for path generation with prescribed timing.
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5.2.1. Phase-I synthesis The input crank motion ð/1 Þ, b1 , m1 and displacement vector ðd1 Þ are prescribed. a1 , u1 , Z2 and Z5 are the free choices. Hence, there will be 16 numbers of solutions. Then the unknowns Z3 , Z4 , Z6 , Z1 and Z8 are determined using Eqs. (4)–(8) respectively. 5.2.2. Phase-II synthesis d2 and k2 are prescribed and m2 is a free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z9 and Z10 could be determined using Eqs. (12) or (13) or (14)–(16) respectively. 5.2.3. Phase-II synthesis (alternative) Here, the displacement vector B2 B3 i.e. d2a or u2 and k2 are prescribed and m2 is chosen freely. Therefore, there will be 1 numbers of solutions. Z7 , Z11 and Z12 are determined using Eqs. (20) or (21) or (22)–(24) respectively. 5.3. Function generation A function generation mechanism is a linkage in which relative motion or forces between links (generally) connected to ground is taken into account. It is required to coordinate the rotation or sliding motion of input and output links for two specified design positions. 5.3.1. Phase-I synthesis In function generation problem, the input and output crank motions (/1 , b1 and m1 ), and Z6 are prescribed. a1 , u1 , Z2 and Z5 are the free choices. Therefore, there will be 16 numbers of solutions. Then the unknowns Z3 , Z4 , Z1 and Z8 are determined as follows: Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ Z6 ðeim1 1Þ ib1
Z5 ðe Let
1Þ þ Z4 ðe
iu1
im1
1Þ ¼ Z6 ðe
1Þ
Z6 ðeim1 1Þ ¼ d01
ð25Þ ð26Þ ð27Þ
where d01 is the displacement vector, B1 B2 . Reducing the Eqs. (25) and (26) to the forms of standard dyad equations [14] as follows: Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ d01
ð28Þ
Z5 ðeib1 1Þ þ Z4 ðeiu1 1Þ ¼ d01
ð29Þ
Z3 ¼
d01 Z2 ðei/1 1Þ ðeia1 1Þ
ð30Þ
Z4 ¼
d01 Z5 ðeib1 1Þ ðeiu1 1Þ
ð31Þ
Z1 and Z8 are determined using Eqs. (7) and (8) respectively. 5.3.2. Phase-II synthesis Writing the loop closure equation for Phase-II (refer Fig. 3),
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Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ Z3 eia1 ðeia2 1Þ
ð32Þ
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ Z3 eia1 ðeia2 1Þ
ð33Þ
Reducing the Eqs. (32) and (33) to the form of standard dyad equation [14], Let
Z3 eia1 ðeia2 1Þ ¼ d2
ð34Þ
Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2
ð35Þ
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ d2
ð36Þ
Here, b2 , k2 and u2 are prescribed and m2 is the free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z9 and Z10 are determined as follows. When d2 (or a2 ) and k2 are prescribed, Z7 ¼
d2 Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð37Þ
When b2 , u2 and k2 are prescribed, Z7 ¼
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð38Þ
Z9 and Z10 could be determined using Eqs. (15) and (16) respectively. 5.3.3. Phase-II synthesis (alternative) The input motion in Phase-II is k2 . The displacement vector B2 B3 is given by d2a . Writing the dyad equations [14] for Phase-II alternative (refer Fig. 4), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ Z4 eiu1 ðeiu2 1Þ
ð39Þ
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ ¼ Z4 eiu1 ðeiu2 1Þ
ð40Þ
All the link lengths except the Z7 , Z11 and Z12 are known from Phase-I. Therefore, it is enough to solve Eq. (39) to determine the link 7. Z11 and Z12 are determined by loop closure equations. Reducing the Eqs. (39) and (40) to the form of standard dyad equation [14], Let
Z4 eiu1 ðeiu2 1Þ ¼ d2a
ð41Þ
Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2a
ð42Þ
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ ¼ d2a
ð43Þ
Here, d2a (or u2 ), /2 , a2 and k2 are prescribed and m2 is the free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z11 and Z12 are determined as follows: When d2a (or u2 ) and k2 are prescribed, Z7 ¼
d2a Z6 eim1 ðeim2 1Þ ðeik2 1Þ
When /2 , a2 and k2 are prescribed,
ð44Þ
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Table 2 Summary of Phase-I, -II and -II (alternative) synthesis of seven-link variable topology mechanism for two finitely separated positions Description
Phase-I
(a) Motion generation (rigid body guidance) Link fixed temporarily DOd Prescribed parameters a1 , u1 and d1 Free choice made /1 , b1 , m1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 , Z6 and Z8 No. of solutions 17 Total no. of solutions (b) Path generation with prescribed timings Link fixed temporarily DOd Prescribed parameters /1 , b1 , m1 and d1 Free choice made a1 , u1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 , Z6 and Z8 No. of solutions 16 Total no. of solutions (c) Function generation Link fixed temporarily DOd Prescribed parameters /1 , b1 , m1 and Z6 Free choice made a1 , u1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 and Z8 No. of solutions 16 Total no. of solutions
Phase-II
Phase-II Alternative
Oa A d2 ðb2 ; u2 =a2 Þ and m2 k2 Z7 , Z9 and Z10 1 18
COc d2 and m2 k2 Z7 , Z11 and Z12 1
Oa A d2 and k2 m2 Z7 , Z9 and Z10 1 17
COc d2 and k2 m2 Z7 , Z11 and Z12 1
Oa A b2 , u2 and k2 m2 Z7 , Z9 and Z10 1 17
COc k2 and u2 m2 Z7 , Z11 and Z12 1
Symbols have the meaning as given in Table 1.
Z7 ¼
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð45Þ
Then the unknowns Z11 and Z12 are determined using Eqs. (23) and (24). Thus, for a given task and the prescribed values the dimensions of the seven-link mechanism with variable topology can be found out using Eqs. (1)–(45). A summary of Phase-I, Phase-II and Phase-II (alternative) synthesis is given in Table 2.
6. Solution space The method yields an infinite number of solutions depending on number of free choices made. In order to obtain practically useful mechanisms, it is necessary to prescribe limits on such quantities as transmission angle [16,20] and max. to min. link length ratio. Then by determining the angle between the two specified dead-center positions of links of mechanism in terms of known angular motions and solving the equations of displacement, one may get the optimum linkage lengths. Thus the solution space can also be reduced. The requirements like the tracing point/coupler point or some link to be moved in a particular fashion (horizontally/vertically or at some angle or direction) makes the synthesis more specific
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thereby reducing the solution space. The solution space is reduced by 12 solutions if one link length is prescribed and by 1 solutions if one angular motion of link is prescribed. On the other hand a synthesis problem should be stated in such a way that no extra restrictions are imposed. It is advantageous not to use up all the available parameters of mechanism by specifying motion requirements but to leave some freedom, which can be used to fulfill transmission angle and other requirements. For example, if for a certain required specifications, two positions can suffice, one should not specify three or more positions, which may reduce the number of available solutions [4].
