Salambô Dago

Jorge Pereda

Sergio Ciliberto

Ludovic Bellon

Erratum: Virtual double-well potential for an underdamped oscillator created by a feedback loop

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

Erratum: Virtual double-well potential for an underdamped oscillator created by a feedback loop.

Salambô Dago, Jorge Pereda, Sergio Ciliberto and Ludovic Bellon * Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

In this note, we correct the expression of the switching rate Γ(B, T ) of the potential barrier B for a system at temperature T expressed in Eq. 8 and derived in appendix B. We also update below all the equations using the switching rate expression. The error is nevertheless small enough to not change any of the other results or conclusions of the paper.

The error came from the assumption that the total energy in the double well potential E was simply distributed according to the Boltzmann probability distribution P (E) = 1 k B T exp{-E/(kBT )}, which would be valid only in an harmonic potential. Indeed, in the double well potential, the motion period T depends on the system energy. Therefore the probability distribution of the energy P (E) is obtained by integrating the Boltzmann canonical distribution on the time range [-T (E)/2 + T (E)/2] (time range required to explore the full phase space). Contrary to the harmonic case where T (E) = T0 is constant, here the energy dependence modifies the expression of P (E).

I. ERRATUM: EQ. 8

In this section we present the erratum regarding the switching rate expression in the case of a bi-quadratic potential U (x) = 1 2 k(|x| -x 1 ) 2 , with an energetic barrier B = 1 2 kx 2 1 , for a system temperature T .

A. Replacement to Eq. 8

Eq. 8 of the paper has to be replaced by Eq. ( 2):

Γ = ω 0 e -βB β ∞ 0 dE e -βE π + 2 sin -1 B E (1) = ω 0 k B T B e -B/(k B T ) ∞ 0 d e -B/(k B T ) π + 2 sin -1 ( -1 2 ) , (2)
with β = 1 k B T , and ω 0 = k/m the angular resonance frequency in a single well of the bi-quadratic potential. Note that in this expression, we extend the definition of the sin -1 function to arguments greater than one, with

sin -1 (x) = π/2 for x > 1.
(3)

B. Replacement to Eq. 14

As the Γ expression was corrected, Eq. 14 also has to be updated. The correct expression of the function g(z) is:

g(z) = 4 √ π 1 z e -z 2 ∞ 0 d e -z 2 π + 2 sin -1 ( -1 2 ) . (4)
It presents a global maximum g * = 0.19 in z * = 0.59. * ludovic.bellon@ens-lyon.fr

II. ERRATUM TO APPENDIX B. SWITCHING RATE

We study a one degree of freedom mechanical system of mass m and position x evolving in a double well potential of potential energy U (x) = 1 2 k(|x| -x 1 ) 2 . The barrier between the two wells is placed in x = 0 and its energy is B = 1 2 kx 2 1 . The system kinetic energy is K = p 2 /(2m), with p = m dx dt the momentum, so that the system Hamiltonian is:

H(x, p) = 1 2m p 2 + U (x). (5)
In the limit of weak damping, the total energy E = H(x, y) of the cantilever is conserved in time. Therefore, the Hamiltonian dynamics can be solved for a given total energy E by integrating the following equation:

dx dt = ± 2 m [E -U (x)]. (6)
The system motion is periodic: the time T (E) to explore the full phase space available for a given E, depends on the value of E with respect to the barrier height B.

• If E > B the system explores both wells every period, so there are 2 barrier crossing every period, with the period being:

T (E > B) = 2 x M -x M dx 2[E -U (x)]/m (7)
where x M = x 1 + 2E/k is the maximum excursion of the cantilever allowed by the criterion U = E -K ≤ E.

• If E < B, then the motion is confined to a single well, there are no switches, and the period is T 0 = 2π/ω 0 . As there are 2 wells, the time to explore the full phase space available for an energy E is

T (E < B) = 2T 0 . (8)
Using the the convention of Eq. ( 3), we can compute for all values of E:

T (E) = 2 ω 0 π + 2 sin -1 B E . (9)
Eq. ( 9) is the same as Eq. B3 of the paper, simply extended to E < B. This time is twice T 0 when E ≤ B or E B, and tends to T 0 for E B.

It remains to compute the energy probability distribution P (E). The first simplistic (and wrong) approach that we originally applied consists in using the Boltzmann distribution P B (E) = exp(-E/k B T )/k B T . The latter would be valid in a single well where the time to explore the full phase space available doesn't depend on the system's energy. But in the double well potential case, the computation of P (E) has to take into account the energy dependency of the motion period [T (E)]. In the next paragraph we derive P (E).

The first step consists in going back to the canonic motion variables of position and momentum, that are distributed at all time from a Boltzmann distribution:

P (x, p) = 1 Z e -βH(x,p) , (10)
with Z = dx dp e -βH(x,p) .

