Erwann Delay

L p ALMOST CONFORMAL ISOMETRIES OF SUB-SEMI-RIEMANNIAN METRICS AND SOLVABILITY OF A RICCI EQUATION

Keywords: Sub-Semi-Riemannian metrics, Ricci curvature, Einstein metrics, Inverse problem, Quasilinear elliptic systems. 2010 MSC : 53C21, 53A45, 58J05, 35J62, 53C17, 53C50

Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M . Under suitable conditions, we show that they are almost conformally isometric in an L p sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in some sense solvable, without assuming any closeness near a special metric.

Introduction

The goal of this note is to prove that the two principal results of D. DeTurck [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] given for positive definite symmetric bilinear form and for special Einstein metrics can be extended significantly in different ways.

Firstly, we can extend the positive definideness condition of the Riemannian metrics to Sub-Semi-Riemmannian metrics with the same rank and signature.

Secondly we are able to replace some particular Einstein metrics of non zero scalar curvature by any parallel Ricci metrics (ie. metrics with covariantly constant Ricci tensor).

Date: January 18, 2107. Let M be a smooth compact manifold without boundary. A Sub-Semi-Riemannian metric G (SSR-metric for short) is a symmetric covariant 2-tensor field with constant signature and constant rank.

Let us now state the first result, interesting by itself, about almost conformal isometries.

Lemma 1.1. Assume that G and G are two smooth SSR-metrics on M with the same rank and signature. Let g be a smooth Riemmannian metrics on M , let p ∈ [1, ∞) and let ε > 0. Then there exist a smooth diffemorphism Φ and a smooth positive function f such that Φ * (f G)-G is ε-close to zero in the L p norm relative to g.

Before going to the application for a Ricci equation, let us introduce some notations. For (M, g) a smooth riemannian manifold, we denote by Ric(g) its Ricci curvature. For a real constant Λ, we consider the operator Ein(g) := Ric(g) + Λg. This operator is geometric in the sense that for any smooth diffeomorphism ϕ, ϕ * Ein(g) = Ein(ϕ * g). We would like to invert Ein. We thus choose E a symmetric 2-tensor field on M , and look for g Riemannian metric such that Ein(g) = E.

(1.1) This is a geometrically natural and difficult quasilinear system to solve, already for perturbation methods. The prescribed Ricci curvature problem has a long history starting with the work of D. DeTurck [START_REF] Deturck | Existence of metrics with prescribed ricci curvature : Local theory[END_REF], [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], [START_REF] Deturck | Metrics with prescribed Ricci curvature of constant rank. I. The integrable case[END_REF], [START_REF] Deturck | Uniqueness and nonexistence of metrics with prescribed Ricci curvature[END_REF], [START_REF] Deturck | Some regularity theorems in riemannian geometry[END_REF], [START_REF] Baldes | Nonexistence of Riemannian metrics with prescribed Ricci tensor, Nonlinear problems in geometry[END_REF], [START_REF] Hamilton | The Ricci curvature equation[END_REF], [START_REF] Ph | Obstruction to prescribed positive Ricci curvature[END_REF], [START_REF]Local solvability of elliptic, and curvature, equations on compact manifolds[END_REF], [START_REF]Etude locale d'opérateurs de courbure sur l'espace hyperbolique[END_REF], [START_REF] Delay | Ricci curvature in the neighbourhood of rank-one symmetric spaces[END_REF], [START_REF]Study of some curvature operators in the neighbourhood of an asymptotically hyperbolic Einstein manifold[END_REF], [START_REF]Sur l'inversion de l'opérateur de Ricci au voisinage d'une métrique Ricci parallèle[END_REF], [START_REF] Delay | Inversion d'opérateurs de courbure au voisinage de la métrique euclidiennne[END_REF],... Motivated by the explosion of studies around the Ricci flow, and recently, some discrete versions thereof (eg. Ein(g i+1 ) = g i ), a renewed interest arises for this kind of natural geometric equations. We invite the reader to look at the nice recent works of A. Pulemotov and Y. Rubinstein [START_REF] Pulemotov | Metrics with prescribed Ricci curvature near the boundary of a manifold[END_REF] and [START_REF] Pulemotov | Ricci iteration on homogeneous spaces[END_REF] for related results. Our contribution here is the following. Theorem 1.2. Assume that M carries a Riemannian metric g with parallel Ricci tensor. Let Λ ∈ R such that Ein(g) is non degenerate, and that -2Λ is not in the spectrum of the Lichnerowicz Laplacian of g. 1 Then for any E ∈ C ∞ (M, S 2 ) with the same rank and signature as Ein(g) at each point of M , there exist a smooth positive function f and a Riemannian metrics g in C ∞ (M, S 2 ) such that

Ein(g) = f E.
The proof goes by combining the Lemma 1.1 , the local inversion result of Proposition 3.1 for weak regular metric (where the conformal 1 Like D. Deturck [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], we may allow an eigenspace spanned by g when Λ = 0 factor f is not required) and a regularity argument. We have then solved the problem up to a positive function f . Here we do not expect that f can be taken equals to one in general, this will be the subject of future investigations.

Parallel Ricci metrics, are (locally) products of Einstein metrics (see eg. [START_REF] Wu | Holonomy groups of indefinite metrics[END_REF]). They exists on the simplest examples of manifolds who do not admit Einstein metrics, like S 1 ×S 2 , Σ g ×S 2 , (g ≥ 1) or Σ g ×T 2 , (g ≥ 2) where Σ g is a surface of genius g. They are also static solutions of some geometric fourth order flows (eg. ∂ t g = ∆ g Ric(g)). Finally they are particular cases of Riemannian manifolds with Harmonic curvature (or equivalently Codazzi Ricci tensor).

Our global result show once again that such metrics with covariantly constant Ricci tensor deserve a particular attention.

L p closeness of some Sub-Semi-Riemannian metrics

We follow the section 3 called "approximation lemma" in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] in order to verify that all the step there can be adapted for SSR-metrics as above. This will prove the Lemma 1.1.

We will keep almost the same notations as in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], just replacing S and R there respectively by G and G here.

Let G and G be as in the introduction, we thus assume they have the same signature and the same rank. For the rest of the section we fix a Riemannian metric g, an ε > 0 and p ∈ [1, +∞). All the measures, volumes, and norms are understood with respect to g.

At each point x ∈ M , the two SSR-metric G and G having the same rank and signature, there exists an orientation preserving automorphism u x of T x M , such that

G x (u x (.), u x (.)) = G x (., .).
For x ∈ M , the following construction can be performed using the gexponential map at x. There exists an open set U x such that : (i) U x is contained in a coordinate neighborhood of x where in this coordinate (centered at 0), up to a positive automorphism u

x of T x M , G is equal to G at x: t u x G x u x = G x ,
(ii) For any positive real α x , the linear change of coordinates

Φ x := √ α x u x
satisfies on U x the estimate (the left hand side of which does not depend upon α x , and vanishes at the origin),

(Φ * x 1 α x G) y -G y p ≤ min ε p 2Vol(M ) , |G y | p . (2.1)
We consider a triangulation of M where each simplex S lies in the interior of some U x with x ∈ S. Since the point x belong to the interior of the simplex S, shrinking α x if necessary, we are sure that Φ x send S into S (the norm of Φ x approaches zero when α x tends to zero).

Let Ω, Ω 1 , Ω 2 , Ω 3 be some open neighbourhoods of the (n-1) dimensional skeleton (composed with union of the boundary of all simplex) with the properties :

Vol(Ω) < ε p 2(max M |G| + 2 max M |G|) p ,
and

Ω 3 ⊂ Ω 3 ⊂ Ω 2 ⊂ Ω 2 ⊂ Ω 1 ⊂ Ω 1 ⊂ Ω.
The rest of the proof in section 3 of [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] is based on triangular inequalities between norms of tensors and can be implemented here without any change. For a better understanding, though, we provide further details of the figure page 368 of [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], specifying the estimates that occur on the differents parts of the simplex, see figure 1. On the picture, we have denoted the error |Φ * (f G) -G| by e :

e = |Φ * (f G) -G|.
On the inner part T of the simplex, e is estimated by (2.1). The transition of the diffeomorphism Φ, on the middle ring R 2 = S ∩ (Ω 1 \Ω 2 ), from Φ x to the identity, still exist because our Φ x = √ α x u x is an orientation preserving map with norm less than 1 as in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF].

G(x, y, z) = -g X ,y,z (x) ⊕ g Y,x,z (y) ⊕ 0 Z ,
where g X ,y,z is a family of Riemannian metrics on X , depending on the parameters y, z and smooth in all of its arguments. Here, any of the three manifolds but one may be reduced to a point.

Solvability of a Ricci type equation

We revisit the section 2 of [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] called "perturbation lemma".

We first need to introduce some operators. The divergence of a symmetric 2-tensor field and its L 2 adjoint acting on one form are (δh

) j := -∇ i h ij , (δ * v) ij := 1 2 (∇ i v j + ∇ j v i ).
The gravitationnal operator acting on symmetric 2-tensors is

G(h) := h - 1 2 Tr g (h)g. The Lichnerowicz Laplacian is 2 ∆ L = ∇ * ∇ + 2 Ric -2 Riem .
It appears in the Linearization of the Ricci operator :

D Ric(g) = 1 2 ∆ L + δ * δ G.
The Hodge Laplacian acting on one forms is

∆ H = ∆ + Ric = ∇ * ∇ + Ric = d * d + dd * .
We also define the following Laplacian

∆ V := 2 δ G δ * = ∇ * ∇ -Ric = ∆ H -2 Ric .
We denote by V its finite dimensionnal kernel, composed of smooth one forms (by elliptic regularity).

