Romain Quiban

Nicolas Grenet De Bechillon

Thomas Touret
email: thomas.touret@ecam.fr

Pierre Navet

Yasser Diab

Jérôme Cavoret

Fabrice Ville

Christophe Changenet

On the interest of a semi-empirical model for the tooth friction coefficient in gear transmissions

Keywords: Friction, boundary regime, mixed regime, two discs, gears

Accurate modelling of friction coefficient is of primary importance in efficiency, vibration and failure analyses of enclosed gear drives. After showing the influence of surface/lubricant interactions on friction, the authors used a semi-empirical model which can take all these aspects into account. Lubricant is modelled as an Eyring-Reynolds fluid and rough surfaces are described with two parameters via a stochastic approach. A specific two-disc machine is used to perform series of friction measurements on smooth and rough discs. Smooth discs allow to operate under full film lubrication and to measure a reference shear stress of lubricant, whereas rough discs reproduce gear tooth roughness and generate a representative value of friction on asperities. The purpose of this present paper is to describe calculations of this physical-based friction coefficient model and to present the experimental process. On the basis of new results, the impact of a surface finishing process is assessed as well as the consequence of calculating friction coefficient based on oil injection instead of local bulk temperature.

Introduction

Sliding friction at tooth contact is one of the main sources of power loss in geared transmissions as well as a potential source of vibration and noise. Its estimation is necessary for engineering purposes especially in a context where environmental concern drives the industry to more severe operating conditions. Accurate modelling of friction coefficient is therefore of primary importance in efficiency, vibration analyses and failure of mechanical transmissions.

In their monograph entitled 'Elasto-hydrodynamic lubrication: the fundamentals of roller and gear lubrication", Dowson & Higginson described in detail about Elasto-Hydrodynamic Lubrication (EHL) from a theoretical, experimental and applied point of view. [START_REF] Dowson | Elasto-Hydrodynamic lubrication: the fundamentals of roller and gear lubrication[END_REF] In chapter 13 of this monograph, they applied their modelling to involute spur gears. They showed that contact conditions and thus film thickness vary along the line of action and gave abacus to evaluate film thickness depending on viscosity, gear ratio and load. In chapter 10, written by J.F. Archard & A.W. Crook, the authors dealt with friction, and they emphasised the importance of thermal aspects to accurately determine oil film thickness and friction which are intimately linked. It must be noticed this interesting sentence at the end of the chapter: 'Now, in addition, the disk machine can be seen as an instrument for study of the rheological properties of oils in conditions of high pressure and transient stress'.

Nowadays, it is obvious that friction coefficient depends on multiple parameters (e.g. load, speed, temperature, surface roughness, solid and fluid physical properties, etc.). Its estimation is therefore complex and true prediction is still at an early stage. [START_REF] Björling | Towards the true prediction of EHL friction[END_REF][START_REF] Zhu | A computer program package for the prediction of EHL and mixed lubrication characteristics, friction, subsurface stresses and flash temperatures based on measured 3-d surface roughness[END_REF] Numerical approach are possible, [START_REF] Pei | Mixed elastohydrodynamic lubrication analysis of line contact with Non-Gaussian surface roughness[END_REF][START_REF] Zhang | Friction prediction of rollingsliding contact in mixed EHL[END_REF] to do so most of the previous work use model based various arrangements of the Reynolds equation. Modelling can include roughness [START_REF] Pei | Mixed elastohydrodynamic lubrication analysis of line contact with Non-Gaussian surface roughness[END_REF] and solve the problem considering gear geometry and operating conditions. [START_REF] Wu | A friction model of partial-Ehl contacts and Its application to power loss in spur gears[END_REF][START_REF] Chimanpure | A transient mixed elastohydrodynamic lubrication model for helical gear contacts[END_REF][START_REF] Li | A transient mixed elastohydrodynamic lubrication model for spur gear pairs[END_REF] They can also be coupled with contact dynamics or include complex rheology behaviour such as shear thinning. [START_REF] Akbarzadeh | Performance of spur gears considering surface roughness and shear thinning lubricant[END_REF] This approach has the advantage of being very flexible and offers the possibility of investigating specific contact problematics. Yet, it can be computer intensive and difficult to set up when one aims at evaluating friction coefficient for industrial purposes such as gear optimisation.