Since time is a natural variable for this problem, we operate a change of variables from (x, p) to (t, E): for any given position and momentum, we can define a unique time and energy, and vice-versa. We consider here that as the system is periodic, the time range is restricted from -1 2 T (E) to + 1 2 T (E) for a given total energy E. The above change is a canonical transformation for the Hamiltonian since the Jacobian determinant is 1:

dx dp = ∂x ∂E ∂p ∂t - ∂x ∂t ∂p ∂E dt dE = dt dE, (12)
The change of variable is therefore straightforward in the expression of the probability:

P (t, E) = 1 Z e -βE , (13)
where P (t, E) is defined only for -1 2 T (E) < t < 1 2 T (E) (0 otherwise), as the motion is periodic. The probability of having an energy E is deduced by integrating on time, leading to:

P (E) = 1 Z e -βE T (E), (14)
with Z = ∞ 0 dE e -βE T (E). (15)
Now we have all we need to compute the crossing rate of the barrier. For a given energy E, the escape rate is 0 if E < B. On the contrary, for E > B the crossing occurs twice every T (E). The mean transition rate in the double potential, that is the rate at which the system crosses the barrier, regardless of the direction is therefore:

Γ = ∞ B dE 2 T (E) P (E) (16) = 2e -βB β ∞ 0 dE e -βE T (E) , (17)
which taking into account Eq. ( 9) is equivalent to Eq. ( 2).

If the barrier B is high (with respect to k B T = 1/β), then the denominator to Eq. ( 17) can be seen as the average time to explore the bottom of the wells, T (0) = 2T 0 . We get in this case a very simple expression matching Kramers' expression:

Γ(B k B T ) = 1 T 0 e -βB . (18)
FIG. 1. switching rate on a bi-quadratic well U

(x) = 1 2 k(|x| - x1) 2 . The barrier height is B = 1 2 kx 2 1 .
The period in a single harmonic well is T0. At low barrier height, Kramers' simple expression is a factor 2 below the result from Eq. ( 2). The latter perfectly matches the switching rate extracted from a numerical simulation of a Langevin dynamics in the potential U (x), using a quality factor Q = 10.

In Fig. 1, we apply Eq. ( 2) and compare the predicted escape rate to the one measured on a numerical simulation of a Langevin dynamics in such a potential : the agreement is excellent. The Kramers approximation holds only in the limit of large barriers.


P Navet

C Changenet

F Ville

D Ghribi

J Cavoret

Thermal modelling of the FZG test rig: application to starved lubrication conditions

Keywords: FZG test rig, starvation condition, gear power loss, gear bulk temperature, thermal analysis, thermal network method

During their lifetime, gears may undergo starved lubrication operating conditions. Starvation condition has an impact on friction between gears that leads to modify gearbox efficiency. In this paper, lubricant starvation tests on an FZG test rig adapted to this condition are presented.

Gears were equipped with embedded thermocouple sensor to measure their operating bulk temperature. An experimental method to isolate test box power loss of FZG machine is proposed. A thermal model of this test rig draws up the power loss distribution into the test box for a nominal condition of lubrication. At last, a numerical method based on experimental results is proposed to estimate gear power loss variation during lubricant starvation.

Introduction

During their lifetime, gears may undergo starved lubrication operating conditions. Starvation condition has an impact on friction between gears that leads to modify gearbox efficiency.

Thermal characterization of gears is usually based on the equilibrium between power loss sources and heat dissipation [START_REF] Höhn | Thermal rating of drives: balance between power loss and heat dissipation[END_REF]. A significant number of experimental studies on gear efficiency and thermal behaviour have been performed on back-to-back gear test rig. Impact of the lubrication method on efficiency and temperature of spur gears is studied by comparing dip and spray lubrication [START_REF] Andersson | Efficiency and temperature of spur gears using spray lubrication compared to dip lubrication[END_REF] 5) is used in a global scuffing criteria [START_REF] Castro | Global and local analysis of gear scuffing tests using a mixed film lubrication model[END_REF]. A thermal model of the FZG test rig has been built to study gear power loss [START_REF] Matins | Power loss performance of high pressure nitrided steel gears[END_REF][START_REF] Martins | FZG coefficient in FZG gears lubricated with industrial gear oils: Biodegradable ester vs. mineral oil[END_REF]. In those studies, gear temperatures are considered equal to oil sump one. However, to estimate more accurately the gearbox efficiency, local temperature variation has to be modelled [START_REF] Changenet | Power loss predictions in geared transmissions using thermal networksapplications to a six-speed manual gearbox[END_REF]. A thermal modelling of the FZG test rig built with the thermal network method is able to determine the bulk temperature of gears for oil bath lubrication condition [START_REF] De Gevigney | Thermal modelling of a back-to-back gearbox test machine: Application to the FZG test rig[END_REF]. Thus, a power loss distribution for lubricated conditions is estimated. However, none of these studies focused on starved lubrication condition. Most of studies concerning lubricant starvation focus on point and elliptical contacts that are more representatives of contacts within rolling element bearings.

Chevalier et al. [START_REF] Chevalier | Film Thickness in Starved EHL Point Contacts[END_REF] characterize numerically the degree of starvation of EHL point contacts with the amount of oil present on the surfaces at the contact inlet. This formalism was extended to elliptical contacts [START_REF] Damiens | Starved Lubrication of Elliptical EHD Contacts[END_REF]. Impact of starvation on contact fatigue was studied by Querlioz et al. [START_REF] Querlioz | Experimental investigations on the contact fatigue life under starved conditions[END_REF].

The aim of this paper is to study the impact of starved lubrication conditions on gear power loss. Firstly, the FZG test rig and the operating conditions are described. In the second part, an experimental method to isolate power loss of the test box is proposed. In the third part, a starvation test performed on a FZG test rig adapted to these conditions is presented. Then, a thermal model of the test rig is built in order to draw up the power loss distribution into the test box for a nominal condition of lubrication. At last, gear power loss variations during starvation are estimated using a numerical method based on experimental results and the thermal model of the FZG test rig adapted to starved condition.

