A Decomposition Method For Modular Dimensional Synthesis Of Planar Multi-Loop Linkage Mechanisms

Abstract

The essence of mechanism synthesis is to find the mechanism for a given motion or task.
There are three customary tasks for kinematic synthesis: function generation, path generation and rigidbody
guidance. The task is often defined by a number of prescribed displacements and orientations called
precision points. Conceptual design of mechanisms has two main stages: (i) Type Synthesis, where the
number, type and connectivity of links and joints are determined, and (ii) Dimensional synthesis, where
the link lengths and pivot positions at the starting position are computed. From the first stage we already
get a mechanism represented by a graph (Pucheta and Cardona, In Mec´anica Computacional, volume
XXVI, proc. of MECOM 2005, Buenos Aires, Argentina). To evaluate its feasibility to fulfill a given
task it must necessarily have dimensions. To this purpose, we implement a strategy developed by Sandor
and Erdman (Advanced Mechanism Design: Analysis and Synthesis, vol. 2, Prentice-Hall, 1984).
This strategy consists in: (a) decomposing the complex mechanism topology into Single Open Chains
(SOCs), (b) solving dimensionally each SOC using complex numbers and the analytical Precision Point
Method, and (c) reassembling the solutions. Decomposition of complex multiloop linkages into single
subsystems was deeply studied for automated kinematic and dynamic analysis. However, its use in automated
synthesis applications is less addressed in the literature. The proposed SOCs Decomposition
algorithm uses the graph structure, the geometry of the prescribed parts and the motion constraints data
imposed on them. The resultant order of SOCs is not unique, there could be many valid orders. The
optimal order will be a compromise between what best satisfies the solvability (number of equations for
linearization required by analytical methods) and what best matches the number of prescribed motion
constraints given by the precision points. In spite of the complexity of this method, it produces multiple
good initial guesses for subsequent optimization stages based on gradient methods which often fail
because of the bifurcating and highly non-linear nature of this inverse problem.
The method was programmed in C++ language under the Oofelie environment (Cardona et al., Engng
Comp, 11:365–381, 1994).