We start with the equivalent of proposition 2.1 in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF].

Proposition 3.1. Let (M, g) be a smooth Riemannian manifold with parallel Ricci curvature. Let Λ ∈ R such that Ein(g) is non degenerate and that -2Λ is not in the spectrum of the Lichnerowicz Laplacian. Let k ∈ N and p > n. Then for any E close to Ein(g) in H k+1,p (M, S 2 ), there exist a Riemannian metrics g in H k+1,p (M, S 2 ) such that

Ein(g) = E.
In [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], the proof of the corresponding proposition is given by a succession of lemmas. We thus revisit them one after the other. Some care is needed because we have to replace Ric and -∆ L there, respectively with Ein and ∆ L + 2Λ here. Furthermore, in our context, the operator ∆ L + 2Λ has no kernel whereas the kernel of ∆ L is nonempty in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF], spanned by g. We clearly have also for any Riemannian metrics g, the Bianchi identity δG(Ein(g)) = 0.

We start with a local study of the action of the diffeomorphim group on the covariant symetrics 2-tensors, near a non degenerate parallel one. The result obtained remains in the spirit of the local study near a Riemmanian metric by Berger, Ebin, or Palais (see eg. the lemma 2.3 of [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF]). Here the metric tensor is replaced by a non degenerate parallel tensor field. Lemma 3.2. Let E be a smooth, non degenerate and parallel symmetric two tensor field. Let X be a smooth Banach submanifold of H k,p (M, S 2 ), whose tangent space at E is a complementary of δ * (H k,p (M, T 1 )). Then for any E close enough to E in H k,p (M, S 2 ), there exist an H k+1,p diffeomorphism Φ close to the identity such that Φ * E ∈ X .

Proof. The tensor field E being parallel, its Lie derivative in the direction of a vector field v is

L v E = 2δ * (Ev).
Locally, the submanifold X can be seen as the image of an immersion

X : U -→ H k,p (M, S 2 ), with X(0) = E. We define T ⊥ to be the set of vector fields v ∈ H k+1,p (M, T 1 ) such that E v is L 2 -orthogonal 3 to ker δ * . Let F : U × T ⊥ × H k,p (M, S 2 ) -→ H k,p (M, S 2 ), defined by F (k, Y, E) = Φ * Y,1 (E) -X(k)
, where Φ Y,1 is the flow of the vector field Y at time 1. We have F (0, 0, E) = 0 and the linearisation of F in the first two variables is

D (k,Y ) F (0, 0, E)(l, X) = 2δ * (E X) -DX(0)l. Now because H k,p (M, S 2 ) = δ * (H k+1,p (M, T 1 )) ⊕ Im DX(0),
and E is non degenerate, then D (k,Y ) F (0, 0, E) is an isomorphism. From the implicit function theorem, for E close to E, there exist k and Y small such that F (k, Y, E) = 0.

Let us recall the lemma 2.5 in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] :

4 Lemma 3.3. For k ≥ 1, we have H k,p (M, S 2 ) = (ker δG ∩ H k,p (M, S 2 )) δ * (V ) ⊕ δ * (H k+1,p (M, T 1 )) ⊕ Gδ * (V ).
The equivalent of the lemma 2.6 in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] becomes (we do not have to quotient by R g because of no kernel for us, and so we do not need to adjust with a constant c). Then for some neigbborood U of 0, F (U ) is a Banach submanifold of H k,p (M, S 2 ) whose tangent space at F (0) = Ein(g) is a complementary space of δ * (H k+1,p (M, T 1 )).

Proof. We have to show that the derivative of F at g is injective and its image has the closed subspace δ * (H k+1,p ) as complementary. A metric with parallel Ricci tensor is a local Einstein product so is smooth. We first show that the spaces Im δ * and ker δG are "stable" (modulo two points of regularity) by ∆ L + 2Λ but also by D Ein(g) (when the metric is Ricci parallel). Indeed, recall that in that case [START_REF] Lichnerowicz | Propagateurs et commutateurs en relativité générale[END_REF]:

δ ∆ L = ∆ H δ,
and the adjoint version :

∆ L δ * = δ * ∆ H .
We deduce that

(∆ L + 2Λ) δ * = δ * (∆ H + 2Λ) = δ * (∆ V + 2 Ein), and
D Ein(g) δ * = 1 2 δ * (∆ H + 2Λ) + 1 2 δ * ∆ V = δ * (∆ V + Ein) = δ * (∆ + Λ)
thus the "stability" of Im δ * by the two operators above. When restricted on the kernel of δ G, we trivially have

D Ein(g) = 1 2 (∆ L + 2Λ),
but also, by linearising δ G Ein(g) = 0 for instance, δ G (∆ L + 2Λ) = 0, then the stability of ker δ G.

We can also remark that with the formula above, if v ∈ V ,

(∆ L + 2Λ)(δ * v) = 2δ * (Ein v) = 2 Ein δ * v and D Ein(g)(δ * v) = δ * (Ein v) = Ein δ * v. For any function u, it is well known that ∆ L (ug) = (∆u)g, so (∆ L + 2Λ)(d * w g) = [(∆ + 2Λ)d * w]g = [d * (∆ H + 2Λ)w]g. We obtain that (∆ L + 2Λ)Gδ * = Gδ * (∆ H + 2Λ). If v ∈ V , we deduce (∆ L + 2Λ)Gδ * v = Gδ * (2 Ein v).
We will complete the proof of proposition 3.1, where now E ∈ H k+1,p , but g and ϕ still cames from Lemma 3.5 so ϕ is a priori not regular enough. If we inspect the pages 364-365 in [START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] we can see that we just have to change Ric(g) by Ein(g) to obtain that ϕ is in fact in H k+2,p . We conclude that (ϕ -1 ) * g ∈ H k+1,p and has E as its image by Ein. At this level we also use that Ein(g) is non degenerate (see equation (2.8) there).

The Theorem 1.2 is now a direct consequence of the Lemma 1.1, the Proposition 3.1 with k = 0, and the regularity result of [START_REF] Deturck | Some regularity theorems in riemannian geometry[END_REF]. Exemple 3.6. Recalling that the Ricci curvature of a product of Riemannian manifolds is the direct sum of the Ricci curvatures of each factors, we see that a product of Einstein manifolds clearly satisfies the assumption of the Theorem 1.2. The simplest example combining the 3 possibilities of Einstein constants is the following. Let us consider three compact Einstein manifolds (X , g -), (Y, g + ), (Z, g 0 ) with Ricci curvatures given by Ric(g -) = -g -, Ric(g + ) = g + , Ric(g 0 ) = 0. Then M = X × Y × Z endowed with g = g -⊕ g + ⊕ g 0 , has parallel Ricci curvature equal to Ric(g) = -g -⊕ g + ⊕ 0.

In this example, the kernel of ∆ L contains the parallel tensors h = c -g -⊕ c + g + ⊕ c 0 g 0 , for any constants c -, c + , c 0 . Here, we only have to choose Λ in order to destroy this kernel and make Ein(g) non degenerate.

1

 1

Figure 1 .

 1 Figure 1. The simplex S with the values of f and Φ, and the estimates of e.

Lemma 3 . 4 . 5 F

 345 Suppose k ≥ 0, p > n, and g satisfies the hypotheses of theorem 1.2 letK = (ker δG ∩ H k+2,p (M, S 2 )) δ * (V ) ⊕ Gδ * (V )and define F : K -→ H k,p (M, S 2 ) by (b) := Ein(g + b).

closed complementary suffice

It seems there is a misprint in the proof this lemma: The Ricci term for δGδ * at the top of page 362 in[START_REF] Dennis | Prescribing positive Ricci curvature on compact manifolds[END_REF] has a different sign.

To avoid ambiguities, we may take any fixed closed complementary W to δ * (V ) in ker δG ∩ H k+2,p (M, S 2 ) instead of the first factor of K

Here also we have to replace the first factor by (∆ L + 2Λ)W when a choice of W was made in the first factor of K.

Acknowledgments : I am grateful to Philippe Delanoë for comments and improvements, and to Alexandra Barbieri and François Gautero for the picture of the simplex.

Assume that -2Λ is not an eigenvalue of ∆ L , then ∆ L + 2Λ is an isomorphism from H k+2,p (M, S 2 ) to H k,p (M, S 2 ). The image of the splitting in Lemma 3.3 by ∆ L + 2Λ produce: 6

The two first factors are the same than the image by D Ein(g) of the corresponding spaces in Lemma 3.3. Let us study the image of third one. For v ∈ V , we compute

We deduce for instance that

Let us define

We now prove that

The fact that F is the sum of the two factors is clear by (3.2). Let w in the intersection of the two factors, so

for some u ∈ H k+1,p (M, T 1 ) and v ∈ V . Because of the decomposition of F, we deduce that Gδ * Ein v = 0 thus (∆ L + 2Λ)Gδ * v = 0, then Gδ * v = 0 and finally, by (3.1), n-2 2 δ * δδ * v = 0 so w = 0. We have obtained

From the Lemma 3.2 with E = Ein(g) and Lemma 3.4 we directly deduce : Lemma 3.5. If E ∈ H k,p and |E -Ein(g)| k,p < ε, then there exist a metric g ∈ H k+2,p and a diffeomorphism ϕ ∈ H k+1,p for which Ein(g) = ϕ * E.