FZG test rig description

Figure 1 presents the FZG test rig machine. This machine is a back-to-back test rig composed of a test box #1 and a slave box #2 both assembled with cylindrical gears. A clutch #3 permits to apply the desired load in the closed loop by torsion of the pinion shaft with a known mass suspended at a lever arm. Thanks to the closed mechanical loop, the motor #4 only ensure kinematics of the system and give mechanical power to compensate power losses. A torque transducer on motor shaft allows to measure total torque loss (Table 1). This FZG test rig is equipped with thermocouple sensors (K type, absolute accuracy of 1.5°C) on fixed parts: housings, oil baths and external ring of rolling element bearings (REB). FZG C30 gears (Table 2) are used for both boxes. Four 6406 bearings are mounted in each gearbox (Table 3). Those are lubricated with an oil bath of a gearbox oil (Table 4, IGOL HYPOID 85W-90) at the level H1 represented on Figure 2.

In this study, one operating condition (Table 5) is selected. The torque applied on pinion is 60.8 Nm (load stage 4 according to standard FZG classification). The rotational speed of the motor is 2000 rpm (3000 rpm at pinion shaft). All the presented tests were started at ambient temperature and no thermal regulation of oil is performed. Thus, each temperature variation comes from power loss generation of the system.

Repeatability tests were performed to verify the measurement reliability of the loss torque measurement (Figure 3). A maximum difference of 0.3 N.m and an average difference of 0.1 N.m on loss torque are observed. The loss torque measurements are therefore repeatable.

Experimental method to estimate the test box power loss

In order to study more specifically the test box, it is necessary to isolate its power loss. One remembers that only the total torque loss of the system can be measured with the torque transducer on the motor shaft. To isolate the test box power loss, the slave box has to be characterized. The two gearboxes of the machine are mounted similarly. The ventilation of the motor is blocked by a wood plate in order to ensure a better thermal symmetry of the machine. Symmetry is verified by comparing the temperature profile of the two oil baths during a test under symmetrical conditions (Figure 4). A maximal difference of 0.3 °C is observed between the two oil baths which allows to consider the machine as symmetrical.

Thus, power losses are evenly distributed between the two boxes that permits to estimate the slave box power loss as the half of total power loss. Figure 5 shows the profile of the slave box torque loss depending on its oil bath temperature. Torque loss can be accurately approximated by a third-degree polynomial function. In this study, the slave box configuration is fixed and so this characterization can be applied on tests for which the test box is mounted with a different configuration. The torque loss of the test box is estimated by subtracting the slave box loss torque from the total torque loss. This method will be used later to isolate the test box power loss for an asymmetrical configuration of the FZG adapted to starvation tests.

Tests on FZG test rig adapted to lubricant starvation

In a context of oil bath submerging largely gears (level H1, Figure 2), gear bulk and oil sump temperatures are close [START_REF] Changenet | Power loss predictions in geared transmissions using thermal networksapplications to a six-speed manual gearbox[END_REF]. However, for marginal lubrication conditions like low oil bath level (level H2, Figure 2) or when starvation occurs, temperature elevation of gears has to be taken into account to estimate more accurately their associated power loss. In order to measure gear bulk temperatures, thermocouple sensors (K type) were stuck directly on the tooth side of the pinion and of the wheel (Figure 6). Hollow shafts were designed in order to route wires to the data loggers. Those are directly fixed to the output of the two shafts (Figure 7). Figure 8 gives embedded gears bulk temperatures and oil bath temperature. Pinion operating temperature is almost 20 °C higher than the oil bath one. This thermal elevation is explained by the low oil level which thermally insulates the pinion from oil cooling.

For starvation tests, the test box was assembled with FZG A20 gears (Table 2). Those gears are largely used in works focusing on scuffing. In fact, their profile shift modifications were chosen to induce an exacerbated sliding speed at the recess. Four greased and sealed REB (Table 3, 6306-2RS1) were mounted within the test box. The grease will ensure lubrication of REB during starved conditions.

The test box is lubricated with a base oil (Table 5; Nycobase 5750) at the low level H2 (Figure 2). This level will prevent washing of the REB grease during lubricated phase. Base oil was chosen in order to focus on the impact of rheological parameters of oil and not to observe the chemical influence of additive package on friction like tribofilm formation [START_REF] Meheux | Influence of slide-to-roll ratio on tribofilm generation[END_REF].

Figure 8 shows the repeatability of embedded gears bulk temperatures and the oil bath temperature during the lubricated phase. The maximal difference of the temperature observed is 1 °C and the average difference is 0.2 °C. Temperature measurements are therefore repeatable.

The procedure performed for a starvation test is:

(1) To perform a test under lubricated regime (operating condition 1) until loss torque is considered almost constant (torque loss measurement does not vary over a range of 0.1

N.m during 5 minutes).

(2) To drain the oil bath out of the test box while temperature of each mechanical part drops to ambient temperature.

(3) To perform a starvation test on residual oil. The test has to be stopped in case of critical increase of torque that could damage the transducer.

Figure 9 shows differences between gear bulk temperatures during lubricated and starved test.

