This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

1.
Bricard, R., 1927, Lec¸ons de Cine´matique, Tome II, pp. 301–311.
2.
Cayley
A.
,
1876
, “
On Three-bar Motion
,”
Proc. London Math. Soc.
, Vol.
VII
, pp.
136
166
.
3.
Darboux, G., 1879, “De l’emploi des fonctions elliptiques dans la the´orie du quadrilate`re plan,” Bull. des Sciences, math. et astronom. 2e serie T. III, pp. 109–128.
4.
Dhingra, A. K., Almadi, A. N., and Kohli, D., 1998, “A Framework for Closed-form Displacement Analysis of 10-link 1-dof Mechanisms,” Proc. ASME Des. Eng. Tech. Conf., Sept. 13–16, Atlanta, GA, paper DETC98/MECH-5885.
5.
Groenman, J. T., 1950, “Behandeling van de koppelkromme met behulp van isotrope coo¨rdinaten,” Ph.D. Thesis, Technische Hogeschool te Delft.
6.
Haarbleicher
A.
,
1933
, “
Application des coordonne´es isotropes a l’e´tude de la courbe des trois barres
,”
J. de l’Ecole Polytechnique
, II serie, Vol.
31
, pp.
13
40
.
7.
Innocenti, C., 1994, “Analytical-Form Position Analysis of the 7-Link Assur Kinematic Chain with Four Serially-Connected Ternary Links,” Vol. 116, No. 2, pp. 622–628.
8.
Innocenti
C.
,
1995
a, “
Polynomial Solution to the Position Analysis of the 7-link Assur Kinematic Chain with One Quaternary Link
,”
Mech. Mach. Theory
, Vol.
30
, No.
8
, pp.
1295
1303
.
9.
Innocenti
C.
,
1995
b, “
Polynomial Solution of the Spatial Burmester Problem
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
117
, pp.
64
68
.
10.
Kinzel
G. L.
, and
Chang
C.
,
1984
, “
The Analysis of Planar Linkages Using a Modular Approach
,”
Mech. Mach. Theory
, Vol.
19
, No.
1
, pp.
165
172
.
11.
Moler
C. B.
, and
Stewart
G. W.
,
1973
, “
An Algorithm for Generalized Matrix Eigenvalue Problems
,”
SIAM J. Numer. Anal.
, Vol.
10
, No.
2
, pp.
241
256
.
12.
Nielsen, J., 1997, “Solving Sets of Nonlinear Equations for the Design and Analysis of Mechanical Systems,” Ph.D. Thesis, Dept. of Mech. Eng., Stanford University.
13.
Nielsen, J., and Roth, B., 1998, “Solving the Input/Output Problem for Planar Mechanisms,” Proc. ASME Des. Eng. Tech. Conf., Sept. 13–16, Atlanta, GA, paper DETC98/MECH-5912.
14.
Primrose
E. J. F.
, and
Freudenstein
F.
,
1963
, “
Geared Five-bar Motion: Part 2-Arbitrary Commensurate Gear Ratio
,”
Trans. ASME Series E (J. Applied Mechanics)
, Vol.
30E
, pp.
170
175
.
15.
Primrose
E. J. F.
,
Freudenstein
F.
, and
Roth
B.
,
1967
, “
Six-bar Motion (Parts I-III)
,”
Archive for Rational Mechanics and Analysis
, Vol.
24
, pp.
22
77
.
16.
Roberts
S.
,
1875
, “
On Three-bar Motion in Plane Space
,”
Proc. London Math. Soc.
, Vol.
VII
, pp.
14
23
.
17.
Sturmfels
B.
, and
Zelevinsky
A.
,
1994
, “
Multigraded resultants of Sylvester type
,”
J. of Algebra
, Vol.
163
, No.
1
, pp.
115
127
.
18.
Wampler, C., 1996, “Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics from a New Perspective,” Proc. ASME Des. Eng. Tech. Conf., Aug. 18–22, Irvine, CA, Paper 96-DETC/Mech-1210.
19.
Wunderlich
V. W.
,
1963
, “
Ho¨here koppelkurven
,”
O¨st. Ing. Arch.
, Vol.
17
, pp.
162
165
.
This content is only available via PDF.
You do not currently have access to this content.