An extensive literature survey of the subject of gear dynamics is undertaken and the increasing recognition of the role of the lubricating oil film in this field, especially as a damping source, is highlighted. The oil film separating the mating surfaces of involute spur gears is analysed assuming hydrodynamic conditions, rigid teeth (as far as the film shape is concerned) and pressure dependent viscosity. Gear tooth mesh stiffness is expressed as a function of the dynamic load and the position of contact. simple model of a pair of spur gears is subjected to a transient response analysis and the behaviour of the lubricating oil film observed. According to the motions of equivalent masses of the apars under these transient conditions damping due to the oil film is determined. The numerical solutions obtained at various operating conditions are combined to form an approximate formula to predict the damping ratio in terms of the dynamic tooth load, rolling speed of the tooth surfaces and the viscosity of the lubricating oil. A digital computer simulation of the dynamic motion of the pair of gears is carried out incorporating the above damping ratio formula. The actual load sharing between the pairs of teeth (when more than one pair of teeth are in mesh), considering the tooth deflections, pitch errors, oil film thicknesses and the differences in mesh stiffnesses, is taken into account. The variations of the total maximum dynamic load and the maximum tooth load are studied under different nominal loads, contact ratios, oil viscosities and pitch errors over a wide range of speeds covering the resonance area. The variations of the dynamic load, individual tooth load, mesh stiffness and the oil film thickness during complete mesh cycles are also analysed under different operating conditions to identify particular areas where high loads and minimum film thicknesses occur. Theoretical results are compared with the experimental results obtained on a back-to-back gear test rig.

The load distribution across the contact line(s) of spur gears is essential for the gear designer to be able to accurately stress gears for a given application. Existing gear standards (eg BS 436, AGMA 218 DIN 3990) use a thin slice (2D) model of the meshing gear teeth to estimate the contact line load distribution. This approach clearly fails to model properly teeth subjected to mal-distributed loads, since the buttressing effect of adjacent tooth sections tends to flatten the load distribution. Non-linear tooth modifications such as crowning and some forms of lead correction are also inadequately modelled. This thesis sets out the theory for a 3D elastic model of widefaced- spur gears that has been implemented on a micro-computer. The required 3D contact line influence coefficients for standard form zero modification spur gears with 18 to 100 teeth have been determined by Finite Element analysis. These theoretical values have been compared with results from experiments carried out on a complete large module (18. Omm) wide-faced spur gear. The effect of the various elemental gear errors (eg pitch, profile, lead) and profile modifications have been investigated using the 3D computer model; the results compared with results predicted by the existing gear design standards. The existing gear standards use 2D tooth compliance values up to 50% less than those obtained in this work, largely due to inadequate modelling of the gear body compliance, which is most significant in gear wheels. Comparison of 3D tooth compliance values shows a large discrepancy between author's results again due to inadequate modelling of the gear body.

