LaMCoS) at INSA Lyon, in partnership with SAFRAN Helicopter Engines (formerly-Turbomeca)

I very much want to thank Jérôme Bruyère for his help and guidance for the beginning of my journey into the gearing world and his availability over the years. Many thanks also to David Dureisseix, for his positive support during these years, as well as his very substantial technical contribution and his agreement be part of this jury. I would also like to thank Siobhan Wegeler, for checking through my English. An amical thought too for Fabrice Ville for his participation in the construction of this project and his much appreciated friendly support.

I am not forgetting my first research companions, Matthieu, Mouhamad and Nina, who welcomed me in the laboratory.

To my later friends Cyril and Charlotte, Marine and Eleanor for their patience and tolerance over the years, especially during the writing of this thesis. Thanks too to all my lab colleagues.

I would like to address my final thanks to my family, especially to my parents whose love (and much needed calm) supported me during those years, as well as my brother and uncle for attending my PhD defence.

Résumé

Ces travaux de thèse ont été réalisés grâce à une collaboration entre SAFRAN Helicopter Engines (anciennement Turbomeca) et le Laboratoire de Mécanique des Contacts et des Structures (LaMCoS) de l'INSA de Lyon (UMR CNRS 5259).

Les boîtes de transmission par engrenages des moteurs d'hélicoptères convoient la puissance mécanique du turbomoteur aux accessoires (pompes, démarreur) et au rotor. Leur conception dépend des nécessités des équipements embarqués, en particulier l'allègement pour réduire la consommation en carburant. Les engrenages haute-vitesse de la transmission sont allégés grâce à des enlèvements de matière dans les corps sous la denture, les voiles-minces.

Un modèle dynamique d'engrenages a été développé pendant ce projet de recherche.

Son approche modulaire permet l'inclusion conjointe des sollicitations dues aux vibrations de Finalement, le modèle est utilisé sur les deux cas académiques validés pour visualiser les effets des corps flexibles plus en détail. Les résultats mettent en avant le rôle des effets centrifuges et des modifications de forme à haute-vitesse. Enfin, l'optimisation massique de l'engrenage grâce au design du voile est examiné et commenté.

List of Figures

Superscripts

T

Matrix transposed

General Introduction

Gears have been used for centuries to transmit power between rotating axes in heavy duty machinery but also precision mechanisms such as clepsydrae and clocks. An archetypal example of the latter is the famous Antikytera machine built by Archimedes which was capable of giving the positions of the sun, the moon and other planets in the solar system.

Research Context

The first gears defined as toothed wheels capable of transmitting power while changing speed, torque and direction of rotation were developed by the Greeks in Alexandria during the 3 rd century BC mostly in the following two areas:

 The strength of classic gear arrangements for heavy duty machines, required in mills and for irrigation.

 Accurate smaller gears used in clocks.

A typical example of heavy duty work machinery is known as "Saqiya", which was originally developed in Egypt and very popular in the 5 th century AD [START_REF] Lewis | Gearing in the ancient world[END_REF]. It was made of a horizontal gear driven by an animal operating a vertical gear lifting up buckets of water.

The main application on early clocks can be illustrated by the works of Archimedes of Syracuse (287-212 BC). Later, more sophisticated mechanisms were set up in order to predict the positions of the sun, the moon and planets.

Fig. I-1: Re-composition of the gears in the Antikythera mechanism from Archimedes [LEW93]

The most famous mechanism of this era is probably the Antikythera machine whose size was that of a shoe box and comprised 30 gears (Fig.

I-1). It was used in

Celestial Mechanics and is one of the oldest known differential systems [START_REF] Lewis | Gearing in the ancient world[END_REF].

These advanced technologies were in use in the Roman and Byzantine empires and then in Western Europe where true mechanical clocks were developed in the 13 th and 14 th centuries.

With the help of missionaries, clocks were produced as of the beginning of the 17 th century in Japan and it has been found that, even if there was no evidence of tooth profile theory at the time, Japanese clocks circa 1688 had some tooth crowning and were characterised by fairly good vibration and noise levels [START_REF] Ueda | Explores for the origin of gears in traditional Japanese clock (Wadokei)[END_REF].

The 19 th century Industrial Revolution fostered the development of both heavy duty and precision gears in order to meet the needs for higher load carrying capacities in heavy machinery This representation assumes that the contact effects remain localised close to the contact zone. In order to eliminate the singularity in the displacement field, the authors supposed that the contact displacement is nil at the tooth neutral fibre thus leading to the following equation for the contact deflection:

   

 

for the pinion and gear respectively , E  : Young's modulus and Poisson's ratio of the material Lundberg [START_REF] Lundberg | Elastische Berührung zweier Halbraüme[END_REF] proposed an alternative 3D approach by considering a band of pressure at the surface of a half elastic space and determining the normal displacement at the centre of the pressure band (contact area).

b. Tooth bending

The tooth bending component is found by assimilating a tooth to a cantilever of variable cross-section [START_REF] Weber | Formänderung und Profilrücknahme bei Gerad-und Schrägverzahnten Antriebstechnik[END_REF] submitted to a lumped force F in the line of action direction (Fig. I-5).

After equating the external load work and the cantilever strain energy, the deflection in the force direction at points on the neutral fibre can be derived as: c.

           

Elastic foundation

In reality, the connection between the teeth and gear bodies cannot be simulated by builtin-edge conditions and an additional displacement needs to be added to account for gear body compliance. Weber [START_REF] Weber | The deformations of loaded gears and the effects on their load carrying capacity[END_REF] and Weber & Banascheck [START_REF] Weber | Formänderung und Profilrücknahme bei Gerad-und Schrägverzahnten Antriebstechnik[END_REF] considered the tooth to be rooted in an elastic half-plane by introducing normal and shear stress distributions at the surface of the half-plane deduced from tooth loading. The general form of the additional deflection brought by body compliance can be expressed under the form:

      2 22
1 cos 1 .

ww fd p p fd fd uu F L M P Q tg b E S S                            (1.3)
With:

fd S : Tooth thickness at critical section L, M, P and Q: constants depending mostly on the Poisson's ratio (for

0.3, 5.2, 1.4 LP     and 0.3 Q  [WBE53]
), they also differ depending on the author

[ANK16]
Velex and Sainsot [START_REF] Sainsot | Contribution of gear body to tooth deflections-A new bidimensional analytical formula[END_REF] expanded the previous modelling strategy by considering that gear bodies could be simulated as elastic rings rather than half-planes. Other classic errors comprise eccentricities which occur when the polar axis of inertia of the gear does not coincide with the axis of rotation thus generating speed transfer modulations between the pinion and the gear along with imbalances and synchronous excitations.

i i i f i i i f f f A C h D X h B h E h F           (1.

Main Response Parameters Transmission Errors

The concept of transmission error was first introduced by Harris in 1958 [START_REF] Harris | Dynamic loads on the teeth of spur gears[END_REF] who suggested that the variations in relative displacements between gears at low speeds were at the root of vibrations developed at higher speeds. This was confirmed by a number of experiments and it has been proved theoretically that, as long as linear behaviour prevails, the time-variations of quasi-static transmission errors (TEs) under load are representative of mesh excitations for a pinion-gear pair [VEL06-07] but also multi-mesh systems [VEL16]. The classic definition of transmission error is "for any instantaneous angular position of one gear, the angular displacement of the mating gear from the position it would occupy if the teeth were rigid and unmodified" [GRE63]. Transmission error can be expressed as an angular deviation but it is also often expressed as a distance by projection on the base plane. No-load and quasi-static transmission errors are frequently used to characterize mesh excitations and the influence of shape and mounting errors whereas dynamic transmission errors and dynamic factors are metrics to quantify gear dynamic response.

In the context of defining optimum tooth profile modification, Harris introduced the notion of design load which corresponds to that particular load or torque for which TEs is theoretically constant, i.e., mesh excitations are minimal. By plotting for one given gear, the transmission error curves at different loads on the same graph, the author generated the so- a.

No-load transmission error

No-load transmission error, also referred to as kinematic transmission error, accounts for assembly and geometrical errors along with tooth shape modifications. In the absence of errors and if tooth flanks are unmodified (perfect involute profile), the no-load transmission error is nil. If the profile and lead are modified no-load transmission error is generally not nil and varies with a period equal to that of meshing. On the other hand, pitch errors, run out generate a no-load transmission error function whose period is the least common multiple of the pinion and gear rotational periods. A typical no-load transmission error signal is shown in dependency can be theoretically found for systems for which torsion is prevalent [START_REF] Velex | On the relationship between gear dynamics and transmission errors[END_REF] while Sainte-Marie et al. [START_REF] Sainte-Marie | A study on the correlation between dynamic transmission error and dynamic tooth loads in spur and helical gears[END_REF] showed that this is no longer the case when bending effects cannot be neglected.

Tooth Modifications -Influence on Dynamic Response

Type of Modifications

As seen before, gear dynamics is influenced by errors and shape deviations which can The tooth flank can also be modified in the helix direction, commonly, crowning and They are usually used to eliminate the overloads at the edge of tooth flanks, possibly created by misalignment [START_REF] Hotait | Experiments on root stresses of helical gears with lead crown and misalignments[END_REF].

Optimisation of Tooth Modifications

Tooth modifications are also used to improve gear dynamic behaviour and a number of The latest evolutions addressed the global dynamic problem including not just the gearshaft-bearing assembly but also the housing contributions [START_REF] Lim | A Review of Gear Housing and Acoustics Literature Dynamics[END_REF]. In the 1990s, Lim et al.

[ coupled the housing with a gearbox interior for dynamic analysis. Rigaud et al. the fluctuations of bearing stiffness due to rollers passage into the loaded zone. However, the contact models for rolling element bearings were not used in dynamic due to their enormous computational needs. To reduce computational costs and allow to cover a speed range (in speedsweeps), the authors proposed an equivalent lumped-parameter model in which gear tooth contact is solved by discretizing contact lines into linear spring distributions.

Flexibility Inclusion

All these models present the same limitation as they use rigid body gears and cannot represent the full flexible behaviour of the lightweight gears that are used in aeronautical transmissions. In an attempt of giving an accurate description of the flexible gear static and dynamic behaviour, many studies have been conducted over the years on these particulars wheels. In order to limit the computational times due to FE models, Eritenel and Parker [ERI12] proposed a model which accounts for the non-linearities at high speeds, such as partial and total contact losses, by introducing a full contact model and also bulk stiffness for rim deflections. (that of the hole passage in the meshing zone) superimpose on the classic mesh period and its harmonics, thus emphasising here too the reality of the interactions between the tooth contact conditions and the structural properties of gear bodies. Based on these observations, it can be anticipated that the dynamic behaviour is likely to be modified too hence justifying extensions of the classic lumped parameter models.

Dynamic Hybrid Models

Thesis Objectives

In the continuation of the previous works on the integration of flexible gears, the research reported in this memoir will focus on the numerical integration of flexibles components in dynamic gear models. In order to be compatible with design constraints in industry, particular attention will be given to limiting computational times so that extensive parameter studies are still possible. From a different perspective, the objective to develop models which can be used together for static and dynamic conditions will be pursued. To this end, an original mesh interface will be presented which is compatible with finite elements for structural parts and elastic foundations models as used in most gear dynamic models for contact simulations. The concept of hybrid models will be presented which makes it possible to combine continuous and discrete models based on the architecture and geometry of the gear system under consideration. By so doing, computational times remain limited and modelling precision can be brought to the parts with maximum flexibility and/or complex geometry only.

Elements of validations will be given based on comparisons with other model results but also experimental evidence from actual helicopter transmissions. Finally, some typical phenomena associated with high speed gears will be presented and discussed.

This chapter is aimed at introducing the main characteristics of an original dynamic model of geared transmissions which makes it possible to account for deformable, thin-rimmed pinions and gears as encountered in lightweight helicopter transmissions.

The classic FE models rely on beam, lumped parameter elements and connection elements representing pinion-gear pairs. This gear element generally comprises two rigid cylinders with all 6 degrees of freedom each, which are connected by an elastic foundation

[MAA96].
The main hypothesis is that all the contacts between the mating teeth are line contacts lying in the theoretical plane of action, tangent to the pinion and gear base circles so that tooth elasticity is transferred to the base plane and leads to a time-varying, possibly nonlinear mesh stiffness matrix. This matrix is determined by slicing all the contact lines into individual cells which are all attributed an individual stiffness ij k .

In the new approach, the rigid pinion or/and gear can be replaced by a flexible body, modelled by Finite Elements and condensed at some master nodes located at the rim and bearing interfaces. This combination leads to a fully modular model while keeping the same mesh stiffness and forcing term modelling. However, the discrete elastic foundations and the FE continuous models are not directly compatible and can generate numerical artefacts when directly connected. In order to solve this compatibility issue, a mortar based interface has been developed which makes it possible to correctly relate the discretised tooth flanks and the gear/pinion FE models.

The elements of theory needed for the development of this enhanced modelling strategy are presented in this chapter along with some technical elements associated with the solution of the dynamic equations and the introduction of centrifugal effects which need to be incorporated in high-speed transmissions.

Modular Gear Model Elements

Classical Gear Model Elements

Fig. II-1: Shaft element

The different motions are uncoupled and lead to a total stiffness matrix as the sum of the individual contributions for axial, bending and torsional displacements (eq. 2.1).

shaft tc Tor Bs Bt shaft tc Tor Bs Bt         K K K K K M M M M M (2.1)
With indexes tc : traction, Tor : torsion, Bs and Bt bending effect (defined in annex 1.)

Bearings and housing are represented by additional lumped stiffness elements at the designated node of the supported shaft elements described above.

b. Gear Body

For many applications with solid gears, it can be approximated as fully rigid or by using beam elements for wide-faced gears for which pinion bending and torsion cannot be neglected.  

