This paper deals with the formulation and validation of a comprehensive algebraic algorithm for the kinematic analysis of slider-crank/rocker mechanisms, which is based on the use of geometric loci, as the fixed and moving centrodes, along with their evolutes, the cubic of stationary curvature and the inflection circle. In particular, both centrodes are formulated in implicit and explicit algebraic forms by using the complex algebra. Moreover, the algebraic curves representing the moving centrodes are recognized and proven to be Jeřábek’s curves for the first time. Then, the cubic of stationary curvature along with the inflection circle are expressed in algebraic form by using the instantaneous geometric invariants. Finally, the proposed algorithm has been implemented in a MATLAB code and significant numerical and graphical results are shown, along with the particular cases in which these geometric loci degenerate in lines and circles or give cycloidal positions.

References

References
1.
Hartenberg
,
R. S.
, and
Denavit
,
J.
, 1964,
Kinematic Synthesis of Linkages
,
McGraw-Hill
,
New York
.
2.
Dijksmann
,
E. A.
, 1976,
Motion Geometry of Mechanisms
,
Cambridge University Press
,
London
.
3.
Hunt
,
K. H.
, 1990,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
New York
.
4.
Erdman
,
A. G.
, and
Sandor
,
G. N.
, 1990,
Mechanism Design: Analysis and Synthesis
,
Prentice Hall
,
Englewood Cliffs, NJ
.
5.
McCarthy
,
J. M.
, 1990,
Introduction to Theoretical Kinematics
,
MIT
,
Cambridge, MA
.
6.
Waldron
,
K. J.
, and
Kinzel
,
G. L.
, 1999,
Kinematics, Dynamics and Design of Machinery
,
John Wiley & Sons
,
New York
.
7.
McCarthy
,
J. M.
, 2000,
Geometric Design of Linkages
,
Springer-Verlag
,
New York
.
8.
Reuleaux
,
F.
, 1963,
The Kinematics of Machinery
,
Dover Publications, Inc.
,
New York
.
9.
Shoup
,
T. E.
, 1991, “
Centrodes of the Slider-Crank Mechanism
,”
Proceedings of the 8th IFToMM World Congress on the Theory of Machines and Mechanisms
, Prague, Vol.
1
, pp.
59
62
.
10.
Wunderlich
,
W.
, 1970,
Ebene Kinematik, Hochschultaschenbuch 447/447a
,
Bibliograohisches Inst.
,
Mannheim
.
11.
Salmon
,
G.
, 1879,
A Treatise on the Higher Plane Curves: Intended as a Sequel to a Treatise on Conic Sections
, Booksellers: Hodges, Foster and Figgis, Dublin, Reprinted by
Kessinger Publishing
,
Whitefish, MT
.
12.
Yates
,
R. C.
, 1952,
Curves and Their Properties
,
The National Council of Teachers of Mathematics
,
Reston, Virginia
.
13.
Hoeltzel
,
D. A.
, and
Chieng
,
W.-H.
, 1990, “
Pattern Matching Synthesis as an Automated Approach to Mechanism Design
,”
ASME J. Mech. Des.
,
112
(
6
), pp.
190
199
.
14.
Jensen
P. W.
, 1992, “
The Polode Synthesis Method
,”
Forsch. Ingenieurwesen
,
58
(
6
), pp.
152
163
.
15.
Pennock
,
G. R.
, and
Kinzel
,
E. C.
, 2004, “
Graphical Technique to Locate the Center of Curvature of a Coupler Point Trajectory
,”
ASME J. Mech. Des.
,
126
(
6
), pp.
1000
1005
.
16.
Wu
,
T. M.
, and
Chen
,
C. K.
, 2005, “
Computer-Aided Curvature Analyses of Planar Four-Bar Linkage Mechanism
,”
Appl. Math. Comput.
,
168
(
2
), pp.
1175
1188
.
17.
