Three-Dimensional Nonlinear Dynamics and Vibration Reduction of Gear Pairs and Planetary Gears

This work aims to provide insight into the three-dimensional vibration of gears by investigating the mechanisms of excitation and nonlinearity coming from the gear tooth mesh. The focus is on gear pairs and planetary gears.;The forces and moments generated at the gear tooth mesh cause three-dimensional relative displacements of contacting gear tooth, which disengage portions of gear tooth surface (partial contact loss) nominally designed to be in contact. While complete tooth disengagement (total contact loss) is the most commonly recognized nonlinearity in gears, partial contact loss is also a source of nonlinearity. A three-dimensional lumped-parameter gear mesh model produces the net force and moment at the gear mesh due to an arbitrary load distribution on the gear tooth surface using a translational and twist spring. Thus, the three-dimensional lumped-parameter model, named the equivalent stiffness model, concisely captures the nonlinear behavior. Both translational and twist stiffnesses depend strongly on spatial displacements at the gear mesh, and so are highly nonlinear and time-dependent. The twist moment periodically fluctuates over a mesh cycle, causing twist vibrations.;With gear pairs, there is a twist vibration mode, where the twist stiffness is active, and a mesh deflection mode, where the translational stiffness is active. The dynamic response is nonlinear due to partial and total contact loss. The dynamic displacements distorts the instantaneous dynamic contact loads compared with the static design contact loads.;To quantitatively assess nonlinear vibrations of gear pairs, a method is developed to give a closed-form analytical expression of the frequency-amplitude curve. Partial contact loss is captured with quadratic and cubic nonlinear terms. The vibration excitation comes from the time-dependent fluctuations due to periodic tooth engagement. The closed-form solution, found using the method of multiple scales, enables immediate calculation of nonlinear dynamic response, stability of the response, and the frequency range of total contact loss.;With planetary gears, modes of vibration are crucial in understanding and reducing vibration. For equally-spaced planetary gears, all vibration modes belong to three types: 1) Rotational-axial modes (named for the displacements of the central members), 2) Translational-tilting modes (named for the displacements of the central members), and 3) Planet modes (only planets are active). This classification is mathematically derived. It depends only on planet spacing, and thus persists for axial asymmetry, e.g., use of helical gears, overhung shafts, different bearings at shaft ends.;Planet spacing and gear tooth counts in planetary gears, when selected based on a set of rules, eliminate some force and moment fluctuation harmonics. It is shown that these fluctuations stem from the relative phase between planet gear meshes. The set of rules that eliminate force and moment fluctuations are derived. The derivation relies solely on the circumferential symmetry, so it is equally valid for static and dynamic conditions, elastic or rigid components, and for axially asymmetric systems.