A thermal kinetics acceleration is observed without oil. In fact, an increase of more than 60 °C on bulk temperatures are measured after 3 minutes (more than 140 %). This increase is due to the fact that no thermal energy can be collected by oil. Figure 10 presents the difference between the total power loss during lubricated and starved condition. Comparing to nominal lubricated conditions, an increase of 800 W after 3 minutes on the total power loss (60 %) is observed. Focussing on the test box power loss using the experimental method described in part 2, the increase of the power loss described above is observable (Figure 11). It confirms that it comes from the test box. Because REB are lubricated by their grease, one can suggest that the increase of power loss in the system may come from an increase of gear tooth friction.

FZG thermal modelling for oil bath lubrication condition

In order to analyse the dissipation sources which may explain the torque evolution presented in Figure 11, a thermal model of the test gearbox was developed. Thermal modelling of transmission systems was led by Changenet et al. [START_REF] Changenet | Power loss predictions in geared transmissions using thermal networksapplications to a six-speed manual gearbox[END_REF] using the thermal network method. This method consists in performing a discretization of the transmission system into isothermal elements at mechanical part scale (bearings, gears, shafts, oil, etc). These elements are connected by thermal resistances defined by an analogy of Ohm's law.

Durand de Gevigney et al. [START_REF] De Gevigney | Thermal modelling of a back-to-back gearbox test machine: Application to the FZG test rig[END_REF] proposed a thermal modelling of the FZG test rig for oil bath lubrication condition. On Figure 12, a representation of the thermal network is given showing elements, thermal resistances and heat sources of the model. In order to evaluate heat exchanges, the model uses four types of thermal resistance:

(1) Convection and radiation are respectively based on Newton's and Stefan-Boltzmann's laws. Housings are assimilated to a plate assembly. This simplification permits to apply classic laws of plates submitted to normal and tangential air flux (9).

(2) Convection with oil is modelled by laws derived from classical formulation of surfaces submitted to oil flow [START_REF] Chevalier | Film Thickness in Starved EHL Point Contacts[END_REF]. Heat transfers between the oil bath and internal surfaces of housings are modelled using the forced convection on flat plates. Heat transfer between gears and oil is modelled using rotating disk law.

(3) Conduction is calculated with classic formulations of heat transfer by conduction.

Bearing rings, shafts and gears are considered as cylindrical elements in order to apply radial conduction law [START_REF] Changenet | Power loss predictions in geared transmissions using thermal networksapplications to a six-speed manual gearbox[END_REF]. ( 4) Striction thermal resistance is based on Blok's work [START_REF] Neurouth | Influence of Rolling Element Bearing Modelling on the Predicted Thermal Behavior of the FZG Test Rig[END_REF]. Heat generated due to friction in gear and REBs contacts are localized on a small area compared to their dimensions. This resistance allows to distribute physically the heat source towards the two elements without assumption on a possible partition coefficient [START_REF] Changenet | Power loss predictions in geared transmissions using thermal networksapplications to a six-speed manual gearbox[END_REF].

In [START_REF] De Gevigney | Thermal modelling of a back-to-back gearbox test machine: Application to the FZG test rig[END_REF], REBs are supposed isothermal and are modelled with a single node. Neurouth et al. [START_REF] Neurouth | Influence of Rolling Element Bearing Modelling on the Predicted Thermal Behavior of the FZG Test Rig[END_REF] developed a complex bearing model discretizing a bearing in 10 nodes within the FZG thermal modelling. They highlighted the difference of temperature between inner and outer rings. This temperature gradient in the bearing influences gear temperature. To model the temperature gradient a simplified approach (Figure 12) is implemented following Niel et al. [START_REF] Niel | Thermomecanical study of high speed rolling element bearing: A simplified approach[END_REF] modelling strategy. Thus, each REB is discretized by four nodes: external and internal ring, rolling elements, and grease (Figure 12).

For lubricated condition, gear power loss is applied to the mesh element of the thermal network given in Figure 12. The model given by the standard (ISO 14179-2) ( 18) is used:

- : mechanical power (W)
-: form factor depending on gear geometry (-) -: Mean friction coefficient along line of action (-) [START_REF] Höhn | Thermal rating of drives: balance between power loss and heat dissipation[END_REF] The friction coefficient along the action line is calculated with the formulation of Castro and Seabra (4). This one is based on Winter and Michaelis equation ( 5) and was derived from experimental FZG C type gear tests [START_REF] Winter | Scoring load capacity of gears lubricated with, epoils[END_REF]. The REB power losses (6306-2RS1) are applied to the central element of each REB (Figure 12). This power loss is modelled with SKF model (19) formalized as: [START_REF] Höhn | Frictional behaviour of synthetic gear lubricants[END_REF] Where and are loss torque respectively induced by rolling and sliding phenomenon. is loss torque by friction between seals and rings. is the loss torque due to drag of rolling elements moving throughout oil. The power loss decrease observed in Figure 13 is strongly linked to the decrease of oil viscosity during temperature rise. Operating temperatures of the system are themselves driven by heat generations. Numerical power loss (Figure 13) and temperatures (Figure 14) are representative of their experimental counterparts. In these figures, marks represent the experimental findings whereas lines are associated with the numerical ones. This means the strong coupling between heat generations and temperatures is accurately performed. This coupling will be all the more important for the modelling of starvation conditions that implies severe temperature variations (Figure 9).