1 1 1 1 1 1 k k k k k k k v w u O                 k k S T Z u S T Z ω (2.2) With k O centre of pinion or wheel, 1, 2 k 
depending on pinion of wheel (Fig. II-2).

For the cases of beam-like deformable bodies, two nodes (hence 12 DOFs) are introduced which are connected by a Timoshenko beam element in order to simulate gear body distortions (mainly the torsion and bending of wide-faced pinions) and their influence on tooth load distributions.

Flexible Body Integration -FE Condensation

For real full flexible gear body, the beam-like model presented above is not sufficient anymore, especially for lightweight webbed wheels with complex body geometries and 3D finite elements models are preferred [START_REF] Mounetou | Non-intrusive measurement of gear dynamic response[END_REF]. shows a wheel modelled by using brick Finite Elements (Fig. II-4). Its mechanical analysis is made through a finite number of equations proportional to the number of nodes present in the system.

Fig. II-4: Dhatt's brick Finite Elements [DHA81]

In the present case, the webbed body can have a particular geometry that requires a large number of elements leading to millions of degrees-of-freedom which, in practice, prevents from the direct use of finite element models. A classic method of model reduction is the condensation of FE models, also referred to as Component Mode Synthesis (CMS) which can simulate the static and modal behaviour of a complex FE structure by using a reduced number of interface nodes. This approach has two main advantages:

61

-The use of interface nodes only considerably reduces the system size and computational cost.

-The possibility of creating modular models by connecting a variety of substructures linked by their respective interface nodes.

In order to keep the interface nodes only, the set of nodes of the complete model is divided into two subsets: a) the boundary and, b) the interior nodes leading to the following partitioned DOF vector ss X , mass and stiffness ( , ss ss MK) matrices: The dynamic modes are the n modes obtained on the sub-structure while the interface nodes remain free (first mode in Fig. II-9).

,                  b ss i bb bi bb bi ss ss ib ii ib ii X X X M M K K MK M M K K (2.

Fig. II-9: First Dynamic mode [WEN98]

The main problems of this type of condensation techniques come from the possible presence of rigid-body modes with a free interface. Herting [START_REF] Herting | A general purpose, multi-stage component modal synthesis method[END_REF] includes these rigid-body modes by temporarily imposing a boundary condition on their displacement so that the system is in equilibrium between the load on the free interface DOFs and the inertial load arising from rigid-body accelerations [START_REF] Tran | Réduction de modèle en dynamique des structures et des systèmes couplés aéroélastiques[END_REF].

The choice between these two techniques mainly depends on the load distribution on the whole structure [START_REF] Herting | A general purpose, multi-stage component modal synthesis method[END_REF]. A system with heavy components, where most of the interface points are constrained, will be better condensed by using the fixed-interface method. A lighter structure, comprising lots of interface nodes that could remain free after the link to the other part of the model will be better condensed with the free-interface method. Besides, the fixedinterface method, easier to implement, is more suited when low frequencies are concerned whereas the free-interface method is more precise for higher frequencies.

Considering the particular case of interest in this memoir, i.e. high-speed light-weight gears connected to the rest of the system by a few nodes only, the Herting free-interface method has been retained, for which the resulting DOF vectors are expressed in terms of the full DOFs system as:

    0               b gh h i i u u H u u u (2.4) With:
, at any cell of the base plane can be expressed as:

Gear Element Presentation -Modular Model

      0 ij ij ij M M M     (2.6)
With 0  and  , respectively the initial tooth separation and normal approach.

The normal approach with respect to rigid-body positions in the normal direction reads (eq. 2.7).

      .. ij ij ij M M M   1 1 2 2 u n u n (2.7)
With , 12 nn the unit outer normal vector to the pinion and gear tooth flanks respectively.

The initial tooth gap [START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF] includes the sum of the normal deviations on both pinion and wheel flanks denoted   ij eM which, conventionally, is positive when material is added to the theoretical flank surfaces and negative when material is removed (profile or lead modifications etc.) (Fig. II-13).

Fig. II-13: Base place with initial tooth gap distribution on contact lines [VEL11]

Under no-load conditions, contacts occur where   ij eM is maximum so that the initial normal separation between the pinion and gear tooth flanks is derived as:

        0 max ij M ij ij M e M e M   (
        1 0 2 T ij ij O MM O                    1 1 1 ij 1 1 2 2 2 ij 2 2 n u n M O ω n u n M O ω (2.9) With       12 T OO  1 1 2 2
Xu ω u ω DOFs at the pinion and gear centres.

The proposed approach consists in replacing one or two rigid cylinders by a deformable body simulated by using finite elements as schematically represented in As for the classic model, the DOF vector associated with the pinion at any potential point of contact can be readily deduced from that at its centre by using the shifting property of screws. On the other hand, the degrees of freedom of interest for the flexible gear model comprise only translational displacements and no rotations (brick elements) and are not defined at the position corresponding to the points of contact on tooth flanks but at the root of these teeth. In these conditions, a specific tooth model needs to be developed to connect the root and flank DOFs.

        1 0 ij T ij ij p O MM M                     ij 1 1 1 1 ij 1 2 2 2 ij p 2 u n ω n M O n u n M M ω (2.

Hybrid Model -Tooth Model Development

  p M       u Z (2.11)
By so doing, the displacement at any point of contact M on the side of this slice can be determined using the shifting property of the moment fields of screws. The situation is therefore similar to that of the classic model with the difference that far more degrees of freedom are involved (those of all the slices in contact) as opposed to the 6 degrees-of-freedom at the centre of the pinion or the gear.

The main difficulty at this stage comes from the way to connect every slice to the deformable body (gear and/or pinion) model and the two following issues will be successively addressed:

a) The definition of the angular degree of freedom Z knowing that the FE model uses brick elements with translational DOFs only.

b) The problem of compatibility between a discrete Winkler foundation with independent stiffness elements (mesh elastic model) and a continuous FE model.

Two-point Connection

The link between every thin slice and a deformable body can conceptually be viewed as a two-point connection, i.e. two points

        1 2 pp pp MM MM         1 2 pp pp u u M M Z u u M M Z (2.12)
In theory these two vector fields must be identical but the connection is not fully compatible (since the circle arc can be distorted and 12 pp MM is not necessarily constant).

However, an approximate formulation is employed which relies on the minimization of the difference between the exact and approximate displacements. The following functional is introduced:

      22 1 , , , 2
J u v w        1 1 2 2 u u u u (2.13)
where , uv and

w are the three components of   p M u 0, 0, 0 0 J J J u v w J              (2.14)
After developing and rearranging the first three equations, one obtains:

    2 p M      
    1 2 p M  12 u u u (2.16)
The minimisation with respect to  (eq. 2.17) combined with the approximate result above (eq. 2.16) finally gives the expression of the angular degree of freedom associated with the thin slice:

        . .0             11 22 p p p p 1 p p p p 2 M M Z u M M Z u M M Z u M M Z u (2.17)       22 1 . 2      12 12 1 2 p p p p p p u -u M M Z M M M M (2.18)
Based on these results the displacement at the point of contact M of the gear in the normal direction of the slice can be derived by using the shifting property of the moment field of screws as:

      . . . p MM     2 2 p 2 u n u n MM Z n (2.19)
NB: similar developments can apply for the pinion but, for the sake of simplicity, only the case of a rigid pinion body and a flexible gear is considered in this section.

The new normal deflection at p M point is then be upgraded for both pinion and wheel sides with the same procedure for each tooth slice ij (eq. 2.10). This leads to an expression that separates the tooth flank line of contact, into two lines lying on rim mesh (Fig. II-17). q  is a segment with quadratic approximation (length L) for both displacement and force fields which can be linked with the finite element and ensures continuity between adjacent segments. The three nodes are denoted

        1 0 11 ( ) [ , , , ] 22 T T T pp O M M M M                1 1 1 1 2 p 2 p 1 2 u ω n n MO n κ n κ u u (2.
1,2 1,2 (1) (2)
, PP NN and The objective is formulated as to find the relation between the displacements of the stepwise and quadratic segments based on the works generated on the two grids. The work associated with the stepwise discretisation is simply:

1,2 (3)
C C C W F Q L  (2.21)
whereas, for the quadratic approximation, it reads:

    T TT q Wd    q q q q N F N Q (2.22)
where q N includes the quadratic shape functions for a line element [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF] such that:

2 4 2 11 22 T LL L L L L L                                    q N (2.
  T C C C F Q L F d    qq NQ (2.24)
hence,

1 T C Qd L       qq NQ (2.25)
which, after integration, finally gives:

  1 5 20 5 30 C Q q Q (2.26)
The demonstration so far was limited to a single DOF per node but can be readily extended to nodes with 3 translational degrees-of-freedom as is the case for the brick finite elements. on the discretised line of contact, for all three ,,

u v w , c Q is extended to the vector   1,2 p ij M u
, and the vector q Q is extended from 3 to 9 DOFs (3 nodes of 3 DOFs each) to the vector

    1,2 1,2,3 p ij N u .
The shape function matrices are duplicated for all directions, and the resulting displacement field is written from eq. 2.25, leading to visualisation Fig. II-20.

        1,2 1,2 1,2,3 p p ij ij MN  u N u (2.27)
with   The exact solution consists in a uniform displacement field and it can be observed that direct collocation induces unrealistic load spikes whereas (2.26) leads to the exact solution thus proving that the proposed methodology is sound.

  

                         2 15 2 26 2 37 2 48 11 1 1 1 1 1 42 11 1 1 1 1 1 42 11 1 1 1 1 1 42 11 1 1 1 1 1 42 PP PP PP PP                                                          (2.28)
Their contribution to the displacement 3 u of discretisation q  of the contact line can therefore be written in the form:

        1,2 1,2 1,2,3 1,2,3 pp ij ij NN  32 u P X (2.29) With     1,2 1,2,
      * 0 T ij ij M M M     m VX (2.30)
With: The global mesh stiffness matrix and forcing term vector can then be written in terms of the structural vector obtained with the new modular approach. Still considering a hybrid model with a rigid pinion body and a sub-structured gear, the global equations of motion are: Modal damping can also be used supposing that there is no energy transfer between different modes thus assuming that, for each mode i , a damping coefficient i c can be defined in terms of the modal mass and stiffness coefficient ( , ii mk) as:

          * int int 11 22 TT T T T T T pp M MM                        1 1 1 2 p 2 p V n n MO n κ I n κ I       p M    
            ** * 0 [] T ij ij ij ij ij ij ij t k M M t k t M      12 12 K V V FV ( 2
          2
2. i i i c k m   ( 2
i i i ee     (2.36)

Centrifugal Effects

For high rotational speeds, flexible bodies are submitted to centrifugal effects which might have a direct impact on rim dynamic behaviour and therefore need to be taken into account in the dynamic model. A number of preliminary simulations have been conducted on full 3D FE models of pinions and gears and the following conclusions have been drawn:

-The influence of centrifugal deformations on gear geometry key parameters, such as pressure angle, plane of action discretisation is negligible. A maximum variation of 1% change has been observed for the most flexible test case at its maximum speed (this point will be examined in the validation part).

-Changes are reported on thin-rimmed, thin-webbed applications only and a rigidbody approach is still valid for solid gears, -Considering the amplitudes of the deformed shapes caused by centrifugal forces, the hypothesis of small deflections remains sound and the meshing equations can be developed using the non-deformed gear geometries. From these hypothesis, the gear geometry parameters are considered constant for any speed [GUO01-LI08.a-13].

The centrifugal effects are extracted from the FE model and condensed along with classical mass and stiffness matrices. This way of extraction allows the system to include the centrifugal nodal displacement [START_REF] Geradin | A new approach to finite element modelling of flexible rotors[END_REF] which is the very dominant component of the studied flexible wheels [START_REF] Mounetou | Non-intrusive measurement of gear dynamic response[END_REF]. It is included into the damped system equation of motions (eq. 2.32 and 2.33) re-written according to equation 2.37 [START_REF] Lalanne | Rotordynamics". Prediction in Engineering[END_REF].

        tt             c M X C X K X F F ( 2

Fig. II-25: Simulation algorithm [VEL96]

The complementary time and contact loop makes it possible to evaluate the instant contact length and uses the instant mesh stiffness and forcing term to proceed to the time-step integration. At each time step, the first loop verifies that the contact load ij P at each segment of the line is compressive and if not, remove the contribution of the segment to the instant mesh stiffness (   0 ij kt ) and forcing term. The second loop controls that there is no interference outside the contact length L determined at the end of the first loop.

Conclusion

In this chapter, the theoretical foundations of a new gear model have been presented

which can connect classic lumped parameter and continuous finite element models which are required when it comes to thin structural elements. Such a modelling strategy leads to so-called hybrid models which can present increasing degrees of precision depending on the gear architecture specificities. More refined models can be introduced where static and dynamic elastic deflections are likely to be prominent whereas simpler (but faster) simulations can be kept for solid members. One of the interests of the methodology is that mesh elasticity and shape deviations are still accounted for by using a classic time-varying, non-linear Winkler foundation model which does not introduce additional degrees of freedom and makes it possible to still use the contact algorithm together with the integration Newmark scheme proposed by Velex et al. [START_REF] Velex | A mathematical model for analysing the influence of shape deviation and mounting errors on gear behaviour[END_REF].

In order to limit the number of unknowns, the finite elements models are reduced by using a condensation method which combines static and dynamic modes. A free-interface modal synthesis technique has been preferred.

One key point is the connection between incompatible models since Winkler The proposed modular model will be first applied to the academic test cases for which the flexible parts have been rigidified by using a very high Young's modulus (

14 2 4.10 / E N m 
) in order to directly compare with the results provided by the classic rigid-body model (providing that the gear body mass and inertia are adjusted). Comparisons will then be extended by simulating the contributions of a deformable gear body (with standard Young's modulus for steel) in order to appraise the static and dynamic influence of gear body compliance.