7. Advantages of the method The dyad technique by Sandor and Erdman permits handling any type of bar linkage mechanism and is extensively used for the synthesis of single degree of freedom mechanisms. Here it has been applied to variable topology mechanisms, which in this case are essentially two degrees of freedom systems. The authors have extended the technique to identify optimal mechanisms in terms of limits on transmission angles. The method reduces the job of synthesizing seven-link mechanism of two degrees of freedom to design a six-bar mechanism with single degree of freedom but of course in two phases. The synthesis of a six-bar mechanism is simpler. Thus the method is simpler compared with other methods of synthesis of seven-link mechanism. The advantages of the method proposed may be enlisted as follows: • The solution is consistent with the definitions of standard kinematic tasks like motion, function and path generation for two, three, four and five prescribed positions resulting in a unified method of synthesis [21]. • The mechanism may be designed for one task in one phase and for a different task in the other phase. It may also be designed for combined tasks in one phase itself. The solution may be modeled with groups [12] of standard form equations for either simultaneous function and path generation or for combined function and motion generation with prescribed timings. • It has codable steps leading to computerization of the method. • Simplicity, ease of application and generality are the attractions of the method. • Unlike the graphical methods, it is not limited by drawing accuracy. The method can be applied to synthesis of mechanisms for three; four or five infinitesimally separated positions or multiply separated positions also. • Solution is applicable to any other link-mechanisms acting as variable topology mechanisms.
8. Limitations of the method • The proposed method is applicable only to complex number approach. • The mechanism synthesized by the method may suffer from branch, Grashof or circuit defects, which can be rectified separately. • The solution does not permit good initial guesses for all possible solutions (free choices).
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9. Conclusion The method is for the synthesis of seven-link mechanism with variable topology for the motion between the any two dead-center positions. In the present work, order is not significant as numbers of design positions are only two but there may be GrashofÕ defect, circuit and branch defects. The proposed method reduces the synthesis of seven-link two degree of freedom mechanism to the synthesis of a six-bar single degree of freedom linkage mechanism but of course in two phases. Different combinations of types of six-link mechanism for variable topology are proposed. An alternative method for Phase-II is also suggested. The method is simple and straightforward. Designer can look up a number of solutions and select particular solution that suits the space or transmission angle requirements.
Appendix A. Numerical example Problem: It is required to synthesize a seven-link mechanism with variable topology for the motion between the dead-center positions and for the following tracer point displacement specifications: Phase-I: from point (72.5868, 218.7403) to the point (102.3319, 206.2695) Phase-II: from point (102.3319, 206.2695) to the point (86.5953, 227.0085) Phase-II (alternative): from point (102.3319, 206.2695) to the point (87.33581374, 199.7030247) Suggest the dimensions of the mechanism for (a) motion generation, (b) path generation with prescribed timings and (c) function generation. Solution: (a) Motion generation Phase-I synthesis: Given that: displacement, d1 ¼ ð29:7451 12:4708iÞ, a1 ¼ 24 (cw) and u1 ¼ 44 (cw) Let /1 ¼ 50 (cw), b ¼ 84 (cw), m1 ¼ 60 (ccw), Z2 ¼ ð11:4890 þ 25:6107iÞ and Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(4), (5), (6), (7), (8),
Z3 Z4 Z6 Z1 Z8
¼ ð25:6301 þ 19:6044iÞ ¼ ð24:1301 þ 14:4668iÞ ¼ ð25:6726 19:5246iÞ ¼ ð30:7543 þ 10:9856iÞ ¼ ð9:0594 þ 53:7541iÞ
i.e., i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:2682 jZ4 j ¼ 28:1344 jZ6 j ¼ 32:2536 jZ1 j ¼ 32:6575 jZ8 j ¼ 54:5122
Phase-II synthesis: Given that: displacement, d2 ¼ ð15:7366 þ 20:7392iÞ, m2 ¼ 36 (cw), b2 ¼ 84 (ccw), a2 ¼ 48 (ccw) and u2 ¼ 10 (ccw)
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Let k2 ¼ 49 (cw) From Eq. (12), Z7 ¼ ð20:4301 3:6894iÞ i.e., jZ7 j ¼ 20:7605 From Eq. (15), Z9 ¼ ð18:5204 þ 14:2777iÞ i.e., jZ9 j ¼ 23:3850 From Eq. (16), Z10 ¼ ð29:4895 þ 57:4435iÞ i.e., jZ10 j ¼ 64:5708 Phase-II Alternative synthesis: Given that: displacement, d2 ¼ ð14:9961 6:5665iÞ, m2 ¼ 43 (cw), u2 ¼ 30 (cw) and /2 ¼ 31 (cw) Let k2 ¼ 43 (ccw) From Eq. (20), Z7 ¼ ð19:6794 3:8593iÞ i.e., jZ7 j ¼ 20:0542 From Eq. (23), Z11 ¼ ð51:1945 þ 5:5756iÞ i.e., jZ11 j ¼ 51:4972 From Eq. (24), Z12 ¼ ð9:0493 62:8535iÞ i.e., jZ12 j ¼ 63:5016 (b) Path generation with prescribed timings By interchanging the values to be prescribed and the values to be free-chosen of motion generation problem above, it results in a problem of path generation with prescribed timings. Phase-I synthesis: Given that: displacement, d1 ¼ ð29:7451 12:4708iÞ, /1 ¼ 50 (cw), b1 ¼ 84 (cw) and m1 ¼ 60 (ccw) Let a1 ¼ 24 (cw), u1 ¼ 44 (cw), Z2 ¼ ð11:4890 þ 25:6107iÞ, Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(4), (5), (6), (7), (8),
Z3 Z4 Z6 Z1 Z8
¼ ð25:6301 þ 19:6044iÞ ¼ ð24:1301 þ 14:4668iÞ ¼ ð25:6726 19:5246iÞ ¼ ð30:7543 þ 10:9856iÞ ¼ ð9:0594 þ 53:7541iÞ
i.e., i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:2682 jZ4 j ¼ 28:1344 jZ6 j ¼ 32:2536 jZ1 j ¼ 32:6575 jZ8 j ¼ 54:5122
Phase-II synthesis: Given that: displacement, d2 ¼ ð15:7366 þ 20:7392iÞ and k2 ¼ 49 (cw) Let m2 ¼ 36 (cw) From Eq. (12), Z7 ¼ ð20:4301 3:6894iÞ From Eq. (15), Z9 ¼ ð18:5204 þ 14:2777iÞ From Eq. (16), Z10 ¼ ð29:4895 þ 57:4435iÞ
i.e., jZ7 j ¼ 20:7605 i.e., jZ9 j ¼ 23:3850 i.e., jZ10 j ¼ 64:5708
Phase-II Alternative synthesis: Given that: displacement, d2a ¼ ð14:9961 6:5665iÞ, k2 ¼ 43 (ccw) and u2 ¼ 30 (ccw)
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Let m2 ¼ 43 (cw) From Eq. (20), Z7 ¼ ð19:6794 3:8593iÞ From Eq. (23), Z11 ¼ ð51:1945 þ 5:5756iÞ From Eq. (24), Z12 ¼ ð9:0493 62:8535iÞ
i.e., jZ7 j ¼ 20:0542 i.e., jZ11 j ¼ 51:4972 i.e., jZ12 j ¼ 63:5016
(c) Function generation Phase-I synthesis: Given that: /1 ¼ 50 (cw), b1 ¼ 84 (cw), m1 ¼ 60 (ccw) and Z6 ¼ ð25:7194 19:4098iÞ Let a1 ¼ 24 (cw), u1 ¼ 44 (cw), Z2 ¼ ð11:4890 þ 25:6107iÞ, Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(27), d01 ¼ ð29:6691 12:5688iÞ (30), Z3 ¼ ð25:7756 þ 20:0979iÞ (31), Z4 ¼ ð23:9707 þ 14:4218iÞ (7), Z1 ¼ ð30:7404 þ 11:5241iÞ (8), Z8 ¼ ð9:2656 þ 53:5943iÞ
i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:6850 jZ4 j ¼ 27:9747 jZ1 j ¼ 32:8295 jZ8 j ¼ 54:3893
Phase-II synthesis: Given that: b2 ¼ 84 (ccw), u2 ¼ 10 (ccw) and k2 ¼ 49 (cw) Let m2 ¼ 36 (cw) From From From From
Eq. Eq. Eq. Eq.