David Guyonneau

Emmanuel Mermoz

Jean Mailhe

Jean-Michel Sprauel

Jean-Marc Linares

Stress optimization and study of the sensitivity to geometric variations of a spur gear tooth profile ´

Keywords: Spur Gear / Tooth profile / Tooth Contact Analysis (T, C, A, ) / Monte Carlo simulation / Uncertainty

This paper presents an approach for obtaining an optimized geometry for the flank of a tooth by minimizing the equivalent contact stress. The stress calculation method is based on Hertz theory. As the majority of tooth profiles are involute, the geometric variation of the flank of the tooth is achieved variationally relative to the involute profile. The optimum profile is obtained by Monte Carlo simulation. During this optimization, a polynomial expression of the tooth geometry is used. The parameters influencing the simulation are the four characteristic contact points. The approach presented has been applied in a representative case. A study of the geometric sensitivity was conducted on the optimized tooth profile. Two different approaches were considered: by Monte Carlo simulation and analytical propagation. The robust and linear nature of the behavior of the tooth profile was demonstrated when it was subjected to geometric variations.

Introduction

In the aeronautical sector, the mechanical transmission of power is achieved through various gear configurations (Fig. 1). Full control of the geometry of tooth a Corresponding author: david.guyonneau@eurocopter.com profiles and their manufacture are indispensible in order to guarantee the reliability of the mechanism. In order to ensure the correct operation of a reduction gearbox, the majority of studies examine the mechanical strength of the gears under load. For helicopter manufacturers, the search for performance involves minimizing weight whilst increasing the efficiency and the service life of the mechanical components constituting, for example, the Main

F C I n contact P 1 R pi R bi R ti W x i y i z i (x i , y i , z i ) Z i α ε t μ θ ω 1 σ eq σ i 1 2 S E
Gearboxes (M.G.B.). A 300 kg M.G.B., including 5 reduction stages, transmitting two megawatts power is shown in Figure 1.

Currently, the sizing and geometric design of meshing surfaces constitute complex strategic domains. In the field of application to helicopter power gearboxes, specific work has been undertaken to extend the service life of gear assemblies and to ensure improved behavior under load. With respect to this problem, work is essentially focused on:

characterization of the material and its various treatments (shot peening, nitriding), elaboration of behavioral laws representing the contact physics, geometric modification evolution of the meshing surfaces.

This paper concerns the third item. In this mechanical environment, the scientific literature proposes numerous studies; a representative sample will be presented in the bibliographical study (Sect. 2). Two study paths are examined in this paper: presentation of a method for generating a tooth profile minimizing the equivalent contact stress during meshing and a study of the sensitivity of this

Bibliography on the geometric generation and sensitivity of gears

Among the literature, the work by Henriot [START_REF] Henriot | Engrenages -Conception -Fabrication -Mise en oeuvre -Ed[END_REF] is a considerable source of information on the design of gears and the characterization of their fields of use. It presents the usual definition of the variables allocated to gears. The work by Henriot [START_REF] Henriot | Engrenages -Conception -Fabrication -Mise en oeuvre -Ed[END_REF] incidentally contributed to the elaboration and compilation of international standards (ISO standard 6336 in particular). The involute tooth profile represents the founding basis for modern gear geometries. The sizing of the teeth is achieved by accounting for the spatial requirements, calculating the contact pressure (pitting) related to the applied loads, estimating the maximum shear stress in the root of the tooth (fracture), minimizing the gearing noise and optimizing the efficiency.

This design environment enables definition of the functional operating conditions of a gear. Our bibliography hinges around three main axes:

the methods for obtaining tooth profiles, a study of the mechanical behavior based on the equivalent contact stress, the influence of geometric variations of the flank of the tooth.

The notion of conjugated profiles is an inescapable aspect in the theory of gears. ISO standard 21771 [START_REF]Gears -Cylindrical involute gears and gear pairs -Concepts and geometry[END_REF] specifies the concept and the geometry of tooth flank. It gives some advice to modify the local tooth profile in order to improve the meshing. The paper by Spitas et al. [START_REF] Spitas | Fast modeling of conjugate gear tooth profiles using discrete presentation by involute segments[END_REF] proposes a method for obtaining combined tooth profiles using the properties of the involute. The work by Ye & Ye [START_REF] Ye | A new method for seeking the optimum gear tooth profilesthe theoretical basis of Wildhaber-Novikov gearing[END_REF] and Simon [START_REF] Simon | Influence of tooth modifications on tooth contact in face-hobbed spiral bevel gears[END_REF] also studies the geometry and the methods for obtaining a gear tooth profile. They introduce local geometric modifications in order to observe their influence on the behavior of the gear during meshing. Faggioni et al. [START_REF] Faggioni | Dynamic optimization of spur gears[END_REF] optimize the geometry of spur gears in order to improve their dynamic behavior. For that purpose, they construct a rheological model using springs and dampers. All these works propose a geometric definition for the flank of a tooth. The paper of Velex et al. [START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF] and Ghribi et al. [START_REF] Ghribi | A contribution to the design of robust profile modifications in spur and helical gears by combining analytical results and numerical simulations[END_REF] purposed also local geometrical modifications of the gear tooth profile. They studied the influence of profile modifications on transmission errors.

Characterization of the loading

A great number of studies were performed to characterize the load applied to gears. Velex et al. [START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF] used an analytical model to define the mechanical solicitations applied to modify spur and helical gear tooth profiles. In similar approach, Pedrero et al. [START_REF] Pedrero | Load distribution model along the line of contact for involute external gears[END_REF] studied an involute tooth profile of an external spur gear under load by characterization of the mechanical behavior along the line of action. The characterization of the stresses in gearboxes has been covered by Osman and Velex [START_REF] Osman | A model for the simulation of the interactions between dynamic tooth loads and contact fatigue in spur gears[END_REF] and Pedrero et al. [START_REF] Pedrero | Contact stress calculation of undercut spur and helical gear teeth[END_REF]. Hwang et al. [START_REF] Hwang | Contact stress analysis for a pair of mating gears[END_REF] examine the identification and analysis of the equivalent contact stress. However, these numerous works [START_REF] Faggioni | Dynamic optimization of spur gears[END_REF][START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF][START_REF] Ghribi | A contribution to the design of robust profile modifications in spur and helical gears by combining analytical results and numerical simulations[END_REF][START_REF] Pedrero | Load distribution model along the line of contact for involute external gears[END_REF][START_REF] Osman | A model for the simulation of the interactions between dynamic tooth loads and contact fatigue in spur gears[END_REF][START_REF] Pedrero | Contact stress calculation of undercut spur and helical gear teeth[END_REF][START_REF] Hwang | Contact stress analysis for a pair of mating gears[END_REF] do not question the geometry of the involute tooth profile. They concentrate on characterization of the mechanical loading at the contact points. Most of these works are based on F.E.M. (Finite Element Method) calculations, but some of them characterize the contact zone using Hertz theory. These various works consider the mechanical aspect of the problem by characterizing the loading on the tooth.

The work by Litvin et al. [START_REF] Litvin | Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis[END_REF] covers the geometric modification of a tooth profile. Their paper has three objectives: to improve the contact, to reduce the transmission error and to characterize the equivalent contact stress. The search for an optimum profile by Litvin [START_REF] Litvin | Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis[END_REF] is common to our paper. Nevertheless, the main difference lies in the methods for obtaining the tooth profile geometries. Indeed, in our paper, the proposed modification is variational relative to the involute without reference to known profiles. In the work by Litvin [START_REF] Litvin | Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis[END_REF], it is based on involute and epicycloidal profiles. The variational aspect is important since it does not presuppose the optimum profile which results from the approach.

Methods for generating tooth profiles 2.3 Geometric sensitivity

In the second part of this paper, a study was made of the sensitivity of tooth profiles to geometric variations. The work by Linares et al. [START_REF] Linares | Impact of geometric uncertainties onto the operating performance of a mechanical system[END_REF] and by Zamponi et al. [START_REF] Zamponi | Impact of geometrical defects on bearing assemblies with integrated raceways in aeronautical gearboxes[END_REF] on the geometric error of mechanical parts demonstrates the impact of loading on the geometric distortion of a mechanical system. In these papers, Zamponi et al. [START_REF] Zamponi | Etude des méthodes de calcul des pressions de contact dans les roulements a ` pistes intégrées des boîtes de transmission aéronautiques[END_REF][START_REF] Zamponi | Contact pressure calculation methodologies in aeronautic gearboxes in the CAD process[END_REF] examine the effect of local distortion (at the point of contact) as well as structural distortion (displacement of the shafts) on its mechanical systems. However, the geometric variations due to the manufacturing process are not considered. They propose original methods for calculating the pressure using analytical, numerical (F.E.M.) or hybrid methods using substitution elements.

The work by Xu et al. [START_REF] Xu | Prediction of mechanical efficiency of parallel-axis gear pairs[END_REF] proposes the estimation of gearbox performance according to the input conditions. These works concern cylindrical gears with straight or helical teeth. This study is based on a contact friction model and concludes with a numerical calculation (F.E.M.) of mechanical efficiency. Xu thus studies the performance of reduction gears according to the geometric design data, the operating conditions and the assembly and manufacturing errors.