Numerical analyses have been performed for each reduction stage of the industrial reduction considered separately and a full FE model of the pinions and gears in contact has been set up and run under quasi-static conditions in order to determine bearing loads and the contact pressure distributions on the mating teeth. On the other hand, experimental measurements were conducted on the entire two-stage system by using displacement probes to estimate gear web dynamic deflections over a range of speeds close to the nominal speed. It is shown that the experimental evidence compare well with the model results thus proving that the proposed numerical simulation is sound and can effectively be used on actual helicopter transmissions.

All the dimensional results are presented in the reference frame of the gear element and FE lumped parameters model represented in Fig. III-1. Given the simple geometry of the web, this accessory gear is well adapted to better apprehend the effect of gear body flexibility.

Static/Quasi-Static Numerical Validation

Bearing displacements have been calculated in static conditions for the three models described above using a time-averaged mesh stiffness function and unmodified tooth shapes.

The results for bearings 1 and 2 on the pinion and gear shafts (labelled B1 and B2) in Tab. III-2

and Tab. III-3 show that all the configurations give very similar radial and tangential displacement amplitudes although some slight deviations (<5%) are reported on the gear bearings when a condensed 3D model is used.

It is also noted that the transmission is lightly loaded in relation to bearing stiffness since the maximum displacement does not exceed 1.6 µm. distributions even for centred rims suggesting that cautions must be taken in the design of thin rim and web assemblies. However, here again, for this spur gear example, the differences remain limited.

Bearing

Fig. III-5: Profile relief parameter on tooth [GHR13]

The extent of modification is frequently expressed as a distance in the base plane and can therefore be normalised with respect to the theoretical length of contact, leading to the nondimensional parameter  defined as:

Tab. III-4: RMS value of quasi-static transmission error comparison for all models

However, the contour plots of the RMS of TEs in Although, some slight difference can be found, all the simulation results are in good agreement thus proving the consistency of the various approaches in this case of rigidified gear body. The initial model response curve exhibits a peak near 1150 rad/s not observed for the hybrid models which, based on modal energy distributions, is related to the gear shaft and bearings whose models are different for the H1 and H2 simulations (3D FE based). Some discrepancies can also be noticed between 1500 and 2000 rad/s which are attributed to amplitude jumps and the difficulty to obtain stabilized time responses in this speed range. In order to investigate this point further, higher and different damping has been considered (Fig.

III-11

). The results confirm that the nature and amplitude of damping reduce the deviations noted above thus confirming the role of dissipation. (0-3000 rad/s) obtained for 10, 50 and 100 modes and proves that most of the response curve is obtained even when using a limited number of modes thus confirming the major contribution of the static deflection. In the rest of this chapter, 50 modes have been kept in order to limit the computational cost while keeping a good precision on dynamic factors.

Power Transmission -Numerical Validation Validation for Rigid Body Gears

Helical Gear Data

In helicopters, turboshafts are connected to the main transmission gearbox via power transmission gears which transfer higher powers and loads than the accessory gearboxes studied in the previous section (

Tab. III-5: Helical gear geometry

These constraints associated with the larger powers to be transmitted and the fact that mass reduction remains crucial, imply that material needs to be removed from gear bodies leading to thin web geometries. For instance, Fig. III-14 shows an academic example based on a former SAFRAN HE gear (wheel) design which will be used for validation purposes in this section. The gear shaft rests on three bearings materialized as dots in the figure and labelled B1, B2 and B3 from left to right in what follows. The pinion (input) associated with this gear is solid and is mounted on a symmetric shaft so that is can be modelled as a rigid cylinder on beam and lumped parameter elements. Simulations are extended to dynamic conditions and the dynamic factor has been calculated over a broad range of speeds. As before, the integration scheme at one given speed is initialized by using the steady-state solution obtained at the previous speed (for speed sweep).

Modal and Rayleigh damping models are employed and comparisons in dynamics will be limited to the so-called initial and H2 (fully sub-structured) models. The 1 st hybrid model, H1

has been validated the same way but, for the sake of concision, its results will not be presented here.

Flexible Web Influence

The Young's modulus of the pinion and gear webs is set to the standard value for steel and the influence of deformable gear bodies is now investigated. Here again, only the initial and H2 (sub-structured pinion and gear) models are compared.

Effect of wheel static deformation

Tab. III-8 shows (in red) the gear bearing loads under quasi-static conditions and proves that gear and pinion deflections influence static bearing load distribution to a large extent 102 particularly on bearings 2 and 3. It can be noticed that the bearing closer to the web is more loaded while the end of the shaft is unloaded. It is concluded that, in this case, web and rim compliance need to be considered in any realistic load simulation. shows the load distribution on both lines of action for a central web and an offset one for quasistatic analysis.

Bearing

Fig. III-16: Local load over a tooth for various web arrangements [LI02.b]

The same analysis has been made for the power gear.

Fig. III-19: Modal comparison between the gear analysis (H1) and full FE model of wheel alone

The results in Fig. III-19 are in line with the conclusions in [START_REF] Mounetou | Non-intrusive measurement of gear dynamic response[END_REF] and the n-diameter modes for one flexible gear or for the entire reduction unit correspond well. However, it can be noticed that the simulation of the complete reduction stage (pinion + gear) leads to more mode shapes than the wheel alone, which is consistent with the fact that adding the pinion shaft should generate additional vibration modes. It is also noticed that, depending on the nodal diameter, the agreement is variable and modes with zero or one nodal diameter tend to exhibit more deviation as opposed to the modes with more diameters.

This supports the observations of Blevins

Fig. III-20: Modes distribution for (a) FE wheel studied (b) plain circular disc [SCH97]

Analytically, the dynamic displacements in circular plates can be described using Fourier series as: at their connections with their supporting shafts thus rendering these modes more sensitive to the shafts. This reasoning probably explains why the agreement between an isolated gear and the full system is poorer when it comes to zero and one diameter modes. On the other hand, 3 nodal diameter modes are self-balanced and they are consequently far less sensitive to the boundary conditions at the gear-shaft interface.

      0 cos sin nn t c s n nn          (3.

Fig. III-27: Scheme of rim expansion effect on gear key parameters

It can therefore be concluded that the influence of centrifugal effects on mesh global parameters is of secondary importance compared with their contributions to the local contact conditions associated with tooth shape modifications.

Industrial Power Transmission Model Presentation -Numerical Validation

Gearbox Presentation

The gearbox under consideration is the two-stage power helical transmission presented which can generate localised overloads if the tooth profiles are unmodified (as is the case here).

On the other hand, one of the basic hypothesis of the hybrid model is that all tooth contacts lie 115 in the base plane thus making it impossible to simulate these pressure peaks at engagement. However, the influence remains very local and it is believed that the hybrid dynamic model can also provide relevant quasi-static pressure distributions.

Fig. III-33: Line of contact pressure on base plane

Further simulations for dynamic conditions have been performed using the hybrid model in order to assess the possible modifications brought by dynamic effects on tooth loading. obtained for a tooth above one of the holes in the gear web and proves that there is no real influence of the holes on tooth loading. Based on these results, it can be concluded that tooth loads and pressures are dependent on centrifugal effects via the web deflections but far less sensitive to vibrations (dynamic behaviour) and the presence of holes (which is certainly not desirable).

Tab. III-10: Bearing load comparison with FE model with wrench inclusion for LST influence

The results in Tab. III-10 are satisfactory with a maximum difference around 13% which prove that the modelling of the bearing and the additional forces and couples generated by LST is realistic.

Modal Behaviour

Prior to running extensive dynamic simulations with the hybrid model, the modes of the substructure shown in Neglecting damping and focusing on one mode n of natural frequency 0 n   , the displacement in the coordinate system attached to the disc can be expressed, as long as interactions between adjacent natural frequencies can be discarded, as:

      00 , cos sin n R t n t        (3.5)
with 0  , natural pulsation of the studied mode.

re-written as:

        0 0 0 0 , sin sin 2 n R t t n t n                (3.6)
When the observer is attached to the fixed frame, the wheel/disc rotational speed is  and, from the fixed frame perspective, the angular coordinate  of the stationary wave on the wheel becomes:

0 t     (3.
            0 0 0 0 0 0 sin sin 2 n F t n t n t                   (3.8)
With F  total displacement expressed from a fixed frame observer Each mode exhibits therefore two apparent frequencies for an observer attached to the fixed frame, depending on its natural frequency and its spatial distribution, i.e. its nodal diameter number n . In case of forced response, a mode is amplified when the excitation frequency coincides with its natural frequency. For conventional gears (with the exception of planetary gears with rotating planet carriers), mesh excitation are attached to the fixed frame and act on a small fraction of the disc periphery. The corresponding fundamental excitation frequency is Zt  with respect to the fixed frame for a gear (simulated as a disc) with Z teeth so that, depending on the coordinate system of the observer, resonances will occur when:

  0 0 in fixed frame in rotating frame Zn Zn             (3.9)
This qualitative demonstration can be easily reversed to find the meshing frequency of excitation as viewed from the rotating frame attached to the disc. In this case, for a known rotational speed, the excitation cannot be separated from the mode spatial repartition, i.e. its diameter number n and it will also lead to two response frequencies in the rotating frame. For each coordinate system, the evolution of the response frequencies for one given mode are plotted as Campbell diagrams (Fig. III-37).

Fig. III-37: Campbell diagrams depending on the chosen frame of expression

This figure illustrates how two critical rotational speeds 1  and 2  , independent of the coordinate system of reference, can be identified which conventionally represent the backward and forward whirl at frequency 0  respectively. It can be noticed that points attached to the wheel (disc) do not respond with a frequency equal to the excitation frequency and that the change from stationary to rotary coordinate systems must be included in the analysis.

For the industrial case treated here, the frequency of excitation in the fixed frame is the gear mesh frequency 22 127

   Z
and the conversion for each gear body nodal diameter order of resonance in the rotary coordinate system is given in Tab. III-11.

Data Acquisition Process

Experimental Device

The full power transmission (Fig.

Tab. III-12: Speed for each mode apparition

The first frequency (R1) can be identified as the forward 8 th nodal diameter mode, +8ND of the first family (no nodal circle). The second one, R2 corresponds to the backward 9 th nodal diameter mode (-9ND) of the same family whereas R3 is identified as -10ND and R4, +9ND.

It is observed that the frequency shift by the rotational speed has placed these two separate modes very close to each other in the nominal speed range, thus justifying the careful analysis of the wheel dynamic behaviour. In the simulation, modal damping factors i  are introduced and estimated from the experimental response peaks using the bandwidth method [LAL84]: It can be observed that, for high frequencies, i.e. high rotational speeds as is the case for the systems under consideration, parameter R a can be ignored so that:

1 2 ii i i i i f fQ        (3.10) with 21 i i i f f f    ,
2 2 R i i iR i b b       (3.13)
The damping matrix can therefore be taken as proportional to the average global stiffness matrix such that:

    R b  CK (3.14)

Dynamic Model

The numerical model which was described in section 1 is limited to the High Speed

Validation -Experimental Comparison

Results Analysis at Significant Speed Steps

Validation -Experimental Comparison

The summation of the amplitudes at all the orders has been performed to be able to It can be noticed that the overall agreement is reasonable thus proving that the simulation and model put in place are sound.

Conclusion

This chapter is dedicated to the validation of the new hybrid model whose main lines are presented in chapter II. To this end, three test cases have been used. The first one is a part of an academic accessory gearbox with a spur gear over a thin straight web. The second case is an academic helical gear unit corresponding to one reduction stage in a power transmission.

The gear is also a webbed wheel but its shape is more complex with an inclined and sided web under a large rim. Finally, an actual helicopter two-stage power transmission has been analysed and both experimental and numerical results have been presented and compared.

Several variants of hybrid models have been considered which combine rigid-body and finite element models of the pinion and/or the gear. A number of comparisons for quasi-static conditions are presented which all prove the validity of the modelling principles and simulation techniques. In particular, it is shown that the bearing displacements and loads found by all the models agree well and that the tooth load distribution given by the hybrid model is very similar to that obtained when using a full 3D finite element model with contact elements.

Comparisons have been extended to dynamic results and, here gain, all the models are consistent and their results agree very well providing that the damping levels and modelling are comparable. It has also been demonstrated that centrifugal effects can strongly influence tooth load patterns in the presence of a thin web/rim.

Finally, the numerical results have been confronted with the experimental measurements from displacement probes attached to the ground over a range of speeds. By taking into account the frequency shift induced by the rotation of the wheel, it is possible to reconstruct the amplitudes at a given frequency and compare data obtained in the fixed and rotary coordinate systems. By so doing, a fairly good agreement between the predictions and measurements is observed which, to a large extent, validate the theoretical developments in the previous chapter.

The The examples treated in this chapter prove the versatility of the modelling strategy and show that the proposed modular approach makes it possible to integrate most of the components in light-weight high-speed transmissions.

Static Deflection Influence -Accessory Gearbox Gear Design -Tooth Optimisation

Helix Angle Inclusion Effect -Web Adequacy

The accessory gear described in chapter III is a thin-rim spur gear mounted on a centred thin-web. The initial validation of the hybrid models H1 and H2 was performed by considering a rigid pinion and wheel (simulated by using a condensed sub-structure with a very large Young's modulus: be observed that the load distribution across the face width is not satisfactory. Because of the axial thrust generated by the helix angle, the web is deformed and tooth contacts are misaligned under load, the effect is more marked as the helix angle increases and can possibly lead to severe overloads at one edge of the face width whereas the opposite one can be unloaded. This effect is correctly captured by the proposed model and proves that it can be used at the early design stage and help define adapted geometries for a given load and type of gear.