(34), (37), (15), (16),
d02 ¼ ð16:3491 þ 20:9678iÞ Z7 ¼ ð20:4521 5:0782iÞ Z9 ¼ ð18:5065 þ 13:7392iÞ Z10 ¼ ð29:7177 þ 58:6725iÞ
i.e., jZ7 j ¼ 21:0731 i.e., jZ9 j ¼ 23:0490 i.e., jZ10 j ¼ 65:7693
Phase-II Alternative synthesis: Given that: k2 ¼ 43 (ccw) and u2 ¼ 30 (ccw) Let m2 ¼ 43 (cw) From From From From From
Eq. Eq. Eq. Eq. Eq.
(41), (44), (45), (23), (24),
d02a ¼ ð12:5449 7:2332iÞ Z7 ¼ ð21:8295 6:5412iÞ Z7 ¼ ð20:7917 5:0482iÞ Z11 ¼ ð51:1806 þ 6:1141iÞ Z12 ¼ ð51:4813 33:8938iÞ
i.e., i.e., i.e., i.e.,
jZ7 j ¼ 22:7885 jZ7 j ¼ 21:3958 jZ11 j ¼ 51:5445 jZ12 j ¼ 61:6371
References [1] S. Rose, Five-bar loop synthesis, Machine Design (October 12) (1961) 189–195. [2] K.L. Ting, G.H. Tsai, Mobility and synthesis of five-bar programmable linkages, in: Proceedings of 9th OSU Applied Mechanisms Conference, Kanas City, MD, 1985, pp. III-1–III-8. [3] K.L. Ting, Five-bar Grashof criteria, Transaction the ASME, Journal of Mechanisms, Transmissions and Automation in Design (December) (1986) 533–537. [4] Y.R. Rawat, Synthesis of variable topology mechanisms, M.Tech., Dissertation, M.E.D., IIT, Bombay, 1996. [5] S.A. Joshi, C. Amarnath, Y.R. Rawat, Synthesis of variable topology mechanisms for circuit breaker applications, in: Proceedings of the 8th NaCoMM Conference, IIT, Kanpur, 1997.
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[6] S.A. Joshi, Variable topology mechanisms for circuit breaker applications, M.Tech. Dissertation, M.E.D, IIT, Bombay, 1998. [7] S. Balli, S. Chand, Synthesis of a five-bar mechanism of variable topology type with transmission angle control, Transactions ASME Mechanical Design, manuscript no. 020401, communicated. [8] S. Balli, S. Chand, Synthesis of a five-bar mechanism with variable topology for motion between extreme positions, Mechanism and Machine Theory 36 (10) (2001) 1147–1156. [9] S. Balli, S. Chand, Five-bar motion and path generators with variable topology for motion between extreme positions, Mechanism and Machine Theory 37 (11) (2002) 1435–1445. [10] S. Balli, S. Chand, Defects in link mechanisms and solution rectification, Mechanism and Machine Theory 37 (9) (2002) 851–876. [11] L.I. Wu, H.S. Yan, Counter examples in transmission angles, in: Proceedings of 10th OSU Applied Mechanism Conference, Oklahoma State University, Stillwater, Oklahoma, 1987, pp. 1–4. [12] A.V. Mohan Rao, A.G. Erdman, G.N. Sandor, et al., Synthesis of multi-loop, dual-purpose planar mechanisms utilizing Burmester theory, in: Proceedings of 2nd Applied Mechanism Conference, Oklohoma State University, Stillwater, Oklahoma, 1971, pp. 7-1 to 7-23. [13] H.S. Yan, L.I. Wu, On the dead-center positions of planar linkage mechanisms, ASME Journal of Mechanisms, Transmissions and Automation in Design 111 (March) (1989) 40–46. [14] A.G. Erdman, G.N. Sandor, Mechanism Design: Analysis and Synthesis, vol-1, Prentice Hall, 1997. [15] S. Balli, S. Chand, A general procedure for four FSP synthesis of defect-free 5-bar variable topology mechanism, Transactions of ASME Mechanical Design, manuscript no. 010317, communicated. [16] S. Balli, S. Chand, Transmission angle in mechanisms A review, Mechanism and Machine Theory 37 (2) (2002) 175–195. [17] S. Balli, S. Chand, Order rectification of five-bar variable topology mechanism, in: Proceedings of the NaCoMM2001 Conference, IIT, Kharagpur, December 2001, pp. 73–80. [18] K. Hain, Applied Kinematics, McGraw Hill, New York, 1967, translated by Adams and others. [19] T.E. Shoup, B.J. Pelan, Design of a four-bar mechanism for optimum transmission angle and optimum structural error, in: Proceedings of 2nd OSU Applied Mechanism Conference, Stillwater, Okla, 1971, pp. 4.1–4.9. [20] A.S. Hall, Kinematics and Linkage Design, Prentice Hall, 1961. [21] A.G. Erdman, Three and four precision point kinematic synthesis of planar linkages, Mechanism and Machine Theory 16 (1981) 227–245. [22] R.E. Philipp, F. Freudenstein, Synthesis of two-degree-of-freedom linkages––a feasibility study of numerical methods of synthesis of bivariate function generators, Journal of Mechanisms 1 (1996) 9–21. [23] Lakshminarayana, R.G. Narayaamurthi, Derivate synthesis of plane mechanisms to generate functions of two variables, Journal of Mechanisms (1970) 249–271. [24] K. Lakshminarayana, A simplified approach to 2 degree-of-freedom linkages, ASME Journal of Engineering for Industry (1973) 584–588. [25] T.S. Mruthyunjaya, A note on synthesis of the general, two degree-of-freedom linkage, Mechanism and Machine Theory (1975) 77–80. [26] S. Chand, S. Balli, Synthesis of seven-link mechanism with variable topology, in: Proceedings of the CSME-MDE2001 Conference, Concordia University, Montreal, Quebec, Canada, 2001. [27] A.H. Soni, Mechanism Synthesis and Analysis, McGraw Hill, New York, 1974.
