30120130405032

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30120130405032

  1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 5, September - October (2013), pp. 279-285 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET ©IAEME DIMENSIONAL SYNTHESIS OF 6-BAR LINKAGE FOR EIGHT PRECISION POINTS PATH GENERATION Dr. Aas Mohammad1, Mr. Yogesh kumar2 1 2 (Mechanical Engg. Dept.F/o Engg. & Tech., Jamia Millia Islamia, New Delhi-25, India) (Mechanical Engg. Dept.F/o Engg. & Tech., Jamia Millia Islamia, New Delhi-25, India) ABSTRACT This work is a specific application of a particular one degree of freedom six-bar planar mechanism. A complex number dyadic loop closure approach to synthesis mechanism for path to eight precision points is considered. In this technique the position equations which provides additional insight into the geometry of the planer linkage, the angle through which the output link oscillates, for each revolution of the input crank as follow a desired path. The input parameters are displaced points on the coupler as in the tracer path, the displaced orientation angle at each position along the path of the input link provides further development of 6-bar linkage with the help of MATLABr2012b. Keywords: Kinematic parameters, tracer points, links orientation, links length, phase angle. 1. INTRODUCTION The pointer (say tracer point) on the coupler exactly is forced to follow a desired path which may be linear or non linear. The work in this method is used to obtain the solution space for eightprecision-point [4] problem by the authors. Freudenstein's paper on path generation by geared five-bars, Hoffpauir [3] investigated the five-bar mechanism utilizing non-circular gearing to coordinate the input cranks. Joshi et Al. [6] and Joshi [2] used the dyad synthesis of a five-bar variable topology mechanism for circuit breaker applications. Chand and Balli proposed a method of synthesis of a seven-link mechanism with variable topology [5]. zhou and ting [7] deal with adjustable slider-crank linkages for multiple path generation. Gadad et. Al. [8] presented combined triad and dyad synthesis of seven-link variable topology mechanism using a ternary link.One has to reiterate the procedure till he gets a practically feasible mechanism that functions satisfactorily requiring a large amount of time to satisfy their constraints. Dhingra and Mani developed computer aided strategy to solve the precision position type function, path and motion generation for six-bar mechanisms [9]. Thus, an analytical method will 279
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME contribute both in theory and practice to eight precision point-path synthesis of a planar 6-bar linkage. 2. STANDARD FORM OF 6-BAR LINKAGE The Stephenson III linkage shown in fig. (2.1.1.) used to demonstrate the six bar synthesis using dyad approach. From fig. (2.1.2.) there are three independent loops; two loops & one triod loop. From fig. (2.1.1.) E & EJ are path cord δJ . For motion generation δJ is prescribed. For path generation with prescribed timing, ψJ is prescribed. For additional function generation, not only ψJ but also θJ or ØJ is prescribed. Fig. 2.1.1. Dyadic Synthesis of Stephension III Six-Bar Mechanism Fig. 2.1.2. Closed Loop Form of Six-Bar Mechanism with Respect to Refrence Plane 280
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME The path point of the coupler link moves along a path from position E to EJ defined in an arbitrary complex coordinate system by R1 and R J as shown in fig. (2.1.2.).All vector rotations are measured from the starting position, positive counter-clockwise fig. (2.1.1.). Suppose that we specify ‘Jth’ position for an unknown dyad by prescribing the values of. R1 , R J , γJ and ψJ as in fig. (2.1.2.). To find the unknown starting position vectors of the dyad, WA, ZA and ZD, a loop closure equation may be derived by summing the vectors clockwise around the loop (1) A0ADEEJDJAJ containing WA eiψJ , ZA eiβJ , ZD eiγJ, R1, RJ, ZD, ZA and WA. WA eiψJ + ZA eiβJ - ZD eiγJ – δJ + ZD - ZA - WA = 0 WA (eiψJ -1) + ZA (eiβJ -1) - ZD (eiγJ -1) = δJ (1) { δJ = RJ - R1 } To find the unknown starting position vectors of the dyad WB and ZF, a loop closure equation may be derived by summing the vectors clockwise around the loop(2) B0BEEJBJ containing WB eiØJ , ZF eiγJ , R1 , RJ , ZF , WB. WB eiØJ + ZF eiγJ – δJ – ZF – WB = 0 WB (eiØJ -1) + ZF (eiγJ -1) = δJ (2) To find other unknown vectors ZE, we consider another closed loop DBED as ZE + ZF + ZD = 0 ZE = - ( ZF + ZD ) (3) To find other unknown vectors ZC, we consider another closed loop ACDA as ZB + ZC – ZA = 0 ZC = ZA - ZB (4) Considering closed loop A0ACC0A0, for finding unknown vector WC WC = C0A0 + WA + ZB (5) 3. EXAMPLE OF SIX-BAR LINKAGE An example is to design a six-bar mechanism with vectors as design variables Z1, Z2 ,Z3, Z4 ,Z5, Z6 ,Z7, Z8 , Z9, Z10 and their orientations at home position with prescribed timing. The desired or target points are eight precision points along a curve which are specified by {(790,670),(619.38,760.31); (462.35,786.22),(359.99,770.73); (338.57,740.94);(541.04,696.60); (836.18,553.29);(888.62,543.54); (789.99,670)} the range of input variables adopted from SAM6.0 are set to θJ, øJ [0,2π] and αJ , ψJ , βJ [0,π]. 281
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME Fig. 3.1. Six-Bar Mechanism with Two Temary Link Having Two Fixed Point Writing down the loop closure equations of independent vector loops E0 F0 G0 GJ FJ E0 yields from fig (3.1) Z6 + Z9 + δJ – Z9 eiβJ – Z6 eiØJ = 0 Z9 eiβJ + Z6 eiøJ = Z6 + Z9 + δJ Z6 (eiøJ -1) + Z9 (eiβJ -1) = δJ (6) Writing down the loop closure equations of independent vector loops A0 B0 C0 G0 GJ CJ BJ A0 yields fig (3.1) Z1 + Z2 + Z8 + δJ – Z8 eiψJ – Z2 eiαJ – Z1 eiθJ = 0 Z1 + Z2 + Z8 + δJ = Z8 eiψJ + Z2 eiαJ + Z1 eiθJ Z8 (eiψJ -1)+ Z2 ( eiαJ -1) + Z1 (eiθJ -1) = δJ (7) Fig. 3.2. Mobility of Six-Bar Mechanism from Initial to Primed Position 282
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME Writing down the loop closure equations of independent vector loops F0 G0 C0 D0 and FJ GJ CJ DJ yields fig (3.1) Z7 + Z9 – Z8 – Z4 = 0 (8) Z7 eiøJ + Z9 eiβJ – Z8 eiψJ – Z4 eiαJ = 0 (9) Subtract eqn (8) from eqn (9), we get Z7 (eiøJ -1) – Z4 ( eiαJ -1) = – { Z9 (eiβJ -1) – Z8 (eiψJ -1)} (10) From fig. (3.1.) Z2 – Z4 = Z3 (11) Z5 + Z7 = Z6 Z5 = Z6 – Z7 (12) Where i = √െ1 and θJ , αJ ,ψJ , ØJ , βJ are respectively the angular displacements of links A0 B0 , B0 C0 ,C0 G0 , E0 F0 and F0 G0 relative to their home position. Equations from (6) to (12) are called “kinematic synthesis equations”. 4. RESULT AND DISCUSSION Kinematic solution obtained for path generation example of 6-bar mechanism is tabled below TABLE 4.1 Design Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 variables Desired 232.4 303.4 371.6 199.0 392.9 336.4 197.9 299.5 282.3 ------links length(mm) Optimized 217.6 298.0 353.6 149.9 386.6 343.0 131.6 166.7 287.0 421.4 links length(mm) 5. CONCLUSION It is well known that the kinematic synthesis problems depending on the prescribed parameters [10] can be categorized into function, motion and path generations. By equating the number of equations to the number of unknowns, one may determine the maximum number of displacements say ‘J’, to be six, four and eight, respectively. The proposed method is straightforward and the benefit of the resulting approach in the complex number field and treat complex variables and their conjugate as independent variables. The optimized design variables configured a mechanism obtained by a solution from MATLABr2012b program. 283