Quasi-Static Optimisation of Transmission Error

Case of Actual Gears -Tooth Profile Deviations

Definition -Local Effects

The previous analyses were conducted using an academic gear example (ideal geometry) and it seems interesting to focus on the influence of tooth shape deviations which, for the most part, cannot be avoided in actual geared transmissions. Accessory gears are usually manufactured with an ISO quality grade of 5/6 and, in order to simulate realistic gears, the profile errors shown in Fig. IV-6, have been superimposed on the linear profile relief trace. Two different defect amplitudes (8 µm and 13 µm peak-to-peak) corresponding to ISO 5 and 7 quality grades have been considered and it is assumed that there is no variation in the tooth face width direction. clearly visible that the improvement on dynamic tooth forces by using tip relief is no longer observed in the presence of profile errors. Here again, two different damping models have been used which both lead to similar conclusions re the influence of profile deviations. As expected, the most severe dynamic response is found for the largest profile error. From a technical viewpoint, it is to be noted that the stabilization of the time response at every speed has been checked over the entire range of speeds

Fig. IV-12: Profile deviation influence on dynamic ratio

Webbed Pinion

To further investigate the effect of gear body flexibility, a webbed pinion whose geometry is close to that of the wheel, is introduced and simulated using the modular H2 model, i.e. comprising both a sub-structured pinion and wheel. The global geometry is unchanged and the only difference comes from the pinion body (Fig. IV-13) with a straight central web.

Fig. IV-13: Webbed pinion geometry

The corresponding dynamic mesh force factor versus pinion speed curves are shown in

Dynamic Contributions of Rotating Webs -Power Transmission Gears

The dynamic participation of rotating webs is studied in this part for a total of 6 rotational speeds listed in Tab. IV-1.

Transmission Modal Behaviour -Effect of Modes

The simulations on the academic power transmission defined in Chapter III have been validated and this particular gear set is used again for the rest of this chapter. This transmission has the particularity of having a helical gear over a slightly inclined web with a large rim (

40 b mm 
).

The study of the industrial test case in Chapter III showed that wheel modes, comprising nodal circles and diameters, are excited by the meshing and that the model was able to capture them. Using the H1 hydric model (rigid pinion body and beam elements for the pinion shaft), the critical speeds 12 ,  listed in Tab. IV-2 have been identified ( 3 is a non-resonant speed).  with the frequency axis normalised with respect to the mesh frequency.

The two first speeds ( 12 , ) exhibit the highest response at the meshing frequency and its second harmonic while the non-resonant speed amplitude is far lower. The number of spatial periods of each mode corresponds well with the identified mode for each graph. It is visible that all the modes are not attached to wheel points but are moving with respect to the wheel body since the modal nodes vary with time. The plots were displayed over a very short amount of time 1/16 dt Tm  and the numerical simulations were performed with 64 time-steps for one mesh period in order to ensure good space and time representations.

The direct modes identified as positive in the Campbell diagram present a shift in the wheel rotation direction whereas the only identified backward mode seems to move in the other direction. This confirms that the modes actually correspond to propagating waves whose properties have been summarized in Tab. IV-4 in terms of rotational speed and corresponding propagation speed in the material. The theoretical propagation speed of a mechanical wave is estimated from [START_REF] Martin | Elements d'acoustique générale : de quelques lieux communs de l'acoustique à une première maitrise des champs sonores[END_REF], by considering the rim circumference as a bar submitted to bending waves as:

Tab. IV-4: Wave speed

The wave velocities as determined by (4.

Tooth Design Constraints Regarding Flexible Wheels

From a design perspective, the conclusion of the previous section is that case B is very likely the most appropriate choice as it reduces the wheel mass substantially while keeping satisfactory dynamic properties. Tab. IV-7 shows the three different speeds (non-resonant cases) which have been considered for this case. The first speed is so low that it prevents from any dynamical effects, the second one would correspond to standard operating conditions and finally, the third one is typical of high-speed applications. Higher modification depths lead to more localised pressure patterns which are still centred on the face width but whose maximum amplitudes increase. An amplitude of 10 µm appears therefore as an optimum value for the example treated beyond which it is not desirable to go as contact stresses would become too high and parts of the flanks would be unloaded.

Flexibility for Fully Webbed Power Transmission Case Webbed Pinion Integration

Pinion Web -Influence on Meshing

In order to further reduce mass, thin-web pinions such as that shown in

Fig. IV-37: Pinion modal data

It is also interesting to note that only a few pinion modes have a frequency below the first meshing harmonic at the maximal rotating speed so that the most part of the dynamic response is caused by the wheel body. The dynamic mesh force factor versus speed has been calculated for a solid and a thin-web pinion and the corresponding responses curves are in Fig.

IV-38 below. Most of the major peaks are conserved with a slight frequency shift suggesting that mass reduction is the dominant phenomenon.

Tab. IV-8: Energy comparison for principal response modes depending on pinion: rigid or webbed

At the lowest response peak around 410 rad/s, the pinion contribution increases between the rigid and the flexible pinion models while bearing contributions decrease. At this low speed, no pinion mode can be triggered (Fig. IV-37) so its contribution is only static. The higher critical speeds are strictly wheel and pinion modes respectively. The wheel mode, at 2490 rad/s is unchanged thus leading to a similar peak with slightly different amplitudes. The second mode at 2900 rad/s is present only for the webbed pinion case. Besides resonance shift, the dynamic factor seems to be higher in the high-speed area for the webbed pinion case. To clarify this issue, a more precise analysis has been conducted on the newly designed gear at the maximum speed 3000 / p rad s  .

Dynamic Optimisation of Lightweight Gear

Web Optimisation -Load Distribution

Having achieved most of the optimisation with respect to mass, web shapes can also be enhanced by refining its geometry as illustrated in Tab. IV-9, test case O standing for the original wheel that has been previously studied. the tooth face width. Above 10 µm, the edges become progressively less loaded which might be interesting in practice providing that the maximum contact stress at mid-tooth span is not too high. Finally, care needs to be taken as crowning is introduced to balance the highly speeddependent centrifugal effects and should therefore be adjusted depending on the operating conditions.

In terms of dynamics phenomena, the dynamic factors for different crowning depths in Fig. show that the larger the crowning, the higher the mesh force oscillations calculated at high speed (3000 rad/s). In such conditions, it can be concluded that an interesting compromise between load pattern quality and dynamic forces would be for a crowning amplitude around 10 µm. the maximum load distribution for the three test cases for each crowning amplitude. The web thickness effect is limited except at one tooth edge for which the limit of contact loss is dependent on the web geometry. Consequently, lead modification needs to be specifically defined according to the particular web geometry and cannot be directly transposed from another design.

Conclusion

This chapter was aimed at analysing the influence of pinion/gear body geometry on the quasi-static and dynamic characteristics of a transmission. The first case study dealt with an accessory spur gear with a straight web which was found to be highly influential on the gear behaviour and needed to be designed accordingly. It has also be shown that profile modifications can substantially improve the meshing conditions and that web geometry and deflections have to be considered when looking for optimum profile relief. Similar conclusions have been drawn when considering a thin-webbed flexible pinion body in addition to a flexible wheel body.

In a second part, the dynamic contribution of flexible webs was investigated on the academic power transmission architecture. The dynamic mesh force curves prove that body modes have an impact on the whole gear dynamic behaviour. The response at some web nodes has been studied at several resonant and non-resonant speeds which indicate that body modes exhibit progressive wave properties. The influence of holes in webs has also been tackled and it has been demonstrated that, in certain conditions, both the static and dynamic responses can be altered when hollow webs are employed.

Finally, the design of light-weight pinions and gears in power transmissions has been examined. It has been found that the modal properties are altered compared with systems with solid pinions. One major result concerns the importance of accounting for centrifugal effects together with a precise elastic web model in high-speed gears since they which can strongly modify tooth load distributions and consequently control the definition of tooth shape modifications.

General Conclusion

The major objective of this PhD was to present the developments associated with some original modular hybrid models of mechanical transmissions with thin-rimmed/webbed pinions and gears. As opposed to what can be found in the literature, a quasi-analytical model is used to describe tooth contacts in terms of elasticity and shape deviations whereas 3D finite elements are employed for all the components with linear behaviour. By so doing, the previous modelling efforts at LaMCoS re instant tooth contacts based on time-varying non-linear Winkler foundations are just adapted to a different modelling environment accounting for thin rims and webs, shafts, bearings, etc. The resulting hybrid models have therefore the capacity to combine lumped parameter + shaft element sub-models (for a pinion shaft for instance) with condensed 3D finite elements to simulate a thin-rimmed gear body and shaft. In these conditions, particular attention has been given to the connection of the lumped and finite element models via discrete elastic foundations. It has been shown that direct collocation generates artificial load spikes on tooth flanks which are caused by the incompatibility between a discrete foundation and a continuous rim model. A mortar-based approach has been developed and implemented which smoothen the numerical load oscillations mentioned above. The corresponding state equations are solved step by step in time by using a Newmark scheme and a unilateral normal contact algorithm which verifies that all contact forces between the tooth flanks are compressive and that there is no contact outside the defined contact area. Such a methodology can simulate the contributions of tooth shape deviations and errors and also capture the amplitude jumps and softening effects caused by contact losses and shocks between the teeth at the major tooth critical frequencies.

The resulting dynamic models have been largely validated by comparisons with both numerical and experimental results thus proving the versatility of the approach and its capacity to simulate actual gear sets. Further investigations concerning specifically the influence of gear body compliance and centrifugal effects have been conducted which highlight the significant contribution of centrifugal effects at high-speeds and the interest of profile and lead modifications in terms of tooth load pattern quality and vibrations. However, uncertainties remain about the simulation of damping which, as expected, is of primary importance since it controls the amplifications at critical speeds. Qualitatively, Rayleigh and modal damping models have their own limitations and quantitatively, damping amplitudes very much rely on empirical data whose range of validity should be critically assessed especially for thin members.

In terms of perspectives, the hybrid approach could be extended to incorporate casings whose flexibility certainly needs to be taken into account for determining tooth load distributions and estimating the noise radiated to the surroundings. The simulations of multimesh gears such as those in accessory gearboxes would also be an interesting research direction since they comprise both thin-rimmed pinions and thin casings.

Annexes 1. Shaft Element -Beam Matrices

The beam elements rely on Timoshenko beam theory [START_REF] Timoshenko | Strength of materials, Part I. Elementary theory and problems[END_REF]. Each element is composed of two nodes (Figure 1) with 6 degrees-of-freedom each (three translations and three rotations).

LL v L L L EI L v L sym L                           K (1.8)         1 22 1 3 2 2

Newmark Integration Scheme

Newmark time-step integration scheme is often used for system resolution in dynamics.

It remains on third order Taylor development of displacement vector t+Δt q at time-step tt  , giving the following expression for both displacement and speed vector:

2 2 t t t             t+Δt t t t t t+Δt t t t
q q q q q q q q q (2.1)

With q DOF vector, t previous time-step and tt  in calculation time-step.

The acceleration derived is considered linear between two time-steps:

t t        t+Δt t t+Δt t t t qq q q q q (2.2)
Giving the following equation when put back into eq. 2.1.

  22 1 2 1 t t t tt                    t+Δt t t
t t+Δt t+Δt t t t+Δt q q q q q q q q q (2. Expressing acceleration and speed at tt  by displacement by introducing eq. 2.3 into 2.4:

2 2 1 1 1 1 1 2 1 11 2 tt tt t t                                                          t+Δt t+Δt t t t t t t M C K q F M q q q C q q q (2.5) With:   2 2 1 11 2 11 . 2 t t tt t                                                    t+Δt t t+Δt t t
t+Δt t+Δt t t t qq q q q q q q q q (2.6)

Practically, the calculation is achieved by calculating t+Δt q by solving the equation 2.5 fully written in terms of displacement, speed and acceleration at time-step t . Then the terms of speed and acceleration are calculated thanks to eq. 2.6, leading to the full system expression ( ,, t+Δt t+Δt t+Δt qqq ) in tt  in terms of its components at t .

Introduction

Les

Revue Bibliographique

Paramètres Principaux de Réponse des Engrenages La principale source d'excitation de l'engrenage vient du passage des dents au contact (Figure 2). La raideur résultante varie en fonction du nombre de dents en prise à chaque instant.

Il existe plusieurs méthodes possibles pour calculer cette raideur. Elles intègrent de façon différente l'ensemble des raideurs de denture, le contact direct entre les dents, la flexion de ces dents au contact, et la fondation, résultant de la rotation de la base.

La seconde source provient des écarts de formes et de montage auxquels peuvent être soumis l'engrenage. Les premiers sont par exemple des déviations de profils ou d'hélice, pour lesquels les profils parfaits des plans de définition ont été altérés par les processus de manufacture. Les écarts de montage peuvent eux prendre la forme de désalignements.

 

      1 0 2 T ij ij O MM O                    1 1 1 ij 1 1 2 2 2 ij 2 2 n u n M O ω n u n M O ω (1) Avec       12 T OO  1 1 2 2
Xu ω u ω Degrés de libertés aux centres du pignon et de la roue.

Le modèle est modifié pour intégration de la roue flexible dans l'élément engrenage.

Pour cela, le noeud central de la roue est remplacé par un modèle EF (Figure 6 Une fois ces problèmes de compatibilité résolus, les points des lignes quadratiques sont liés aux noeuds des éléments finis de jante par les fonctions de formes des éléments finis de briques réduits à la surface de ces éléments selon Dhatt [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF]. Le lien entre les lignes de contact et la roue EF est ainsi complet (eq. 5).

      * 0 Cette procédure peut aussi être faite pour le pignon par symétrie dans le cas où le pignon et la roue sont tous les deux souples et condensés.

T ij ij M M M     m VX (5)

Intégration des Effets Dynamiques

La raideur instantanée d'engrènement est ensuite calculée par intégration. Le modèle dynamique résultant est résolu par intégration temporelle grâce à un schéma de Newmark couplé à un algorithme de contact [START_REF] Velex | A mathematical model for analysing the influence of shape deviation and mounting errors on gear behaviour[END_REF] qui permet de prendre en compte les potentielles pertes de contact, partielles ou totales, provenant des vibrations dynamiques de l'engrenage.