Another approach is based on analytical formulations. In order to cope with the complexity of the contact problem, most of them are based on semi-empirical model that requires, to some extent, an empirically fitted part. Disc tribometers or two-disc machines [START_REF] Ville | On the Two-disc machine: a polyvalent and powerful tool to study fundamental and industrial problems related to elastohydrodynamic lubrication[END_REF] are suited to reproduce tooth contacts as they can generate line or elliptical contacts with high heat flow density, similar to the ones that occur in gears. Main parameters influencing lubricated contacts such as geometry, operating conditions, material properties and environment can be modified easily in a wide range. Various empirical friction laws based on two-disc machine measurements can be found in the literature. [START_REF] Kuzmin | Coefficient of friction in heavily loaded contact[END_REF][START_REF] Misharin | Influence of the Friction Condition on the Magnitude of the Friction Coefficient in the Case of Rollers with Sliding[END_REF][START_REF] Höhn | Frictional behaviour of synthetic gear lubricants[END_REF][START_REF] Benedict | Instantaneous coefficients of gear tooth friction[END_REF][START_REF] Donoghue | Friction and temperature in rolling sliding contacts[END_REF][START_REF] Kelley | Lubrication of involute gearing[END_REF][START_REF] Drozdov | Friction and scoring under the conditions of simultaneous rolling and sliding of bodies[END_REF][START_REF] Matsumoto | The New Estimation Formula of Coefficient of Friction in Rolling-Sliding Contact Surface under Mixed Lubrication Condition for the Power Loss Reduction of Power Transmission Gears[END_REF] One of their limitations comes from the specimen surface roughness, which is often different from the one of actual gears. In fact, most of the gear teeth are cross grounded with roughness oriented in the axial direction. A part of the abovementioned studies used circumferentially grounded discs [START_REF] Benedict | Instantaneous coefficients of gear tooth friction[END_REF][START_REF] Donoghue | Friction and temperature in rolling sliding contacts[END_REF][START_REF] Kelley | Lubrication of involute gearing[END_REF][START_REF] Drozdov | Friction and scoring under the conditions of simultaneous rolling and sliding of bodies[END_REF][START_REF] Matsumoto | The New Estimation Formula of Coefficient of Friction in Rolling-Sliding Contact Surface under Mixed Lubrication Condition for the Power Loss Reduction of Power Transmission Gears[END_REF] which leads to significantly different interactions between asperities. The other part [START_REF] Kuzmin | Coefficient of friction in heavily loaded contact[END_REF][START_REF] Misharin | Influence of the Friction Condition on the Magnitude of the Friction Coefficient in the Case of Rollers with Sliding[END_REF][START_REF] Höhn | Frictional behaviour of synthetic gear lubricants[END_REF] used smooth discs which are not able to reproduce mixed lubrication. Castro and Seabra [START_REF] Castro | Coefficient of friction in mixed film lubrication: gears versus twin-discs[END_REF] showed that the evaluation of friction between gear teeth, at a single contact point or along the meshing line, requires a mixed film lubrication model able to analyse the contact in full film and boundary regime and balancing between these two extreme conditions. This approach has the advantage of being simpler to solve numerically. On the other hand, it requires to generate experimental results consistent with the targeted application.

In this paper, the authors used an existing semiempirical model of the mixed friction coefficient in nonconformal rolling-sliding contact [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] to highlight how an analytical approach, as well as numerical ones, can model the impact of roughness features, contact temperatures, physical and chemical interactions on the friction coefficient. This friction is considered as the sum of both fluid and asperities contribution. The model is implemented on a two-disc machine with operating conditions which covers gear applications. A series of friction coefficient measurements are performed on both smooth and rough discs. Smooth discs allow to operate under full film lubrication. In this condition, the two-disc machine is used to measure a reference shear stress of the lubricant. [START_REF] Smith | Lubricant behavior in concentrated contactsome rheological problems[END_REF][START_REF] Johnson | Observations of viscoelastic behaviour of an elastohydrodynamic lubricant film[END_REF][START_REF] Johnson | Shear behaviour of elastohydrodynamic Oil films[END_REF][START_REF] Hirst | Elastohydrodynamic lubrication at high pressures[END_REF][START_REF] Houpert | Rheological and thermal effects in lubricated E.H.D. Contacts[END_REF][START_REF] Evans | The rheological properties of elastohydrodynamic lubricants[END_REF][START_REF] Gupta | Viscoelastic effects in MIL-L-7808-type lubricant, part I: analytical formulation[END_REF][START_REF] Sottomayor | Experimental evaluation and numerical simulation of MIL-L-23699 traction curves: in memorium of georges tourlonias[END_REF] Rough discs reproduce roughness magnitude and orientation of gear teeth to estimate a representative value of friction on asperities. A discussion is given on the effect of surface finishing process on a rough surface and how it is considered in the present model.

In gear application, kinematics is such that sliding speed is important and corresponding thermal phenomena cannot be neglected for friction prediction. [START_REF] Olver | Gear lubrication-a review[END_REF][START_REF] Changenet | Power loss predictions in geared transmissions using thermal networks-applications to a Six-speed manual gearbox[END_REF][START_REF] Liu | Crucial role of solid body temperature on elastohydrodynamic film thickness and traction[END_REF] The local rise in temperature influences both oil film thickness and friction at tooth contact. Isaac [START_REF] Isaac | Thermal analysis of twin-disc machine for traction tests and scuffing experiments[END_REF] showed that bulk temperature has an influence on friction. In practice, gear bulk temperature is seldom known and the temperature used as a reference for tooth friction prediction is often either the one of oil jet or oil bath, depending on lubrication solution. This subject is discussed in light of new results and an adaptation of the existing friction model is presented.

Theory and calculation

Evolution of friction coefficient and associated physical phenomena are different depending on the value of reduced oil film thickness in the contact or hydrodynamic roughness parameter of Tallian [START_REF] Tallian | The theory of partial elastohydrodynamic contacts[END_REF] :

Λ = h c σ (1)
with Λ the reduced oil film thickness, h c the isothermal oil film thickness at the contact centre under fully flooded conditions, σ the composite roughness defined such as

σ = Rq 2 1 + Rq 2 2 ,
and Rq 1 , Rq 2 the root mean square (RMS) roughness of surface 1 and 2. Gear contacts operate mostly in mixed lubrication regimes. [START_REF] Olver | Gear lubrication-a review[END_REF][START_REF] Martin | A review of friction predictions in gear teeth[END_REF][START_REF] Hamrock | Fundamentals of fluid film lubrication[END_REF] Diab et al. model [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] In the present study, the authors use the semi-empirical model of sliding friction coefficient of non-conformal rolling-sliding contacts developed by Diab et al. [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] This model has the advantage of including a complete set of equations to simulate a mixed lubricated contact. This is why it was chosen as it gives acceptable results under various operating conditions. In the original model, a strong coupling was implemented between the gear dynamic model and the evaluation of the friction coefficient. The originality of this paper is to show that it is possible to decouple the dynamic solving from the analytical friction evaluation while obtaining a reliable friction coefficient. It also includes a discussion of the surface finish and how thermal aspects are included in the model based on recent studies.

