Nonlinear dynamics of multi-mesh gear systems

Multi-mesh gear systems are used in a variety of industrial machinery, where noise, quality, and reliability lie in gear vibration. The complicated dynamic forces at the gear meshes are the source of vibration and result from parametric excitation and tooth contact nonlinearity. The primary goal of this work is to develop mathematical models for multi-mesh gearsets with nonlinear, time-varying elements, to conduct numerical and analytical studies to understand parametric and nonlinear gear dynamic behaviors, such as parametric instabilities, frequency response, contact loss, and profile modification, and to provide guidelines for practical design and troubleshooting.; First, a nonlinear analytical model considering dynamic load distribution between individual gear teeth is proposed, including the influence of variable mesh stiffnesses, profile modifications, and contact loss. This model captures the total and partial contact loss and yields better agreement than two existing models when compared against nonlinear gear dynamics from a validated finite element benchmark. Perturbation analysis finds approximate frequency response solutions for the system operating in the absence of contact loss due to the optimized system parameters. The closed-form solution is validated by numerical integration and provides guidance for optimizing mesh phasing, contact ratios, and profile modification magnitude and length.; Second, the nonlinear, parametrically excited dynamics of idler and counter-shaft gear systems are examined. The periodic steady state solutions are obtained using analytical and numerical approaches. With proper stipulations, the non-smooth tooth separation function that determines contact loss and the variable mesh stiffness are reformulated into a form suitable for perturbation. The closed-form solutions from perturbation analysis expose the impact of key parameters on the nonlinear response. The analysis for this strongly nonlinear system compares well to separate harmonic balance/continuation and numerical integration solutions. The expressions in terms of fundamental design quantities have natural practical application.; Finally, this work studies the influences of tooth friction on parametric instabilities and dynamic response of a single-mesh gear pair. A mechanism whereby tooth friction causes gear tooth bending is shown to significantly impact the dynamic response. A dynamic translational-rotational model is developed to consider this mechanism together with the other contributions of tooth friction and mesh stiffness fluctuation. An iterative integration method to analyze parametric instabilities is proposed and compared with an established numerical method. Perturbation analysis is conducted to find approximate solutions that predict and explain the numerical parametric instabilities. The effects of time-varying friction moments about the gear centers and friction-induced tooth bending are critical to parametric instabilities and dynamic response. The impacts of friction coefficient, bending effect, contact ratio, and modal damping on the stability boundaries are revealed. The friction bending effect on the nonlinear dynamic response is examined and validated by finite element results.