Eléments de Validation Numérique et Expérimentale

This

  PhD research work has been successfully conducted thanks to the (ANRT, 2014/1067) via a CIFRE grant. It is the result of the partnership between SAFRAN Helicopter Engines (ex-Turbomeca) and the Contact and Structure Laboratory (LaMCoS, UMR CNRS 5259) of INSA de Lyon, directed by Daniel Nelias. I would like first to express my gratitude toward Philippe Velex, Professor at INSA de Lyon for his constant support over these PhD years. It was a real honour and pleasure to be able to learn from his extensive knowledge and experience, both in general mechanics and specific gear dynamics, without forgetting his very helpful English support. None of this work could have been achieved without his accurate guidance. I also want to thank my collaborators at SAFRAN HE for welcoming me in the Transmissions Department. I would especially like to thank Philippe Cutuli for his guidance, Lionel Renault and Patricio Almeida for their help every time I needed it, despite their own work loads and Antoine Mounetou for his supervision and his warm welcome to SAFRAN HE. I want to express my gratitude to Wim Desmet and to Karsten Stahl for the time they spent reviewing this work. I am also grateful at Anne Tanguy and Roger Serra for accepting to be members of this jury.

  l'engrenage et de la nouvelle flexibilité des voiles-minces. Il dérive d'un modèle à paramètres concentrés, comprenant des arbres en poutre, des paliers et carters sous forme de raideurs additionnelles et un élément d'engrenage rigide inclus par son noeud central. Hypothèse est faite que tous les contacts sont situés sur les lignes de contact du plan d'action. Ces lignes sont discrétisées selon des tranches-minces dans les dents et la déviation normale des cellules est recalculée à chaque pas de temps selon la déflexion de la denture. Le nouveau modèle remplace l'engrenage rigide par une modélisation EF du pignon et/ou de la roue condensée sur les noeuds de jante. Une interface lie les raideurs du plan d'action discrétisé aux éléments finis du corps d'engrenage. L'élément prend donc en compte à la fois les sollicitations de l'engrenage et le comportement statique et modal des corps flexibles en dynamique. Des comparaisons sont faites avec des données numériques et expérimentales. Elles attestent de la capacité du nouveau modèle à prédire le comportement dynamique des engrenages flexibles à hauts régimes de rotation. Ces résultats intègrent entre autres des données locales et globales en dynamique.

Fig. I

  Fig. I-1: Re-composition of the gears in the Antikythera mechanism from Archimedes [LEW93] ..... Fig. I-2: Turboshaft from SAFRAN Helicopter Engines (ex-Turbomeca) [MOU14] .......................... Fig. I-3: Meshing process for gears with one or two teeth in contact during meshing [VEL88] .......... Fig. I-4: Hertz pressure .......................................................................................................................... Fig. I-5: Tooth parameter definition for stiffness calculation [WEB53] ............................................... Fig. I-6: Profile deviation definition [HEN78] ...................................................................................... Fig. I-7: Lead deviation definition [HEN78] ......................................................................................... Fig. I-8: Pitch deviation definition [HEN78] ........................................................................................ Fig. I-9: Cumulated pitch deviations [HEN78] ..................................................................................... Fig. I-10: Inclination and deviation due to mounting deviations [SUZ86] ........................................... Fig. I-11: Harris map [HAR58] ............................................................................................................. Fig. I-12: Typical no-load transmission error signal, acc. to Munro [MUN89] .................................... Fig. I-13: Comparison between Kubo's experimental results [KUB72] and numerical predictions, acc. to Ozguven and Houser [OZG88.b] ...................................................................................................... Fig. I-14: Tip and root relief [ISO21771] .............................................................................................. Fig. I-15: Helix modification definition: (a) helix crowning and (b) helix relief [ISO21771] .............. Fig. I-16: Analytical MasterCurve [BRU13] ......................................................................................... Fig. I-17: Dynamic tooth load factor versus speed -Influence of profile modifications (--: without profile modification) [BRU15] .............................................................................................................. Fig. I-18: First spring-mass model from Tuplin [OZG88.a] ................................................................. Fig. I-19: (a) Two-degree-of-freedom torsional model of a gear pair and (b) its equivalent single degree-of-freedom system (acc. to Özgüven and Houser [OZG88.a]) .................................................. Fig. I-20: Schematic representation of a finite element model of a geared rotor system, acc. to Kahraman et al. [KAH92] ..................................................................................................................... Fig. I-21: Definition of shape deviations on the base plane, acc. to Velex and Maatar [VEL96] ......... Fig. I-22: Finite element model of a realistic housing geometry, acc. to Zhou et al. [ZHO14] ............ Fig. I-23: Finite element model of a spur gear pair, acc. to Parker et al. [PAR00.a] ............................ Fig. I-24: Contact zone with analytical solution for inner part and finite element solution for outer part [PAR00.a] .............................................................................................................................................. Fig. I-25: Rotating ring scheme with space-fixed discrete stiffness [COO14]...................................... Fig. I-26: Eigenvalues of a rotating ring on elastic foundation (a) alone and (b) with a single discrete stiffness element for varying rotating speed [COO14] .......................................................................... Fig. I-27: Dynamic ratio versus increasing rotation speed for both solid and thin-rimmed gears [BET07] ................................................................................................................................................. Fig. I-28: Example of mode shape for a large rim, thin-webbed wheel found by FEM [MOU14] ....... Fig. I-29: Calculated (vertical line) and measured Eigen frequencies [MOU14] .................................. Fig. I-30: FEM model wheel for (a) mesh load and (b) centrifugal analysis [LI02.a-08.a] .................. Fig. I-31: Tooth contact stresses for two types of webs at 0 min -1 [LI13]............................................. Fig. I-32: Tooth contact stresses for quasi-static analysis with centrifugal effect at

  Fig. I-1: Re-composition of the gears in the Antikythera mechanism from Archimedes [LEW93] ..... Fig. I-2: Turboshaft from SAFRAN Helicopter Engines (ex-Turbomeca) [MOU14] .......................... Fig. I-3: Meshing process for gears with one or two teeth in contact during meshing [VEL88] .......... Fig. I-4: Hertz pressure .......................................................................................................................... Fig. I-5: Tooth parameter definition for stiffness calculation [WEB53] ............................................... Fig. I-6: Profile deviation definition [HEN78] ...................................................................................... Fig. I-7: Lead deviation definition [HEN78] ......................................................................................... Fig. I-8: Pitch deviation definition [HEN78] ........................................................................................ Fig. I-9: Cumulated pitch deviations [HEN78] ..................................................................................... Fig. I-10: Inclination and deviation due to mounting deviations [SUZ86] ........................................... Fig. I-11: Harris map [HAR58] ............................................................................................................. Fig. I-12: Typical no-load transmission error signal, acc. to Munro [MUN89] .................................... Fig. I-13: Comparison between Kubo's experimental results [KUB72] and numerical predictions, acc. to Ozguven and Houser [OZG88.b] ...................................................................................................... Fig. I-14: Tip and root relief [ISO21771] .............................................................................................. Fig. I-15: Helix modification definition: (a) helix crowning and (b) helix relief [ISO21771] .............. Fig. I-16: Analytical MasterCurve [BRU13] ......................................................................................... Fig. I-17: Dynamic tooth load factor versus speed -Influence of profile modifications (--: without profile modification) [BRU15] .............................................................................................................. Fig. I-18: First spring-mass model from Tuplin [OZG88.a] ................................................................. Fig. I-19: (a) Two-degree-of-freedom torsional model of a gear pair and (b) its equivalent single degree-of-freedom system (acc. to Özgüven and Houser [OZG88.a]) .................................................. Fig. I-20: Schematic representation of a finite element model of a geared rotor system, acc. to Kahraman et al. [KAH92] ..................................................................................................................... Fig. I-21: Definition of shape deviations on the base plane, acc. to Velex and Maatar [VEL96] ......... Fig. I-22: Finite element model of a realistic housing geometry, acc. to Zhou et al. [ZHO14] ............ Fig. I-23: Finite element model of a spur gear pair, acc. to Parker et al. [PAR00.a] ............................ Fig. I-24: Contact zone with analytical solution for inner part and finite element solution for outer part [PAR00.a] .............................................................................................................................................. Fig. I-25: Rotating ring scheme with space-fixed discrete stiffness [COO14]...................................... Fig. I-26: Eigenvalues of a rotating ring on elastic foundation (a) alone and (b) with a single discrete stiffness element for varying rotating speed [COO14] .......................................................................... Fig. I-27: Dynamic ratio versus increasing rotation speed for both solid and thin-rimmed gears [BET07] ................................................................................................................................................. Fig. I-28: Example of mode shape for a large rim, thin-webbed wheel found by FEM [MOU14] ....... Fig. I-29: Calculated (vertical line) and measured Eigen frequencies [MOU14] .................................. Fig. I-30: FEM model wheel for (a) mesh load and (b) centrifugal analysis [LI02.a-08.a] .................. Fig. I-31: Tooth contact stresses for two types of webs at 0 min -1 [LI13]............................................. Fig. I-32: Tooth contact stresses for quasi-static analysis with centrifugal effect at 1 40 000 min  [LI13] .................................................................................................................................................... Fig. I-33: Wheel geometry for the hollow web study [HEI15] ............................................................. Fig. I-34: Instantaneous mesh stiffness for axisymmetric lightweight wheel (steady), and lightweight gear with holes (varying) [HEI15] ........................................................................................................

Fig. II- 1 :bF  ) and 2 nd ( 1 bM

 11 Fig. II-1: Shaft element ......................................................................................................................... Fig. II-2: Gear body for (a) rigid gear with single node and (b) beam type gear body, with 1, 2 k  depending on pinion or wheel ............................................................................................................... Fig. II-3: Finite Element model of a gear flexible wheel ...................................................................... Fig. II-4: Dhatt's brick Finite Elements [DHA81] ................................................................................. Fig. II-5: Simple example of a beam with a single interface node [WEN98] ....................................... Fig. II-6: First ( 1, 0 bb x   ) and second ( 0, 1 bb x   ) constraint (static) modes [WEN98] .... Fig. II-7: First dynamic mode [WEN98] ............................................................................................... Fig. II-8: First ( 1

Fig. III- 13 :

 13 Fig. III-13: Number of mode choice in the condensed wheel influence on the dynamic ratio.............. Fig. III-14: Power transmission wheel geometry, including the web .................................................... Fig. III-15: Dynamic ratio comparison for (a) modal dependent damping and (b) Rayleigh damping101 Fig. III-16: Local load over a tooth for various web arrangements [LI02.b] ...................................... Fig. III-17: Load distribution for (a) flexible web and (b) when compared to rigid and initial model maximum contact line load when passing through the base plane ...................................................... Fig. III-18: Typical web flexion mode, characterised as 0 nodal circle and 3 nodal diameters (displayed on wheel + shaft) ................................................................................................................................. Fig. III-19: Modal comparison between the gear analysis (H1) and full FE model of wheel alone ... Fig. III-20: Modes distribution for (a) FE wheel studied (b) plain circular disc [SCH97] .................. Fig. III-21: Example of 3 rd nodal diameter theoretical shape for   cos 3 [JAC97] .......................... Fig. III-22: Circular plate mode shape for (a) no and (b) 1 nodal diameter mode [JAC97] ................ Fig. III-23: Dynamic ratio in case of the flexible wheel for (a) modal dependant damping and (b) Rayleigh damping................................................................................................................................ Fig. III-24: Comparison for centrifugal effect influence with (a) modal dependant damping and (b) Rayleigh damping for clarity ............................................................................................................... Fig. III-25: Rim pressure (a) without and (b) with centrifugal effects (id. for both hybrid model) .... Fig. III-26: FE wheel model deformation (norm and direction) or (a) full wheel and (b) rim cut ...... Fig. III-27: Scheme of rim expansion effect on gear key parameters ................................................. Fig. III-28: Test Transmission Architecture (Interest: drive and intermediate gear)........................... Fig. III-29: FE intermediate train (HST wheel + LST pinion) prepared for condensation process ..... Fig. III-30: Industrial FE model of the HST ........................................................................................ Fig. III-31: Intermediate pinion tooth boundary condition.................................................................. Fig. III-32: Tooth Flank analysis with quasi-static FE model (centrifugal load for 8820 w rpm  ) ............................................................................................................................................................. Fig. III-33: Line of contact pressure on base plane ............................................................................. Fig. III-34: Meshing pressure of the hybrid model H1 on dynamic run for (a) above material and (b) above a hole meshing conditions ......................................................................................................... Fig. III-35: FE wheel modal comparison to interferometry test (starting to the 2 nd nodal diameter (ND) mode for each circle family (NC)) ...................................................................................................... Fig. III-36: 9 th diameter mode as example: Rim mode of interest ....................................................... Fig. III-37: Campbell diagrams depending on the chosen frame of expression .................................. Fig. III-38: Angular position of Displacement Probes (DP) for Numerical model / Experiment comparison .......................................................................................................................................... Fig. III-39: Radial and axial position of Displacement Probes (DP) devices for rim radial displacement measure ................................................................................................................................................ Fig. III-40: (a) Experimental results and (b) extracted scheme for each DP ....................................... Fig. III-41: Contribution of Rayleigh damping parameters depending on frequency i  ...................

Fig. III- 42 :

 42 Fig. III-42: Numerical data gathering scheme: Rim Node (RN) ......................................................... Fig. III-43: Rim Node response over a full wheel revolution ............................................................. Fig. III-44: Fast Fourier Transform of the Rim Node response over a full revolution versus tooth number (127 being meshing order) ..................................................................................................... Fig. III-45: Campbell diagram drawn with the RN, (0 NC corresponding to the 8 th , 9 th and 10 th nodal diameters of the family) ...................................................................................................................... Fig. III-46: Identification of the speeds of interest .............................................................................. Fig. III-47: Full frequency analysis with FFT for case A and B .........................................................

