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
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IJMET
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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
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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
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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].
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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
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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.
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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
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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.
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