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Swerea KIMAB is one of Europe’s leading institutes for metallic materials and excels in many different areas. KIMAB’s main advantage is its open internal structure where all groups and projects can share information and knowledge between the sections. After moving to new premises it was noted that the surface fatigue test rig that had been used had started to leak oil and was no longer suitable for new projects. Because surface fatigue testing will be of importance in the future for the development of new alloys that will replace existing alloys in for example gears, this thesis was created as a foundation for how a new test rig should be acquired.This thesis goal is to describe KIMAB’s requirements in a test rig and to do a market screening over existing solutions for test rigs. Thereafter a designed concept shall be developed for KIMAB’s specific requirements and describe how it should be made. These different parts will be the ground for how KIMAB should continue in the acquisition of a new rig.The base of the thesis is a literature study in surface fatigue, its mechanics and how these can be affected to give the desired test scenario. This information is used to make a market screening for suitable test rigs that fulfils the requirements. Thereafter a concept generation is made and evaluated during a meeting on KIMAB. The chosen concept will then be designed to be user friendly, robust and as reliable as possible.When the final concept is done and quotes from manufacturers have been gathered for most of the parts in the design, a time and cost estimation was made to give the reader the chance of deciding which alternative is the most suitable for KIMAB. This choice has to be made with regard to future projects and how the market will develop for the ordering of surface fatigue testing. / Swerea KIMAB är ett av Europas ledande institut för metalliska material med spetskompetens inom ett flertal områden. KIMAB’s stora fördel är dess öppna interna struktur där alla grupper och projekt kan dela information och kunskap emellan avdelningarna. Efter att KIMAB flyttade till nya lokaler uppmärksammades det att den kontaktutmattningsrigg de använt under flera år började läcka olja och inte längre var optimal för dagens projekt. Eftersom kontaktutmattning kommer att bli ett viktigare inslag i framtiden då nya legeringar kommer att ersätta befintliga i exempelvis kugghjul, så skapades detta examensjobb som en grund för hur en ny rigg skall införskaffas.Detta examensjobb har som mål att beskriva KIMAB’s behov i en testrigg samt att göra en marknadsundersökning efter befintliga riggar. Därefter skall ett förslag designas på hur en testrigg som är anpassas just för KIMAB’s behov skall kunna byggas. Dessa delar skall sedan ligga som grund för hur KIMAB skall gå vidare i införskaffandet av en ny rigg.Examensarbetets grund ligger i en litteraturstudie i kontaktutmattningsskador samt mekanismerna bakom dessa skador och hur de kan påverkas i ett test scenario. Denna kunskap användes för att undersöka marknaden efter lämpliga riggar som kan uppfylla kraven. Därefter skapades ett antal koncept som utvärderades under ett möte på KIMAB. Det koncept som valdes har därefter designats för att vara så användarvänligt och tillförlitligt som möjligt.När konceptet var färdigställt och prisuppgifter hämtats in på merparten av delarna så har en kostnads och tidskalkyl utförts för att låta läsaren avgöra vilket alternativ som passar bäst för KIMAB. Detta val måste baseras på framtida projekt och hur marknaden ser ut för beställning av kontaktutmattningsprover.

Ce travail de thèse, réalisé dans le cadre d’une collaboration avec la société Hispano-Suiza (groupe SAFRAN) porte sur la modélisation et l’analyse des vibrations et surcharges dynamiques sur les engrènements d’un réducteur aéronautique multi-étage constitué de plusieurs cascades d’engrènements situés dans des plans décalés. Une modélisation originale est proposée qui repose sur le couplage entre la résolution des équations du mouvement et la prise en compte des conditions de contact instantanées sur l’ensemble de tous les engrènements. Le modèle gère les phases relatives entre engrènements ainsi que l’influence des modifications de corrections de forme des dentures. En s’appuyant sur de nombreux exemples d’application, il est montré que des corrections de profil linéaires et symétriques adaptées permettent d’abaisser très sensiblement le niveau de surcharges dynamiques sur une plage de vitesse de rotation importante. Il est également confirmé que les efforts dynamiques sur les engrènements sont fortement corrélés aux amplitudes des variations temporelles des erreurs de transmission locales pour un fonctionnement quasi-statique sous charge. Considérant différentes architectures de transmissions, des optimisations conduites grâce à un algorithme génétique montrent que les paramètres optimaux de corrections de profil sont à choisir au voisinage d’une ‘courbe maîtresse modifiée’, initialement définie pour un seul engrènement et dont les propriétés semblent pouvoir être extrapolées à des systèmes avec plusieurs engrènements simultanés. Les résultats de simulations prouvent que l’ensemble des corrections sélectionnées sur les courbes maîtresses modifiées améliorent sensiblement le comportement dynamique des systèmes d’engrenages multi-étages. En revanche, pour des systèmes soumis à des niveaux de charge différents, des corrections optimales courtes semblent plus adaptées en termes de dynamique de l’engrènement tandis que des corrections optimales longues semblent plus efficaces en ce qui concerne le niveau de contraintes en pied de dents. / This research work, conducted in cooperation with Hispano-Suiza (SAFRAN group), is focused on the modelling and analysis of vibrations and dynamic loads in aeronautical multi-mesh gears comprising several spatial gear arrangements (idler gears, several pinions on one shaft). An original model is presented which relies on the simultaneous solution of the equations of motion and the instant contact conditions for all the tooth contacts and all the meshes. The phasing between the various meshes along with tooth shape modifications are integrated in the simulations. Based on a number of simulation results, it is shown that linear symmetric profile modifications can substantially reduce dynamic tooth loads over of broad range of speeds. It is also confirmed that dynamic mesh forces are strongly correlated with the time-variations amplitudes of local quasi-static transmission errors under load. Considering several gear arrangements, tooth profile optimisations have been performed using a genetic algorithm which indicate that optimum reliefs always lie in the vicinity of the so-called ‘modified Master Curve’ initially defined for a single pinion-gear pair. This finding suggests that the concept of Master Curve can probably be extended to more complex gear systems. The simulation results prove that all the optimal profile modifications on the Master Curve improve the dynamic tooth loading conditions in multi-mesh gears. However, for systems submitted to several load levels, short optimal reliefs seem preferable whereas long optimal reliefs are more effective in terms of root stresses.