Fig. III- 48 :Fig. IV- 2 :Fig. IV- 6 : 2  pinion mode and standard response 3  , 4  and 5  3 

 482623453 Fig. III-48: Frequency analysis for the case C .................................................................................... Fig. III-49: Frequency analysis for the case D .................................................................................... Fig. III-50: Sum of all nodal diameter signals ..................................................................................... Fig. IV-1: Load at contact for (a) 10 b   and (b) 20 b   helix angle ........................................ Fig. IV-2: Optimisation with regards to the Master Curve for (a)

Fig. IV- 32 :

 32 Fig. IV-32: Meshing pressure depending on speed for tooth spot over hollow web for case study B Fig. IV-33: Linear modification along lead ......................................................................................... Fig. IV-34: Effect of a 10 µm linear lead modification on pressure ................................................... Fig. IV-35: Webbed pinion geometry ................................................................................................. Fig. IV-36: Pressure distribution or (a) plain and (b) webbed pinion ................................................. Fig. IV-37: Pinion modal data ............................................................................................................. Fig. IV-38: Dynamic ratio comparison for webbed pinion inclusion .................................................. Fig. IV-39: Pressure distribution at 3000 /

Fig. IV- 41 :

 41 Fig. IV-41: Web thickness influence on wheel modal behaviour -FE analysis ................................. Fig. IV-42: Flexion mode example: 2 circles, 3 diameters (17 922 Hz for Case O) ........................... Fig. IV-43: Effect of the web on quasi-static load distribution for (a) maximum contact line load and (b) maximum pressure ......................................................................................................................... Fig. IV-44: Maximum pressure over a meshing period for the three test cases at 3000 / p rad s  ............................................................................................................................................................. Fig. IV-45: Meshing pressure comparison for both thinner web cases at 3000 /

  Nowadays, gears are widely used in all kinds of applications ranging from car gearboxes, precision watches and high-technology aeronautical transmissions. SAFRAN Helicopter Engines (formerly-Turbomeca) is one of the world leaders for turboshaft development and production in the field of aeronautics. The gears in helicopters have two distinct roles: a) transmit the power from the main motor shaft to the helicopter rotor and, b) operate the various accessories (starter, pumps, etc.). The prime movers can rotate up to 60 000 rpm and mass reduction constraints often lead to light-weight, thin-rimmed or thin-webbed gears to reduce fuel consumption. In this particular context of light-weight gears at high rotational speeds, a research partnership has been established between SAFRAN HE and the Contact and Structural Mechanics Laboratory (LaMCoS) of INSA Lyon with the objective of developing specific gear dynamic models adapted to helicopter transmissions. In this memoir, Chapter I synthesises the research works on gear dynamics found in the open literature covering the definition of the major parameters and modelling strategies which, most of the time, rely on lumped parameter models of a pinion-gear pair. Some recent extensions towards the simulation of flexible gear bodies in statics and dynamics are also presented. An original modular dynamic hybrid model is introduced in Chapter II which combines lumped parameter, shaft and 3D finite elements and makes it possible to account for thinrimmed/web gears with minimum computational cost. Chapter III presents some elements of model validation based on simulation results but also experimental measurements on actual helicopter transmissions. Satisfactory agreements are observed which, to a large extent, validate the theoretical and numerical developments in Chapter II.Finally, Chapter IV focuses on the influence of web flexibility on the static and dynamic behaviour of helicopter gears. The impact on the definition of profile modifications and the contributions of gear body compliance to the critical speeds are examined in detail. Finally, the possibility of further mass reductions by modifying web designs is investigated. This PhD work focuses on the dynamic behaviour of thin webbed-rimmed gears such as those used in high-speed mechanical transmissions. The literature on the topic is sparse as most of the research effort has been directed towards the quasi-static and dynamic analysis of solid gears whereas the influence of gear body flexibility has received far less attention. This chapter presents a survey of the research works on gear dynamics with emphasis being placed on the contributions concerning light-weight, thin-rimmed gears. The phenomena at the root of gear vibrations are described and the classic models in gear dynamics are presented. Finally, the specific modelling efforts for thin-rimmed gears are described and commented upon.

Fig. I- 3 :

 3 Fig. I-2:Turboshaft from SAFRAN Helicopter Engines (ex-Turbomeca)[START_REF] Mounetou | Non-intrusive measurement of gear dynamic response[END_REF]

Fig. I- 4 :

 4 Fig. I-4: Hertz pressure

Fig. I- 5 :

 5 Fig. I-5: Tooth parameter definition for stiffness calculation[START_REF] Weber | Formänderung und Profilrücknahme bei Gerad-und Schrägverzahnten Antriebstechnik[END_REF]

F

  angle between the tooth centre line and tooth flank intersection with root circle constant that can be found in[START_REF] Sainsot | Contribution of gear body to tooth deflections-A new bidimensional analytical formula[END_REF] Shape DeviationOther sources of excitation arise directly from the manufacturing and finishing processes. Gear teeth are cut and often heat treated and may finally be finished[START_REF] Henriot | Traité théorique et pratique des engrenages, Tome I : Théorie et technologie[END_REF]; all these operations inevitably induce shape distortions and errors with respect to the ideal geometry even after grinding or superfinishing. a. Profile and lead deviations Profile and lead deviations appear on tooth flanks. Profile deviations represent the difference between the theoretical and actual tooth profiles in the direction normal to the tooth trace as illustrated in Fig. I-6.

Fig. I- 6 :

 6 Fig. I-6: Profile deviation definition[START_REF] Henriot | Traité théorique et pratique des engrenages, Tome II : Etude complète du matériel[END_REF]

Fig. I- 7 :

 7 Fig. I-7: Lead deviation definition[START_REF] Henriot | Traité théorique et pratique des engrenages, Tome II : Etude complète du matériel[END_REF]

Fig. I- 9 :

 9 Fig. I-9: Cumulated pitch deviations[START_REF] Henriot | Traité théorique et pratique des engrenages, Tome II : Etude complète du matériel[END_REF]

Fig. I- 10 :

 10 Fig. I-10: Inclination and deviation due to mounting deviations[START_REF] Suzuki | Influence of misalignment on vibration of helical gears[END_REF]

called

  Harris maps which provide a clear overview of the system behaviour in terms of profile modifications and load (cf. Fig. I-12).

Fig

  Fig. I-11: Harris map [HAR58]

Fig

  Fig. I-12.

Fig. I- 12 :

 12 Fig. I-12: Typical no-load transmission error signal, acc. to Munro[START_REF] Munro | The D. C. component of gear transmission error[END_REF]

Fig

  Fig. I-13: Comparison between Kubo's experimental results [KUB72] and numerical predictions, acc. to Ozguven and Houser [OZG88.b]

  modify local load distributions and generate corner contacts [HOU06]. Profile and helix modifications are usually introduced to avoid shocks at engagements and improve the load distributions on tooth flanks. They can be introduced separately or merged as topological modifications [ISO21771] where the tooth flank surface is modified so as to give the proper load distribution during the meshing process [CON71-73-LIT91-92]. Profile modifications, for example tip and tooth relief (Fig. I-14), are often used to reduce overloads at engagement [MAA97].

Fig. I- 14 :

 14 Fig. I-14: Tip and root relief[ISO21771]

  helix relief at both ends are used (Fig. I-15).

Fig. I- 15 :

 15 Fig. I-15: Helix modification definition: (a) helix crowning and (b) helix relief[ISO21771]

  research works have been conducted on the definition of optimum tooth modifications with regard to the time variations of transmission errors. Tavakoli and Houser [TAV86] presented a numerical model to calculate static transmission error under load including an algorithm to minimize the amplitudes of the various TEs harmonics. Bahk and Parker [BAH13] developed an analytical method based on a perturbation method to study the influence of tooth modifications spur planetary gear. The authors found that the best modifications obtained when considering the whole system were different from those derived when considering individual meshes isolated from the rest of the mechanical system. Bruyère and Velex [VEL11-BRU13-15] employed an analytical approach to the definition of optimum profile relief and proposed a so-called Master Curve (Fig. I-16) in the depth versus extent of relief plane which gives the families of symmetric linear relief minimising transmission error variations for spur and helical gears transmitting a range of loads. It has been confirmed by numerical simulations that the relief on the Master Curve improves the dynamic response of gears as illustrated in Fig. I-17.

Fig

  Fig. I-16: Analytical MasterCurve[START_REF] Bruyère | Derivation of optimum profile modifications in narrow-faced spur and helical gears using a perturbation method[END_REF]

3 .

 3 Usual lumped Parameter Gear Dynamic Models Models depending on meshing stiffness description Following the presentation by Özgüven and Houser [OZG88], the simplest models are limited to rigid-solids (usually the pinion and the gear) connected by a mesh stiffness function whereas the flexibility of the surrounding elements (shafts, bearings, etc.) is discarded. The first mathematical models date back to the 1950's, Tuplin [TUP50] introduced the first spring-mass model, based on an equivalent constant mesh stiffness function and wedges of various shapes were introduced at the base of the spring to simulate gear errors (Fig. I-18). The model was used for transient excitations, without taking its periodicity into account and therefore could only estimate the dynamic ratio below any resonance.

Fig. I- 18 :

 18 Fig. I-18: First spring-mass model from Tuplin [OZG88.a]

Fig. I- 20 :

 20 Fig. I-20: Schematic representation of a finite element model of a geared rotor system, acc. to Kahraman et al. [KAH92]

Fig. I- 21 :

 21 Fig. I-21: Definition of shape deviations on the base plane, acc. to Velex and Maatar [VEL96]

[

  RIG96-99] coupled a hybrid lumped parameter / finite element model of a gear-shaft-bearing system with a finite element model of the casing. Transmission error excitations were considered and the authors concluded that any change in the elastic properties of the gearbox components modifies its dynamic behaviour, thus justifying the need for an accurate model of the global gearbox. Abbes et al. proposed an alternative method based on sub-structuring [ABB05-07] and developed a coupled structural-acoustic model to predict the acoustic pressure [ABB08]. Zhou et al. [ZHO14] used the finite element discretization to analyse the vibrations of a gearbox housing in Fig. I-22. The dynamic response was analysed independently of the housing and the corresponding dynamic bearing forces were used as excitations sources for the housing model.

Fig. I- 22 :

 22 Fig. I-22: Finite element model of a realistic housing geometry, acc. to Zhou et al. [ZHO14]

  Fig. I-23: Finite element model of a spur gear pair, acc. to Parker et al. [PAR00.a]

  Parker and Cooley simulated flexible gears by introducing rotating elastic rings resting on constant stiffness elastic foundations while keeping lumped stiffness elements for mesh elasticity.

Fig

  Fig. I-25:Rotating ring scheme with space-fixed discrete stiffness[START_REF] Cooley | Vibration of high-speed rotating rings coupled to spacefixed stiffnesses[END_REF]

Fig. I- 27 :

 27 Fig. I-27: Dynamic ratio versus increasing rotation speed for both solid and thin-rimmedgears[START_REF] Bettaieb | A static and dynamic model of geared transmission by combining substructures and elastic foundations -application on thin-rimmed gears[END_REF]

Fig

  Fig. I-31: Tooth contact stresses for two types of webs at 0 min -1 [LI13]

Fig

  Fig. I-32: Tooth contact stresses for quasi-static analysis with centrifugal effect at 1 40 000 min  [LI13]

Fig. I- 33 :

 33 Fig. I-33: Wheel geometry for the hollow web study[START_REF] Heirman | A gear contact model to analyse the dynamics of transmissions with lightweight, flexible gears[END_REF]

  -Shaft, Bearing and Housing The initial gear model is a lumped parameter model based on the developments of Velex and Maatar [VEL96] which comprises standard Timoshenko 2-node shaft elements with 6 degrees of freedom per node [TIM55] (Fig. II-1).

Fig. II- 2 :

 2 Fig. II-2: Gear body for (a) rigid gear with single node and (b) beam type gear body, with 1, 2 k  depending on pinion or wheel

Fig. II- 3 :

 3 Fig. II-3: Finite Element model of a gear flexible wheel

3 )

 3 With : b and i respectively standing for boundary and interior DOFs From this choice of interior and boundary nodes, numerous technics of condensation are possible which are usually separated into a) fixed-interface condensations and, b) free-interface condensations. Both methodologies rely on shape functions of static and dynamic modes. For the sake of illustrating these notions, the simplified example proposed in [WEN98] is developed below. Fig. II-5 shows a simple beam model composed of two beam elements and a single interface node with two DOFs ( , bb x  ) at its centre (Fig. II-5).

Fig. II- 5 :

 5 Fig. II-5: Simple example of a beam with a single interface node[START_REF] Wensing | On the dynamic of ball bearings[END_REF]

Fig

  Fig. II-6: First (

Fig. II- 7 :

 7 Fig. II-7: First dynamic mode[START_REF] Wensing | On the dynamic of ball bearings[END_REF]

FigM

  Fig. II-8: First (

u 5 )

 5 relative function calculated[START_REF] Herting | A general purpose, multi-stage component modal synthesis method[END_REF] This relation between the interior and boundary DOFs makes it possible to keep the boundary nodes only thus enabling direct DOF couplings in modular systems comprising several sub-structures. The resulting condensed matrices can then be expressed in terms of the boundary static and modal DOFs as: With b and n respectively boundary and modal DOFs. 64 It is also interesting to note that in this particular free-interface formulation, the boundary and modal DOFs are statically uncoupled [HER85]. Following [BET06-COO15-NOG09-BOG12], the gear teeth are discarded from the substructure representing the wheel (Fig. II-3) and the interface nodes are those lying on the gear body rim and one node located at the centres of the bearing supporting the gear shaft as illustrated in Fig. II-10.

Fig

  Fig. II-10: Sub-structure master nodes ( 1, 2 k  depending on pinion or wheel)

Fig

  Fig. II-11: Initial gear element with rigid body approach

Fig

  Fig. II-12: Discretised base plane of the gear

  Fig. II-14: Modular model with (a) only flexible gear and (b) flexible pinion and gear

O

  and the tooth root ij p M of the gear for each cell ij of the discretised base plane.