Numerical method to estimate power loss variation during a starvation test

Firstly, the thermal modelling of the FZG is modified to consider starved conditions. In fact, these conditions imply modifications on heat exchanges within the system. The oil bath node is replaced with the internal air. The gear rotation generates internal air flow responsible for forced convection with housing internal surfaces and gear flanks. The heat transfer coefficient applied on housing surfaces is estimated using the following formulation of the Nusselt number corresponding to the case of flows over flat plates [START_REF] Schlichting | Boundary Layer Theory[END_REF]:

For [7]
Else [START_REF] Martins | FZG coefficient in FZG gears lubricated with industrial gear oils: Biodegradable ester vs. mineral oil[END_REF] The heat transfer coefficient applied on gear flanks is estimated using the following formulation of the Nusselt number corresponding to the case of a disk rotating in a motionless fluid (10): for dry air and assumed constant versus temperature REBs are greased and sealed to prevent starvation. Power loss in seals is considered as constant. Without oil bath, the churning loss is suppressed and the windage power loss is neglected because of the small dimensions and low operating speeds. Figure 15 is a diagram describing a numerical method to estimate the variation of gear power losses during a starvation test.

is the power loss applied to gear. is the gear power losses under lubricated condition used as initial power loss value. In this case, a mean friction coefficient of 0.03 was calculated with equation 3 that corresponds to a power loss of 160 W (equation 1). is the experimental power loss of the test box determined with the experimental method presented in part 4 (Figure 11) . is the sum of all power loss sources calculated in the thermal network. ε is the precision desired on the solution (here ε W .

As expected, the numerical total power loss of the test box is accurately determined by the method for 4 iterations (Figure 16). The main hypothesis of the method is that only gear power losses are impacted by starvation.

To verify this assumption, the power loss increase due to starvation is numerically assigned to seals. Gear power loss is modelled using equation [START_REF] Höhn | Thermal rating of drives: balance between power loss and heat dissipation[END_REF] by considering it was not affected by starvation conditions. Extra seal power loss is obtained by adapting the method described in Figure 16. At initialisation step, seal power loss value is calculated using equation [START_REF] Winter | Scoring load capacity of gears lubricated with, epoils[END_REF]. For each iteration, ΔP is added to until experimental power loss is reached. This calculated power loss is shared between the four seals of the test box considering account their rotational speed. Indeed, according to equation [START_REF] Winter | Scoring load capacity of gears lubricated with, epoils[END_REF], seals associated to the secondary Accepted Manuscript shaft dissipate 1.5 times more than those mounted on the primary one. Figure 18 highlights differences between numerical and experimental temperatures for this modelling. It confirms that the power loss increase during starvation occurs for gears.

Figure 19 presents gear power loss variations during starvation tests. A severe increase more than 900 W that represents a rise of 575 % concerning gear power loss is observed after 3 min 20 s of starvation test. However, focusing on experimental power loss values, the variation observed between lubricated and starved condition is 800 W (part 4, Figure 11). To explain this difference, Figure 20 draws up the variation power loss source distribution for lubricated and starved condition tests. For the lubricated condition, a low variation is observed during 3 minutes because the thermal kinetics is very slow. However, under starved conditions, thermal kinetics is quick because of the severe increase of gear tooth friction losses. A decrease of 100 W is observed, by adding variations of bearing and churning losses comparing a lubricated and a starved test. These are actually dissipated by gears. This highlights the importance of a model including a strong coupling between temperatures and heat generations.

Conclusion and perspective

The aim of this paper was to study the impact of starved lubrication conditions on gear power loss. Firstly, the FZG test rig and the operating conditions were described. In the second part, an experimental method to isolate power loss of the test box was proposed. The torque loss of the slave box was characterized from test performed with the FZG test rig for a symmetrical configuration. This torque is used to isolate the test box power loss during starvation tests. To perform these last ones, modifications of the FZG test rig and a specific procedure were presented. The main conclusions linked to the tests are:

-Thermal kinetics is severely accelerated under starved conditions. An increase of more than 60 °C on gear bulk temperatures is measured after 3 minutes which corresponds to an increase of more than 190%.

-Power losses of the system increase significantly during starvation. An increase of 800 W is observed after 3 minutes that represents an increase of 60 % in comparison with nominal lubricated conditions.

To analyze the dissipation sources with this power loss evolution, a thermal model of the test rig is proposed. This model, based on thermal network method, allows a strong coupling between power losses and temperatures. For a starvation context involving severe temperature variations, this coupling is essential to estimate accurately gear power loss variations. These gear power loss increases are estimated using a semi-inverse method with the previous thermal model adapted to starved conditions. Gear power loss estimated thanks to the numerical model highlights a severe rise of 900 W during the starvation that represents an increase of 575 %

In sight, determination of gear power loss variations during starvation allows to go further in the field of starvation conditions. In fact, works oriented to starved contact behaviour may be undertaken. For instance, film thickness reduction analyses may be performed based on an expansion of the semi-inverse method. This analysis could lead to predict scuffing under starved conditions.