All physical parameters used in the following equations are mean values over the contact area.

Normal load (F n ) is decomposed into two parts (i) one supported by surface roughness asperities referred as asperity part with subscript 'a' and (ii) one supported by oil film referred to as fluid part with subscript 'f' (equation ( 2)). In the same way, apparent contact area of Hertz 36 (A 0 ) is decomposed into an actual contact area (A a ) and the complementary fluid area (A f ) (equation ( 3)).

F n = F n.a + F n.f (2)
A 0 = A a + A f (3)
Thereby, a mixed friction coefficient formula (equation ( 5)) is expressed in terms of shear stresses, using equations ( 2)-( 3) and Coulomb's law [START_REF] Coulomb | Théorie Des machines simples[END_REF] (equation ( 4)).

μ = F t F n = τ p (4) μ = 1 p A a A 0 τ a + 1 - A a A 0 τ f (5)
with μ the friction coefficient, p the mean contact pressure, τ a the shear stress from asperities and τ f the shear stress in fluid film.

Fluid contribution

To take into account the non-Newtonian behaviour in EHL, [START_REF] Hirst | Non-Newtonian behaviour in elastohydrodynamic lubrication[END_REF] fluid shear stress (τ f , equation ( 6)) is reformulated here in terms of a Newtonian shear stress (τ N , equation ( 7)) [START_REF] Newton | Philosophiae naturales principia mathematica[END_REF] and a reference shear stress, called Eyring stress (τ E , equation ( 8)). [START_REF] Eyring | Viscosity, plasticity, and diffusion as examples of absolute reaction rates[END_REF][START_REF] Moore | The behaviour of lubricants in elastohydrodynamic contacts[END_REF] This last one depends on both pressure and temperature. [START_REF] Evans | The rheological properties of elastohydrodynamic lubricants[END_REF] Experimental constants C 0 , C p , C T in Houpert et al. formula [START_REF] Houpert | Rheological and thermal effects in lubricated E.H.D. Contacts[END_REF][START_REF] Forster | Viscoelastic effects in MIL-L-7808-type lubricant, part II: experimental data correlations[END_REF] (equation ( 8)) are determined using smooth discs to carry out several full film EHL traction curves at different operating., taking into account thermal effects occurring in traction tests. [START_REF] Evans | The rheological properties of elastohydrodynamic lubricants[END_REF] Lubricant pressure in the contact (p f , equation ( 10)) is derived from both load and area sharing (equations ( 2) and ( 3)). Note that smooth disc surfaces have sufficiently low roughness amplitude compared to lubricant film thickness to avoid any contact between asperities (h c /σ >7 ,A a /A 0 = 0). In this case, the whole load is lifted by fluid film, inducing fluid pressure equals to Hertz mean pressure. 36

τ f = τ E sin h -1 τ N τ E (6) τ N = η (T c ,p f ) v s ϕ T h c (7
)

τ E = C 0 e (C p p f ) e C T 1 Tc -1 T 40 (8) ϕ T = 1 -13.2( p 0 / E ′ )L 0.42 T 1 + 0.213(1 + 2.23 |SRR| 0.83 )L 0.64 T ( 9
)
p f = F n.f A f = p -p a • A a / A 0 1 -A a / A 0 ( 10
)
with versus the sliding speed, h c the isothermal fully flooded central film thickness suggested by Hamrock and Dowson [START_REF] Hamrock | Isothermal elastohydrodynamic lubrication of point contacts part IIIfully flooded results[END_REF][START_REF] Brewe | Simplified solution for elliptical-contact deformation between Two elastic solids[END_REF] and Φ T the thermal reduction due to inlet backflow, revised by Gupta et al. [START_REF] Gupta | Viscoelastic effects in MIL-L-7808-type lubricant, part I: analytical formulation[END_REF] Oil dynamic viscosity evolution with temperature η(T ) is calculated with Vogel model [START_REF] Vogel | Das temperaturabhaengigkeitsgesetz Der viskositaet von fluessigkeiten[END_REF] (equation ( 11)) and evolution with pressure η(p) with Roelands model [START_REF] Roelands | The viscosity-temperature-pressure relationship of lubricating oils and Its correlation with chemical constitution[END_REF] (equation ( 13)). Following the EHL problem, [START_REF] Dowson | Elasto-Hydrodynamic lubrication: a survey of isothermal solutions[END_REF] film thickness is calculated with lubricant properties at contact conjunction inlet temperature (T b ). [START_REF] Liu | Crucial role of solid body temperature on elastohydrodynamic film thickness and traction[END_REF][START_REF] Greenwood | Elastohydrodynamic lubrication[END_REF] Whereas, shear stress is calculated with lubricant properties at contact temperature (T c ) and pressure (p f ). [START_REF] Olver | Prediction of traction in elastohydrodynamic lubrication[END_REF] η(T , p atm ) = η 40 e β((1/T )-(1/T 40 )) ( 11)

