A new analytical model to predict the transversal deflection under load of stepped shafts

Abstract

Shaft deflection is a common phenomenon in machine design that has an important influence on the behavior of many transmission elements supported by the shafts, like gears, pulleys, sprockets, etc. This deflection can be estimated efficiently by using 1D models associated to beam theories, but machine shafts are usually stepped shafts and it has been demonstrated that classical beam theories do not predict accurately the deflection of this type of shafts. Thus, in this work, an equivalent model of the shaft to be used in conjunction with the Timoshenko beam theory has been proposed to improve the accuracy of the computed deflections. The new model substitutes the steps of the shaft by a linear variation of the diameter of the cross-section, removing the discontinuities caused by these steps. The slope of the linear variation provides new variables that are optimized locally and globally for the best coincidence with the deflection obtained with a realistic finite element model of the shaft. Then, two different approaches (with locally and globally optimized slopes) are proposed and their accuracy is investigated with 56 cases of study and with the analysis of a realistic gear drive. The results demonstrate the higher accuracy of the proposed model and the improvement with respect to the direct application of the Timoshenko beam theory.