Symbol

  Fernandes et al. (3) modelled power loss of the FZG test rig under steady-state operating conditions to estimate experimentally the friction coefficient between gears. Another formulation of friction coefficient derived from FZG tests (4) based on Michaelis et al. formulation (

  along line of action (GPa) -: Rolling speed along line of action (m.s-1 ) -: Arithmetic mean height of roughness (μm) B: Beginning, end of line of action -N(x) : Normal force along line of action (N) -: Sliding speed along line of action (m.s-1 ) [3]

  Seal power losses are applied to the shaft elements of the thermal network. They are estimated considering Freudenberg formulation (20): speed (rpm) Churning power losses are applied to the oil bath node (Figure 12) during oil bath lubrication using Changenet et al. model (9): -: Oil density (kg.m -3 ) -: Submerged surface (m 2 ) -: Pitch radius (m) -: Rotational speed (rad.s -1 ) -: Dimensionless drag coefficient defined by LePrince et al. (21) to take oil aeration into account [6]

Figure 17

 17 presents correlation between experimental and numerical temperatures. The thermal modelling is able to predict the global variations of temperature during the starvation test. It should be noticed a severe increase of the pinion bulk temperature between 2.5 and 3 min. Further experimental analyses show scuffing. This phenomenon induced local and transient thermal elevations. This specific model cannot predict it as gears are assumed to be isothermal.

  of REB (N.mm) Torque loss due to sliding for REB (N.mm) Torque loss due to rolling for REB (N.mm) Torque loss due to seals for REB (N.mm) Torque loss due to drag for REB (coefficient along the meshing line (-) Oil density (kg.m -3 )

Figure 2 .Figure 3 .

 23 Figure 2. Oil levels

Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 .Figure 14 .Figure 15 .

 456789101112131415 Figure 4. Oil bath temperatures; operating condition 1

Figure 16 .Figure 17 .Figure 18 .Figure 19 .Figure 20 .

 1617181920 Figure 16. Numerical and experimental total power loss of the test box

Table 1 .

 1 Torque transducer characteristics

	Torque	Characteristics
	transducer
		Nominal Torque	Voltage	Sensitivity on	Accuracy	Absolute
	T22	(N.m)	output (V)	voltage (V)	class	accuracy (N.m)
		50	± 5	± 0.2	0.5	0.25

Table 2 .

 2 Parameters of gears

	Accepted Manuscript

Table 3 .

 3 Parameters of bearings

	Oil	Parameters
		Density at 40 °C (kg.m -3 )	(mm 2 /s)	(mm 2 /s)
	IGOL HYPOID 85W-90	900	187	17.1
	NYCOBASE 5750		24	5

Table 4 .

 4 Oil parameters

	Gears	Pinion	Maximal normal	Maximal hertzian	Pinion Speed	Input power
	Parameters	torque (Nm)	load (N)	pressure (GPa)	(rpm)	(kW)
	Operating	60.8	1800	0.65	3000	19
	condition 1

Table 5 .

 5 Operating conditions Figure 1. Drawing of a FZG test rig

Acknowledgments

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article. The authors would like to thank the ANRT (CIFRE n°2018/0789).

Nomenclature


F Cavacece

L Frache

D Tonazzi

N Bouscharain

D Philippon

G Le Jeune

Y Maheo

F Massi

Roller bearing under high loaded oscillations: life evolution and accommodation mechanisms

High loaded oscillating bearings are used on several industrial fields such as robotics, wind power, aeronautics and aerospace. These bearings are subjected to extremely high local contact pressures and relatively low oscillatory speed. In this work, endurance tests on commercial single row spherical roller bearings enable to propose a degradation scenario for the bearing lifecycle. Topographical analyses carried out at each phase of the bearing life allowed to understand the degradation evolution at the interface between the inner-ring raceway and rolling elements. The analysis of the experimental results, with the support of numerical stress analysis, led to the understanding of the involved accommodation mechanisms.

Then, a comparison with the wear processes in deep-groove ball bearings is provided.

Introduction

Rolling contacts between the rolling elements and raceways guarantee to support load while permitting, with the lowest possible friction losses, the relative rotation between two solids of the mechanical system [START_REF] Harris | Rolling bearing analysis[END_REF] [2] [START_REF] Sadeghi | A Review of Rolling Contact fatigue[END_REF]. Nowadays, bearings are expected to support extreme working conditions. In fact, the unceasing increase of both design optimization and system power transmitted involve a reduction of the contact surfaces between contacting pairs and extremal local contact solicitations [START_REF] Houpert | A Uniform Analytical Approach for Ball and Roller Bearings Calculations[END_REF] [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF] [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF].

The analysis is here focused on high loaded oscillating bearings. These working conditions are specific to aeronautical, naval and aerospace applications. High load leads to extreme contact pressures [START_REF] Johns | Roller bearings under radial and eccentric loads[END_REF] between rollers and raceways, mainly of the inner-ring, while the slow oscillating motion causes repetitive mechanical solicitations on surfaces [START_REF] Massi | Degradation of high loaded oscillating bearings: Numerical analysis and comparison with experimental observations[END_REF] combined with a variable way and speed of rotation.

Due to these operating conditions, the circulation of a lubricant and the establishment of a full elastohydrodynamic film d. A grease lubrication has to be employed; the thickener hold the oil near to the contact and then releases it under specifics conditions like high pressure, even in very low speed regimes [START_REF] Booser | Minimum oil requirements of ball bearings[END_REF] [START_REF] Baker | Grease bleeding-a factor in ball bearing performance[END_REF] . It is supposed to assure a better lubrication than oil [START_REF] Lugt | A Review on Grease Lubrication in Rolling Bearings[END_REF] [12] [START_REF] Cann | An analysis of the mechanisms of grease lubrication in rolling element bearings[END_REF], at least for full rotation motion.

