This work investigates steady deformations, measured spectra of quasi-static ring deformations, natural frequencies, vibration modes, parametric instabilities, and nonlinear dynamics of high-speed planetary gears with an elastically deformable ring gear and equally-spaced planets. An analytical dynamic model is developed with rigid sun, carrier, and planets coupled to an elastic continuum ring. Coriolis and centripetal acceleration effects resulting from carrier and ring gear rotation are included. Steady deformations and measured spectra of the ring deflections are examined with a quasi-static model reduced from the dynamic one. The steady deformations calculated from the analytical model agree well with those from a finite element/contact mechanics (FE/CM) model. The spectra of ring deflections measured by sensors fixed to the rotating ring, space-fixed ground, and the rotating carrier are much different. Planet mesh phasing significantly affects the measured spectra. Simple rules are derived to explain the spectra for all three sensor locations for in-phase and out-of-phase systems. A floating central member eliminates spectral content near certain mesh frequency harmonics for out-of-phase systems. Natural frequencies and vibration modes are calculated from the analytical dynamic model, and they compare well with those from a FE/CM model. Planetary gears have structured modal properties due to cyclic symmetry, but these modal properties are different for spinning systems with gyroscopic effects and stationary systems without gyroscopic effects. Vibration modes for stationary systems are real-valued standing wave modes, while those for spinning systems are complex-valued traveling wave modes. Stationary planetary gears have exactly four types of modes: rotational, translational, planet, and purely ring modes. Each type has distinctive modal properties. Planet modes may not exist or have one or more subtypes depending on the number of planets. Rotational, translational, and planet modes persist with gyroscopic effects included, but purely ring modes evolve into rotational or one subtype of planet modes. Translational and certain subtypes of planet modes are degenerate with multiplicity two for stationary systems. These modes split into two different subtypes of translational or planet modes when gyroscopic effects are included. Parametric instabilities of planetary gears are examined with the analytical dynamic model subject to time-varying mesh stiffness excitations. With the method of multiple scales, closed-form expressions for the instability boundaries are derived and verified with numerical results from Floquet theory. An instability suppression rule is identified with the modal structure of spinning planetary gears with gyroscopic effects. Each mode is associated with a phase index such that the gear mesh deflections between different planets have unique phase relations. The suppression rule depends on only the modal phase index and planet mesh phasing parameters (gear tooth numbers and the number of planets). Numerical integration of the analytical model with time-varying mesh stiffnesses and tooth separation nonlinearity gives dynamic responses, and they compare well with those from a FE/CM model. Closed-form solutions for primary, subharmonic, superharmonic, and second harmonic resonances are derived with a perturbation analysis. These analytical results agree well with the results from numerical integration. The analytical solutions show suppression of certain resonances as a result of planet mesh phasing. The tooth separation conditions are analytically determined. The influence of the gyroscopic effects on dynamic response is examined numerically and analytically. / Doctor of Philosophy / Planetary gears in aerospace applications have thin ring gears for reducing weight. These lightweight ring gears deform elastically when transmitting power. At high speed, Coriolis and centripetal accelerations of planetary gears become significant. This work develops an analytical planetary gear model that takes account of an elastically deformable ring gear and speed-dependent gyroscopic (i.e., Coriolis) and centripetal effects. Steady deformations, measured spectra of quasi-static ring deformations, natural frequencies, vibration modes, parametric instabilities, and dynamic responses of planetary gears with equally-spaced planets are investigated with the analytical model. Steady deformations refer to quasi-static deflections that result from applied torques and centripetal acceleration effects. These steady deformations vary because of periodically changing mesh interactions. Such variation leads to cyclic stress that reduces system fatigue lives. This work evaluates planetary gear steady deformations with the analytical model and studies the effects of system parameters on the steady deformations. Ring deflections measured by sensors fixed to the rotating ring gear (e.g., a strain gauge), space-fixed ground (e.g., a displacement probe), and the rotating carrier have much different spectra. The planet mesh phasing, which is determined by gear tooth numbers and the number of planets, significantly influences these spectra. Simple rules are derived that govern the occurrence of spectral content in all the three measurements. Understanding these spectra is of practical significance to planetary gear engineers and researchers. Planetary gears have highly structured modal properties due to cyclic symmetry. Vibration modes are classified into rotational, translational, and planet modes in terms of the motion of central members (sun and carrier). The central members have only rotation for a rotational mode, only translation for a translational mode, and no motion for a planet mode. Translational modes have two subtypes, rotational modes have only one subtype, and planet modes may not exist or have one or more subtypes depending on the number of planets. For each subtype of modes, all planets have the same motion with a unique phase relation between different planets and the elastic ring gear has unique deformations. Understanding this modal structure is important for modal testing and resonant mode identification in dynamic responses. Sun-planet and ring-planet mesh interactions change periodically with mesh frequency. These mesh interactions are modeled as time-varying stiffnesses that parametrically excite the planetary gear system. Parametric instabilities, in general, occur when the mesh frequency or one of its harmonics is near twice a natural frequency or combinations of two natural frequencies. Closed-form expressions for parametric instability boundaries that bound the instability region are determined from the analytical model. Certain parametric instabilities are suppressed as a result of planet mesh phasing. Near resonances, vibration can become large enough that meshing teeth lose contact. The analytical model is extended to include the tooth separation nonlinearity. Closed-form approximations for dynamic responses near resonances are determined from the analytical model, and these analytical results compare well with those from numerical simulations of the analytical model. Tooth separation conditions are analytically determined. The influences of planet mesh phasing and Coriolis acceleration on dynamic responses near resonances are investigated numerically and analytically.

