Approach for the Calculation of Transmission Ratios and Their Errors in 4-Bar Mechanisms, Considering the Precision Variations by Dimensional Tolerances

This paper presents research and theoretical development of a mathematical model that, first, allows us to understand how the positional exactitude of the output link of a four-bar mechanism depends on the manufacturing dimensional tolerances. To find this dependence, the total differentials of the kinematic constraint functions that govern the field of positions must be determined for each kinematic cycle of the mechanism under consideration. These total differentials lead to a system of equations whose solution gives the positional errors of the movable output links as a function of the manufacturing dimensional errors and an incidence matrix that varies with each one of the positions of the input element. On the other hand, the theoretical transmission ratio between the output velocities with respect to the input velocity of the articulated kinematic chain is defined, and for determining the total errors in each ratio, the total differential of each one of them is calculated, showing a clear dependence with respect to the positional errors of the output links (previously defined) of the mechanism. The sum of the theoretical transmission ratio and its respective error provides the real transmission ratio. Furthermore, the described methodology allows for determining the sensitivity (influence coefficients) in the transmission ratios due to errors inherent in the link lengths. Finally, the presented analytical approach is numerically implemented through an example of articulated parallelogram design, principally characterizing in graphic form the transmission ratios in their regions of permitted movements and blocking positions, for a specific IT degree of precision of the bilateral dimensional tolerances of their functional geometric parameters, with the objective of analyzing every aspect related to the performance of the mechanisms. This formalism is validated through three particular design cases using a CAD model in a simulation module of kinematic motion analysis; additionally, the evolution of the transmission angle is discussed. The methods and conclusions proposed in this document also leave open the way as future work to study separately the magnitudes and signs of the positional errors and the transmission ratio, or even the influence coefficients themselves, in order to assign the most convenient degree of IT precision for each link in the mechanism with the purpose of reducing errors in the designs and obtain better efficiency in the transmission ratio.