Fig

  Fig. II-15: Rigid plate tooth model

2 pM

 2 situated on the root circle arc of a slice at the same distance from p M are supposed to be rigidly fixed to the finite element grid surface at the rim (Fig.II-16). Such a methodology makes it possible to transfer both translations and rotations (limited to that around Z here) between the tooth and gear body models.

Fig 2 pM

 2 Fig. II-16: Tooth rotation inclusion on a thin slice for tooth root displacement

  Fig. II-17: Two-point connection for tooth width (only point   2 p ij M

Fig

  Fig. II-18: Projection of discretised segment on FE mesh

Fig

  Fig. II-19: Different discretisation for the segment, from c for discrete line of contact to q for the quadrating FE matching segment

c 2 pM

 2 is an initial stepwise segment of length L which stems for the discrete contact lines and is associated with the two-point connection model each segment centred at points (nodes) 1,on the surface of the rim finite element model (Fig. II-17). It is characterised by a uniform force per unit of contact length c F and a constant displacement c Q .

  P N. The corresponding force per unit of contact length at the three nodes of the segment is denoted q F whereas the displacements at the same nodes are embodied in vector q Q .

  Fig. II-20: New node place on mesh after mortar

Fig

  Fig. II-21: Patch test problem with discrete elements (DE) and finite element (FE)

Fig

  Fig. II-23: Node repartition for brick element surface

2 X

 2 Fig. II-24: Scheme of interface summary

  .31) with ij k local stiffness at each contact cell of the base plane, calculated from Weber and Banaschek analytical model of contact and tooth flexion stiffness [WEB53], removing the foundation stiffness which is directly included in the tooth model developed in section 2.



  modal damping factor, typically between 0.01 and 0.1 .In the particular case of gears, a composite formula can be used based on the works of Umezawa[START_REF] Umezawa | Vibration of power transmission helical gears : (the effect of contact ratio on the vibration)[END_REF] and Ankouni. Umazawa experimentally found that a damping factor around 0.07 was representative of the dissipation at the gear mesh whereas structural modes were characterised by lower damping factors around 0.01. It is therefore proposed to employ a weighted average of the two characteristic amplitudes using the percentages of modal strain energy in the gear mesh i e such that:

Algorithm

  The system obtained eq. (2.37) can exhibit nonlinearities associated with momentary partial or total losses followed by shocks between the mating teeth arising from large vibration amplitudes. The solving solution has first been introduced by Velex and Maatar, it relies on the combination of a Newmark time step integration (annex 2.) and a contact algorithm illustrated Fig.

  foundations rely on discrete stepwise segments which do not naturally connect with 3D 20node finite elements. A 3-step methodology is proposed: a) a two-point connection between every thin-slice of the Winkler foundation (thin or elemental tooth) and the rim finite element grid, b) a mortar based technique to introduce continuity and spread the displacement field between the segments of one contact line and, c) the shape functions of reduced 8-node elements to calculate the displacements anywhere at the surface of the finite element grid and relate them to nodal displacements.Another improvement on the previous model is the introduction of centrifugal effects in the state equations which are shown to be influential in high-speed thin members such as those in helicopter transmissions. aimed at presenting elements of validation of the dynamic model presented in chapter II based on numerical simulations of helicopter transmissions, namely accessory gears and power transmissions. Academic examples of accessory gearbox and power transmission will be considered along with an actual industrial two-stage helicopter power transmission. For confidentiality reasons, the data and results related to the industrial case study have been normalised and dimensionless parameters will be used. The following three modelling configurations will be employed:-rigid pinion / flexible condensed gear, (further referred to as 'Hybrid 1: H1') -sub-structured pinion and gear, ('Hybrid 2: H2') and finally the previously developed rigid pinion and gear model which has already been largely validated by comparison by experimental evidence, further referred to as initial.

Fig

  Fig. III-1: Gear element coordinate system

Fig. III- 2 :

 2 Fig. III-2: Geometry for (a) pinion and (b) wheel of the spur test case

  Fig. III-3.a shows the tooth load distribution in N/mm over the face width b , for a tooth passing in the base plane as given by the hybrid model H1. The usual sharp variation in the line of action direction for spur gears is observed which corresponds to the transition between two tooth pairs and one single one. It can be noticed that, because of the bearing asymmetry on the pinion shaft, the load distribution across the face width is uneven. All the models give similar results as illustrated in Fig. III-3.b where slight load oscillations are observed when using finite elements. The maximum difference between the maximum tooth load distributions is around 3.5% and the full finite element based model appears as slightly more flexible with more pronounced axial asymmetry.

Fig. III- 3 :

 3 Fig. III-3: (a) Local load on contact line and (b) maximum local load for different modular models in case of rigid gear

Fig. III- 4 :

 4 Fig. III-4: Local load maximum for hybrid models webbed wheel



  Fig. III-6 shows the geometrical deviation at point M ( 00 () eM ) (cf. chapter II) which stems from a symmetric linear profile modification (same profile modification on the pinion and gear tooth tips) for which two parameters only E and  are required.

Fig. III- 6 :

 6 Fig. III-6: Profile relief parameters in the profile direction (or line of action direction) [BRU14]

Fig

  Fig. III-7: Analytical MasterCurve comparison with numerical sweep from the initial model[VEL96] and genetic algorithm[START_REF] Dhafer | Robust Optimization of Gear Tooth Modifications Using a Genetic Algorithm[END_REF]

Fig

  Fig. III-8: Quasi-static transmission error depending on wheel flexibility (rigid and flexible condensed wheel with model H1: one node pinion and condensed wheel)

  Fig. III-9: Comparison with rigid body Master Curve for (a) 2 nd hybrid model H2 with rigid bodies and (b) 1 st hybrid model H1 with flexible wheel

Fig

  Fig. III-10: Comparison with modal dependent damping -Rigidified gear body

Fig

  Fig. III-11: 2nd hybrid model H2 comparison to initial depending on Rayleigh damping definition with (a) D1: 16 25.0 , 2.5 10 RR a s b s    

Fig

  Fig. III-12: Comparison between model with flexible wheel in both hybrid models for (a) modal dependant damping and (b) Rayleigh damping D1 with 16 25.0 , 2.5 10 RR a s b s    

  order of magnitude on the pinion). Their design is therefore different since helical gears are preferred and thicker rims / tooth face widths are employed (Tab. III-5).

Fig

  Fig. III-14: Power transmission wheel geometry, including the web

Fig. III-

  Fig. III-15 presents the response curves obtained by the two models (initial and H2 with rigidified pinion and gear) when using a) modal damping factors and b) Rayleigh's damping. Itcan be observed that the critical speeds are globally at the same speeds with some slight frequency shifts. The amplitudes are comparable and the majority of the differences are attributed to the chosen damping model. However, some peaks emerge in the low-medium speed range which can be attributed to the flexible shafts modelling, i.e., beam elements as opposed to 3D finite elements thus suggesting some degree of interaction between shaft vibrations and dynamic tooth loading.

  Fig. III-15: Dynamic ratio comparison for (a) modal dependent damping and (b) Rayleigh damping

  Li has shown [LI02] that the gear and rim flexibility has also a local impact. Fig. III-16

  Fig. III-17.a shows the line of contact load distribution over a passing into the mesh contact for the flexible wheel (H1). The global shape is the one that can be expected for a helical gear, even if it is more in a triangular shape than what is usual for rigid gears.

Fig. III- 17 :

 17 Fig. III-17: Load distribution for (a) flexible web and (b) when compared to rigid and initial model maximum contact line load when passing through the base plane

Fig

  Fig. III-18: Typical web flexion mode, characterised as 0 nodal circle and 3 nodal diameters (displayed on wheel + shaft)

  [BLE79] and Schmiechen [SCH97] who studied the patterns of circular plate modal behaviour. The good correspondence between such patterns and thin webbed gears is illustrated Fig. III-20 in which ND is the number of nodal diameters and NC, the number of nodal circles.



  Fast Fourier Transform components and n the nodal diameter number as illustrated in Fig. III-21 for the theoretical 3 nodal diameter mode.

Fig

  Fig. III-21: Example of 3 rd nodal diameter theoretical shape for

  Fig. III-22: Circular plate mode shape for (a) no and (b) 1 nodal diameter mode[START_REF] Jacquet-Richardet | Bladed Assemblies Vibration[END_REF]

  Fig. III-25: Rim pressure (a) without and (b) with centrifugal effects (id. for both hybrid model)

  Fig. III-26: FE wheel model deformation (norm and direction) or (a) full wheel and (b) rim cut

in

  Fig. III-28. It is composed of the High Speed Train (HST), including the drive and intermediate gears, and the Low Speed Train (LST) comprising the pinion on the intermediate gear and the output gear. Comparisons will be performed on the HST.

Fig. III- 28 :Rim

 28 Fig. III-28: Test Transmission Architecture (Interest: drive and intermediate gear)

Fig

  Fig. III-30: Industrial FE model of the HST

Fig. III- 32 :

 32 Fig. III-32: Tooth Flank analysis with quasi-static FE model (centrifugal load for 8820 w rpm  )

  Fig. III-34.a shows the resulting pressure distribution obtained at the nominal speed (same scale as in Fig. III-33). The dynamic pressures are similar to what was found in quasi-static conditions but more evenly distributed over the tooth flank. Fig. III-34.b corresponds to the results

  Fig. III-34: Meshing pressure of the hybrid model H1 on dynamic run for (a) above material and (b) above a hole meshing conditions

  Fig. III-29 are compared with the experimental results obtained on the real system by interferometry [MOU14]. Fig. III-35 displays the percentages of deviation between the calculated and measured frequencies on the real system for several identified mode shapes in terms of nodal circles (NC) on the gear web and pinion body (both on the intermediate shaft). The vertical blue lines represent the limits of the operating mesh frequency on stage 1 which will be used further in the comparisons between experimental and simulated displacements.

Fig

  Fig. III-35: FE wheel modal comparison to interferometry test (starting to the 2 nd nodal diameter (ND) mode for each circle family (NC))

Fig 0 

 0 Fig. III-36: 9 th diameter mode as example: Rim mode of interest

7) with 0 

 0 an arbitrary initial angular coordinate on the disk. Equation 3.6 in the rotating frame leads to the following expression in the fixed frame.

  -28) was tested over the frequency range corresponding to the operating conditions (defined by the two vertical blue lines in Fig.III-35) and emphasis was placed on the mode shapes with significant rim deformations as they are likely to interfere with the meshing conditions.

Fig

  Fig. III-38: Angular position of Displacement Probes (DP) for Numerical model / Experiment comparison

Fig. III- 39 :

 39 Fig. III-39: Radial and axial position of Displacement Probes (DP) devices for rim radial displacement measure

  Fig. III-40: (a) Experimental results and (b) extracted scheme for each DP

Fig. III-

  Fig. III-40.b is a dimensionless variant of the previous graph which is introduced for comparisons purposes with simulation results and will be used in the rest of this chapter. The vertical scale has been defined by setting the maximum value to 1.0.

b

  bandwidth and i Q , system amplification factor for mode i .A damping matrix can be derived by using a Rayleigh damping as Rayleigh coefficients;   K , the averaged overall stiffness matrix. For one mode i , one obtains [LIU95], [CHO03]: frequency of the modes i . 124 (3.12) is plotted in Fig. III-41, in order to separate the contribution of R a and R b depending on the frequency range.

Fig

  Fig. III-41: Contribution of Rayleigh damping parameters depending on frequency

  Fig. III-42: Numerical data gathering scheme: Rim Node (RN)

Fig

  Fig. III-44: Fast Fourier Transform of the Rim Node response over a full revolution versus tooth number (127 being meshing order)

Fig

  Fig. III-45: Campbell diagram drawn with the RN, (0 NC corresponding to the 8 th , 9 th and 10 th nodal diameters of the family)

Fig. III- 46 :

 46 Fig. III-46: Identification of the speeds of interest

Fig 2 Z 2 Z

 22 Fig. III-47: Full frequency analysis with FFT for case A and B

Fig. III- 49 :

 49 Fig. III-49: Frequency analysis for the case D

  compare the numerical simulation results with the experimental displacement probe responses over the range of speeds under consideration Fig. III-50. Two graphs are presented: a) the first one corresponds to the direct summation of the spectrum amplitudes and, b) to the summation of filtered spectra in order to eliminate some of the numerical noise produced by the simulation process. To this end, the contributions in the non-resonant areas have been eliminated before summations were performed.

  Fig. III-50: Sum of all nodal diameter signals

  signal analysis at several main speed shows that the model successfully place the main critical speeds in frequency. The resulting curve confirms the results coherence. It also points out the issues of the signal frequency analysis detail and the noise which is present between resonances, both due to limitation in the simulation precision for time calculations reason. Despites these imprecisions, it is demonstrated that the model validation can be done with success. It has been shown in the previous chapters that the proposed hybrid modelling technique was effective and led to results on load distributions, dynamic factors and transmission errors which agree well with other simulation results and experimental findings. This chapter is aimed at illustrating the potential of the models by specifically analysing the influence of flexible members on the design of high-speed gears. Three examples are successively examined: a) a gear from an accessory gearbox whose body deflections are significant and which is considered with the objective of optimising transmission error under load. b) dynamic effects in terms of modal behaviour or high speed phenomena are investigated on a helicopter power transmission with one gear exhibiting several web modes in the operating speed range.c) finally, the coupling between the pinion and gear flexibility is studied by considering a thin webbed pinion introduced in the power transmission and its impact on its modal and high speed behaviour.