In addition to the work by Xu et al. [START_REF] Xu | Prediction of mechanical efficiency of parallel-axis gear pairs[END_REF], two other papers examine sensitivity. These are the paper by Linares et al. [START_REF] Linares | Uncertainty of reference frames characterized by real time optical measurements: Application to computer Assisted Orthopaedic Surgery[END_REF] and that by Bruyère et al. [START_REF] Bruyère | Statistical tolerance analysis of bevel gear of tooth contact analysis and Monte Carlo simulation[END_REF]. Linares uses Monte Carlo simulation and Analytical Propagation. The work developed by Bruyère et al. [START_REF] Bruyère | Statistical tolerance analysis of bevel gear of tooth contact analysis and Monte Carlo simulation[END_REF], proposes a statistical analysis of the functional surfaces of a bevel gear. This sensitivity study looks at the impact of tolerance choices on the quality and manufacturing costs of the meshing surfaces of a reduction gear.

In conclusion, the objective of our work is to propose a new gear tooth profile which minimizes the equivalent contact stress, and to study the robustness under stress of this new geometry according to manufacturing errors. The use of error propagation methods enabled investigation of the robustness under stress of the optimized tooth profile according to the geometric variations due to the capabilities of the production means.

Method for optimizing the tooth profile under stress

The approach used demands the establishment of a simplified geometry of the gear (Fig. 3). In our resolution method, the gear is a spur gear. The potential zone of contact is shaded in Figure 3, and the line of contact, which is characteristic of an involute tooth profile, is defined by the segment between points A (input point) and D (output point). It is determined by the functional head radii of the pinion (indexed 1) and the wheel (indexed 2), and is inclined by the angle of pressure α.

In the involute tooth profile configuration, the contact path must be included into [T 1 T 2 ] segment. The nominal localization of the contact entry point A is derived from a previous minimization of the maximum equivalent contact stress. The coordinate of point D is derived from the contact ratio. The basis pitch defines the contact distance between two successive teeth and permits to set points B and C which correspond to the transition of one to two teeth under contact.

Only four inputs are required in the method developed in this paper:

-Pressure angle, "α", which characterizes the angle of inclination with the line of contact. -Distance between centers, "a", the distance between the centers of rotation, O 1 and O 2 . -Reduction ratio, "i", determined by the number of teeth on the pinion and the wheel (Eq. ( 1)), and -Contact ratio, "ε t ", which is used to determine the distance between A and D in Figure 4 (Eq. ( 2)).

Z i = (-1) n Z1
(1)

2 2π
with n = 1 for an external gear and n = 0 for an internal gear.

ε t = (θ E -θ S ) Z 1 (2)
where θ S , θ E are the angular positions of the pinion at input (Start: S) and output (End: E). The pitch circle radius (Eq. ( 3)) and the basis radius (Eq. ( 4)) are directly linked to three of them.

i 1 R p1 = 1 -i a and R p2 = |1 -i| a (3) R bi = R pi cos (α) (4)
However, these parameters do not suffice for the establishment of the entire operating environment of the gear.

In order to supplement this initialization data, some other characteristics are required, such as the width of the teeth (b), the input power (P 1 ) or the rotation speed of the pinion (ω 1 ) are required in order to establish the operating conditions of the gear considered. The characteristics of the gear studied are presented in Table 1.

Once these data has been defined, it is possible to progress to using the profile optimization method. The flowchart for this method is shown in Figure 4.

In this Figure 4, our method is principally composed of three steps. First, the initial configuration corresponding to a perfect involute tooth profile is considered. The in-depth stresses are computed using a generalized analytical Hertz contact method allowing to define the maximal equivalent stress during meshing. As a first approach, only the quasi-static behavior is considered. Secondly, a modified tooth profile is generated randomly using a Monte Carlo simulation algorithm. The in-depth stresses are then recomputed to the actual best solution. Only enhanced configuration is saved. At the end of the process, the optimized tooth profile parameters are displayed.

This procedure is detailed in the next paragraphs.

Evolution of the tooth profiles

The optimization procedure involves examining the contact during meshing. The geometric definition of the tooth profile meets the constraints of gear theory. These are essentially of two types, non-interference of the teeth and compliance with the conjugation of the profiles. The choice of modifying the profile variationally relative to involute enables partial avoidance of the constraints. There is no modification to the profile in the width of the tooth. The curvature C y in the width of the tooth is part of the input data provided in Table 1. This curvature is retained.

Equation [START_REF] Simon | Influence of tooth modifications on tooth contact in face-hobbed spiral bevel gears[END_REF] shows the expression of the involute tooth profile of the pinion, in Cartesian coordinates centered on pitch point I:

⎨ ⎫ ⎧ ⎪ ⎪ ⎪ ⎩ cos (α) cos (θ) ⎧ x C1 (θ) ⎬ ⎨ Rb 1 sin (θ) -θ cos (α + θ) ⎩ y C1 (θ) ⎭ = ⎪ Rb 1 cos (α) + θ sin (α + θ)-Rp 1 θ ∈ [θ S ; θ E ] (5)
with θ S , θ E the angular positions of the pinion at the contact input (point A) and output (point D).

⎨ ⎫ ⎧ ⎪ ⎪ ⎪ ⎩ - cos (α) - cos (α) ⎧ x C2 (θ) ⎬ ⎨ Rb 2 sin (iθ) +iθ cos (α+iθ) ⎩ y C2 (θ) ⎭ = ⎪ Rb 2 cos (iθ) -iθ sin (α+iθ) +Rp 2 θ ∈ [θ S ; θ E ] (6)
here, i defines the reduction ratio. These initial involute profiles for the pinion (Eq. ( 5)) and the wheel (Eq. ( 6)) are slightly modified. For their purpose, the initial functions are subsequently multiply by weighting functions. These functions are approximated by power series expansions. In a spur gear with a contact ratio between 1 and 2, there are four characteristic points during meshing: the input point (A), the output point (D) and two transition points from one to two teeth under contact (B and C). For this reason, the power series expansion functions were restricted to 4th degree polynomials. This avoids excessive oscillations of the tooth profiles. The coefficients of the polynomials are selected randomly by Monte Carlo simulation (Sect. 3.3).

The only variable in these functions is the angular position of the pinion θ. The analytical expression of the modified profile is proposed in Equation [START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF]. The reasoning adopted is analogous for the geometric evolution of the wheel. In the remainder of this paper, the development of the approach and the presentation of results are provided only for the active profile of the pinion. see equation [START_REF] Velex | Some analytical results on transmission errors in narrow-faced spur and helical gears: Influence of profile modifications[END_REF] next page.

In these equations, angle θ is the only variable. For θ = 0, the point of the tooth profile is at I at the pitch radius. With this single variable, it is possible to reconstruct the tooth profile in 2D. To do this, the extreme contact positions must be defined. As the contact ratio is defined, θ E is deduced from θ S (Eq. ( 2)).

Calculation of the equivalent contact stress

The geometric optimization of the tooth profiles is achieved by minimizing the equivalent contact stress in accordance with Von Mises criterion. To undertake this stress calculation during meshing of the pinion with the wheel, it is necessary to know the contact loading at any time.

The rheological data are defined by Young's modulus E and Poisson's coefficient ν for the materials used. The loading is deduced from a nominal load, relative to the input power P 1 , the rotational speed of the pinion ω 1 and the contact radius at the time considered R C (θ). The load F (θ) is distributed according to the number of teeth under contact n contact . Isotorque sharing is selected for that purpose. This assumption permits fast search of the optimal solution. This best configuration is finally validated using F.E.M. modeling. This number is determined from the contact ratio and the current angular position

⎨ ⎬ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ cos (α) " cos (α) # ⎧ x C1 (θ) ⎫ ⎪ Rb1 sin (θ) -θ cos (α+θ)fx (θ) ⎩ yC1 (θ) ⎭ = ⎩ Rb1 cos (θ) + θ sin (α + θ) -Rp1 fy (θ) , θ ∈ [θ S ; θ E ] ⎨ ⎬ ⎪ ⎪ ⎨ ⎪ ⎩ cos (α) cos (α) ⎧ x C1 (θ) ⎫ ⎧ Rb 1 sin (θ) -θ cos (α + θ)1 + c 1x θ + c 2x θ 2 + c 3x θ 3 + c 4x θ 4 ⎩ yC1 (θ) ⎭ = ⎪ Rb1 cos (θ) + θ sin (α + θ) -Rp11 + c1yθ + c2yθ 2 + c3yθ 3 + c4yθ 4 , θ ∈ [θ S ; θ E ] (7)
of the tooth profile (Eq. ( 8)). Generally, the contact ratio is between 1 and 2, thus:

⎧ ⎪ ⎨ ⎪ ⎩ Z Z 1 1 ⎪ θ (rd) ∈ θ E ; θ S - 2π ∪ θ E + 2π ; θ S ⇒ n contact = 2 Z Z ⎪ θ (rd) ∈ θ S - 2π ; θ E + 2π ⇒ n contact = 1 1 1 (8)
Knowing the input torque C 1 (Eq. ( 9)) and the number of teeth in contact n contact , the load evenly distributed between the teeth can be obtained according to θ (Eq. ( 10)).

ω C 1 = P 1 (9) 1 C 1 F (θ) = n contact (θ) R C (θ) (10) Z Z 1 1
The meshing simulation enables observation of the contact between the pinion and the wheel. The characterization of the contact is expressed in 100 angular positions of one pinion tooth. These 100 positions are ranged between θ = -π and θ = π . In Figure 5, the calculation of the equivalent curvature is an interesting intermediate result for two reasons.

First, the curvature only depends on the geometry of the profiles under contact. The equivalent curvature at the point of contact is related to the angular position of the pinion. Secondly, this equivalent curvature is involved in the calculation for the equivalent contact stress (Fig. 6).