Ces travaux de thèse sont le fruit de la collaboration entre la société Hispano-Suiza et le LaMCoS de l’INSA de Lyon. Dans le cadre du développement de nouveaux systèmes de propulsion, l’implantation d’un train planétaire / épicycloïdal entre la turbine et l’hélice semble être une voie intéressante pour atteindre les performances souhaitées en terme de rendement. L’augmentation des puissances transmises et la réduction des masses embarquées dans les applications aéronautiques tendent à rendre les composants de plus en plus déformables. Lors de ces travaux de recherche, un modèle dynamique de trains planétaires a été développé, incorporant les effets des erreurs de montage, des écarts de forme et des sous-ensembles flexibles. Une approche à paramètres concentrés est utilisée, intégrant notamment des éléments spécifiques d’engrenage et des éléments d’arbre. Pour les éléments d’engrenage, le formalisme des fines tranches juxtaposées est employé pour représenter les dentures. Une raideur élémentaire et un écart normal sont attribués à chacune des tranches et sont réactualisés à chaque pas de temps en fonction de la cinématique des composants et des déviations du profil des dentures. Les déformations de la couronne sont introduites à l’aide d’un anneau discret composé de poutres droites couplé aux éléments d’engrenage. Les dentures double-hélice sont modélisées par deux éléments d’engrenage d’angles d’hélice opposés liés par une poutre de Timoshenko. Finalement, les équations du mouvement sont résolues pas à pas dans le temps par un schéma de Newmark combiné à un algorithme de contact normal, permettant de prendre en compte les pertes de contact partielles ou complètes. Dans un premier temps, un certain nombre d’éléments de validation est présenté et comparé à des résultats tirés de la littérature. Afin d’asseoir notre modélisation, l’influence des erreurs de positionnement des satellites, du décalage des hélices, des erreurs de pas et des déformations de la couronne sur les distributions de charge est abordée pour différentes configurations de train planétaire. Dans un deuxième temps, l’optimisation des corrections de denture dans le but de réduire les vibrations est investiguée. Les corrections de profil sont introduites sur les engrènements de façon (i) linéaire et symétrique en tête de dents et (ii) identique pour tous les satellites mais (iii) différente selon le flanc actif. Dans ce contexte, les corrections sont tout d’abord optimisées vis-à-vis des efforts dynamiques d’engrènement à l’aide d’un algorithme génétique. Puis, leurs performances sont analysées en fonction du couple transmis et de la vitesse de rotation. Par la suite, un critère « équivalent » est dérivé, vérifié et utilisé pour étudier l’influence du décalage des hélices et d’une correction longitudinale parabolique sur ces corrections de profil optimales. Enfin, une sous-structure du porte-couronne est introduite et son impact sur les distributions de charge est exploré. / This research work was conducted at the Contact and Structural Mechanics Laboratory (LaMCoS) of LaMCoS - INSA Lyon (UMR CNRS 5259) in partnership with Hispano-Suiza (SAFRAN group). In the context of new turbo jet engine developments, a promising technological solution consists in inserting a planetary / epicyclic gear train between the turbine and the propeller which, in theory, can improve the system performance, especially in terms of efficiency. Increasing power densities and mass reduction constraints lead to more compliant structures which need to be analyzed from a dynamic viewpoint. The present work deals therefore with the dynamic modelling of planetary / epicyclic gears and the effects of assembly errors, tooth shape deviations and deformable structural components. A lumped parameter approach has been favored which combines rigid-body gear elements, beam and lumped parameters elements. A thin-slice model has been used to simulate the time-varying elastic properties of gear teeth with an elemental stiffness and a normal deviation functions attributed to every discrete cell on the contact lines (thin slice) and updated at each