Synthesis of a planar seven-link mechanism with variable topology for motion between two dead-center positions Shrinivas S. Balli a b
a,*
, Satish Chand
b
Department of Mechanical Engineering, Basaveshwar Engineering College, Bagalkot 587 102, Karnataka, India Department of Mechanical Engineering, M.N. National Institute of Technology, Allahabad 211 004, UP, India Accepted 18 March 2003
Abstract An analytical method of synthesis of a planar seven-link mechanism with variable topology for motion between two dead-center positions is suggested. The method is useful to reduce the solution space. The proposed method is non-iterative. A complex number, dyadic approach of synthesis of mechanism for motion, path (with prescribed timing) and function generation is considered. An alternative choice for Phase-II is also suggested. 2003 Elsevier Ltd. All rights reserved. Keywords: Synthesis; Seven-link mechanism; Variable topology; Dead-center positions
1. Introduction A seven-link mechanism has two degrees of freedom. Various methods are proposed to synthesize it. A seven-bar linkage with variable topology operates in two phases. In each phase, a link adjacent to the permanently fixed link of seven-bar linkage is fixed temporarily and the resulting portion acts like a six-bar mechanism with single degree of freedom. In the beginning, an overview of the available literature on the variable topology mechanism has been given to form the basis of the method developed in the present work. Rose [1], Ting and Tsai [2] and Ting [3] with the help of graphical methods made indirect reference of five-bar variable topology mechanism. Rawat [4] established a synthesis technique for a five-bar variable
*
Corresponding author. Tel.: +91-8354-433025/426397; fax: +91-835-420504/425102. E-mail addresses: ssballi@rediffmail.com (S.S. Balli), satishchand@rediffmail.com (S. Chand).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00077-6
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topology mechanism operating in two phases. Joshi et al. [5] and Joshi [6] used the dyad synthesis of a five-bar variable topology mechanism for circuit breaker applications. Balli and Chand deal with various aspects like transmission angle control [7], motion between extreme positions [8,9], defects and solution rectification [10,15,17] of five-bar variable topology mechanism. Chand and Balli proposed a method of synthesis of a seven-link mechanism with variable topology [26]. In these methods, the arbitrarily prescribed or chosen values lead to an infinite number of solutions to a given problem resulting in a very large domain of solution space. The solutions thus obtained may be infeasible. One has to reiterate the procedure till s/he gets a practically feasible mechanism that functions satisfactorily requiring a large amount of time. The criterion of motion of mechanism between its dead-center positions may be used to reduce the solution space. Therefore, the objective of the problem undertaken is to develop a non-iterative analytical method of solution to determine the links of a seven-link mechanism with variable topology for motion between two dead-center positions of the mechanism. 2. Variable topology of seven-link mechanism Any mechanism with five or more links and with two or more degrees of freedom can be made to act as variable topology mechanism operating in two or more phases [4–6]. The planar sevenlink mechanism is shown in Fig. 1. Phase-I, Phase-II and Phase-II (alternative) shown in Figs. 2–4 respectively are discussed in the following paragraphs. 2.1. Phase-I In Phase-I, the link DOd is temporarily fixed and therefore, linkage becomes a six-bar mechanism with single degree of freedom. It is a combination of two four-bar linkages in series. Oa A1 is
Fig. 1. A planar seven-link mechanism.
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Fig. 2. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-I.
Fig. 3. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-II.
the input link. B is the possible path tracer point. Suffix 1 and 2 of alphabets in Fig. 2 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-I. It is to be noted that D is a temporarily fixed pivot. Oa and Oc are the permanently fixed pivots.
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Fig. 4. A planar seven-link mechanism with variable topology at two of its dead-center positions, Phase-II (alternative).
2.2. Phase-II Once the above six-bar portion of seven-bar mechanism, reaches the position 2, the link DOd is released to move and the link Oa A is fixed temporarily, thus switching on to the Phase-II. In Phase-II also, it is a single degree of freedom, six-bar mechanism. Link DOd is the input link. B is the possible path tracer point. Suffix 2 and 3 of alphabets in Fig. 3 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-II. It is to be noted that D is no more a fixed pivot whereas A2 is a temporarily fixed pivot. Oc and Od are the permanently fixed pivots.
2.3. Phase-II (alternative) Instead of link Oa A of seven-link mechanism, if link COc is fixed temporarily, then also it results in six-bar linkage mechanism (Fig. 4) with single degree of freedom. Link DOd is the input link. B is the possible path tracer point. Suffix 2 and 3 of alphabets in Fig. 4 show the two finitely separated positions of the six-bar portion of the seven-bar variable topology mechanism in Phase-II (alternative). Again it is to be noted that D is no more a fixed pivot whereas C2 is a temporarily fixed pivot. Oa and Od are the permanently fixed pivots. Phase-I, Phase-II and Phase-II (alternative) might be interchanged i.e. instead of Phase-I being synthesized first, Phase-II synthesis might be made first and Phase-I later. Likewise, Phase-II (alternative) synthesis might be made first and Phase-I later and so on. Accordingly the values are prescribed or chosen and the unknowns are determined.
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3. Description of the problem In dead-center positions, a mechanism becomes instantaneous structure [27]. However, the mobility can be retained if its input link is changed properly. It is required to move the mechanism in both the phases between two dead-center positions. The six-bar portion of seven-link variable topology mechanism could be in dead-center position in more than one way [13]. 3.1. Transmission angle in planar six-bar mechanism Hain [18] has identified the different cases of six-bar linkage such as: (i) Six-bar linkage consisting of two four-bar linkages in series having two transmission angles. In this case, the positions of the mechanism in which the extreme values of transmission angle occur are dependent only on the dimensions of the basic four-bar linkage and on the position of the fixed pivot. (ii) Six-bar linkage with coupler drive (a) with one transmission angle and (b) with two transmission angles 3.2. Transmission angle in dead-center position of mechanism The dead-center position of a planar four-bar mechanism is defined as the position when transmission angle becomes zero [18,27]. The dead-center position of six-bar linkage and transmission angle are discussed in [13]. For symmetric 2-DOF, seven-link mechanism in a position of symmetry the transmission angle is equal to 0 or 180. Then the mechanism will be in a locking position. If such a linkage is required, it is necessary to provide an auxiliary drive [22]. A second driving link is used in dead-center position and then there is generally another transmission angle at another point of linkage so that it is possible to overcome the dead lock. This driving link may be a link, which gives up some of its kinetic energy to overcome dead-center position. Spring, flywheel, lever may be used for the purpose [18]. In the phases discussed above, the six-bar portion of seven-link mechanism is considered to be a combination of two four-bar linkages in series. The two transmission angles resulting are merged to be a single transmission angle [11]. The value of transmission angle in each position of each phase is equal to either 0 or to 180 (Figs. 2–4). Thus, all the positions represent the dead-center positions of mechanism. 3.3. Solution steps The problem to be solved consists of the following steps: iii(i) To identify the link to be fixed temporarily and input link in each phase of operation. ii(ii) To recognize the type of mechanism in each phase (Stephenson, Watt or any other). i(iii) To write down the standard dyad equations for the motion between position 1 and position 2 of Phase-I and also between position 2 and position 3 of Phase-II. i(iv) To identify the values to be specified, values to be chosen freely and the unknowns based on the task to be performed.