  Fig. IV-1: Load at contact for (a) 10 b   and (b) 20 b   helix angle

Fig. IV- 2 Fig. IV- 2 :

 22 Fig. IV-2 shows the level curves of transmission error for the same two examples of helical gear in the presence of symmetric linear profile modifications (the depth at tooth tip is E and the extent in the profile direction is characterised as a percentage of the active line of action  ). The Master Curves as defined by Bruyère et al. [BRU13-15] are superimposed which correspond to the family of optimal relief in the case of solid gears. It can be noticed that, in the case of helical gears, the web flexibility is influential as it moves the optimum relief area towards the deeper profile modifications and needs to be accounted for in the definition of tooth micro-geometry. It can also be observed that, for a deformable web, the performance in terms of transmission error time variations is downgraded for a larger helix angle as opposed to what is often encountered with solid gears.

Fig. IV- 3 :

 3 Fig. IV-3: Quasi-static load distribution on contact line for (a) original gear (b) gear modified with tip relief thanks to the Master Curve

Fig. IV- 5 :

 5 Fig. IV-5: Definition of machining deviations of gears[START_REF] Li | Finite element analyses for contact strength and bending strength for a pair of spur gears with machining errors, assembly errors and tooth modifications[END_REF]

Fig. IV- 6 :

 6 Fig. IV-6: Profile deviation   ij eM for a line of contact through the base plane

  Fig. IV-7: Contact line load for (a) Aeronautic ISO standard and (b) lower quality gear

Fig

  Fig. IV-8: Dynamic ratio comparison for deviated gears

Fig

  Fig. IV-10: Profile modification and deviation definition

Fig. IV-

  Fig. IV-12 shows the dynamic ratio of modified gears for both modal dependent (a) and (b) Rayleigh damping models. The response curves are globally close to those calculated with linear tip relief; however, the amplitudes jumps previously observed when profile defects were introduced have almost disappear showing that parabolic modifications can be superior to linear relief re dynamic mesh forces.

Fig

  Fig. IV-11: Quasi-static load distribution with parabolic modifications and profile deviations

Fig

  Fig. IV-14 below for the two damping models. The specific influence of the thin web pinion seems to be reduced to a shift in the response peaks towards the lower speeds typical of a reduction in the system global stiffness.

Fig

  Fig. IV-15: Quasi-static transmission error depending on pinion model and rigidity

Fig

  Fig. IV-16: Profile optimisation for model with flexible bodies

  -1: Display of the several speeds for rotational wheel analysis

Fig. IV-

  Fig. IV-17 shows the dynamic ratio Fast Fourier Transform for the three speeds of interest 12 ,  and

Fig

  Fig. IV-17: Dynamic ratio for the three dynamic conditions

Fig

  Fig. IV-18: Web node definition for web displacement

Fig. IV- 19 : 2  and 3 Z 2 (b) Case 3  2  3 

 19232323 Fig. IV-19: Web node response for

Fig. IV- 22 :

 22 Fig. IV-22: StopBand filter to eliminate wheel rotation frequency in WN signal

Fig. IV- 23 : 4 (b) 5  , 3ND for speed 4  and two modes for 5 

 234545 Fig. IV-23: Web response phasing after post-processing

5  4  and 5 )Fig

 545 Fig. IV-25: Mode shape of full FE model modal analysis

2 )

 2 Fig. IV-28: Hollow web geometry with regard to wheel rim

Fig 5 Tab. IV- 5 : 3 

 553 Fig. IV-29: Test Case C: Test Case B + secondary holes

3 

 3 Fig. IV-31: Dynamic ratio response depending on test case for

Figand 3  6 Fig

 36 Fig. IV-32 displays the contact pressure patterns for Case B obtained at the three speeds in Tab. IV-7 for two different conditions: a) a tooth between two holes and, b) a tooth just above a hole. For

Fig. IV- 33 :Fig

 33 Fig. IV-33: Linear modification along lead

  Fig. IV-35 can be introduced. The pinion shaft is hollow with a larger bore under the pinion and a web thickness less than half of the rim width in order to soften the contact conditions and improve tooth load distributions.

Fig. IV- 35 :

 35 Fig. IV-35: Webbed pinion geometry

Fig

  Fig. IV-38: Dynamic ratio comparison for webbed pinion inclusion

Fig

  Fig. IV-39: Pressure distribution at

Fig

  Fig. IV-40: Centrifugal effect FE calculation for 3000 / p rad s 

  -9: Wheel web geometry thickness modification As clear from Tab. IV-9, the material removal has consequences on both the wheel flexibility and mass. To assess the resulting impact, comparative modal tests have been conducted on three cases using the FE model of the wheel alone prior to condensation. The results in Fig. IV-41 are indexed by Nodal Circle and Diameter number and prove that the modifications have a limited impact on the wheel web modal behaviour.

Fig

  Fig. IV-41: Web thickness influence on wheel modal behaviour -FE analysis

Fig

  Fig. IV-42: Flexion mode example: 2 circles, 3 diameters (17 922 Hz for Case O)

Fig

  Fig. IV-44: Maximum pressure over a meshing period for the three test cases at 3000 / p rad s 

Fig

  Fig. IV-46: Crowning definition in lead direction

Fig

  Fig. IV-47: Crowning influence on dynamic pressure distribution

Fig

  Fig. IV-48: Dynamic ratio for original gear O with various crowning at 3000 / p rad s 

  -49: Maximum load over a passage through the base plane comparison for various crowning values at 3000 / p rad s  This fact is confirmed by the flank pressure distribution plotted for test case E in Fig. IV-50. The variations for an increasing crowning depth are clearly visible with a more centred pressure distribution with a significant amplification of the maximum pressure combined with pressure relief near the tooth edges. Other shapes of lead modifications such as elliptical crowning or linear + parabolic modification could have been more effective in distributing the load on the teeth more evenly and reducing overloads in the tooth central part.

  -50: Test Case E: meshing pressure distribution depending on crowning at

Figure 1 :

 1 Figure 1: Shaft element -Beam definition

3 )F

 3 With  and  ponderation coefficients. They are defined for the system unconditional stability, system matrices and its full definition, it can be written at tt  under the following expression: global matrices of the system and t+Δt the time dependant 2 nd member vector which is known at tt  .

  boîtes de transmission par engrenages des moteurs d'hélicoptère convoient la puissance mécanique du turbomoteur aux accessoires (pompes, démarreur) et au rotor. Leur conception dépend des nécessités des équipements embarqués, en particulier l'allègement pour réduire la consommation en carburant. Les engrenages haute-vitesse de la transmission sont allégés grâce à des enlèvements de matière dans les corps sous la denture, les voiles-minces. La nouvelle flexibilité de ces engrenages nécessite le développement de nouveaux outils capables d'intégrer à la fois la modélisation dynamique couplée des vibrations issues du passage des dents de l'engrenage et le comportement statique et modal de ces roues à voiles-minces. Les travaux de recherche présentés dans ce manuscrit ont pour but le développement d'un modèle hybride intégrant l'ensemble des effets évoqués plus haut afin de réaliser une simulation réaliste haute-vitesse des vibrations d'une transmission d'hélicoptère.

Figure 2 :

 2 Figure 2: Engrènement, illustration du nombre de dents en prises

L

  'ensemble de ces déviations perturbent les contacts, introduisant une autre source d'excitation dans l'engrenage. L'engrenage en fonctionnement est caractérisé par des paramètres de réponse particuliers. Le premier est l'erreur de transmission, dont la notion a été introduite par [HAR58]. Elle est définie à partir «d'une position instantanée d'une des roues, comme étant le déplacement angulaire de la dent au contact depuis la position qu'elle occuperait si les dents étaient rigides et non modifiées ». Harris a été le premier à établir que les variations de l'erreur de transmission à faible vitesse étaient responsables des vibrations de l'engrenage à un régime plus élevé. Un autre paramètre courant est le coefficient dynamique représentant la charge à la denture normalisée par la charge statique transmise par l'engrenage. Son maximum permet généralement d'estimer les vitesses critiques de l'engrenage (Figure 3).

Figure 3 :

 3 Figure 3: Comparaison des résultats expérimentaux de Kubo [KUB72] et les prédictions numériques selon [OZG88]

Figure 4 : 3 .

 43 Figure 4: Modèle EF pour (a) la charge à l'engrènement et (b) effets centrifuges [LI02-08.a]

Figure 5 :

 5 Figure 5: (a) Elément d'engrenage et (b) plan d'action discretisé associé

  .a) condensé par la technique de sous-structuration sur les noeuds de la jante et aux paliers (Figure 6.b). La sousstructuration réduit la taille d'un modèle EF en ne conservant que les degrés de libertés choisis. Elle modélise le comportement statique et modal de la roue jusqu'à une fréquence donnée. La sous-structuration à interfaces libres de Herting [HER85] est utilisée ici.

Figure 6 :

 6 Figure 6: Modèle hybride d'engrenage avec roue flexible

Figure 7 :Figure 8 :

 78 Figure 7: Schéma de la tranche mince de dent (a) seule et (b) avec les deux points pour attache à la jante EF ( ij M étant noté M )

Figure 9 :

 9 Figure 9: Evolution de la discrétisation du segment de la ligne de contact

2 X

 2 les degrés de liberté des noeuds maîtres de la sous-structure pied de dent après l'interface.

  est ajouté sous forme d'amortissement visqueux dépendant des modes et de leur participation énergétique selon Ankouni [ANK14-16] et Umezawa [UME85] ou, sous forme d'amortissement de Rayleigh par des coefficients multipliant les matrices de raideur et masse [CHO03]. Dans les composants flexibles condensés, les effets centrifuges prennent eux la forme d'un vecteur des efforts centrifuges. Il est démontré dans la partie 4 (Validation) que ces effets sur les paramètres du système et notamment sur les données engrenages sont négligeables.

  Table 1: Paramètres de définition de la boîte accessoire Le pignon (Figure 10.a) est plein et possède un arbre non-symétrique. La roue est centrée sur son arbre et sur ses roulements, elle possède un voile-mince centré sous sa jante.

Figure 10 :

 10 Figure 10: Plans de définition des corps d'engrenages

Figure 11 :

 11 Figure 11: Comparaison des coefficients dynamiques pour les corps rigides

Figure 12 :

 12 Figure 12: Coefficients dynamiques avec les corps souples pour les modèles hybrides

Figure 13 :

 13 Figure 13: Plan de définition de la roue

Figure 14 :

 14 Figure 14: Comparaison du coefficient dynamique avec la roue condensée rigidifiée

Figure 15 :Figure 16 :Table 3 :

 15163 Figure 15: Influence des effets centrifuges pour (a) le coefficient dynamique et (b) les déplacements de la jante ( 3000 / p rad s  )

Figure 17 :

 17 Figure 17: FFT des déplacements radiaux d'un point du voile à 8820 w rpm 

Figure 18 :

 18 Figure 18: Validation expérimentale des déplacements du voile par comparaison des signaux

Figure 19 :

 19 Figure 19: Charge à la denture pour 20  

Figure 20 :

 20 Figure 20: Corrections de profil et Master Curve [BRU15] avec (a) droit (b) 20 

Figure 21 :

 21 Figure 21: Comparaison avec des corrections de profils "optimales" en cas d'erreur de profil

Figure 22 :

 22 Figure 22: FFT du déplacement radial pour une résonance à

Figure 23 :

 23 Figure 23: Déformée de la roue sur le mode étudié (2 250 Hz)

Figure 24 :

 24 Figure 24: Analyse du déplacement de la déformée au cours du temps en filtrant la FFT sur la fréquence d'intérêt ( 1/16  dt Tm )

  Allègement d'une Transmission ComplèteLe modèle avec le pignon et la roue condensés permet l'étude d'une transmission complètement allégée. Le pignon rigide de la transmission de puissance académique est remplacé par un pignon à voile mince lui-aussi. Les modes propres du pignon étant à haute fréquence, l'engrènement est principalement affecté par sa déformée statique. Elle change, par exemple, la répartition énergétique des modes du système. Néanmoins, les modes du voile de la roue restent découplés. L'étude menée à haute vitesse ( la déformation du pignon sous effets centrifuges participe fortement à la distribution de pression à la denture (Figure25).

Figure 25 :

 25 Figure 25: Effet de l'inclusion d'un pignon à voile-mince

Figure 26 :

 26 Figure 26: Amélioration de la distribution de charge à haute vitesse grâce à l'inclusion d'un bombé d'hélice

  Considering rigid pinion and gear bodies with one central node each, the normal deflection can be expressed in terms of the degrees of freedom at the central node 1

						O and 2 O
	as:
						2.8)
	With	max	M		  ij	

eM maximum of   ij eM over all potential contact points ij M

  Les validations du modèle sont faites pour les deux versions : H1: pignon rigide et roue condensée, H2 : pignon + roue condensés. Tout d'abord, les engrenages académiques sont rigidifiés par un haut module d'Young dans la sous-structure ( au modèle numérique précédemment développé et validé au LaMCoS [BAU02]. Enfin, la flexibilité des corps d'engrenages est étudiée. La dernière validation a été faite sur une boîte industrielle d'engrenage par comparaison de données numériques et expérimentales. La première validation du modèle est réalisée sur une transmission académique de boite accessoire, dont les données sont détailléesTable 1. Le couple moteur du pignon est

		14 4.10	/	2	). Puis les deux
	modèles sont comparés
	E		N m

Boite Accessoire -Validation Numérique

Table 4 :

 4 Propagation des ondes dans la roue

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI127/these.pdf © [B. Guilbert], [2017], INSA Lyon, tous droits réservés

Scientific Contributions ......................................................................... Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI127/these.pdf © [B. Guilbert], [2017], INSA Lyon, tous droits réservés

n = -n Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI127/these.pdf © [B. Guilbert], [2017], INSA Lyon, tous droits réservés

Acknowledgements

Acknowledgements Acknowledgements ..................................................................................... Abstract .......................................................................................................

Chapter II Numerical Model Presentation

La déflexion normale à la denture s'écrit avec les translations aux points

Avec :

Conference Publications and Oral Presentations