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.

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A handheld heavy-duty nut runner, commonly used to assemble windmills and oil pipe lines, and capable of producing 4100 Nm of torque, experiences low cycle fatigue and usually fails after 20 000 cycles at the specified torque. A full assembly Finite element model of the last stage of the four-stage planetary gearbox is constructed and simulated over one complete load cycle. The results from the simulation is compared with, and used to verify a KISSsoft simulation of the same model. Using the Finite Element model, a parametric optimization is performed using a full factorial design. The results show that misalignment issues are difficult to prevent due to the planetary gearbox design. Comparing the two models shows similar characteristics and stress levels but that local differences are common. A proposed design improvement results in better load distribution in the planet – ring interaction, which was previously impaired compared to the planet – sun interaction due to deformation induced misalignment. The result shows that by balancing the rotational stiffness of the side 1 and side 2 carrier pin mountings, it is possible to reduce the contact misalignment and improve the load distribution in the gearbox. / En handhållen mutterdragare vars användningsområde innefattar bland annat montering av vindkraftverk och oljeledningar, producerar ett vridmoment om 4100 Nm. På grund av det här havererar vanligtvis verktyget av utmattning vid ca 20 000 cykler, något som tros vara kopplat till vinkelfel som uppkommer vid deformation av verktygets växellåda. Vinkelfelen resulterar i att lastfördelningen mellan kugghjulen blir skev och spänningskoncentrationer uppstår. Finita elementmetoden används för att undersöka uppkomsten av vinkelfelen och en komplett modell av hela det sista steget i den fyrstegade planetväxellådan undersöks. Simuleringen jämförs med en liknande modell i KISSsoft, dels för att bekräfta resultatet från simuleringen, dels för att undersöka skillnader och svagheter i de båda modellerna. FE-modellen används även för att bygga upp en parametrisk optimering baserat på faktoriell design. Resultatet visar att vinkelfel är svårt att motverka på grund av växellådans design och konfiguration. Jämförelsen av de två simuleringsmodellerna uppvisar liknande karaktärsdrag och spänningsnivåer men att lokala skillnader finns mellan de båda modellerna. Optimeringen resulterar i en föreslagen designförändring som visar sig förbättra lastfördelningen i planet – ring – interaktionen utan att påverka lastfördelningen i planet – sol – interaktionen. Det här är att föredra eftersom lastfördelningen mellan planet och sol är bättre än lastfördelningen mellan planet och ring. Resultatet visar också att det är möjligt att minimera vinkelfelet mellan kontaktytorna, och förbättra lastfördelningen i växellådan genom att balansera rotationsstyvheten på var sida om planeten i planetbäraren.

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The vibration of complex mechanical systems that include coupled rotating and stationary bodies motivates this work. A semi-analytical model is developed for high-speed, compliant, rotating bodies. Exploiting the axisymmetry of the rotating body, the developed semi-analytical model only discretizes the two-dimensional radial cross-section; Fourier series are used in the circumferential direction. The corresponding formulation for thin-walled, axisymmetric shells is given. Even though the body is axisymmetric, its deflection as well as external forces, constraints, and supports acting on the body are allowed to be asymmetric. These asymmetric elements can be stationary or rotating. The model includes Coriolis and centripetal effects. The prestress (or geometric) stiffness matrix that arises from external forces and constant centripetal acceleration has additional terms compared to the literature, and these terms can significantly change the natural frequencies. Discrete stiffness-damper elements, elastic foundations, and constraint equations are used to couple the rotating body to other rotating and stationary bodies. The model is developed in a stationary reference frame to avoid time-dependent coefficients in the equations of motion when coupled to stationary components. Surface constraints are developed using equivalent force relations between multiple points on the surface and a reference node. Discrete stiffness-dampers, asymmetric elastic foundation, and asymmetric constraints introduce non-axisymmetry in the system. The speed-dependent natural frequencies and complex-valued vibration modes, presence of multiple Fourier harmonics in each mode, changes to critical speeds, divergence and flutter instability phenomena, and eigenvalue veering are investigated for spinning systems with asymmetric features. The developed semi-analytical model is used for rotationally periodic systems, for example, planetary gears. Rotationally periodic systems consist of multiple vibrating, rotating central components and substructures. The model is developed in a reference frame rotating with the central component that supports the substructures. Structured modal properties of the cyclically symmetric systems and diametrically opposed systems are investigated. The modes of the spinning system are categorized into translational-tilting, rotational-axial, and substructure modes. Time-varying coupling elements act as parametric excitation in the system. Large strain energy in the coupling elements lead to large parametric instability regions. The analytical closed-form expression of the parametric instability bandwidth obtained using a perturbation method compares well with numerical results from Floquet theory. / Doctor of Philosophy / Complex mechanical systems, for example, mechanical transmission, consist of coupled rotating and stationary bodies. The vibrations of rotating bodies are transmitted to the other bodies through coupling elements. To reduce weight of the system, the rotating bodies are made thin-walled resulting in increased flexibility of the body. The existing lumped parameter/rigid body models do not account for the flexibility of these rotating bodies. Conventional three-dimensional finite element models lead to a large number of degrees of freedom in the system, increasing the computational cost. We aim to develop a computationally efficient model to analyze the dynamics and vibration of complex mechanical systems. Most rotating bodies can be approximated as axisymmetric. The axisymmetric property of the rotating body is harnessed to reduce the three-dimensional model of the body to a two-dimensional radial cross-section using Fourier series in the circumferential direction. This reduces the system degrees of freedom. Coriolis, centripetal, and prestress effects are included in the model. Discrete stiffness-dampers, elastic foundations, and constraint equations couple the rotating body to other rotating and stationary bodies. Non-axisymmetric coupling elements and forces introduce asymmetry in the system. The system model for these asymmetric systems are developed in a stationary reference frame to avoid time-dependent coefficient equations of motion. Flexible stationary bodies alter the natural frequencies and vibration modes of the system. Instabilities, critical speeds, effects of asymmetry on the natural frequencies and vibration modes of the system are investigated. The model is extended for rotationally periodic systems, for examples, planetary gears and bearings. This model is developed in the reference frame that rotates with the central component that supports substructures. Structured modal characteristics are observed for the rotationally periodic systems. Changing contact conditions act as a source of parametric excitation in systems. Parametric resonances occur when natural frequencies of vibration with large strain energy in the coupling elements sum to the excitation frequency. Parametric instability regions obtained using an analytical equation compare well with numerical results.