β = ln η 100 η 40 / 1 T 100 - 1
T 40 (12) η(T , p) = η(T , p atm ) e (ln(η 40 )+9.67)•((1+p/1.98×10 8 ) z -1) As slip increases during the traction test, heat generation tends to increase contact temperature (T c, equation ( 15)). This thermal effect is shown in Figure 1, where lubricant shear stress evolution with shear rate is correlated to contact temperature in two cases (i) lightly loaded isothermal and (ii) heavily loaded nonisothermal. Contact temperature is estimated from a reference temperature representative of discs. In the initial study, [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] the reference temperature is set to oil jet temperature. However, it may not be accurate as it does not account for disc thermal environment and associated heat conduction and convection. Here, the local bulk temperature is used [START_REF] Isaac | Thermal analysis of twin-disc machine for traction tests and scuffing experiments[END_REF] and a comparison between both methods is shown in section 4. The contact temperature is then estimated adding flash temperature, ΔT f . [START_REF] Blok | The flash temperature concept[END_REF] This local temperature elevation depends on the heat produced (equation ( 16)), contact area and a thermal resistance (R th , equation ( 17)) due to the striction phenomenon around the moving contact. An iterative process is necessary to compute friction coefficient as friction value depends on contact temperature and vice versa. 29

T c = T b + ΔT f = T b + Q R th A 0 (15) Q = F n μ v s (16)
R th = 0.636 2 2a √ πk 1 ρ 1 c 1 v 1 + πk 2 ρ 2 c 2 v 2 (17)

Contribution of asperities

The second contribution of Diab et al. [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] model is the friction due to interaction between asperities of both surfaces. Corresponding shear stress is given by equation (18). The friction coefficient on asperities (µ a ) depends on surface and lubricant physicochemical properties and interactions. Its value is determined from a series of traction force measurements on rough discs.

τ a = p a μ a 1 -e -5.a.|SRR| m 2 /m 0 √ ( 18
)
with p a mean pressure on asperities, SRR the slide-to-roll ratio, and m 0 , m 2 the zeroth and second spectral moment of a roughness profile. These last correspond to the square of the RMS of height and slope of a profile (known as Rq 2 and Rdq 2 in stylus measuring devices). Spectral moments can be estimated for an equivalent contact composed of two anisotropic rough surfaces with the following equations [START_REF] Sayles | Thermal conductance of a rough elastic contact[END_REF][START_REF] Francis | Application of spherical indentation mechanics to reversible and irreversible contact between rough surfaces[END_REF][START_REF] Manners | Some observations on Persson's Diffusion theory of elastic contact[END_REF] :

m 0 = Rq 2 1 + Rq 2 2 ( 19
)
m 2 = Rdq 2 1.s Rdq 2 1.t + Rdq 2 2.s Rdq 2 2.t (20)
indices 1 and 2 refer to disc surfaces, and indices 's' for sliding direction and 't' for transverse direction. Roughness parameters are obtained by the following standards. [START_REF]ISO 4287 -Geometrical Product Specification (GPS) -Surface Texture: Profile Method -Terms, Definitions and Surface Texture Parameters[END_REF][START_REF]ISO 4288 -Geometrical Product Specification (GPS) -Surface Texture: Profile Method -Rules and Procedures for the Assessment of Surface Texture[END_REF][START_REF]Geometrical Product Specification (GPS) -Filtration -Part 21: Linear Profile Filters: Gaussian Filters[END_REF] To describe pressure on asperities summits, different approaches dealing with statistical formulations were proposed. [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF][START_REF] Ms | The statistical analysis of a random, moving surface[END_REF][START_REF] Bush | The elastic contact of a rough surface[END_REF][START_REF] Nayak | Random process model of rough surfaces[END_REF] As load supported by asperities increases, the real contact area will accordingly increase because of asperity elastic deformations. A consequence of the real contact area contact being proportional to the load is that the real contact pressure is nearly constant (equation ( 21)).

p a = 0.1E ′ m 2 π ( 21
)
with E ′ the reduced elastic modulus. [START_REF] Hamrock | Isothermal elastohydrodynamic lubrication of point contacts part IIIfully flooded results[END_REF] The real contact area of rough elastic bodies is given by 62

A a A 0 = 1 2 • 1 2 1 -erf ϕ T h c 2 m 0 √ (22)
Test rig, specimens and methods

Test rig and specimens

As it was already underlined in the previous section, the above-mentioned model requires some experiments to determine several coefficients. The two-disc machine used in this study is presented elsewhere. [START_REF] Ville | On the Two-disc machine: a polyvalent and powerful tool to study fundamental and industrial problems related to elastohydrodynamic lubrication[END_REF][START_REF] Gupta | On the traction behavior of several lubricants[END_REF] Experiments were carried out with typical aeronautical lubricant (Table 1) and steel discs (Table 2). Some of the discs were smooth (Ra = 0.02 µm) while roughness of the others was similar to that of gears, i.e. Ra ≈ 0.3 µm, with a transverse grinding direction (Figure 2).

