Nonlinear dynamics of single-stage gear transmission

In this work, model tests were carried out, which are mainly focused on the identification of zones in which the motion of gear transmission occurs chaotically. To identify chaotic areas a procedure was used to estimate the largest Lyapunov exponent. The formulated gear model includes variable meshing stiffness and gear backlash characteristics, which was approximated by a logarithmic function. The method of bond graphs was used to derive differential equations of motion, on the basis of which the effect of the error of cooperation between gears and the average load transmitted by the gear was examined. It has been shown that the load conditions affecting the gears have a significant effect on the character of the model solution that maps the gear dynamics. This influence is particularly noticeable in the bifurcation diagrams of steady states. On this basis, the values of the control parameter ω have been determined, for which the Poincaré cross-sections assume the geometric structure of the chaotic attractor. When plotting Poincaré cross-sections, additionally a histogram was taken into account of where the trajectory most often intersects the control plane. Such a complemented image of the graphic cross-section provides important information on the evolution of chaotic attractors.