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ii(v) To solve the equations of motion in each phase separately for the link lengths. i(vi) To retain the link lengths of Phase-I while solving the equations in Phase-II. (vii) To find out the total number of solutions possible in all, by the method. 3.4. Assumptions made In the present discussion, when the coupler and driving link are inline (i.e., when the mechanism is at dead-center positions), the angle between the coupler and the follower link (transmission angle) at these extreme positions is also either maximum or minimum [16] (at which the specified extreme values of transmission angles also occur). In the foregoing derivation, the max. transmission angle is to be 180 and the min. transmission angle is 0. When the transmission angle is 0 or 180 i.e. too much deviated from the ideal one (90), the mechanism exhibit poor transmission characteristics. If the transmission angle l reaches 180, the mechanism will be self-locked and the further motion is seized. It is desirable for some of the applications like toggling mechanism, circuit breakers, etc. The limits of transmission angle l are to be fixed depending upon the application. 4. Types of six-bar linkage and dead-center positions From Figs. 2–4 one can understand that the designer may get different types of six-bar linkage mechanism in different phases depending upon the link fixed temporarily. For Phase-II, an alternative choice (Fig. 4) of fixing the middle link temporarily is also possible if the original sevenlink mechanism has three fixed pivots [26]. Yan and Wu [13] discussed the different configurations of six-bar mechanisms in their dead-center positions. According to them [13] Watt-I six-bar linkage has seven dead-center positions, Watt-II six-bar linkage has seven dead-center positions, Stephenson-I six-bar linkage has seven dead-center positions, Stephenson-II six-bar linkage has five dead-center positions and Stephenson-III six-bar linkage has seven dead-center positions. If the six-bar portion of seven-link variable topology mechanism is other than these Watt or Stephenson types, the different dead-center positions are to be identified. Symmetric seven-bar linkage [22], Type-I and Type-II plane seven-link mechanisms, special configurations like co-axial and the cyclically symmetric mechanisms [23–25] may also be considered for the variable topology synthesis. These may yield different types of six-bar linkages in two phases after fixing temporarily a link adjacent to permanently fixed link. 5. Synthesis It is required to synthesize a planar seven-link mechanism (shown in Fig. 1) with variable topology. One can have three options as follows: ii(i) one end link is fixed temporarily, i(ii) other end link is fixed temporarily, (iii) middle link is fixed temporarily.
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Table 1 Convention followed to denote the angles in Phase-I and Phase-II Link Phase-I Vector representation and From 1st position to 2nd angle between two different position (Suffix-1, Fig. 2) position of link
Phase-II From 2nd position to 3rd position (Suffix-2, Fig. 3)
Phase-II Alternative From 2nd position to 3rd position (Suffix-2, Fig. 4)
Oa Oc –Z1 Oa A–Z2 –/ AB–Z3 –a BC–Z4 –u COc –Z5 –b BD–Z6 –m DOd –Z7 –k Oc D–Z8 Oc A2 –Z9 Oc Od –Z10 Oa C2 –Z11 Od C2 –Z12 Displacement vector
Fixed link Temporarily fixed a2 u2 b2 m2 – – Temporarily fixed Fixed link – – B2 B3 ¼ d2
Fixed link /2 a2 u2 Temporarily fixed m2 k2 – – Fixed link Temporarily fixed Temporarily fixed B2 B3 ¼ d2a
Fixed link /1 a1 u1 b1 m1 Temporarily fixed Temporarily fixed – Fixed link – – B1 B2 ¼ d1
Sign convention
CCW motion +ve
CW motion )ve
Any two of the above three can be used for two phases of a seven-link mechanism with variable topology and the remaining one can be used as an alternative. For the effective transmission of motion/force, these transmission angles are to be controlled so that they vary within some reasonable range [19]. Instead of landing into unnecessary problems while controlling the two transmission angles simultaneously, the method of Hain [18] or Wu and Yan [11] of reducing the two transmission angles into one and controlling this transmission angle within some feasible region is followed. For the dead-center positions of six-bar portion of mechanism, this combined transmission angle becomes 0 or 180. It is assumed that the mechanism moves from dead-center position 1 to the dead-center position 2 in Phase-I and from dead-center position 2 to the deadcenter position 3 in Phase-II. In the present case, as soon as the mechanism moves from one deadcenter position to the other, it stops and then switches on to the Phase-II. So there is no question of overcoming the dead lock and hence, no auxiliary drive is needed. The input motion in Phase-I is /1 . 1 The displacement vector B1 B2 is given by d1 . Writing the dyad equations [14] for Phase-I (refer Fig. 2), Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ d1 Z5 ðe
ib1
iu1
1Þ þ Z4 ðe
1Þ ¼ d1
Z6 ðeim1 1Þ ¼ d1
1
Conventions followed to denote link angles are given in Table 1.
ð1Þ ð2Þ ð3Þ
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5.1. Motion generation In motion generation mechanisms, the body to be guided usually is a part of a floating link. Hence, the location of tracer point on the coupler and the coupler orientation are the part of design specifications as the entire motion of the coupler link is to take place. It requires that an entire body be guided through a prescribed motion order. 5.1.1. Phase-I synthesis In the standard dyad Eqs. (1)–(3), in motion generation, the coupler point motions (a1 and u1 ) and displacement vector ðd1 Þ are prescribed. /1 , b1 , m1 , Z2 and Z5 are the free choices. Hence, there will be 17 numbers of solutions. Then the unknowns Z3 , Z4 , Z6 , Z1 and Z8 are determined as follows: Z3 ¼
d1 Z2 ðei/1 1Þ ðeia1 1Þ
ð4Þ
Z4 ¼
d1 Z5 ðeib1 1Þ ðeiu1 1Þ
ð5Þ
Z6 ¼
d1 ðeim1 1Þ
ð6Þ
Z1 ¼ Oa Oc ¼ Z2 þ Z3 Z4 Z5
ð7Þ
Z 8 ¼ Oc D ¼ Z 4 þ Z 5 Z 6
ð8Þ
5.1.2. Phase-II Synthesis The input motion in Phase-II is k2 . The displacement vector B2 B3 is given by d2 . Writing the dyad equations [14] for Phase-II (refer Fig. 3), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2 Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ d2 ia1
Z3 e ðe
ia2
1Þ ¼ d2
ð9Þ ð10Þ ð11Þ
All the link lengths except the link Z7 , Z9 and Z10 are known from Phase-I. Eq. (9) is solved to determine the link 7 and Z9 and Z10 are determined by loop closure equations. Here, d2 and m2 are prescribed and k2 is a free choice. Therefore, there will be 1 numbers of solutions. Instead of d2 , b2 and u2 or a2 may also be prescribed. Then the unknowns Z7 , Z9 and Z10 are determined as follows: when d2 and m2 are prescribed, Z7 ¼
d2 Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð12Þ
Or when b2 , u2 and m2 are prescribed, Z7 ¼
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
or when a2 and m2 are prescribed,
ð13Þ
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Z7 ¼
Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
1279
ð14Þ
from loop closure equations, Z9 ¼ Oc A2 ¼ Z2 ei/1 Z1
ð15Þ
Z10 ¼ Oc Od ¼ Z5 þ Z4 Z6 Z7
ð16Þ
5.1.3. Phase-II synthesis (alternative) Here, m2 , the displacement vector B2 B3 i.e. d2a or u2 are prescribed and the input motion k2 is chosen freely. Therefore, there will be 1 numbers of solutions. Then the unknowns Z7 , Z11 and Z12 are determined as follows. Writing the dyad equations [14] for Phase-II (refer Fig. 4), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2a i/1
i/2
1Þ þ Z3 e ðe
iu1
iu2
1Þ ¼ d2a
Z2 e ðe Z4 e ðe
ia1
ia2
1Þ ¼ d2a
ð17Þ ð18Þ ð19Þ
When d2a is prescribed, Z7 ¼
d2a Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð20Þ
Or when is /2 and a2 prescribed, Z7 ¼
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð21Þ
When u2 is prescribed, Z7 ¼
Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð22Þ
Z11 ¼ Oa C2 ¼ Z1 þ Z5 eib1
ð23Þ
Z12 ¼ Od C2 ¼ Z7 þ Z6 eim1 Z4 eiu1
ð24Þ
5.2. Path generation with prescribed timing In path generation a point on a floating link is to trace a path defined with respect to the fixed frame of reference. If the path points are to be correlated with either time or input link positions, the task is called path generation with prescribed timings. For this purpose, location of a tracer point on the coupler and the position of input link are the part of the design specifications. The position of the input link must correspond to each position of the tracer point on the coupler. Path of tracer point (motion of coupler point) and rotation of one crank are to be prescribed. The values to be prescribed and the values to be free chosen in case of motion generation mechanism are to be interchanged if the mechanism is required for path generation with prescribed timing.
