Dynamic Balancing of Two-DOF Parallel Mechanisms Using a Counter-Mechanism

Abstract

Dynamic balancing generally involves the static balancing of a mechanism using countermasses followed by the dynamic balancing of the inertia using counter-rotations. This approach requires that the statically balanced mechanism have a constant inertia for any configuration. Two of the main drawbacks of dynamic balancing are a significant increase in mass and actuation inertia. In this paper, static balancing strategies are optimized regarding the addition of mass and actuation inertia using Lagrange multipliers. The results are optimal mass-inertia curves which are akin to Pareto curves. Optimal static balancing rules are obtained and a comparison of balancing strategies shows that relaxing the constant inertia constraint may significantly reduce the total mass and actuation inertia. Then, a counter-mechanism is introduced in order to dynamically balance a mechanism with variable inertia. The conditions for which the counter-mechanism matches the inertia of the main mechanism for any configuration are derived. The significant influence of the radius of gyration of the counter-inertias on the optimal mass-inertia curves is revealed. Additionally, the advantages of counter-mechanisms over counter-rotations are demonstrated. Finally, examples of dynamically balanced mechanisms and a prototype are presented in order to illustrate the concepts.