In order to calculate the equivalent curvature, it is necessary to know the curvature, at the contact point of the pinion and the wheel. Knowing the equation of profile considered (Eq. ( 7)), the analytical expression for the curvature (Eq. ( 11)) is obtained by derivation.

0 00 0 00 0 0 h i 3/2 γ C1 (θ) = ± x C1 (θ) y C1 (θ) -y C1 (θ) x C1 (θ) (11) (x C1 (θ)) 2 + (y C1 (θ)) 2
The curvature of the wheel is calculated similarly from the expression of its tooth profile. Knowing the curvature of the pinion and the wheel, it is easy to calculate the equivalent curvature at each point of contact (Eq. ( 12)). Equation [START_REF] Hwang | Contact stress analysis for a pair of mating gears[END_REF] shows the expression for this parameter.

γ Ceq (θ) = γ C1 (θ) + γ C2 (θ) ( 12
)
The calculation of the maximum equivalent stress in the sub-layer is established for all angular contact positions.

In Figure 6, the equivalent stresses of the initial involute profile and the optimized profile are presented according to the angular position of the pinion. The 4 characteristic points A, B, C and D, of the mesh are identified.

Variable θ characterizes the angular position of the pinion. It is the common variable between the expression for the contact stress and the equation for the associated tooth profile. The Monte Carlo simulation modifies the geometry of the tooth profile to achieve an optimized profile. The procedure selects the best profile. A summary of the Monte Carlo simulation parameters is provided in next paragraph.

Optimization

In the search for the optimum profile, only the maximum equivalent Von Mises stress by Hertz contact is examined along the flank of the teeth (Sect. 3.2). In fact, the profile providing the most benefit is adopted. The coefficients of the tooth profile are picked randomly by Monte Carlo simulation. The number of samples "N " of the Monte Carlo simulation is a parameter to be defined by the designer. The aim is to obtain the parametric equation of tooth profiles which minimizes the equivalent contact stress. Thus the variation ranges of the coefficients are simulation parameters. In this Monte Carlo simulation, apart from the random picking of coefficients for the polynomial equation of the tooth flanks, the main parameter to be defined concerns the number of samples. Several tests have been carried out to define the value providing the best stress benefits. The result of this study is shown in Figure 7.

An estimate of the optimum number of samples for stress has been conducted at points B and C where the stress is at its maximum. Indeed, these two points are the most constraining in terms of the mechanical strength of the teeth. This is explained by the fact that there is just one tooth in contact between these two points. In Figure 7, the benefit ratio relative to the initial involute stress is represented for different numbers of samples. A number of 20 000 samples has been chosen.

All the criteria for the tooth profile optimization procedure are shown. Next paragraph shows and comments on the results obtained.

Results

At the end of the simulation, the tooth profiles providing the best stress benefits were adopted. Figures 5 and6 show the results of the stress optimization. Several comments should be made. Minimizing the curvature (Fig. 5) results in a decrease in the stress (Fig. 6) during meshing. This stress study defines the loading to which the teeth will be subjected during their operating phase.

A reduction in the curvature maximizes the equivalent radius. The applied load is better distributed over the contact surface. More the curvature decreases more the stress is reduced. As shown in Figure 5, the equivalent curvature of the optimized profile is always less than the involute curvature except for the pitch point. In fact, regarding point I, which characterizes the pitch circle of the gears, the curvatures are equal since both the involute and optimized curvatures are tangent at that point. This curvature is fixed by the pressure angle at the pitch circle. The stress at this point for involute and optimized profiles is still the same. A strong reduction in the curvature of the optimized profile is observed at the contact input and output. However, the two zones requiring less curvature are located in the transition zones, where the contact goes from one to two teeth (at B and C). These zones are where the sudden variations in load occur which in particular cause micro pitting.

The calculation of the equivalent Von Mises stress is performed using an algorithm based on a generic Hertz model which considers both the curvatures introduced and a friction field. The method is based on the complete analytical approach of Hills et al. [START_REF] Hills | Mechanics of Elastic Contacts[END_REF] which defines the whole three dimensional stress profile of the contact zone. It permits defining the maximum sub-surface equivalent stress using Von Mises criterion.

The point of contact on the pitch circle (I (0, 0, 0)) is characterized by angle θ = 0. The stress calculation is performed for 100 contact points distributed along the tooth profile. These points are distributed identically for the involute and the optimized profiles. The equivalent stress at point I will still have the same strength. This is a direct consequence of the equal curvature at the pitch circle. In fact, the reduction in stress is present, most particularly, in the transition zones from one to two teeth. In our simulation result, the loading is defined with a Coulomb friction coefficient and isotorque distribution. The choice of this configuration was motivated by its restrictive and representative aspect. Indeed, the deterioration of the contact in a gear assembly increases if the contact is dry, simulating loss of the oil film, whence the choice of Coulomb friction. In the case of isotorque distribution of the load, the contact transition from one to two teeth occurs instantly and suddenly on the flank of the teeth. This provides a better picture of the stress benefits at the characteristic points A, B, C and D. The stress of the optimized profile is maximum in the vicinity of I.

In the transition zones from one to two teeth in contact, the stress is slightly diminished. The reduction in stress is of around 40 MPa at B and C, but 500 MPa at the input (A) and 200 MPa at the output (D). For the loading case considered, the stress benefit is some 14% during meshing.

The tooth profile minimizing the stress during meshing is obtained. The active profile of the pinion tooth differs from the conventional involute profile (Fig. 8).

Table 2 shows the values of the various coefficients for the polynomial expressions for the pinion and the wheel obtained by the tooth optimization procedure.

Study of the sensitivity of the equivalent stress to geometric variations

This study was conducted to test the robustness of the optimized profile to the geometric variations due to machining imperfections. Two studies were considered at the characteristic points: propagation by Monte Carlo method and analytical propagation of uncertainties.

In the Figure 9, the localization of four characteristics points (a, b, c and d) on the pinion tooth profile is represented. During the machining, these points will be perturbed by the production means. A dispersion zone of 0.01 mm is represented on the flank of an optimized tooth in Figure 10. In order to re-use the polynomial expressions for the tooth profile, the choice was made to introduce this geometrical dispersion at the four characteristic points a, b, c and d of the tooth profile. These dispersion zones (W (x ;y ;y ;y ;y ) = 0.01 mm) applied to our two models (pinion and wheel) are defined in Figure 10. The results of the two studies are presented in the paragraphs below.

Propagation of uncertainties by Monte Carlo

The procedure adopted to propagate the geometric uncertainties by the Monte Carlo simulation is detailed in the flowchart in Figure 11. In our study, the geometric variations were introduced at the characteristic points of the tooth profile in order to calculate the average values and the error bar at two standard deviations (2σ) of the maximum equivalent contact stress during meshing. To that end, the five coordinates of the previously defined characteristic points, vary randomly within the dispersion zone of manufacturing. The method also demands that all the variables are independent. This is the reason why the evolution of the geometric parameters was chosen in a Cartesian coordinate system.

The uncertainty propagation study was conducted on the optimized tooth profile. In that geometry, the variables are: Two laws of distribution were considered, one uniform and the other Gaussian. The results of these two simulations are presented in Table 3.

Calculation by analytical propagation

The second uncertainty propagation study was intended to confirm the results obtained via the Monte Carlo simulation. This analytical propagation method is based on G.U.M. standard [START_REF]Guide to the expression of uncertainty in measurement[END_REF]. The propagated uncertainty is derived from the product of the Variance-Covariance matrix (Eq. ( 13)) and the Jacobian of the stress function:

Var (σ eq ) = JCov (x a ; y a ; y b ; y c ; y d ) J T (13) σ = W √ (14) σ =
Both types of distribution were considered, i.e.: uniform (Eq. ( 14)) and Gaussian (Eq. ( 15)). The expression of the standard deviation differs for each law.

Equation ( 16) details expression 13. The independence of the variables used in this method results in a diagonal Variance-Covariance matrix. The result of this calculation is the variance of the equivalent stress. It enables the determination of the Standard Deviation (σ eq ) of the equivalent stress during meshing according to the five chosen simulation parameters.

/ / / / / ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ / / a / / / ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
Var (σ eq ) = ∂σ eq ∂x a ∂σ eq ∂y a ∂σ eq ∂y b ∂σ eq ∂y c ∂σ eq ∂y d

The results obtained by analytical propagation are presented in Table 4.

Discussion of the results

By studying Tables 3 and4, for both types of distribution considered: uniform and Gaussian, close similarities were found. The degree of uncertainty were calculated for a risk of 5%. The results obtained for the degree of error of the maximum equivalent stress was small (less than 1.5% of the mean value). This shows the low sensitivity of the equivalent stress to geometric variations (machining imperfections, wear, etc.). The results from the two methods prove the robustness and local linearity of the geometric definition of the optimum tooth profile for the evaluation criterion adopted.

Conclusion

The new mechanical requirements for gears compel designers to propose new innovative tooth geometries. These geometries must meet service life criteria under ever more demanding loads. Through this paper, our work was aimed at two objectives:

to propose a method for optimizing the tooth profile by equivalent stress, to guarantee the suitability and the robustness of the profile at manufacturing variations.

The design tool proposed is a methodology which converges towards an optimum tooth profile. The overall dimensions and the loading of the gear are input data. The approach shows the existence of profiles, other than involute, which provide a useful response for increasing the service life of gearboxes. This method agrees with the work already undertaken on the geometric modification of tooth profiles, such as that by Litvin [START_REF] Litvin | Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis[END_REF] in particular. The method presented in this paper thus enables identification of a geometric tooth profile in a given operating configuration. This tool, which is an aid for gear design, provides an original approach for the geometric design of a reduction gear. The gear adopted provides better behavior than an involute tooth profile for a given loading.