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5.2.1. Phase-I synthesis The input crank motion ð/1 Þ, b1 , m1 and displacement vector ðd1 Þ are prescribed. a1 , u1 , Z2 and Z5 are the free choices. Hence, there will be 16 numbers of solutions. Then the unknowns Z3 , Z4 , Z6 , Z1 and Z8 are determined using Eqs. (4)–(8) respectively. 5.2.2. Phase-II synthesis d2 and k2 are prescribed and m2 is a free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z9 and Z10 could be determined using Eqs. (12) or (13) or (14)–(16) respectively. 5.2.3. Phase-II synthesis (alternative) Here, the displacement vector B2 B3 i.e. d2a or u2 and k2 are prescribed and m2 is chosen freely. Therefore, there will be 1 numbers of solutions. Z7 , Z11 and Z12 are determined using Eqs. (20) or (21) or (22)–(24) respectively. 5.3. Function generation A function generation mechanism is a linkage in which relative motion or forces between links (generally) connected to ground is taken into account. It is required to coordinate the rotation or sliding motion of input and output links for two specified design positions. 5.3.1. Phase-I synthesis In function generation problem, the input and output crank motions (/1 , b1 and m1 ), and Z6 are prescribed. a1 , u1 , Z2 and Z5 are the free choices. Therefore, there will be 16 numbers of solutions. Then the unknowns Z3 , Z4 , Z1 and Z8 are determined as follows: Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ Z6 ðeim1 1Þ ib1
Z5 ðe Let
1Þ þ Z4 ðe
iu1
im1
1Þ ¼ Z6 ðe
1Þ
Z6 ðeim1 1Þ ¼ d01
ð25Þ ð26Þ ð27Þ
where d01 is the displacement vector, B1 B2 . Reducing the Eqs. (25) and (26) to the forms of standard dyad equations [14] as follows: Z2 ðei/1 1Þ þ Z3 ðeia1 1Þ ¼ d01
ð28Þ
Z5 ðeib1 1Þ þ Z4 ðeiu1 1Þ ¼ d01
ð29Þ
Z3 ¼
d01 Z2 ðei/1 1Þ ðeia1 1Þ
ð30Þ
Z4 ¼
d01 Z5 ðeib1 1Þ ðeiu1 1Þ
ð31Þ
Z1 and Z8 are determined using Eqs. (7) and (8) respectively. 5.3.2. Phase-II synthesis Writing the loop closure equation for Phase-II (refer Fig. 3),
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Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ Z3 eia1 ðeia2 1Þ
ð32Þ
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ Z3 eia1 ðeia2 1Þ
ð33Þ
Reducing the Eqs. (32) and (33) to the form of standard dyad equation [14], Let
Z3 eia1 ðeia2 1Þ ¼ d2
ð34Þ
Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2
ð35Þ
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ ¼ d2
ð36Þ
Here, b2 , k2 and u2 are prescribed and m2 is the free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z9 and Z10 are determined as follows. When d2 (or a2 ) and k2 are prescribed, Z7 ¼
d2 Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð37Þ
When b2 , u2 and k2 are prescribed, Z7 ¼
Z5 eib1 ðeib2 1Þ þ Z4 eiu1 ðeiu2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð38Þ
Z9 and Z10 could be determined using Eqs. (15) and (16) respectively. 5.3.3. Phase-II synthesis (alternative) The input motion in Phase-II is k2 . The displacement vector B2 B3 is given by d2a . Writing the dyad equations [14] for Phase-II alternative (refer Fig. 4), Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ Z4 eiu1 ðeiu2 1Þ
ð39Þ
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ ¼ Z4 eiu1 ðeiu2 1Þ
ð40Þ
All the link lengths except the Z7 , Z11 and Z12 are known from Phase-I. Therefore, it is enough to solve Eq. (39) to determine the link 7. Z11 and Z12 are determined by loop closure equations. Reducing the Eqs. (39) and (40) to the form of standard dyad equation [14], Let
Z4 eiu1 ðeiu2 1Þ ¼ d2a
ð41Þ
Z7 ðeik2 1Þ þ Z6 eim1 ðeim2 1Þ ¼ d2a
ð42Þ
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ ¼ d2a
ð43Þ
Here, d2a (or u2 ), /2 , a2 and k2 are prescribed and m2 is the free choice. Therefore, there will be 1 numbers of solutions. Z7 , Z11 and Z12 are determined as follows: When d2a (or u2 ) and k2 are prescribed, Z7 ¼
d2a Z6 eim1 ðeim2 1Þ ðeik2 1Þ
When /2 , a2 and k2 are prescribed,
ð44Þ
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Table 2 Summary of Phase-I, -II and -II (alternative) synthesis of seven-link variable topology mechanism for two finitely separated positions Description
Phase-I
(a) Motion generation (rigid body guidance) Link fixed temporarily DOd Prescribed parameters a1 , u1 and d1 Free choice made /1 , b1 , m1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 , Z6 and Z8 No. of solutions 17 Total no. of solutions (b) Path generation with prescribed timings Link fixed temporarily DOd Prescribed parameters /1 , b1 , m1 and d1 Free choice made a1 , u1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 , Z6 and Z8 No. of solutions 16 Total no. of solutions (c) Function generation Link fixed temporarily DOd Prescribed parameters /1 , b1 , m1 and Z6 Free choice made a1 , u1 , Z2 and Z5 Unknowns Z1 , Z3 , Z4 and Z8 No. of solutions 16 Total no. of solutions
Phase-II
Phase-II Alternative
Oa A d2 ðb2 ; u2 =a2 Þ and m2 k2 Z7 , Z9 and Z10 1 18
COc d2 and m2 k2 Z7 , Z11 and Z12 1
Oa A d2 and k2 m2 Z7 , Z9 and Z10 1 17
COc d2 and k2 m2 Z7 , Z11 and Z12 1
Oa A b2 , u2 and k2 m2 Z7 , Z9 and Z10 1 17
COc k2 and u2 m2 Z7 , Z11 and Z12 1
Symbols have the meaning as given in Table 1.
Z7 ¼
Z2 ei/1 ðei/2 1Þ þ Z3 eia1 ðeia2 1Þ Z6 eim1 ðeim2 1Þ ðeik2 1Þ
ð45Þ
Then the unknowns Z11 and Z12 are determined using Eqs. (23) and (24). Thus, for a given task and the prescribed values the dimensions of the seven-link mechanism with variable topology can be found out using Eqs. (1)–(45). A summary of Phase-I, Phase-II and Phase-II (alternative) synthesis is given in Table 2.