Hwang
,
W. M.
, and
Fan
,
Y. S.
, 2008, “
Polynomial Equations for the Loci of the Acceleration Pole of a Slider Crank Mechanism
,”
Mech. Mach. Theory
,
43
(
2
), pp.
123
137
.
18.
Maleki
,
M.
,
Babahaji
,
A.
, and
Mohtat
,
A.
, 2009, “
Further Analytical Investigation Into Centrodes of the Planar FBL: Symbolic Representation, Double Points, Tacnodes, Degenerate Forms
,”
Mech. Mach. Theory
,
44
(
4
), pp.
739
750
.
19.
Figliolini
,
G.
, 2001, “
On the Motion Geometry and Kinematic Analysis of Slider-Crank Mechanisms
,”
Proceedings of the 8th IFToMM International Symposium on Theory of Machines and Mechanisms
, Bucharest, Vol.
I
, pp.
129
134
.
20.
Figliolini
,
G.
,
Conte
,
M.
, and
Rea
,
P.
, 2008, “
Analysis and Synthesis of Slider-Crank Mechanisms for Automatic Machines
,”
CD Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Brooklyn, New York, DETC Paper No. 2008-49863.
21.
Jeřábek
,
V.
, 1907, “
On Certain Circular Curves of Fourth Degree With Double Tangent Point
,”
Časopis pro Pěstování Matematiky a Fysiky
,
36
(
3
), pp.
233
239
(in Czech).
22.
Bottema
,
O.
, 1961, “
Some Remarks on Theoretical Kinematics
,”
Proceedings of the International Conference for Teachers of Mechanisms
,
F. R.
,
Erskine Crossley
, ed.,
Yale University/The Shoe String Press
,
North Haven, CT
, pp.
159
167
.
23.
Veldkamp
,
G. R.
, 1963,
Curvature Theory in Plane Kinematics
,
J. B. Wolters
,
Groningen, The Netherlands
.
24.
Di Benedetto
,
A.
, and
Pennestrì
,
E.
, 1993,
Introduction to the Kinematics of Mechanisms
,
Casa Editrice Ambrosiana
,
Milan
, Vol.
2
(in Italian).
25.
Hall
,
A. S.
, 1961,
Kinematics and Linkage Design
,
Waveland Press, Inc.
,
Prospect Heights, IL
.
26.
González-Palacios
,
M. A.
, and
Angeles
,
J.
, 1991, “
SIXPAQ: A Comprehensive Software Package for Analysis and Synthesis of Six-Bar Dwell-Linkages
,”
ASME International Computers in Engineering Conference
, Santa Clara, CA, Vol.
1
, pp.
309
314
.
27.
Beyer
,
R.
, 1963,
The Kinematic Synthesis of Mechanisms
(English version by
Kuenzel
H.
),
Chapman and Hall LTD
,
London
.
28.
Hirschhorn
,
J.
, 1962,
Kinematics and Dynamics of Plane Mechanisms
,
McGraw-Hill
,
New York
.
29.
Myszka
,
D. H.
, and
Murray
,
A. P.
, 2010, “
Slider-Cranks as Compatibility Linkages for Parametrizing Center-Point Curves
,”
ASME J. Mech. Rob.
,
2
(
2
),
021007
.
30.
Quian
,
Y.
,
Cao
,
Y.
,
Yuan
,
W. L.
, and
Zhou
,
H.
, 2011, “
Forward Kinematics Simulation Analysis of Slider-Crank Mechanism
,”
Adv. Mater. Res.
,
308–310
, pp.
1855
1859
.
31.
Tari
,
H.
, and
Su
,
H.-J.
, 2010, “
Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages
,”
ASME J. Mech. Des.
,
132
(
8
),
081003
.
32.
Gregory
,
D. B.
, and
Cavalca
,
K. L.
, 2011, “
Analysis of the Dynamics of a Slider–Crank Mechanism With Hydrodynamic Lubrication in the Connecting Rod–Slider Joint Clearance
,”
Mech. Mach. Theory
,
46
(
10
), pp.
1434
1452
.
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