In literature, models have been proposed to describe continuous rolling bearing damage evolution [START_REF] El-Thalji | A descriptive model of wear evolution in rolling bearings[END_REF] [15] [16] [17] [18] [START_REF] Williams | Wear modelling: analytical, computational and mapping: a contiuum mechanics approach[END_REF]. During bearing lifetime, the damage evolves and there is a transition from a commercially acceptable rate to an unacceptable one. Voskamp [START_REF] Voskamp | Material response to rolling contact loading[END_REF], analyzing the topography of the rolling contact subsurface, has identified three fundamental steps in bearing damage evolution: running-in,

steady-state and instability. El-Thalji [START_REF] El-Thalji | A descriptive model of wear evolution in rolling bearings[END_REF] proposed a more detailed model of bearing damage scenario, which consists in five phases: running-in, steady-state, damage initiation, damage evolution and damage growth.

However, these models all the operative conditions and cannot fully account for the interaction and the competition of the different mechanisms at the origin of the damage [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF].

On the other hand, oscillating and discontinuous motion bearings have been not much investigated in the literature and only few works are addressed to these working conditions. Some models have been derived

for oscillating motions to estimate the rolling bearing lifetime [START_REF] Harris | Rolling Bearing Analysis[END_REF] [START_REF] Houpert | Bearing life calculation in oscillatory applications[END_REF]. Berthier et al. [START_REF] Berthier | Wear mechanism in oscillating bearings[END_REF], performing an experimental analysis on oscillating journal bearings, highlighted the complexity of the degradation scenarios, the role of the third body flows and the existence of more than one wear mechanism. Friction and wear are largely affected by the rheology of the third-body at the contact [23] [24].

Recently, some studies on high loaded oscillating ball bearings have been carried out [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF] [6] [8] [22] [25].

In [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF] , the evolution of greased and ungreased contact surfaces over the bearing whole lifetime has been investigated. These studies demonstrate the role of the grease that increases considerably the bearing lifetime, protecting the bearing contacting surfaces from further superficial damage. It has been established

that the failure of that ball bearings is caused by the accumulation of plastic deformations in the subsurface of the most loaded balls, while the raceways is preserved by the third body. In [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF], a 3D finite element model has succeeded in evaluating the stress and strain distributions at the contact interface in deepgroove ball bearings. Its simulations confirmed the experimental results obtained in [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF] : a sub-surface stress and strain zone in the ball has been observed, and a plasticization of the inner ring raceway as well.

In [START_REF] Ghezzi | Damage evolution and contact surfaces analysis of high-loaded oscillating hybrid bearings[END_REF], under the same boundary conditions, experimental tests on hybrid deep-groove ball bearing has been achieved. The use of a hard-ceramic balls instead of softer steel balls enables to enhance the bearing lifetime and modifies the failure mode, caused now by the destruction of the inner ring raceway surface by the ceramic balls. The comparison between hybrid and steel-steel bearings has shown how bearing materials can modify accommodation mechanisms within bearing lifecycle and so its lifetime.

In this context, this work focuses on high-loaded oscillating roller bearings. The aim is to identify the accommodation mechanisms involved during the roller bearing whole life, in order to build the degradation scenarios. A comparison between the results of this work and previous ones is then performed, in order to analyze the tribological response of different technologies operating under the same oscillating high loaded conditions.

First, the experimental set-up and the methodology is presented; then, the results of the experimental campaign are presented, analyzing the curves of the degradation evolution and the transformation of the contact topographies. Finally, the comparison between deep-groove ball bearings and roller bearings is discussed.

Roller bearing damage evolution

In order to recover the evolution of the bearing life, endurance tests have been performed on a dedicated test bench, and then interrupted tests has been done on each identified phases of the bearing life. In order to compare the wear evolution between ball bearings and roller bearings, the experimental protocol and the test parameters have been kept constant, with respect to previous works [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF] [6] [8] [22] [25].

Here, the experimental tests have been carried out on single row spherical roller bearings, with a pitch diameter of 17.1 mm and a roller of (maximum) diameter of 4.7 mm and length of 5.5 mm (Figure 1-a). The rings and rollers are both made of 100Cr6 steel. The bearing is 100% filled with an aeronautical grease Aeroshell 33. The set-up is conceived to apply high radial load on the bearing outer ring and oscillatory motion on the inner ring, which rotates with the shaft. The oscillatory motion is assured by a crank-rod system, actuated by an electrical engine. The radial load is applied using dead weights and a lever arm. A more detailed description of test bench can be found in [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF]. Applied on the inner ring, the oscillation motion of an amplitude of ± 20° avoids overlapping of the rolling traces and then simplifies the interpretation of the surface observations. In order to accelerate bearing ageing without overheating of the contact, the oscillation frequency is set to 5 Hz.

On the bearing outer ring, the radial load is applied by dead weights and maintained at a constant value of 10 800 N. This load is set in order to obtain a maximum contact pressure of 4 GPa [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF]. The load is supported by the roller situated at the top of the tested bearing (Figure 2,Table 1). The number of rollers that support the radial load depends on the bearing geometry. During the oscillating cycle, the magnitude of the radial load varies between a minimum and a maximum value as a function of the rolling element instantaneous position. Figure 2 shows the theoretical static radial load distribution between the rolling elements. In addition, the name of the most loaded rollers and the respective rolling marks are indicated. In Figure 2, Ri and Tj indicate respectively the i-th Roller and j-th rolling Trace. Table 2 summarizes the test parameters. Three main sensors are used to follow bearing evolutions during the whole test. A proximity sensor ( in Figure 1) model TQ-401 (VibroMeter® Meggitt SA, Fribourg, Switzerland) to measure the relative displacement between the inner and the outer rings (RD). A bi-axial force transducer (model FX2-2, SIXAXES SA, Argenteuil, France) to record the resistant torque (RT) and the normal load. And then a thermocouple touching the outer ring surface to prevent overheating.