6. Solution space The method yields an infinite number of solutions depending on number of free choices made. In order to obtain practically useful mechanisms, it is necessary to prescribe limits on such quantities as transmission angle [16,20] and max. to min. link length ratio. Then by determining the angle between the two specified dead-center positions of links of mechanism in terms of known angular motions and solving the equations of displacement, one may get the optimum linkage lengths. Thus the solution space can also be reduced. The requirements like the tracing point/coupler point or some link to be moved in a particular fashion (horizontally/vertically or at some angle or direction) makes the synthesis more specific
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thereby reducing the solution space. The solution space is reduced by 12 solutions if one link length is prescribed and by 1 solutions if one angular motion of link is prescribed. On the other hand a synthesis problem should be stated in such a way that no extra restrictions are imposed. It is advantageous not to use up all the available parameters of mechanism by specifying motion requirements but to leave some freedom, which can be used to fulfill transmission angle and other requirements. For example, if for a certain required specifications, two positions can suffice, one should not specify three or more positions, which may reduce the number of available solutions [4].
7. Advantages of the method The dyad technique by Sandor and Erdman permits handling any type of bar linkage mechanism and is extensively used for the synthesis of single degree of freedom mechanisms. Here it has been applied to variable topology mechanisms, which in this case are essentially two degrees of freedom systems. The authors have extended the technique to identify optimal mechanisms in terms of limits on transmission angles. The method reduces the job of synthesizing seven-link mechanism of two degrees of freedom to design a six-bar mechanism with single degree of freedom but of course in two phases. The synthesis of a six-bar mechanism is simpler. Thus the method is simpler compared with other methods of synthesis of seven-link mechanism. The advantages of the method proposed may be enlisted as follows: • The solution is consistent with the definitions of standard kinematic tasks like motion, function and path generation for two, three, four and five prescribed positions resulting in a unified method of synthesis [21]. • The mechanism may be designed for one task in one phase and for a different task in the other phase. It may also be designed for combined tasks in one phase itself. The solution may be modeled with groups [12] of standard form equations for either simultaneous function and path generation or for combined function and motion generation with prescribed timings. • It has codable steps leading to computerization of the method. • Simplicity, ease of application and generality are the attractions of the method. • Unlike the graphical methods, it is not limited by drawing accuracy. The method can be applied to synthesis of mechanisms for three; four or five infinitesimally separated positions or multiply separated positions also. • Solution is applicable to any other link-mechanisms acting as variable topology mechanisms.
8. Limitations of the method • The proposed method is applicable only to complex number approach. • The mechanism synthesized by the method may suffer from branch, Grashof or circuit defects, which can be rectified separately. • The solution does not permit good initial guesses for all possible solutions (free choices).
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9. Conclusion The method is for the synthesis of seven-link mechanism with variable topology for the motion between the any two dead-center positions. In the present work, order is not significant as numbers of design positions are only two but there may be GrashofÕ defect, circuit and branch defects. The proposed method reduces the synthesis of seven-link two degree of freedom mechanism to the synthesis of a six-bar single degree of freedom linkage mechanism but of course in two phases. Different combinations of types of six-link mechanism for variable topology are proposed. An alternative method for Phase-II is also suggested. The method is simple and straightforward. Designer can look up a number of solutions and select particular solution that suits the space or transmission angle requirements.
Appendix A. Numerical example Problem: It is required to synthesize a seven-link mechanism with variable topology for the motion between the dead-center positions and for the following tracer point displacement specifications: Phase-I: from point (72.5868, 218.7403) to the point (102.3319, 206.2695) Phase-II: from point (102.3319, 206.2695) to the point (86.5953, 227.0085) Phase-II (alternative): from point (102.3319, 206.2695) to the point (87.33581374, 199.7030247) Suggest the dimensions of the mechanism for (a) motion generation, (b) path generation with prescribed timings and (c) function generation. Solution: (a) Motion generation Phase-I synthesis: Given that: displacement, d1 ¼ ð29:7451 12:4708iÞ, a1 ¼ 24 (cw) and u1 ¼ 44 (cw) Let /1 ¼ 50 (cw), b ¼ 84 (cw), m1 ¼ 60 (ccw), Z2 ¼ ð11:4890 þ 25:6107iÞ and Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(4), (5), (6), (7), (8),
Z3 Z4 Z6 Z1 Z8
¼ ð25:6301 þ 19:6044iÞ ¼ ð24:1301 þ 14:4668iÞ ¼ ð25:6726 19:5246iÞ ¼ ð30:7543 þ 10:9856iÞ ¼ ð9:0594 þ 53:7541iÞ
i.e., i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:2682 jZ4 j ¼ 28:1344 jZ6 j ¼ 32:2536 jZ1 j ¼ 32:6575 jZ8 j ¼ 54:5122
Phase-II synthesis: Given that: displacement, d2 ¼ ð15:7366 þ 20:7392iÞ, m2 ¼ 36 (cw), b2 ¼ 84 (ccw), a2 ¼ 48 (ccw) and u2 ¼ 10 (ccw)
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Let k2 ¼ 49 (cw) From Eq. (12), Z7 ¼ ð20:4301 3:6894iÞ i.e., jZ7 j ¼ 20:7605 From Eq. (15), Z9 ¼ ð18:5204 þ 14:2777iÞ i.e., jZ9 j ¼ 23:3850 From Eq. (16), Z10 ¼ ð29:4895 þ 57:4435iÞ i.e., jZ10 j ¼ 64:5708 Phase-II Alternative synthesis: Given that: displacement, d2 ¼ ð14:9961 6:5665iÞ, m2 ¼ 43 (cw), u2 ¼ 30 (cw) and /2 ¼ 31 (cw) Let k2 ¼ 43 (ccw) From Eq. (20), Z7 ¼ ð19:6794 3:8593iÞ i.e., jZ7 j ¼ 20:0542 From Eq. (23), Z11 ¼ ð51:1945 þ 5:5756iÞ i.e., jZ11 j ¼ 51:4972 From Eq. (24), Z12 ¼ ð9:0493 62:8535iÞ i.e., jZ12 j ¼ 63:5016 (b) Path generation with prescribed timings By interchanging the values to be prescribed and the values to be free-chosen of motion generation problem above, it results in a problem of path generation with prescribed timings. Phase-I synthesis: Given that: displacement, d1 ¼ ð29:7451 12:4708iÞ, /1 ¼ 50 (cw), b1 ¼ 84 (cw) and m1 ¼ 60 (ccw) Let a1 ¼ 24 (cw), u1 ¼ 44 (cw), Z2 ¼ ð11:4890 þ 25:6107iÞ, Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(4), (5), (6), (7), (8),
Z3 Z4 Z6 Z1 Z8
¼ ð25:6301 þ 19:6044iÞ ¼ ð24:1301 þ 14:4668iÞ ¼ ð25:6726 19:5246iÞ ¼ ð30:7543 þ 10:9856iÞ ¼ ð9:0594 þ 53:7541iÞ
i.e., i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:2682 jZ4 j ¼ 28:1344 jZ6 j ¼ 32:2536 jZ1 j ¼ 32:6575 jZ8 j ¼ 54:5122
Phase-II synthesis: Given that: displacement, d2 ¼ ð15:7366 þ 20:7392iÞ and k2 ¼ 49 (cw) Let m2 ¼ 36 (cw) From Eq. (12), Z7 ¼ ð20:4301 3:6894iÞ From Eq. (15), Z9 ¼ ð18:5204 þ 14:2777iÞ From Eq. (16), Z10 ¼ ð29:4895 þ 57:4435iÞ
i.e., jZ7 j ¼ 20:7605 i.e., jZ9 j ¼ 23:3850 i.e., jZ10 j ¼ 64:5708
Phase-II Alternative synthesis: Given that: displacement, d2a ¼ ð14:9961 6:5665iÞ, k2 ¼ 43 (ccw) and u2 ¼ 30 (ccw)
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Let m2 ¼ 43 (cw) From Eq. (20), Z7 ¼ ð19:6794 3:8593iÞ From Eq. (23), Z11 ¼ ð51:1945 þ 5:5756iÞ From Eq. (24), Z12 ¼ ð9:0493 62:8535iÞ
i.e., jZ7 j ¼ 20:0542 i.e., jZ11 j ¼ 51:4972 i.e., jZ12 j ¼ 63:5016
(c) Function generation Phase-I synthesis: Given that: /1 ¼ 50 (cw), b1 ¼ 84 (cw), m1 ¼ 60 (ccw) and Z6 ¼ ð25:7194 19:4098iÞ Let a1 ¼ 24 (cw), u1 ¼ 44 (cw), Z2 ¼ ð11:4890 þ 25:6107iÞ, Z5 ¼ ð7:5169 þ 19:7627iÞ From From From From From
Eq. Eq. Eq. Eq. Eq.
