Graphical Abstract Figure
Abstract
This paper deals with the hitherto unexplored problem of determining the -positions of the first-kind of the coupler link for all types of 4R four-bar linkages. Geometrical proofs for the existence and number of -positions are given based on geometrical inequalities. The number of -positions are enumerated for each circuit and branch of all types of linkages. Analytical as well as geometric methods for the determination of the configuration of the linkage at an asymptotic configuration are also presented. -positions of a 3R1P four-bar linkage are also discussed.
References
1.
Bottema
, O.
, and Roth
, B.
, 1979
, Theoretical Kinematics
, North-Holland Publishing Company
, Amsterdam
.2.
Veldkamp
, G. R.
, 1963
, Curvature Theory in Plane Kinematics
, J.B. Wolters
, Groningen
.3.
Chan
, C. L.
, and Ting
, K.-L.
, 2019
, “Extended Camus Theory and Higher Order Conjugated Curves
,” ASME J. Mech. Rob.
, 11
(5
), p. 051009
. 4.
Chan
, C. L.
, and Ting
, K.-L.
, 2019
, “Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves
,” ASME J. Mech. Rob.
, 12
(1
), p. 011018
. 5.
Shiwalkar
, P. B.
, Moghe
, S. D.
, and Modak
, J. P.
, 2022
, “Novel Methodology for Inflection Circle-Based Synthesis of Straight Line Crank Rocker Mechanism
,” ASME J. Mech. Rob.
, 14
(5
), p. 055001
. 6.
Veldkamp
, G. R.
, 1983
, “The Instantaneous Motion of a Line in a T-Position
,” Mech. Mach. Theory
, 18
(6
), pp. 439
–444
. 7.
Goehler
, C. M.
, Stanišić
, M. M.
, and Perez
, V. M.
, 2004
, “A Generalized Parameterization of T1 Motion and Its Application to the Synthesis of Planar Mechanisms
,” Mech. Mach. Theory
, 39
(11
), pp. 1223
–1244
. 8.
Kimbrell
, J. E.
, and Hunt
, K. H.
, 1981
, “Coupler Point-Paths and Line-Envelopes of 4-Bar Linkages in Asymptotic Configurations
,” Mech. Mach. Theory
, 30
(6
), pp. 897
–912
. 9.
Dan
, A.
, Krishna
, K. R.
, and Saha
, S. K.
, 2023
, “Static Stability of Planar Contacting Systems: Analytical Treatment in Euclidean Space
,” ASME J. Mech. Rob.
, 16
(8
), p. 081009
. 10.
Krishnamurty
, S.
, and Turcic
, D. A.
, 1988
, “A General Method of Determining and Eliminating Branching in Planar Multiloop Mechanisms
,” ASME J. Mech. Trans. Autom. Des.
, 110
(4
), pp. 414
–422
. 11.
Chase
, T. R.
, and Mirth
, J. A.
, 1990
, “Circuits and Branches of Single-Degree-of-Freedom Planar Linkages
,” ASME J. Mech. Des.
, 115
(2
), pp. 223
–230
. 12.
Barker
, C. R.
, and Jeng
, Y.-R.
, 1985
, “Range of the Six Fundamental Position Angles of a Planar Four-Bar Mechanism
,” Mech. Mach. Theory
, 20
(4
), pp. 329
–344
. 13.
Mallik
, A. K.
, Ghosh
, A.
, and Dittrich
, G.
, 1994
, Kinematic Analysis and Synthesis of Mechanisms
, CRC Press
, Boca Raton, FL
.14.
Williams
, R. L.
, and Reinholtz
, C. F.
, 1986
, “Proof of Grashof’s Law Using Polynomial Discriminants
,” ASME J. Mech. Trans. Autom. Des.
, 108
(4
), pp. 562
–564
. 15.
Hall
, H. S.
, and Stevens
, F. H.
, 1921
, A School Geometry
, 2nd ed., Macmillan and Company, Limited
, London
.16.
Barker
, C. R.
, 1985
, “A Complete Classification of Planar Four-Bar Linkages
,” Mech. Mach. Theory
, 20
(6
), pp. 535
–554
. 
