On the theories of triangular sets
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A new algorithm for discussing Gröbner bases with parameters
Journal of Symbolic Computation
Factoring polynomials via polytopes
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Complexity of the resolution of parametric systems of polynomial equations and inequations
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Solving parametric polynomial systems
Journal of Symbolic Computation
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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.