The study of the propagation of geometric errors concludes on the robustness of the equivalent stress with respect to the capability of the manufacturing means. The propagation behavior is locally linear. In other words, the stress variation during meshing is due to factors other than simply the geometric design of the flank of the teeth. Indeed, the dispersion of the characteristic points has negligible effect on the stress variations. Monte Carlo Simulation and Analytical Propagation enabled evaluation of the influence of these geometrical deviations.

The conclusions of this paper show that there is a viable and credible alternative to the involute tooth profile.

Fig. 2 .

 2 Fig. 2. Involute spur gear example.

Fig. 3 .

 3 Fig. 3. Simplified geometry of the gear.

Fig. 4 .

 4 Fig. 4. Flowchart optimization method.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Equivalent curvature for involute and optimized gear tooth profile.

Fig. 7 .

 7 Fig. 7. Optimal simulation number.

Fig. 8 .

 8 Fig. 8. Involute tooth profile and optimized tooth profile.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Localisation of characteristics points on the tooth profile of pinion, a, b, c and d.

Fig. 11 .

 11 Fig. 11. Flowchart of propagation method by Monte Carlo.

-

  The abscissa X and ordinates Y of the input contact point on the flank of the tooth: a. -The ordinates Y of the three other characteristic points on the flank of the tooth: -The two points of switchover to a single point of contact: b and c. -The output contact point: d.

(

  x a ;y a ;y b ;y c ;y d ) (x a ;y a ;y b ;y c ;y d ) 2 3 W (x a ;y a ;y b ;y c ;y d ) (xa;ya;yb;yc;yd) 4

⎛ (σ x a ) 2

 2 y d ) 2 ∂σ eq ∂y d ⎠

Table 1 .

 1 Spur Gear Characteristics of the case study.

		Parameters	Abbreviations Values	Units
		Center distance	a	150	mm
		Number of teeth-pinion	Z 1	31	/
		Number of teeth-wheel	Z 2	89	/
		Module	m	2.5
		Pressure angle	mm
		Face width	α	22 •	deg
	a	Crowning in the width	b	65	mm
		Contact ratio	C y	0.01	mm
		Power entrance	ε T P 1	1.7 1900	/ kW
		Speed rotation entrance	ω 1	23 000 rot.min -1

Table 2 .

 2 Coefficients optimized tooth profile of pinion and wheel.

		Pinion coefficients		Wheel coefficients
	C 1X	-0.591	C 1Y	-0.445	C 1X	-0.566	C 1Y	-0.398
	C 2X	18.041	C 2Y	12.463	C 2X	16.488	C 2Y	11.749
	C 3X -32.028 C 3Y	-18.954 C 3X	-13.462 C 3Y	1.749 -
	C 4X -51.940 C 4Y	-80.826 C 4X	-46.871 C 4Y	119.486

Table 3 .

 3 Results of the Monte Carlo simulation.

	Type	Stress mean	Standard deviation	95% confidence interval
	Uniform distribution	1369.2 MPa	10.8 MPa	±21.5 MPa [1347.7; 1390.7]
	Gaussian distribution	1368.4 MPa	7.0 MPa	±14.1 MPa [1354.3; 1382.5]

Table 4 .

 4 Results of the analytical propagation.

	Type	Stress mean	Standard deviation	95% confidence interval
	Uniform distribution	1368.7 MPa	10.7 MPa	±21.3 MPa [1347.4; 1390.0]
	Gaussian distribution	1368.7 MPa	9.2 MPa	±18.5 MPa [1350.2; 1387.3]

Résumé -Optimisation en contrainte et étude de sensibilité aux variations géométriques d'un profil de denture d'engrenage. Cet article propose une démarche permettant d'obtenir une géométrie optimisée d'un flanc de denture par la minimisation de la contrainte équivalente au contact. La méthode de calcul de la contrainte est fondée sur la théorie généralisée de Hertz. Comme la majorité des profils de denture sont en développante de cercle, l'évolution géométrique du flanc de denture est réalisée de manière variationnelle par rapport au profil en développante de cercle. Le profil optimal est obtenu par une simulation de Monte Carlo. Lors de cette optimisation, une expression polynomiale de la géométrie de la denture a été utilisée. Les paramètres pilotant la simulation sont les quatre points caractéristiques de contact. La démarche proposée a été mise en oeuvre sur un cas représentatif. Sur le profil de denture optimisé, une étude de sensibilité géométrique a été menée. Deux approches ont été envisagées : par simulation de Monte Carlo et par la méthode de propagation analytique. La robustesse et la linéarité du comportement du profil de denture soumis à des variations géométriques sont démontrées.Mots clés :Engrenages cylindriques / Profil de denture / Tooth Contact Analysis (T.C.A.) / Simulation de Monte Carlo / Incertitudes


Julien Chaves-Jacob

Jean-Marc Linares

Jean-Michel Sprauel

Improving tool wear and surface covering in polishing via toolpath optimization

Keywords:

Polishing operations are commonly carried out manually, thus inducing variability on the surface quality. The aim of this paper is to automate the polishing of free-form surfaces in order to obtain high quality surfaces. Tool wear and toolpath surface covering have a great impact on surface properties. The current work proposes therefore a toolpath which optimizes both tool wear and surface covering. This toolpath is composed of an optimized elementary pattern repeated along a 5-axis carrier trajectory. Usually, trochoid patterns are used. Non uniform wear of the tool and uneven probability density function of the surface covering are the main inconvenients of such pattern. So, this paper proposes two optimized patterns: Spade and Triangular. Both of them lead to uniform tool wear. Our paper also demonstrates that the second solution provides a uniform probability density function. All presented computations are validated experimentally.

Introduction

Biomedical implants need a high finishing quality, in particular for the roughness of friction surfaces. Femoral parts of knee prostheses hold these problems. Manufacturing of such workpiece attracts the interest of different research teams in the world. [START_REF] Hilerio | 3D measurements of the knee prosthesis surfaces applied in optimizing of manufacturing process[END_REF] present the Product Life Management of knee prostheses and a study of the polishing stages. The manufacturing process of this biomedical part is presented in Fig. 1. First, the kinematic and functional requirements are specified in the Computer Aided Design (CAD) model. Next, due to the fact that implants are commonly realized in cobalt-chrome alloy, difficult to machine, lost-wax casting is generally selected to obtain the rough of geometry. Sometimes, before starting polishing, a rough-machined is also realized. Following, the prosthesis is polished with a succession of abrasive tools. After manufacturing, the implant undergoes many cleanings and sterilizations.

In the implant process planning, polishing operations are critical and correspond to the longest phases of the finish process. This stage needs accurate positioning of the prosthesis to the abrasive tool. For that reason it remains commonly still manual. This problem is recurrent for a great number of types of parts like molds, die sets, flow components (impellers, inducers, etc.), biomedical implants (knee, hip, etc.), etc. Polishing operations are however needed to obtain the required finish quality. These stages are labor intense and dangerous to health of the operator. Indeed, small metal airborne particles are produced by this process and may be inspired by the operator. [START_REF] Lison | Experimental research into the pathogenesis of cobalt/hard metal lung disease[END_REF] present a study to understand the pathogenesis of lung disease produced by cobalt airborne particles (standard alloy used for prostheses). Furthermore, the geometrical result is greatly depending on the skill of the operator. Therefore, many works try to automate the polishing operation to stabilize the implant quality and reduce the production costs. The first problem is to find a machine able to emulate hand movements. Some special machines have been designed by numerous research teams. For example, [START_REF] Lee | A robust trajectory tracking control of a polishing robot system based on CAM data[END_REF] developed a 2-axis robot designed to be mounted on a 3-axis CNC machine. For polishing assistance, [START_REF] Hocheng | Fundamental study of ultrasonic polishing of mold steel[END_REF] propose a specific machine using ultrasonic axial vibrations of the tool. [START_REF] Wu | New polishing technology of free form surface by Márquez[END_REF] used a grinding center with an elastic ball end tool to polish free-form surfaces. [START_REF] Charlton | Surface and form metrology of polished "freeform" biological surfaces[END_REF] designed an industrial 7-axis machine dedicated to polishing operations. [START_REF] Liao | Modeling and control of automated polishing/deburring process using a dual-purpose compliant toolhead[END_REF] present a compliant toolhead for polishing and deburring operations. The presented head is mounted on a robotized tripod. [START_REF] Hung | Reduction of tool wear during hydrodynamic polishing: a 'rock-and-roll' polishing strategy[END_REF] suggest the use of a 'rock-and-roll' polishing toolpath strategy to get homogeneous tool wear. The presented method uses a ball-tool in rotation around the tool-axis. Furthermore, an additional rotation around the center of the ball-tool is applied to homogenize the tool wear along the spherical surface.

These machines are however only dedicated to polishing operations. In the same way, the use of 5 and 6-axis common industrial robots has been proposed. The advantage of this solution is its low cost. Moreover, the degrees of freedom of the robot offer a great accessibility to any free form surface. However, the accuracy of such machine is limited to about 0.1 mm. The use of a machine dedicated to polishing imposes preparing the surfaces with 5-axis milling. In such process planning, polishing problems may then appear due to the loss of reference frame of the workpiece during its dismounting and next mounting. Thus, [START_REF] Denkena | Prediction of contact conditions and theoretical roughness in manufacturing of complex implants by toric grinding tools[END_REF] suggest the use of one single CNC machine for both rough milling and polishing. In fact, such process planning is the most accurate way to know the localization of the real geometry before polishing.