Methods

A series of traction tests are conducted for both smooth and rough disc pairs. First, smooth disc pair measurements allow to determine the values of C 0 , C p and C T coefficients from Eyring stress formula (equation ( 8)). Indeed, smooth disc roughness amplitude is sufficiently low in front of lubricant film thickness to ensure that traction is mainly due to fluid shearing. Then, rough disc traction curves allow to determine a value of friction on asperities (µ a ) of the surface/lubricant. The values of these parameters are computed using an algorithm that minimises the difference between calculations and experimental results. Measurements are performed for (i) three rolling speeds, (ii) three contact pressures and (iii) three lubricant injection temperatures. This forms a total of 2 × 27 traction tests allowing accurate estimations of experimental coefficients, while operating conditions are chosen to encompass gear operating conditions (Table 3). Disc bulk temperatures, for friction estimation, are obtained via a dedicated thermal model of the two-disc machine. [START_REF] Isaac | Thermal analysis of twin-disc machine for traction tests and scuffing experiments[END_REF]

Results and discussion

Global overview of lubrication regimes encountered during the test campaign is shown in Figure 3 Figure 4 shows a number of experimental traction curves obtained from the two-disc machine [START_REF] Isaac | Thermal analysis of twin-disc machine for traction tests and scuffing experiments[END_REF] as well as results given by equations ( 5) to (22), using values from Table 4. Influence of oil injection temperature, normal load and average speed is highlighted. It appears that there is a good agreement between experiments and calculations.

Effect of a surface finishing process

Another pair of rough discs were used, with the same geometrical and material properties as that of the previous ones, but with a surface finishing process. The process significantly reduces roughness slopes as it can be seen in the profiles measured in the sliding direction of the standard rough disc (Figure 5(a)) and the one with finishing process (Figure 5(b)). The process was sufficiently mild so that roughness amplitudes and principal direction remain close to the one of the standard rough surfaces (Ra rough ≈ 0.35 μm, Ra finish ≈ 0.30 μm). Roughness parameters values of the finished disc pair are shown in Table 5. The main difference with the standard rough discs (Table 2) comes from the average roughness slope which is greatly reduced on the discs with a finishing process, whereas amplitude parameters values remain close.

Traction force was measured for the 27 operating conditions presented in Table 3. The friction coefficient appears to be reduced with the finished discs, especially for low film thickness conditions. Figure 6 shows measured and predicted mixed lubricated traction curves of standard and finished rough discs at oil injection temperature of (a) 40°C and (b) 100°C. At high oil temperature, the film thickness separating the two surfaces of the lubricated contact is reduced and thus reveals the friction contribution of asperities.

In the case of the finished disc, the measured friction coefficient is reduced by 20% compared to the standard rough discs, as shown in Figure 6(b). The present model accounts for the surface finishing process. The reduction of roughness slopes induces a lower pressure on asperities (equation ( 21)) and thus a lower contribution of asperities for the same reduced film thickness. In fact, more of the load is carried by the fluid. However, with this type of surface finish, there is a tendency to change the tribochemical properties of the surface. Thus the parameter that models the contact between surface asperities (μ a ) was also impacted. The best fit for this parameter is equal to 0.10 for the finished discs, compared to 0.11 for the standard ones. Therefore if this law is applied to gears, the process used to manufacture rough discs should be similar to the actual gear of the final application.

Effect of oil reference temperature for computation

Figure 7 shows a traction test measurement and associated numerical prediction using either oil jet or disc bulk temperature as a reference for friction calculation. It can be noticed that measurements are not exactly symmetrical for positive and negative sliding. This is due to disc warming as heat generated by friction dissipates through it. Friction values for negative sliding are slightly reduced as those measurements are operated on hotter discs. Traction curve prediction using oil injection temperature is strictly symmetrical due to constant oil jet temperature during the traction test Results remain close to experimental data as long as disc temperatures are close to the oil one, otherwise, a significant shift is seen, typically of 30% error for temperature difference superior to 5°C in this case. Traction curve prediction using disc bulk temperature is close to measurements, with a maximum error of 10%. This shows that using oil supply temperature to estimate friction coefficient can lead to significant errors depending on disc bulk and oil supply temperature difference.

Influence of surface and lubricant on friction coefficient

As it has been pointed out in previous sections, surface finishing and temperatures (oil properties modification) have an impact on the friction coefficient. To confirm these trends, numerous friction coefficients were measured with different Experimental values were compared to Diab and Benedict and Kelley laws, the latter being traditionally used in industry [START_REF] Benedict | Instantaneous coefficients of gear tooth friction[END_REF] (equation ( 23)). where F n is normal load, l total contact length, η 0 oil dynamic viscosity at contact inlet, versus sliding speed, v r sum rolling speed ( = 2v e )a n da 1 , b 1 best-fitc o e f ficients.

μ BK = a 1 Log 10 b 1 (F n / l) η 0 v s v 2 r ( 23
)
Benedict and Kelley found a 1 = 0.0127 and b 1 = 0.0297 S.I. As shown in Figure 8, friction coefficient values do not lie on Benedict and Kelley master curve. This means that using only oil viscosity (equation ( 23)) to describe the influence of the surface and the lubricant is not sufficient. The influence of roughness amplitude is noticeable as higher friction coefficients were measured during the test campaign with rougher discs (A and B in, Ra ≈ 0.8 μm), whereas lower friction coefficients were measured during the test campaign with smoother discs (F, Ra ≈ 0.1 μm). Finally, test campaigns C and D with the same discs and the same type of lubricant but with or without an additive package suggest that surface/lubricant interactions have a significant influence on friction. Therefore, the use of a model able to take into account surface, lubricant and surface/lubricant interactions are necessary. This is underlined by the computed values of C 0 , C p , C T , µ a coefficients (Table 6). The black solid lines shown in Figure 8 represent the master curves obtained with the coefficient from Table 6.