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 6. REFERENCES [1] G. Erdman, G.N. Sandor, Mechanism Design: Analysis and Synthesis, Vol. I, second ed., Prentice-Hall Inc.,Englewood Cliffs, NJ, 1991. [2] 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. [3] Hoffpauir C. R. Path Generation by five-bar Mechanisms. Master's Thesis, Louisiana State University (1964).MCPHATE A. J. Non-uniform Motion Band Mechanism. ASME, Publication 64-Mech-17 (1964). [4] Mcphate A. J. Non-uniform Motion Band Mechanism. ASME, Publication 64-Mech-17 (1964). [5] S. Chand, S.S. Balli, Synthesis of seven-link mechanism with variable topology, in: Proceedings of the CSME-MDE-2001 Conference, paper section no. WA-8, Concordia University, Montreal, Que., Canada, 2001. [6] S.A. Joshi, Variable topology mechanisms for circuit breaker applications, M. Tech. Dissertation, M.E.D, IIT,Bombay, 1998. [7] H.Zhou and K.L.Ting,”Adjustable slider-crank linkages for multiple path generation,” mechanism and machine theory,vol. 37, no.5, pp.499-509,2002. [8] G.M.Gadad, U.M.Daivagna, and S.S.Balli, “triad and dyad synthesis of planer seven-link mechanism with variable topology,” in proceedings of the 12th national conference on machine and mechanisms, PP.67-73, 2005. [9] A.K. Dhingra, N.K. Mani, Computer-aided mechanism design: a symbolic-computing approach, Computer Aided Design 25 (5) 300–310, 1993. [10] A. Erdman, G. Sandor, S. Kota, Mechanism Design: Analysis and Synthesis, 4th edn.Prentice-Hall, Englewood Cliffs, NJ, 2001. [11] Dr R. P. Sharma and Chikesh Ranjan, “Modeling and Simulation of Four-Bar Planar Mechanisms using Adams”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 429 - 435, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [12] Chikesh Ranjan and Dr R. P. Sharma, “Modeling Modeling, Simulation & Dynamic Analysis of Four-Bar Planar Mechanisms using CATIA V5R21”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 444 - 452, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. APPENDIX Example: First loop closure equation (6) for six bar mechanism written as Z6 (eiøJ -1) + Z9 (eiβJ -1) = δJ For eight precision points, say (J = 0,1,2………..8); First loop closure equation is formulated in the form as Z6 (eiø1 -1) Z6 (eiø2 -1) Z6 (eiø3 -1) Z6 (eiø4 -1) Z6 (eiø5 -1) + + + + + Z9 (eiβ1 -1) Z9 (eiβ2 -1) Z9 (eiβ3 -1) Z9 (eiβ4 -1) Z9 (eiβ5 -1) = = = = = (13) (14) (15) (16) (17) δ1 δ2 δ3 δ4 δ5 284
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME Z6 (eiø6 -1) + Z9 (eiβ6 -1) = δ6 Z6 (eiø7 -1) + Z9 (eiβ7 -1) = δ7 Z6 (eiø8 -1) + Z9 (eiβ8 -1) = δ8 (18) (19) (20) Second loop closure equation (7) is in the form as Z8 (eiψJ -1)+ Z2 ( eiαJ -1) + Z1 (eiθJ -1) = δJ For J = 0,1,2,3,……………….8 Z8 (eiψ1 Z8 (eiψ2 Z8 (eiψ3 Z8 (eiψ4 Z8 (eiψ5 Z8 (eiψ6 Z8 (eiψ7 Z8 (eiψ8 -1)+ Z2 ( eiα1 -1)+ Z2 ( eiα2 -1)+ Z2 ( eiα3 -1)+ Z2 ( eiα4 -1)+ Z2 ( eiα5 -1)+ Z2 ( eiα6 -1)+ Z2 ( eiα7 -1)+ Z2 ( eiα8 -1) + Z1 (eiθ1 -1) + Z1 (eiθ2 -1) + Z1 (eiθ3 -1) + Z1 (eiθ4 -1) + Z1 (eiθ5 -1) + Z1 (eiθ6 -1) + Z1 (eiθ7 -1) + Z1 (eiθ8 -1) -1) -1) -1) -1) -1) -1) -1) = = = = = = = = (21) (22) (23) (24) (25) (26) (27) (28) δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8 Third loop closure equation (10) for six-bar mechanism Z7 (eiøJ -1) – Z4 ( eiαJ -1) = – { Z9 (eiβJ -1) – Z8 (eiψJ -1)} For J = 0,1,2,……………..8 Z7 (eiø1 -1) – Z4 ( eiα1 Z7 (eiø2 -1) – Z4 ( eiα2 Z7 (eiø3 -1) – Z4 ( eiα3 Z7 (eiø4 -1) – Z4 ( eiα4 Z7 (eiø5 -1) – Z4 ( eiα5 Z7 (eiø6 -1) – Z4 ( eiα6 Z7 (eiø7 -1) – Z4 ( eiα7 Z7 (eiø8 -1) – Z4 ( eiα8 -1) = -1) = -1) = -1) = -1) = -1) = -1) = -1) = – { Z9 (eiβ1 -1) – { Z9 (eiβ2 -1) – { Z9 (eiβ3 -1) – { Z9 (eiβ4 -1) – { Z9 (eiβ5 -1) – { Z9 (eiβ6 -1) – { Z9 (eiβ7 -1) – { Z9 (eiβ8 -1) – Z8 (eiψ1 – Z8 (eiψ2 – Z8 (eiψ3 – Z8 (eiψ4 – Z8 (eiψ5 – Z8 (eiψ6 – Z8 (eiψ7 – Z8 (eiψ8 -1)} -1)} -1)} -1)} -1)} -1)} -1)} -1)} (29) (30) (31) (32) (33) (34) (35) (36) First we solve equations from (13) to (20) ; equations from (21) to (28) then equations from (29) to (36) ; we get optimized value of Z6, Z9 ;Z1, Z2 ,Z8 and Z4 ,Z7 by using MAT LAB Program. After finding the optimized value of Z1, Z2, Z4, Z6 ,Z7, Z8, Z9 we can find Z3 and Z5 from equation (11) and (12). The length of frame which is represented by vector Z10 is generated during graphical vector closed loop of optimized value of links representation on design software like AUTOCAD etc. From the above solution ZJ = a + ib {a = real value; b = imaginary value} Where ZJ is the vector representation of given kinematic link as shown in figure (3.1.). The optimized dimensions of links length ZJ (say Z1, Z2 ,Z3, Z4 ,Z5, Z6 ,Z7, Z8 , Z9, Z10) are expressed in Table (4.1) with the help of MAT LAB program. 285

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