Modelling of the gear backlash

Abstract

In this paper, model tests were carried out, which mainly focused on the numerical mapping of the characteristics of the gear backlash. In particular, the effect of the approximation function on the value of the largest Lyapunov exponent was investigated. The generated multi-coloured maps served as a criterion for verifying the results of the model tests. The analysis involved polynomial functions of the third degree, its modified structure, and the logarithmic equation. As a pattern to which the results of model tests were derived, the mathematical model of the gear was used, in which the characteristics of the backlash were modelled with a non-continuous function describing the so-called dead zone. We show that the dependencies described by polynomials imprecisely describe the dynamics of a single-stage gear transmission mechanism. Additionally, the value of the logarithmic coefficient, which approximates the backlash characteristics, for which the Poincare cross section corresponds with its model counterpart, is determined. The coefficient of the logarithmic function was optimized on the basis of bifurcation diagrams, which were used to determine its horizontal asymptote. The elimination of discontinuities significantly simplifies computer simulations and increases their effectiveness without losing information about the phenomena occurring in the gear transmission.