This emphasises the importance of experimental coefficients to accurately qualify the lubricant-material interaction in a mixed lubricated contact.

Conclusion

Analysis of numerous traction measurements on two-disc machines with different discs and lubricants showed the influence of surface, lubricant and surface/ lubricant interactions on friction. In addition to usual roughness amplitude parameters (e.g. Ra, Rq, Rz,e t c . ) to describe a surface, it was shown that roughness slopes (Rdq) can have a significant influence on friction. Likewise, it was shown that lubricant cannot be fully described by its viscosity value. Instead, it may be more accurate to also use surface/lubricant-related parameters.

The mixed friction model of Diab et al. [START_REF] Diab | Prediction of power losses Due to tooth friction in gears[END_REF] was used to predict two-disc machine traction curves. This model is suited to predict gear friction loss as it is developed to reproduce rough disc behaviour which is representative of gear tooth roughness. Although the model is based on simplifying assumptions, in particular the use of constant parameter over the contact, the results are close to those measured. The present model shows average errors ranging from 12.7% to 20.8% regarding experimental data on a broad range of oils, materials and operating conditions.

The temperature to be used for the prediction of friction coefficient was discussed. It appears that using oil temperature can lead to error in friction prediction if this temperature is significantly different from the disc bulk ones.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

z = 1 .

 1 98 × 10 8 α ln (η 40 / (6.315 × 10 -5 )) (14) with η 40,100 lubricant dynamic viscosity values (respectively, at 40 °C and 100 °C) at ambient pressure, T 40 = 313.15 K, T 100 = 373.15 K, α, β the lubricant pressure-viscosity and temperature-viscosity coefficients, respectively, and z the corresponding pressure coefficient of Roelands model.

Figure 1 .

 1 Figure 1. Effect of contact temperature on lubricant shear stress evolution with shear rate. (a) Lightly loaded (isothermal) and (b) heavily loaded (non-isothermal).

Figure 4 .

 4 Figure 4. Experimental and numerical traction curves were obtained for smooth and rough discs. Influence of oil jet temperature, normal load and average speed. Reference condition: T inj = 80°C, p 0 = 1.6 GPa, v e = 20 m/s.

  discs and oils to analyse if these coefficients depend on the surface and the lubricant. Friction coefficient values were obtained from direct traction measurements on a two-disc test rig (see paragraph 3). The range of experimental conditions covered is: η 0 = [0.046-0.18] Pa.s ([5-200] cSt); v s = [1-6] m/s, v e = [6-30] m/s and F n /l = [150-1100] N/mm (p 0 = [0.7-1.9] GPa).

Figure 5 .

 5 Figure 5. Measured roughness profiles of (a) rough disc surface, Rq = 0.52, Rdq = 0.089 and (b) rough disc surface with a finishing process, Rq = 0.40, Rdq = 0.042. Only a portion of the measured length is shown here for visual purpose.

Figure 6 .

 6 Figure 6. Effect of a surface finishing process on traction curve at v e = 10 m/s and p 0 = 1.9 GPa. (a) T inj = 40°C (Λ = 1.0), (b) T inj = 100°C (Λ = 0.4).

Figure 7 .

 7 Figure 7. Traction test prediction using either oil injection or disc bulk temperature, t inj = 80°C, p 0 = 1.9 GPa, v e = 30 m/s.

Figure 8 .

 8 Figure 8. Two-disc friction coefficient measurements are displayed in Benedict-kelley chart. 14 .

Table 1 .

 1 Lubricant data.

	Lubricant type		MIL-PRF-23699
	Cinematic viscosity Density	ν 40-100 24.2; 4.97 cSt ρ 15 975 kg/m 3
	Pressure-viscosity coefficient	α	16 GPa -1
	Temperature-viscosity coefficient β	3169 K

Table 2 .

 2 Discs data.

		Smooth		Rough
		Sliding		Transverse
	Grinding direction	(circumferential)	(axial)
	Material	Nitrided steel
	E' (GPa)	231
	L (mm)	10
	R x (mm)	35
	R y (mm)	∞	200	∞	200
	Ra (µm)	0.02	0.02	0.42	0.29
	Rq (µm)	0.03	0.03	0.52	0.36
	Rdq s (rad)	0.027	0.029	0.089	0.070
	Rdq t (rad)	0.033	0.029	0.026	0.025

Table 3 .

 3 Operating condition during the test campaign. Specimens surface topographies. Figure3. Measured friction coefficient at SRR =+20% for each traction curves (table3).

	Figure 2.
	Lubricant temperature	T inj	40; 80; 100°C
	Maximum pressure	p 0	1.2; 1.6; 1.9 GPa
	Average speed	v e	10; 20; 30 m/s
	Slide-to-roll ratio	SRR	-20% to + 20%

Table 4 .

 4 Experimental coefficients from equations (8) and(18) for nitrided steel material MIL-PRF-23699 lubricant.

	C 0 (MPa)	C p (GPa -1 )	C T (K)	µ a (-)	Mean error
	1.00	1.60	1500	0.11	12.7%

Table 5 .

 5 Additional discs data.

		Rough discs with surface finish
	R x (mm)	35
	R y (mm)	∞	200
	Ra (µm)	0.32	0.26
	Rq (µm)	0.40	0.33
	Rdq s (rad)	0.042	0.053
	Rdq t (rad)	0.022	0.025

Table 6 .