The major problem of automated polishing is the control of the contact pressure between the tool and the workpiece. This pressure is essential in the material removal rate. To monitor this pressure, it is necessary to control the surface geometry and the contact force between the tool and the surface. Using a CNC machine, the workpiece is not dismounted between rough and polishing stages. In consequence, the polished surface geometry is accurately controlled. Two methods are employed to monitor the contact force:

-Using a load cell to control the force applied to the workpiece.

The penetration of the tool is then directly linked to the selected force. [START_REF] Tsai | Efficient automatic polishing process with a new compliant abrasive tool[END_REF] control the load applied to a compliance tool using a polishing force control loop. [START_REF] Nagata | CAD/CAM-based position/force controller for a mold polishing robot[END_REF] employ an industrial robot driven in force to polish molds. [START_REF] Huissoon | Automated polishing of die steel surfaces[END_REF] propose to polish surfaces with a flexible abrasive disk mounted on a load cell. This sensor is used to drive the tool position in force. -Defining the constitutive law characterizing the relation between the tool displacement and the contact force. This law can be obtained through an experimental approach or theoretical calculations. [START_REF] Tsai | Efficient automatic polishing process with a new compliant abrasive tool[END_REF] use a polishing experiment, instrumented with a load cell to determine this law. On the other hand, lots of studies propose a Hertz contact model to determine the constitutive law between the tool displacement and the contact force. For example, [START_REF] Roswell | Modelling and analysis of contact stress for automated polishing[END_REF] implement this model to find the stress in the Hertz contact between the tool and the polished surface.

Another problem is to define the polishing toolpath. Its aim is to cover the entire surface. To solve this problem, [START_REF] Tam | Robotic polishing of free-form surfaces using scanning paths[END_REF] propose a linear scanning of the parametric space of the surface. After, the authors optimize the steps between the different scanning toolpath segments to get a uniform covering of the surface. This method reduces the overlapping of the segments, thus lowering their number. To boost the overlapping, Márquez et al. (2005) propose to use an elementary square pattern with two diagonals. Furthermore, fractal curves can be employed to cover the entire surface. Commonly Hilbert's curves are therefore taken. [START_REF] Chen | Generation of fractal toolpaths for irregular shapes of surface finishing areas[END_REF] propose a method to compute Hilbert's curves on complex surfaces. In this paper, the authors test this method on a large range of geometries. Pessoles and Tournier (2009) use Hilbert's curves as a carrier to generate polishing toolpaths. The surface covering may be resolved using an elementary pattern repeated along a carrier toolpath. Usually, a trochoid pattern is selected, because the related movement looks like human polishing trajectory. For example, [START_REF] Tsai | Efficient automatic polishing process with a new compliant abrasive tool[END_REF] used a trochoid pattern in polishing toolpaths along a linear carrier toolpath. This type of pattern leads to a great number of loops which many times cross the same surface area. The trochoid pattern gives interesting results but its choice is arbitrary.

The aim of current paper is now to propose different toolpaths to automate the polishing operations of workpieces. A mathematical optimization of the polishing toolpath pattern will thus be presented to get homogeneous tool wearing and uniform surface covering. In a first section, the employed polishing process will be detailed. Next, the tool wear induced by trochoid patterns in flank polishing will be analyzed. This study will highlight the non uniform wear of the tool using such pattern. After, a Spade pattern will be proposed to correct this problem. Subsequently, a similar approach will be used to get a pattern which leads to a uniform covering of the polished surface. The results of experiments carried out to demonstrate the relevance of the presented developments will finally be presented.

Polishing process on CNC

The proposed polishing toolpaths can be used on any 5-axis CNC machine or industrial robot. The defined experimental process is illustrated in Fig. 2. The polishing operation is carried out using the flank of a cylindrical abrasive tool.

The proposed 5-axis polishing toolpath is a repetition of elementary loops with a small feed movement. This movement is presented in Fig. 3. The polishing toolpath is composed of a carrier toolpath and an elementary pattern along the tool axis. First, the carrier toolpath is computed using a simple linear interpolation. It composed of a set of points M i , which may be computed using an industrial Computer Aided Manufacturing (CAM) system. After, a five degree polynomial interpolation, as presented by [START_REF] Chaves-Jacob | Increasing of surface quality in friction free-form surfaces of knee prosthesis[END_REF], is used to smooth the carrier toolpath and get a trajectory with continuous slopes and curvatures. Nevertheless, with this polynomial interpolation, all the points M i still lie on the smoothed toolpath. Eq. ( 1) presents the expression of this polynomial interpolation. In these relations, the parameter u polynomial defining the position of a point along the 3D carrier trajectory is a normalized factor varying from 0 to 1.

2 4 3 5 X carrier = a 0 + a 1 • u polynomial + a 2 • u polynomial + a 3 • u polynomial + a 4 • u polynomial + a 5 • u polynomial 2 4 3 5 2 4 3 5 Y carrier = b 0 + b 1 • u polynomial + b 2 • u polynomial + b 3 • u polynomial + b 4 • u polynomial + b 5 • u polynomial (1) Z carrier = c 0 + c 1 • u polynomial + c 2 • u polynomial + c 3 • u polynomial + c 4 • u polynomial + c 5 • u polynomial
Between two consecutive points, M i and M i+1 , of the smoothed carrier toolpath, the polishing toolpath is then computed. It is composed of elementary patterns, repeated N times between points M i and M i+1 . In a first time, the pattern curve is defined in a 2D space, (U1, U2) in Fig. 4. M is the current point of the pattern. It is defined by the coordinates U1 M (u pattern ) and U2 M (u pattern ). The pattern curve is thus described by a parameter u pattern , varying from 0 to N. Each loop induces a feed progression of parameter a in the 2D space. In this study, the elementary patterns were based on trochoid curves. Eq. ( 2) presents the coordinates U1 M (u pattern ) and U2 M (u pattern ) for such trochoid patterns.

U1 M (u pattern ) = R • [1 -cos(2 • u pattern )] + a • u pattern U2 M (u pattern ) = -R • sin(2 • u pattern )
The elementary patterns are driven by four parameters. Two parameters, R and a, define the pattern geometry: R is the half amplitude of the tool in its axial direction and a is the 2D longitudinal feed progression by loop. Fig. 5 presents the 6 possible morphologies of the elementary trochoid pattern. This morphology depends on the ratio of R and a. For polishing patterns, the value of R is usually selected greater than a to be in the first three cases of Fig. 5. The two other parameters define the density of the pattern along the carrier toolpath. They consist of the number N of loops between two consecutive points of the carrier toolpath (M i and M i+1 ) and the distance between these two points.

The polishing pattern, first defined in a 2D space, is then projected to the 3D carried toolpath. For that purpose, the variable u polynomial defining the coordinate of the tool along the carrier trajectory is derived from the position U1 M (u pattern ) in the 2D diagram, using following Eq. ( 3).

U1 M (u pattern ) polynomial a • N u = (3)
This link between the 2D space of the elementary patterns driven by u pattern and the 3D parameter u polynomial is also shown at the bottom of Fig. 4. The additional oscillating movement of the tool, in the direction of the rotation axis corresponds just to the displacement U2 M (u pattern ) as defined in the 2D diagram.

The next step is the computation of the inclination of the tool throughout the carrier toolpath trajectory. A simple linear interpolation of the components (I; J; K) of the tool axis direction vectors (2) (V i , V i+1 ), defined by the CAM system, for points M i and M i+1 is used for that purpose (Eq. ( 4)). Fig. 3 illustrates the used notation for the tool axes computation.

V = (1 -u polynomial ) • V i + u polynomial • V i+l = (I v ; J v ; K v ) (4)
Thereafter, the 3D polishing toolpath is computed using Eq. ( 5). This equation is obtained moving the tool along its axis to describe the elementary pattern loops. This polishing toolpath is decomposed into small linear steps and then smoothed using a five degree polynomial interpolation. To finish, the CNC code is generated by the developed program. The polishing toolpath thus obtained is

Optimization of the tool wear

Non uniform tool wear with trochoid patterns

In a first time, an elementary trochoid pattern is studied to highlight the difference of wear along the tool axis. The trochoid pattern is defined by Eq. (2) and the example, corresponding to the parameters R = 1 and a = 0.5, is presented in Fig. 6.

In our polynomial interpolations, it is the parameter u pattern which is used to drive the CNC axes. This variable is thus directly linked to the system clock and therefore proportional to time. For that reason, the wear of any axial point of the tool just depends on the whole time spent at a given position U2 M in the toolpath. Due to the symmetry of the movement, only a half period of the trochoid pattern had to be studied (Eq. ( 7)). To define the probability density function (PDF) along the tool axis it is then necessary to inverse this function. The inverse function of U2 M is presented in Eq. ( 8) and at the top of Fig. 7.

v ⊂ [-R; R] U2 -1 (v) = 2 • arcsin R (8)
The PDF of the wear is finally computed by deriving and normalizing Eq. ( 8). This leads to the expression given in Eq. ( 9). The PDF is plotted in Fig. 7. It expresses the time ratio spent by any tool point at the same given axial polishing position. Its value is directly linked to the wear of the tool. As clearly shown by Fig. 7, a significant share of the polishing time relates to the two end points of the tool. This obviously demonstrates that trochoid patterns lead to non uniform wear of the tool.

