This work analytically derives design rules to suppress certain harmonics of planet mode response in planetary gear dynamics through mesh phasing. Planet modes are one of three categories of planetary gear vibration modes. In these modes, only the plantes deflect while the carrier, ring, and sun gears have no motion (Lin, J., and Parker, R. G., 1999, ASME J. Vib. Acoust., 121, pp. 316-321; J. Sound Vib, 233(5), pp. 921-928). The dynamic mesh forces are not explicitly modeled for this study; instead, the symmetry of planetary gear systems and gear tooth mesh periodicity are sufficient to establish rules to suppress planet modes. Thus, the conclusions are independent of the mesh modeling details. Planetary gear systems with equally spaced planets and with diametrically opposed planet pairs are examined. Suppression of degenerate mode response in purely rotational degree-of-freedom models achieved in the limit of infinite bearing stiffness is also investigated. The mesh phasing conclusions are verified by dynamic simulations of various planetary gears using a lumped-parameter analytical model and by comparisons to others' research.
Gear noise and vibrations remain key concerns in many transmission applications. The dynamic loads or errors at the gear mesh excite the components in a gear box. Typical transfer path for the noise and vibration includes gears, shafts, bearings, and the housing or the base connected to the housing. Low frequency vibrations propagate through structures as structure-borne noise, and higher frequency vibrations typically manifest as air-borne noise. The housing due to its large surface area acts as a good radiator of sound. Thus, it is imperative that the dynamic analysis of the entire gearbox, including the housing, is essential in understanding its noise and vibration characteristics.Lumped parameter models are commonly used to model gear dynamics, where various components of the gear box are represented by lumped inertias and stiffness. The gear mesh excitation, which is the source for the dynamics, has to be provided externally either as time-varying mesh stiffness or as the static transmission error [1]. General purpose finite element software may also be used to solve gear dynamics. But it requires refined meshes near the contact zone for accurate gear tooth contact modeling. Moreover the local refinement needs to keep moving as the gears rotate, thus making it practically infeasible to solve the time domain gear ABSTRACTIn this paper we present a time-domain dynamic analysis of a helical gear box with different housing models using a unique finite element-contact mechanics solver. The analysis includes detail contact modeling between gear pairs along with the dynamics of gear bodies, shafts, bearings, etc. Inclusion of the housing in the dynamic analysis not only increases the fidelity of the model but also helps estimate important NVH metrics, such as dynamic load and vibration transmission to the base, sound radiation by the gearbox, etc. Two different housing models are considered. In the first, the housing is represented by a full FE mesh, and in the second, the housing is replaced by a reduced model of condensed stiffness and mass matrices. Component Mode Synthesis (CMS) methods are employed to obtain the reduced housing model. Results from both the models are successfully compared to justify the use of reduced housing model for further studies.Steady state sound radiation by the gear box housing is then studied in the frequency domain using a boundary element solver. The housing frequency response, which is the boundary condition for the acoustic analysis, is estimated using two different methods. In one method, the response is computed from the generalized coordinates and component modes using modal superposition, in the other the bearing dynamic loads are used to perform forced response analysis on the full FE mesh of the housing. Thus, a template for end-to-end solution to predict radiated noise from a gear box is established.
Vibration induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The two-dimensional finite element model is developed from a unique finite elementcontact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing conclusions, however, are not valid in the chaotic and period-doubling regions.
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