Kinematic synthesis of the mechanism for static balancing of an input torque in three positions

The aim of this work is to design the links-spring mechanism for balancing, in the three positions of the operating range, a rotary disc subjected to a torque. An energy-related approach towards the conditions of the mechanical system balance for a discrete number of positions leads to the formulation of a task of searching for a four-bar linkage which guides a coupler point through the prescribed positions, where, at the same time, geometrical conditions (specifying the spring tension) and kinematic conditions (defining the radial component of the tension change rate) are satisfied. The finitely and infinitesimally separated position synthesis is considered, however, only a component of the coupler point velocity is essential. A general method was proposed for determining the four-bar mechanism geometry. Mechanism inversion was applied in order to reduce the number of designed variables and simplify the solution method. The system of complex algebraic equations defines the problem. Linear, symbolic transformations and a systematic search technique are utilized to find multiple local optimal solutions. The problem is solved using Mathematica software.