M d v R 2 v ⊂ [-R; R] d(U2 -1 (v)) = 1 • 1 • p 1 (9) 1 -(v/R)

Uniform tool wear with Spade patterns

Section 3.1 demonstrated that trochoid patterns induce an intensified wear of the two tool end points. In this section, a Spade pattern will be presented to correct this wear problem. To obtain a homogeneous wear of the tool, it is obviously necessary to get a uniform PDF along the tool axis (Eq. ( 10)). This distribution leads to M a linear expression of U2 -1 (Eq. ( 11)). Fig. 8 shows the wear density function of the Spade pattern.

M = v ⊂ [-R; R] d(U2 -1 (v)) 1 dv 2R v ⊂ [-R; R] M 1 U2 -1 (v) = 4R • v
After, the loop parameters selected for the trochoid patterns were preserved to compute the wear optimized toolpath, presented in Eq. ( 12). The shape of the related elementary pattern, presented in Fig. 9, looks like a Spade. distorted in direction U2 to get a uniform PDF of this variable. Now, the density of the pattern has to be optimized in direction U1.

Elementary trochoid pattern covering study

The coordinate U1 M of the trochoid pattern is therefore analyzed. Due to the symmetry of the displacement, just a half period is considered for u pattern ⊂ [0; 0.5] in Eq. ( 2). Its PDF is presented in Eq. ( 13) and Fig. 10. It shows that, the trochoid pattern induces a non uniform covering in direction U1. Fig. 10 highlights that the trochoid pattern leads to a larger covering at the beginning and the end of the studied interval which corresponds to the center positions of the tool.

a v ⊂ 0; 2R + 2 1 M 1 d(U1 -1 (v)) (10) dv = N • (2 R • sin(2 • u pattern ) + a) with v = R • [1 -cos(2 • u pattern )] + a • u pattern where u pattern ⊂ [0; 0.5] ( 13
) (11) 4.

Covering optimization

To get a homogeneous covering of the surface, a uniform PDF of the elementary pattern has to be obtained in direction U1. This gives the relationships of Eq. ( 14). The oscillating amplitude, 2R, and the Section 3 presented a Spade pattern, distorting the trochoid pattern, to obtain homogeneous wear of the tool, along its axis. The same type of optimization is now applied to this polishing toolpath to get a uniform surface covering. For that purpose, the elementary patterns must have a uniform PDF in the two main directions (U1, U2) of displacement. However, since the two components (U1, U2) are independent variables, the joint surface PDF corresponds to the product of two independent functions, thus allowing separated optimization. In Section 3, the trochoid pattern was already feed a of the elementary trochoid pattern are however preserved, thus leading to the optimized expression of U1 M presented in Eq. ( 15). The resulting pattern is named Triangular pattern. accurately characterized through optical measurements using a chromatic confocal sensing CMM. The resolution of this CMM is about 10 nm. The profiles are measured along all the active part of the tool. An example of result is shown in Fig. 13. The vertical lines at the end of the measured profiles correspond to the extremities of the tool. As expected (see Section 3.1), the tool which machined a trochoid patterns presents an over-wear at the end of the used area. On the contrary, Spade and Triangular patterns generate uniform tool wear. These measures validate the optimization of toolpath in tool wear.

u pattern ⊂ [0; 0.5] M v ⊂ [0; 2R + a/2] d(U1 -1 (v))/dv = 1/ N • (4R + a) (14) U1 M (u pattern ) = 4R • u pattern + a •

Experimental validation

Subsequently, the polished surfaces were measured on a 2 mm × 4 mm area. The length of these zones were set to the Fig. 12 presents the CAD model of the workpiece used for the experimental validation. The polished surface is a part of friction surface of unicompartmental knee prostheses. First, the surface is rough machined with a flat end mill. Next an abrasive ball tool is used to semi-finishing. A polynomial toolpath is therefore used to reduce the surface irregularities. The finishing is realized with a polishing tool composed of emery grain (size around 50 mm) linked by a supple rubber. The geometry of the polishing tool is a cylinder of 32 mm in diameter and height. During the polishing operation, lubricant is employed. The flow used is a grinding lubricant with a low fat content and a pressure of around 5 bars. The surface is polished using the six carrier toolpaths presented in Fig. 12. The polishing patterns have a radius R of 12 mm and a feed a of 6 mm. The polishing toolpath is computed to have one loop per 2.5 mm.

To highlight the non uniform wear of the tool and the surface covering, three surfaces were polished. The first is polished with the trochoid pattern, the second with the Spade and the third with a Triangular toolpath. Fig. 13 presents the three employed tools and the measured profiles along the tool axes. These profiles are direction of the carrier toolpath and scanned with a step of 4 mm. In the other direction, the measured step was 10 mm. The measured data are then filtered by a smooth three degree polynomial in the two directions. The results of these roughness measurements are presented in the maps at the right of Fig. 14. They clearly show that triangular patterns lead to the lowest roughness amplitude. To evaluate the quality of polished surfaces it is usual to test its reflectivity using the manufactured part as a mirror for a fixed image. The left images of Fig. 14 show the result of such test. The image reflected by the surface polished with the trochoid pattern is more disturbed compared to the surface realized with the triangular toolpath. This is correlated to the roughness results. The reflection perturbations are generated by facets on the polished surface. These facets are induced by the non uniform PDF of the tool covering on the surface using trochoid or Spade patterns. The facets generated by the Spade patterns are smaller compared to those produced by the trochoid toolpath. This result can be explained by the fact that Spade patterns have been optimized in one of the two directions. Finally, Triangular patterns give the best surface quality with a uniform tool wear.

Conclusions

This paper developed an optimal toolpath for surface polishing. First a trochoid pattern was considered since it is commonly employed. In fact, this toolpath creates loops which cross many times the same surface area and its trajectory looks like human polishing movements. The aim of this current work was however to optimize the geometry of the elementary pattern to get homogeneous wear of the tool and uniform covering of the surface.

Our study has also proposed a method to compute 3D polishing toolpaths for 5-axis machines and industrial robots. The proposed method uses the flank of cylindrical abrasive tools. The toolpath was obtained adding elementary patterns on a carrier toolpath. The developed method generates a CNC program using a five degree polynomial interpolation to obtain a smooth toolpath. Furthermore, this work has studied the tool wear by analyzing the probability density function (PDF) along the axial oscillating movement. This analysis showed that the trochoid pattern produces a heterogeneous wear of the tool. In this classical toolpath, in fact, the end points of the tool are the most loaded. Thereafter, a Spade shape pattern has therefore been proposed to generate homogeneous tool wear. Furthermore, it is also necessary to get a uniform covering of the surface. A Triangular pattern has been proposed for that purpose. Mathematical computations have thus demonstrated that this scheme is an optimal solution to obtain high surface quality. Indeed, Triangular patterns simultaneously lead to homogeneous tool wear and uniform surface covering.

Subsequently, three surfaces were polished using trochoid, Spade and Triangular patterns. The tools wear were measured after the polishing operation and corroborate the theoretical developments. Thereafter, the polished surfaces were measured with an optical CMM. The quality of the surface obtained with the Triangular pattern is significantly better in comparison to the surfaces polished with the trochoid and Spade patterns. This improvement is obtained thanks to the uniform covering PDF.

Fig. 1 .

 1 Fig. 1. Process planning of a knee prosthesis femoral part.

Fig. 2 .

 2 Fig. 2. Polishing process on a 5-axis milling machine.

Fig. 3 .

 3 Fig. 3. Polishing toolpath computation.

Fig. 4 .

 4 Fig. 4. Elementary pattern in a 2D space.

Fig. 5 .

 5 Fig. 5. Morphologies of the elementary patterns.

Fig. 6 .

 6 Fig. 6. Example of trochoid pattern.

  u pattern ⊂ [-0.25; 0.25] (7) U2 M (u pattern ) = -R • sin(2 • u pattern ) continuous in position, slope and curvature. Eq. (6) presents one line of the CNC program for a SIEMENS 840D. In this equation, (X p , Y p , Z p ) are the end point coordinates of the line and a i , b i and c i are the coefficients of the five degree polynomial curve of the toolpath. The coefficients a 0 , a 1 , b 0 , b 1 , c 0 and c 1 are not provided in the program since they are directly linked to the current and end coordinates, and are thus computed by the CNC machine.

Fig. 7 .

 7 Fig. 7. Tool wear PDF using a trochoid pattern.

Fig. 8 .

 8 Fig. 8. Tool wear PDF using a Spade pattern.

Fig. 9 .

 9 Fig. 9. Wear optimized patterns, Spade toolpath.

Fig. 10 .

 10 Fig. 10. PDF of the trochoid pattern in U1 direction.

Fig. 11 .

 11 Fig. 11. Surface covering of the three elementary patterns.

  u pattern U1 M (u pattern ) = -4R • (u pattern -1) + a • u pattern if u pattern ⊂ [0; 0.5] (15) if u pattern ⊂ [0.5; 1] The 3 types of elementary patterns are presented in Fig. 11. This figure highlights the homogeneous covering of the Triangular optimized pattern. Fig. 11 also synthesizes the steps of the trochoid pattern optimization. The optimization of the trochoid pattern by the Triangular pattern can surprise the reader because this pattern induces slope discontinuities. In reality, these discontinuities are suppressed by the five degree polynomial smoothing applied to the polishing toolpath. The presented developments have been experimentally tested and provided interesting results.

Fig. 12 .

 12 Fig. 12. Polished surface.

Fig. 13 .

 13 Fig. 13. Experimental tool wear profile comparison.

Fig. 14 .

 14 Fig. 14. Comparison of polished surfaces.