Implementação de modelo de folga de backlash dinâmico em engrenagens para análise de dinâmica de rotação

Abstract

Gearboxes are essential components in industrial rotating machinery, but nonlinear phenomena such as dynamic backlash and time-varying mesh stiffness (TVMS) introduce high complexity to their overall structural response. Thus, it becomes necessary to develop tools capable of adequately representing system behavior and aiding in the understanding of the involved phenomena. In this context, the objective of this study was to develop a mathematical model capable of interpreting such effects and relating them to rotational dynamics. To this end, a nonlinear meshing model was coded in Python and integrated into the Rotordynamic Open-Source Software (ROSS) library, based on the Finite Element Method (FEM). The original formulation of 3 degrees of freedom (DOF) per gear was expanded to 6 DOF per node, enabling the inclusion of geometric parameters and helical gear forces. Additionally, a hyperbolic tangent smoothing technique was incorporated to handle numerical discontinuities and optimize the convergence of time integrators. To solve the differential equations of motion, the Newmark method combined with the Newton-Raphson algorithm and adaptive time stepping was used. A simplified analytical model was also developed to determine the natural frequencies of the gear pair. The results were validated against literature, showing natural frequencies perpendicular and parallel to the Line of Action (LOA). Bifurcation diagrams confirmed clear transitions between periodic and chaotic motion regimes at high speeds. Finally, the integrated methodology was applied to a real practical case of a large-scale industrial gearbox operated by Petrobras (input rotation of 1953 rpm and output of 12225 rpm) employing the full mesh stiffness profile calculated by ROSS. For this case, the smoothing tool indicated a reduction of approximately 7.3% in total computational processing time without loss of physical representativeness of the dynamic system. It is concluded that the developed approach represents a solution for mechanical design and fault diagnosis in complex industrial transmissions