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Dynamic balancing of planar mechanisms using toric geometry. (English) Zbl 1169.14034

Summary: A mechanism is statically balanced if for any motion, it does not apply forces on the base. Moreover, if it does not apply torques on the base, the mechanism is said to be dynamically balanced. In this paper, a new method for determining the complete set of dynamically balanced planar four-bar mechanisms is presented. Using complex variables to model the kinematics of the mechanism, the static and dynamic balancing constraints are written as algebraic equations over complex variables and joint angular velocities. After elimination of the joint angular velocity variables, the problem is formulated as a problem of factorization of Laurent polynomials. Using tools from toric geometry including toric polynomial division, necessary and sufficient conditions for static and dynamic balancing of planar four-bar mechanisms are derived.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B99 Polytopes and polyhedra
14Q99 Computational aspects in algebraic geometry
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