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and
Machine Theory
Mechanism and Machine Theory 43 (2008) 676–694
ate/mechmt
A procedure to count the number of planar mechanisms
subject to design constraints from kinematic chains
Chih-Ching Hung a, Hong-Sen Yan b, Gordon R. Pennock c,*
a Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
b Department of Mechanical Engineering, DaYeh University, Datsuen, Chamghua 515, Taiwan, ROC
c School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA
Received 20 September 2006; received in revised form 6 June 2007; accepted 9 June 2007
Available online 10 August 2007
Abstract
This paper presents a systematic procedure to count the number of planar mechanisms subject to design constraints
from the candidate kinematic chains. The procedure is based on well-known principles that can be found in graph theory
binatorial mathematics. A link-path is employed to include the design constraints that a number of specified joints
must correspond to a number of specified links. Also, modified permutation groups, generating function, and Polya’s the-
ory are used to count the number of non-isomorphic mechanisms with the required design constraints. Then the pattern
inventory is used for the conceptual design of the identified mechanisms. The procedure can identify all of the non-isomor-
phic mechanisms in a specified kinematic chain. In addition, the procedure can be used to determine the isomorphic mech-
anisms in a straightforward manner. Three practical examples are included in the paper to illustrate the systematic nature
of the proposed procedure; namely: the differential-type south pointing chariot, the Watt kinematic chain, and a variable-
stroke engine.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Conceptual design; Design constraints; Candidate kinematic chains; Non-isomorphic mechanisms; Graph theory; Combina-
torial mathematics; Linkage path