Un modèle de trains planétaires est proposé afin de tenir compte de l’influence d’erreurs de position et d’excentricités en lien avec d’éventuels montages ‘flottants’ sur le comportement dynamique d’une transmission. La formulation théorique repose sur le formalisme des torseurs de déplacements infinitésimaux pour simuler à la fois les erreurs géométriques et les degrés de liberté du modèle. Une des propriétés principales de cette approche est que la géométrie des engrènements et les excitations correspondantes sont couplées aux degrés de liberté, conduisant ainsi à des excitations complexes présentant des modulations d’amplitude et de phase. Les résultats de simulation sont comparés avec des mesures sur banc d’essai et un très bon accord est obtenu en terme de partage de charge entre les satellites, validant ainsi le modèle de contact développé. Enfin, des résultats d’études paramétriques portant sur le rôle de certaines erreurs ainsi que sur l’apport éventuel de solaire et/ou satellites flottants dans des applications grandes vitesses concluent ce travail de thèse. / A dynamic model of planetary gears is presented which accounts for planet position errors and eccentricities for either rigid mounts or floating members. The theoretical formulation relies on infinitesimal generalised displacement screws which can simulate both errors and deflections. A unique feature of this model is that mesh properties (geometry and excitations) are coupled with the degrees-of-freedom thus leading to complex frequency and amplitude modulated excitation sources. For a number of planetary gears, it is found that the simulated load sharing between the planets compare well with the experimental evidence thus validating the contact modelling strategy. Finally, the results of extensive parameter analyses are displayed which illustrate the role of certain errors along with the interest and drawbacks of floating sun-gears or planets in high-speed applications.

Dans le contexte environnemental actuel, l’amélioration des performances énergétiques des transmissions mécaniques par engrenage est un réel challenge. De part, leur compacité et leur arrangement axisymétrique, les transmissions mécaniques de type réducteur à trains épicycloïdaux sont de plus en plus répandues dans divers applications (éolien, aéronautique,…). Il est généralement admis que les principales sources de dissipation de puissance dans de telles transmissions sont dues au frottement aux dentures, au mode de lubrification (barbotage ou injection de lubrifiant), au piégeage d’huile entre les dents lors de l’engrènement et à la ventilation des mobiles. Il est également à noter que les pertes de puissance générées par une transmission ne peuvent être découplées de son comportement thermique. En effet, les échauffements locaux dans la transmission ont un impact sur les propriétés du lubrifiant, qui ont elles-mêmes une influence sur les pertes de puissance. A partir de ce postulat, le travail présenté dans ce manuscrit propose un modèle numérique permettant de quantifier les différents postes de pertes de puissance générées dans un réducteur épicycloïdal, lubrifié par injection d’huile, pour une application aéronautique. / In the current environmental context, gearbox efficiency has become a major issue. Because of their compactness and axi-symmetric arrangement, planetary gearboxes are widely used in several applications (such as wind, aerospace…). It is generally accepted that total power losses in such gear transmissions can be decomposed into the contributions of the friction between teeth, the lubrication process (oil splash or jet), the oil trapping during meshing and tooth windage. It is noticed that power losses produced by a mechanical transmission cannot be dissociated from its thermal behavior. Indeed, the local warmings impact lubricant physical properties and these last have influence on power losses in return. Based on this postulate, the purpose of this work is to develop a numerical model in order to estimate the power losses generated into a jet lubricated planetary gear train, for an aeronautical application.

Due to their large torque-speed ratio and transmission efficiency, planetary gears are widely used in the automotive industry. However, high amplitude vibrations remain their critical weakness, which limits their usage especially when new strict noise legislations come into action. A new approach to handle the instability problems of planetary gears encountered in real industrial context is presented in this work. First, the dynamic response of a planetary gear failing to pass the noise regulations is theoretically investigated through an analytical model. The equations of motion were solved using the Spectral Iterative Method. The observed experimental results correlated well with those from the developed model. In order to limit the resonance phenomena, impacts of different macro and micro-geometry modifications were analytically investigated: quadratic teeth profile, different planets positioning, different number of teeth and number of planets. Optimum modifications were retrieved and are expected to be tested experimentally on a test bench and on the truck. Finally, the analytical model’s limits and sensitivity to different parameters were investigated in order to certify its reliability, and suggestions for improvements were presented.