Abstract
Thin-walled gears, designed for aeronautical applications, have shown very rich dynamics that must be investigated in advance of the design phase. One of the signatures of their dynamics is coupling due to the meshing teeth which stand-alone gear models cannot capture. This paper aims to investigate the dynamics of thin-walled gears considering time-varying coupling due to the gear meshing. Each gear is modelled with lumped parameters according to a local rotating reference system and the coupling is modelled by a traveling meshing stiffness. The set of equations of motion is solved by the non-linear Method of Multiple-Time-Scales (MMTS). MMTS is a very powerful technique that is widely used to solve perturbation problems in many fields of mathematic and physics. In the analyzed numerical test case, the relevance of gear coupling is demonstrated as well as the capability of the MMTS to capture the fundamental features of the system dynamics. In this study the analytical methodology, which uses MMTS, allows for the calculation of the forced response of the system made of two meshing gears despite the presence of a parametric quantity, i.e., the mesh stiffness. The calculation is performed in the frequency domain using modal coordinates, which ensures a fast computation. The result is compared with time domain analysis for validation purposes.
Highlights
The need to identify a non-linear methodology for a dynamic study of two meshing gears moves from the evidence of some critical resonances occurring during operations, which cannot be investigated by analyzing a single gear considered as a stand-alone component, but it requires the analysis of the overall system which can be made of two or more than two meshing gears, where time-varying parameters and non-linearities appear in the equations of motion
The phenomenon of dynamic coupling can be experimentally verified in industrial applications, in particular for aeronautical applications where the gears, having specific mechanical characteristics and working at critical speed regimes, show mutual interactions, which largely affect the forced response of the system
The methodology developed here applies the Method of Multiple-Time-Scale (MMTS) to compute the frequency response of a single mesh gear pair, modelled with lumped parameters, and investigate the dynamic coupling, which is established between the gears, verifying the mutual interactions and resonances induced by the phenomenon
Summary
The need to identify a non-linear methodology for a dynamic study of two meshing gears moves from the evidence of some critical resonances occurring during operations, which cannot be investigated by analyzing a single gear considered as a stand-alone component, but it requires the analysis of the overall system which can be made of two or more than two meshing gears (planetary system), where time-varying parameters and non-linearities appear in the equations of motion. Most of the works focused on the combined effect of mesh stiffness variation and backlash between the meshing teeth, which affects largely the response of the meshing gears, developing non-linear methodologies for the iterative and numerical computation of the response [2,3,4,5,6,7,8,9]. The methodology developed here applies the Method of Multiple-Time-Scale (MMTS) to compute the frequency response of a single mesh gear pair, modelled with lumped parameters, and investigate the dynamic coupling, which is established between the gears, verifying the mutual interactions and resonances induced by the phenomenon. Numerical examples of forced response are reported, based on test cases Upon these test-case analyses, the methodology is validated by means of direct time integration (DTI) of the non-linear equations of motion
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