 6 Impact of the lubricant and material pair on the calibration of diab coefficients.LubricantDisc material Ra (μm) C 0 (MPa) C p (GPa -1 ) C T (K) µ a (-) Mean error

	A ISO VG 320 mineral oil	Carburised steel	≈ 0.80	1.84	1.12	235	0.17 20.8%
	B ISO VG 60 mineral oil	Carburised steel	≈ 0.80	3.26	3.24	294	0.11 13.9%
	C MIL-PRF-23699 synthetic base oil Nitrided steel	≈ 0.35	0.52	2.03	1505	0.11 17.7%
	D MIL-PRF-23699 synthetic oil	Nitrided steel	≈ 0.35	1.00	1.60	1500	0.11 12.7%
	E MIL-PRF-23699 synthetic oil	Nitrided steel + finish proc. ≈ 0.30	1.00	1.60	1500	0.10 13.1%
	F MIL-PRF-23699 synthetic oil	Carburised steel	≈ 0.10	0.78	1.47	472	0.09 17.0%

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.


Vincent Bérot

Anton Eremin

Antoine Fauchois

Jeanne Dechamp

Luminita Schneider

Aziza Chermak

Antoine Faycal

Baptiste Sellem

Thibault Orriere

Marion Favier

Hepatitis A, hepatitis B and HPV vaccine needs and coverage in MSM initiating HIV PrEP in a sexual health clinic in Paris

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Hepatitis A, hepatitis B and HPV vaccine needs and coverage in MSM initiating HIV PrEP in a sexual health clinic in Paris

AUTHORS

V. Bérot 1 , A. Eremin 1 , A. Fauchois 1 , J. Dechamp 1 , L. Schneider 1 , A. Chermak 1 , A. Faycal 1 , B.

Sellem 1 , T. Orriere 1 , M. Favier 1 , R. Tubiana [START_REF]Vaccine recommendations for human papillomavirus infections in men [Recommandations vaccinales contre les infections à papillomavirus humains chez les hommes[END_REF][START_REF]Élargissement de la vaccination contre les papillomavirus aux garçons[END_REF] , MA Valantin 1,2 , V. Pourcher 1,2 , E. Todesco [START_REF]Élargissement de la vaccination contre les papillomavirus aux garçons[END_REF][START_REF] Brandl | Are men who have sex with men in Europe protected from hepatitis B?[END_REF] , G.

CONTRIBUTORS

VB, RP, JD, CK contributed to the study design and developing analysis plan. AF, JD, RA, CK, VB, AE contributed to the analysis and interpretation of data. VB, AE wrote the first draft of the manuscript and subsequent drafts after revisions. CK reviewed all versions of the manuscript; VB, AE, AF, JD, LS, AC, AF, BS, TO, MF, RT, MAV, VP, ET, GM, RP, CK contributed to the recruitment of participants, reviewed the final version of the manuscript and contributed to the interpretation of the data.

Pre-exposure prophylaxis (PrEP) with tenofovir disoproxil/emtricitabine (TDF/FTC) is a powerful tool to prevent HIV acquisition and provides an opportunity to offer comprehensive prevention services, including assessment for sexually transmitted infections and evaluation of immune status towards vaccine-preventable viral infections such those due hepatitis A virus (HAV), hepatitis B virus (HBV) and human papillomavirus (HPV). In addition to the programme in girls aged 11-19, French guidelines recommended HPV vaccination for MSM ≤ 26 years old in February 2016 [START_REF]Vaccine recommendations for human papillomavirus infections in men [Recommandations vaccinales contre les infections à papillomavirus humains chez les hommes[END_REF] and for all boys aged 11-19 in December 2019 [START_REF]Élargissement de la vaccination contre les papillomavirus aux garçons[END_REF] . There are no restrictions on HPV vaccine use beyond these age limits, but the cost is not covered by the French Health Service. In real-life settings, suboptimal vaccination coverage against HPV as well as HBV has been reported among European MSM [START_REF] Brandl | Are men who have sex with men in Europe protected from hepatitis B?[END_REF][START_REF] Petit | Men having sex with men and the HPV vaccine in France: A low vaccine coverage that may be due to its infrequent proposal by physicians[END_REF] .

Our objective was to evaluate HAV, HBV and HPV vaccine needs and coverage in individuals initiating PrEP in a sexual health clinic in Paris. In this observational retrospective single-centre study, we reviewed all individuals who initiated PrEP between January 1 st , 2016 and December 31 st , 2020 with ≥ 1 year of follow-up after PrEP initiation. At baseline, we assessed the presence of HAV and HBV antibodies and HPV vaccination status. Immune protection against HAV and HBV was defined as the presence of anti-HAV IgG index S/CO > 1.00 and anti-HBs IgG >10 International Units/L (IU/L), respectively. HPV vaccination status at baseline was assessed through the participants' recall.

Subsequently, we assessed vaccine prescription by physicians for non-immune and unvaccinated participants, followed by a review of completion of vaccination. Vaccination schedules were considered complete after 2 doses for HAV with a time interval of 0 and 6 months, 3 doses for HBV with an interval of 0, 1, 6 months, and 3 doses for HPV with an interval of 0, 2, 6 months. Contrary to HAV and HBV vaccines, HPV vaccine was not accessible in the sexual health center and had to be purchased from a private pharmacy. Finally, we assessed overall HAV, HBV immune status combining immune protection acquired in the past or by vaccination after PrEP initiation and HPV vaccine coverage. HPV vaccine completion was analysed by ≤ 26 y.o. or > 26 y.o. age groups. If any information was missing, individuals were contacted by phone or email to determine whether vaccination had been performed and if not, the reason why. All clinical, biological and prescription data are routinely documented in an electronic health record (NADIS), for which all patients gave consent for the collection and use of their anonymized data after approval by the CNIL (French Data Protection Authority; CNIL authorisation number: 2085881). Statistical data are presented with total numbers and proportions and compared by a chi-square test. A P-value <0.05 was considered statistically significant.