Les roulements à billes sont l’un des composants les plus importants et les plus critiques dans les turbomachines ou dans les éoliennes. Les butées à billes rencontrées dans les pieds de pales d’éoliennes doivent supporter des chargements très élevés, avec des bagues très fines par rapport aux dimensions du palier. Le roulement à quatre points de contact à haute vitesse est un autre exemple de bagues minces, où la cinématique interne est fortement liée à la géométrie des pistes qui elle, dépend de la rigidité des bagues et des logements. Pour cette application, les pistes intérieures et extérieures sont archées et bien souvent constituées de deux demi-bagues. La souplesse de ces dernières ainsi que celle du logement modifie la géométrie interne et l’interaction entre les composants. Il est proposé dans cette thèse un modèle permettant de dimensionner des roulements à billes à quatre points de contact, principalement dans le domaine d'application des turbines aéronautiques. Ce modèle est capable de rendre compte des déformations globales des bagues et de leur logement et environnement proche. Un ensemble de travaux existants et différentes possibilités envisagées pour la mise en place d’un modèle de roulement à bagues déformables est présenté pour définir une stratégie de couplage efficace entre un modèle analytique et un modèle éléments finis. La prise en compte de la souplesse des bagues s’appuie sur la résolution préalable d’un problème semi analytique de modélisation avec bagues rigides. Ensuite un couplage entre les résultats de ce modèle et un modèle éléments finis est réalisée pour prendre en compte la souplesse des bagues. Des choix sont nécessaires pour ce couplage, notamment sur la modélisation des contacts billes/bagues par l’utilisation de forces nodales pour simuler fidèlement ces contacts. Plusieurs méthodes sont ainsi évaluées pour calculer au mieux la nouvelle géométrie de la bague, en observant son comportement lorsqu'elle est soumise au contact d'une bille. Finalement, cette souplesse est intégrée au modèle semi analytique pour comparer le comportement d'un roulement à bagues rigides à celui d'un roulement à bagues souples. Des premiers résultats numériques sur une géométrie académique montrent des variations des grandeurs internes du roulement (angles de contact, ellipse de contact) ainsi qu’une meilleure répartition du chargement. Des essais ont été réalisés pour valider expérimentalement le modèle développé dans cette thèse. Les comparaisons par mesures du déplacement axial des bagues et des ondulations en surface des bagues montrent que la souplesse du support n’est pas négligeable, même dans le cas de bagues larges. Egalement, ces essais ont démontré la pertinence du couplage entre un modèle analytique et un modèle éléments finis pour rendre compte des déformations de bagues de roulements à billes à quatre points de contact. / Ball bearings are one of the most important and most critical part in turbomachine and wind turbine. They require a careful design in order to create reliability and economic relevance, which leads to compact bearings with high dynamic and static load capacity. Then ball bearing encountered in wind turbine must carry high loads, with thin rings regarding mean diameter of the bearing. High speed four point contact ball bearing is another example of thin rings, where internal kinematics is highly linked to raceway geometry, and raceway geometry depends on rings and housing stiffness. For this application, internal and external ring are arched and frequently made of two parts. There stiffness change the internal geometry and interaction with bearings components. It implies a change in load distribution and internal speed. As a consequence bearing and housing stiffness is an important parameter in order to estimate the admissible loads for the bearing. This thesis propose a model for the four point contact ball bearing, mainly for aeronautical turbine engine. This model can account for structural ring deformation as well as of housing deformation. Some existing work and different possibility for such a model are presented in order to define a coupling strategy between an analytical model and a finite element model. The accounting for ring stiffness rely on the resolution of a rigid ring semi analytical model. Then a coupling between this results and finite element results is done in order to account for ring stiffness. Some choices are made for the coupling, especially on ball/ring interaction by using nodal forces to model contact with fidelity. Some methods are evaluated to compute new ring geometry due to contacts with balls. Finally this stiffness is integrated in the semi analytical model in order to compare the behavior of rigid ring bearing with deformable ring bearing. First numerical results on an academic bearing shows change in internal parameter (contact angle, contact ellipse) and a better load distribution. Some experimental tests are made in order to validate the model presented in this thesis. Comparison on axial displacement and ring surface undulation shows that housing stiffness is not negligible even with large ring bearing. This tests show the relevance of a coupling between an analytical model and a finite element model in order to account for ring deformation in four point contact ball bearing.

Grandes bague de roulement utilisés dans éolienne sont l'un des éléments de transmission de charge importantes de ces machines tournantes. Ces roulements fonctionnent grâce à des cycles de charge et de la fréquence et de l'expérience des défis complexes tribologiques sévères. Le coût de remplacement de ces paliers est très élevé et conduit aussi à quantité importante de temps d'arrêt. Il est donc important de comprendre certains des principaux problèmes de conception et tribologiques de ces roulements. Quatre points type de roulement de l'anneau de contact de rotation a été considéré comme une base de référence pour cette étude pour démontrer les questions de contact de troncature et d'échec de la cage pour les roulements de hauteur. Un palier de contact à deux points de remplacement est proposé d'éliminer le contact troncature et de réduire la force de la cage accumulation. Les méthodes de conception et d'analyse démontré dans cette étude peuvent être facilement étendus à lacet paliers ainsi que d'autres grands roulements utilisés dans l'industrie. / Large slewing ring bearings used in wind turbine are one of the important load transmitting elements of these rotating machines. These bearings operate through complex load and frequency cycles and experience severe tribological challenges. The cost of replacement of these bearings is very high and also leads to significant amount of down-time. It is therefore important to understand some of the major design and tribological issues in these bearings. Four-point contact slewing ring bearing type has been considered as a baseline for this study to demonstrate contact truncation and cage failure issues for pitch bearings. An alternate two-point contact bearing is proposed to eliminate contact truncation and reduce the cage force build-up. The design and analysis methods demonstrated in this study can be easily extended to yaw bearings as well as other large bearings used in the industry.