time step with respect to the meshing course and the instant positions of the teeth. Ring-gear deformations are introduced via a model of elastic annulus discretized into straight beam elements and connected to the gear elements. Double-helical gears are simulated by linking two gear elements of opposite hands by Timoshenko beam elements. The possibility of helix stagger is implemented by shifting the helix positions in the base plane. Finally, the equations of motion are solved step by step in time by combining a Newmark scheme and a normal contact algorithm which makes it possible to account for partial and total instant contact losses. A number of comparisons with benchmark results from the literature are presented which prove that the proposed theoretical and numerical developments are sound and can actually be used to simulate the influence of planet position errors, helix stagger, pitch errors and ring-gear deformations. The optimization of tooth shape modifications, i.e. profile and lead modifications, re dynamic mesh forces in planetary gears is tackled. Using a genetic algorithm, optimum profile modifications are derived and compared with some analytical results in the literature. Their performance over a range of loads and speeds is assessed for helical and double helical gears with rigid and flexible ring-gears. A quasi-static “equivalent” criterion based on local transmission errors is presented and commented upon. Having proved its relevance, a number of results are derived concerning the influence of helix stagger and lead crowning superimposed on optimum profile modifications. Finally, a deformable ring-gear support is introduced using a sub-structuring technique and its contribution in terms of tooth load distribution is examined.

This diploma thesis primarily deals with the transmission error issue which is one of the dominant sources of vibration in gear pairs and transmission systems. The vibrations subsequently generate noise which is often subjected to increasingly stricter demands across the industry, including the automotive one. It turns out that reducing the peak-to-peak value of the transmission error has a beneficial effect on the vibro-acoustic properties of gears and gear pairs. This thesis aims to determine the transmission error under static conditions, since a gear pair with a low static transmission error is a good assumption for a low transmission error even under dynamic effects. The resulting values of the transmission error can be influenced already during the design of the gear macro-geometry. It is also suitable to apply micro-geometric adjustments – modifications to the gear teeth. For this reason, the search part of the thesis is dedicated to theoretical knowledge, especially concerning the geometry of gears, modifications of teeth and the overall transmission error and its determination. The transmission error can be determined in several ways, including a technical experiment. However, due to time and financial reasons, this is not always possible, and therefore, the possibility of using numerical simulations is offered. In this thesis, the approach using stress-strain quasi-static contact analysis using the finite element method in Ansys Workbench software is used. The advantage is, among other things, a good comparability of results. The input to the FEM analysis is 3D CAD geometry – in this case, it is specifically a helical gear pair with parallel axes. The model/assembly of this gear pair is created in PTC Creo software fully parametrically, so it is possible to generate arbitrary gear pair configurations by changing the input parameters, which significantly saves time. At the end of this diploma thesis, the stress-strain analysis of various gear configurations is evaluated, with respect to the equivalent stress and contact pressure. Furthermore, the static transmission error – its graphs and peak-to-peak values – is determined from FEM analyses for different gear geometry, including tooth modifications, and for various loading torques. Last but not least, the effects of contact/overlap ratio and centre distance are evaluated.