A total of 591 PrEP users were analysed. All were MSM with a median age of 33 years (IQR 28-41), including 118 participants (20%) aged ≤ 26 y.o.

At baseline (Table 1), 57.7% (341/591) of PrEP users were immune against HAV and 73.4% (434/591) against HBV. Vaccines were prescribed for 93.2% (233/250) of HAV non-immune and 87.2% (137/157) of HBV non-immune participants. Vaccination was completed in 85.8% (200/233) and in 91.2% (125/137) individuals with an HAV and HBV vaccine prescription, respectively. Our results are consistent with other studies where HAV vaccination rates were high, especially among PrEP users [START_REF] Burrell | Hepatitis A vaccine uptake among men who have sex with men from a time-limited vaccination programme in Melbourne in 2018[END_REF] .

With regards to HPV, only seven of the 591 (1.2%) individuals had been vaccinated before PrEP initiation, including 4/118 (3.4%) individuals aged ≤ 26. The prescription rate by physicians remained low throughout the study period at 26% (152/584) for all ages and 39.5% (45/114) for those ≤ 26 years. These results are in agreement with those of other studies which report infrequent HPV vaccination prescription by physicians [START_REF] Petit | Men having sex with men and the HPV vaccine in France: A low vaccine coverage that may be due to its infrequent proposal by physicians[END_REF][START_REF] Wheldon | Physician Communication Practices as a Barrier to Risk-Based HPV Vaccine Uptake Among Men Who Have Sex with Men[END_REF] . Following prescription, the HPV vaccine completion rate was 54.6% (83/152) including 64.4 % (29/45) in participants aged ≤ 26 years. Of 69 individuals who did not complete HPV vaccination despite prescription, 5 (7%) participants did not respond to the questionnaire, 64 (93%) reported the following reasons: forgetting to go to a pharmacy for vaccine delivery (n=29), not feeling at risk (n=20), lost prescription (n=6) and vaccine cost (n=9, all > 26 y.o).

Several factors may explain our findings: recentness of the French guidelines, vaccine cost and lack of motivation for HPV vaccination [START_REF] Nadarzynski | Vaccine acceptability, uptake and completion amongst men who have sex with men: A systematic review, meta-analysis and theoretical framework[END_REF] , as one third of the participants described not feeling at risk for this 77 viral oncogenic disease. 78

Finally, combining immunity acquired in the past or by vaccination after PrEP initiation, the overall 79 immune protection rate for these 591 MSM initiating PrEP was 91.5% for HAV, 94.6% for HBV and 80 15.2% for HPV, including 28% in the ≤ 26 years age group and 12% in the > 26 years age group. 81

Given the high burden of HPV-attributable lesions in MSM compared to heterosexual men 8 , a change 82 in prevention approaches is required. Greater vaccine promotion against sexually transmitted viruses, 83 including vaccination in PrEP guidelines, and expanding the age criteria for HPV vaccination in MSM 84 as recommended in the UK [START_REF]Recommendations Chapter 18a, Human papillomavirus, in Immunisation against infectious disease -'The Green Book[END_REF] would help to improve targeted vaccination campaigns in this at-risk 85 population [START_REF] Wheldon | Physician Communication Practices as a Barrier to Risk-Based HPV Vaccine Uptake Among Men Who Have Sex with Men[END_REF][START_REF] Nadarzynski | Vaccine acceptability, uptake and completion amongst men who have sex with men: A systematic review, meta-analysis and theoretical framework[END_REF][START_REF] Chin | The Role of Healthcare Providers in Promoting Human Papillomavirus Vaccines among Men Who Have Sex with Men: A Scoping Review[END_REF] . 86

Non-immune (HAV/HBV) or nonvaccinated (HPV) at PrEP initiation

Vaccine prescription rate in case of no prior immunity (HAV, HBV) or vaccination (HPV)

Vaccine completion rate after prescription at PrEP initiation

Overall immune protection (HAV, HBV) * and HPV vaccine coverage

Table 1. Baseline immune status, vaccine prescription and uptake in PrEP users (n=591)

Monsel 1 , 2 AFFILIATIONS 1 .

 121 R. Agher 1 , R. Palich 1,2 , C. Katlama 1,Sorbonne University, Infectious Diseases Department, Pitié-Salpêtrière Hospital, AP-HP, Paris, France. 2. Pierre Louis Epidemiology and Public Health Institute (iPLESP), INSERM U1136, Paris, France 3. Sorbonne University, Virology Department, Pitié-Salpêtrière Hospital, AP-HP, Paris, France. CORRESPONDING AUTHOR Pr Christine Katlama, MD Service de Maladies Infectieuses et Tropicales, hôpital Pitié-Salpêtrière, 47-83 boulevard de l'hôpital, 75013 Paris, France Tel: +330142160130 Email: christine.katlama@aphp.fr TRANSPARENCY DECLARATION No competing interest.