Dynamic balancing of linkages by algebraic methods

A mechanism is statically balanced if, for any motion, it does not exert forces on the base. Moreover, if it does not exert torques on the base, the mechanism is said to be dynamically balanced. In 1969, Berkof and Lowen showed that in some cases, it is possible to balance mechanisms without adding additional components, simply by choosing the design parameters (i.e. length, mass, centre of mass, inertia) in an appropriate way. For the simplest linkages, some solutions were found but no complete characterization was given. The aim of the thesis is to present a new systematic approach to obtain such complete classifications for 1 degree of freedom linkages. The method is based on the use of 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 factorisation of Laurent polynomials. Using computer algebra, necessary and sufficient conditions can be derived. Using this approach, a classification of all possible statically and dynamically balanced planar four-bar mechanisms is given. Sufficient and necessary conditions for static balancing of spherical linkages is also described and a formal proof of the non-existence of dynamically balanced spherical linkage is given. Finally, conditions for the static balancing of Bennett linkages are described.