(27), d01 ¼ ð29:6691 12:5688iÞ (30), Z3 ¼ ð25:7756 þ 20:0979iÞ (31), Z4 ¼ ð23:9707 þ 14:4218iÞ (7), Z1 ¼ ð30:7404 þ 11:5241iÞ (8), Z8 ¼ ð9:2656 þ 53:5943iÞ
i.e., i.e., i.e., i.e.,
jZ3 j ¼ 32:6850 jZ4 j ¼ 27:9747 jZ1 j ¼ 32:8295 jZ8 j ¼ 54:3893
Phase-II synthesis: Given that: b2 ¼ 84 (ccw), u2 ¼ 10 (ccw) and k2 ¼ 49 (cw) Let m2 ¼ 36 (cw) From From From From
Eq. Eq. Eq. Eq.
(34), (37), (15), (16),
d02 ¼ ð16:3491 þ 20:9678iÞ Z7 ¼ ð20:4521 5:0782iÞ Z9 ¼ ð18:5065 þ 13:7392iÞ Z10 ¼ ð29:7177 þ 58:6725iÞ
i.e., jZ7 j ¼ 21:0731 i.e., jZ9 j ¼ 23:0490 i.e., jZ10 j ¼ 65:7693
Phase-II Alternative synthesis: Given that: k2 ¼ 43 (ccw) and u2 ¼ 30 (ccw) Let m2 ¼ 43 (cw) From From From From From
Eq. Eq. Eq. Eq. Eq.
(41), (44), (45), (23), (24),
d02a ¼ ð12:5449 7:2332iÞ Z7 ¼ ð21:8295 6:5412iÞ Z7 ¼ ð20:7917 5:0482iÞ Z11 ¼ ð51:1806 þ 6:1141iÞ Z12 ¼ ð51:4813 33:8938iÞ
i.e., i.e., i.e., i.e.,
jZ7 j ¼ 22:7885 jZ7 j ¼ 21:3958 jZ11 j ¼ 51:5445 jZ12 j ¼ 61:6371
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[6] S.A. Joshi, Variable topology mechanisms for circuit breaker applications, M.Tech. Dissertation, M.E.D, IIT, Bombay, 1998. [7] S. Balli, S. Chand, Synthesis of a five-bar mechanism of variable topology type with transmission angle control, Transactions ASME Mechanical Design, manuscript no. 020401, communicated. [8] S. Balli, S. Chand, Synthesis of a five-bar mechanism with variable topology for motion between extreme positions, Mechanism and Machine Theory 36 (10) (2001) 1147–1156. [9] S. Balli, S. Chand, Five-bar motion and path generators with variable topology for motion between extreme positions, Mechanism and Machine Theory 37 (11) (2002) 1435–1445. [10] S. Balli, S. Chand, Defects in link mechanisms and solution rectification, Mechanism and Machine Theory 37 (9) (2002) 851–876. [11] L.I. Wu, H.S. Yan, Counter examples in transmission angles, in: Proceedings of 10th OSU Applied Mechanism Conference, Oklahoma State University, Stillwater, Oklahoma, 1987, pp. 1–4. [12] A.V. Mohan Rao, A.G. Erdman, G.N. Sandor, et al., Synthesis of multi-loop, dual-purpose planar mechanisms utilizing Burmester theory, in: Proceedings of 2nd Applied Mechanism Conference, Oklohoma State University, Stillwater, Oklahoma, 1971, pp. 7-1 to 7-23. [13] H.S. Yan, L.I. Wu, On the dead-center positions of planar linkage mechanisms, ASME Journal of Mechanisms, Transmissions and Automation in Design 111 (March) (1989) 40–46. [14] A.G. Erdman, G.N. Sandor, Mechanism Design: Analysis and Synthesis, vol-1, Prentice Hall, 1997. [15] S. Balli, S. Chand, A general procedure for four FSP synthesis of defect-free 5-bar variable topology mechanism, Transactions of ASME Mechanical Design, manuscript no. 010317, communicated. [16] S. Balli, S. Chand, Transmission angle in mechanisms A review, Mechanism and Machine Theory 37 (2) (2002) 175–195. [17] S. Balli, S. Chand, Order rectification of five-bar variable topology mechanism, in: Proceedings of the NaCoMM2001 Conference, IIT, Kharagpur, December 2001, pp. 73–80. [18] K. Hain, Applied Kinematics, McGraw Hill, New York, 1967, translated by Adams and others. [19] T.E. Shoup, B.J. Pelan, Design of a four-bar mechanism for optimum transmission angle and optimum structural error, in: Proceedings of 2nd OSU Applied Mechanism Conference, Stillwater, Okla, 1971, pp. 4.1–4.9. [20] A.S. Hall, Kinematics and Linkage Design, Prentice Hall, 1961. [21] A.G. Erdman, Three and four precision point kinematic synthesis of planar linkages, Mechanism and Machine Theory 16 (1981) 227–245. [22] R.E. Philipp, F. Freudenstein, Synthesis of two-degree-of-freedom linkages––a feasibility study of numerical methods of synthesis of bivariate function generators, Journal of Mechanisms 1 (1996) 9–21. [23] Lakshminarayana, R.G. Narayaamurthi, Derivate synthesis of plane mechanisms to generate functions of two variables, Journal of Mechanisms (1970) 249–271. [24] K. Lakshminarayana, A simplified approach to 2 degree-of-freedom linkages, ASME Journal of Engineering for Industry (1973) 584–588. [25] T.S. Mruthyunjaya, A note on synthesis of the general, two degree-of-freedom linkage, Mechanism and Machine Theory (1975) 77–80. [26] S. Chand, S. Balli, Synthesis of seven-link mechanism with variable topology, in: Proceedings of the CSME-MDE2001 Conference, Concordia University, Montreal, Quebec, Canada, 2001. [27] A.H. Soni, Mechanism Synthesis and Analysis, McGraw Hill, New York, 1974.