A set of endurance test was performed on roller bearings and, among them, two tests are presented here, corresponding to the two possible life evolutions recovered during the tests. Figure 3 shows the resistant torque (RT) and the relative displacement (RD) measured during two different tests until the set limit of the resistant torque is reached (bearing end of life). The resistant torque is a result of the local frictional interaction, the rolling element kinematics and the surface evolution. The local friction coefficient of the greased contact, for new surfaces, has been measured on a dedicated tribometer and has been estimated at about 0.1. The curves have been normalized along the x-axis with respect to the duration of each single test. The relative displacement between the two rings is measured along the radial (loading) direction, and an increase of its value (reduction of the gap between the rings) corresponds to a degradation of the rolling interfaces. The presence of a running-in phase;

The presence of a first plateau, i.e. a steady-state phase, where the value of the RT remains constant;

The presence of a relative fast increase of the RT in the final part of the bearing life, which leads to its end.

On the contrary, a difference has been identified in the possible presence of a second steady-state phase, after the first plateau and just before the fast final increase of the torque, point 5 in Figure 4. In Figure 3, the different ageing step can be identified. At the beginning of the test, from point 0 to point 1 (Figure 4), the RT increases slightly before decreases again. This transitory represents the running-in phase and it lasts on average approximately 1,000 cycles. After, from point 1 to point 3 (Figure 4) the signal stabilizes, there is the main steady-state phase where the RT amplitude remains constant. This stationary phase is interrupted at the point 3 (Figure 4), when the amplitude starts to increase in a faster way. At the end of this first plateau and after the increase in the RT, two possible scenarios occur. In one case, a second steady-state is present, point 5 (Figure 4), followed by a further increase leading to the final damage. In the other case the RT continues to increase quickly to the end of the bearing life, without the second plateau.

In the final phase of both cases, a sudden decrease and subsequent increase in the signal is present.

Starting from this scheme, interrupted tests have been carried out in order to understand the accommodation mechanisms involved during each step. The following phases of the degradation scenario have been analyzed: running-in, from point 0 to point 1 (Figure 4); first steady-state, from point 1 to point 3 (Figure 4); damage initiation, from point 3 to point 4 (Figure 4); second steady-state, around the point 5 (Figure 4); damage growth and bearing end of life, from point 5 to point 7 or point 4 to point 6 (Figure 4).

At each point, loaded rollers and inner ring raceway have been analyzed with a SEM, Scanning Electron Microscope (model FP 2014/23 QUANTA 600, FEI, Brno, Czech Republic).

Surface topography evolution

In this section the evolution of the surface topography in the key points of the curves, shown in Figure 3, will be presented. Surface topography analysis has been analyzed using a SEM and because stresses at the outer-raceway contact are lower (due to the concave contact), the contact of the roller with the inner-ring will be focused.

The analysis of surface topography will be supported by the results of numerical simulations. A 3D finite element model has been developed and solved using a commercial code. Because of the high computational effort due to the contact non-linearities, the analysis is focused on the most loaded roller, taking in account the inner and outer ring sectors. Moreover, fine meshing, of about 10 µm size, has been developed at the contact surfaces, ensuring a reliable distribution of the contact stress and strain. Based on an elastic-plastic material behavior, the numerical simulation consists in a loading phase followed by the simulation of a cycle of oscillation of amplitude ±10° and characterized by a frictional coefficient defined experimentally and equal to 0.1. A static analysis has been performed with a ramp of the loading force on the outer ring and the oscillation imposed at the inner surface of the inner ring, as imposed into the experiments by the oscillation of the shaft. The reader is referred to reference [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF] for the details on the numerical approach.

Initial surface topography

The point 0 (Figure 4) corresponds to a new bearing. Figure 5 presents the topographies of the inner-ring raceway and the rollers. They are the reference topography of the contact surfaces, needed for the understanding of the superficial evolutions. The roller and the inner ring raceway present a different initial topography, due to different machine processes. As shown in Figure 5, the raceway surface presents regular and parallel machining grooves while the roller presents a smoother surface with several local superficial defects.

Accepted Manuscript

Running-In

From point 0 to point 1 (Figure 4), the running-in represents the initial part of the bearing evolution scenario. From the RT signal, we can distinguish the running-in phase in two steps:

Macroscopic running-in. The inner ring track shows macroscopic plastic deformation (Figure 6a, Zone 2) and the contact between raceway and rollers becomes more conform, increasing the bearing resistant torque and producing a decrease of the ring distance. The contact pressure, calculated by the 3D elasto-plastic model, ranges between 3 and 4.5 GPa during the bearing oscillation, along the highest loaded contact trace between roller and race.

Microscopic running-in. In this phase the interaction between the third and the first bodies lead to the plasticization of the machining grooves [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF] [START_REF] Ghezzi | Damage evolution and contact surfaces analysis of high-loaded oscillating hybrid bearings[END_REF]. A smoother area starts to appear (Figure 6a, Zone 1). This stage of running-in provides a slight decrease in the RT.

Considering the most loaded roller ("R1"), the analysis of surface topography shows the presence of rolling (local sticking) and sliding contact regions are well recognizable. In Figure 6b, numerical results concerning the state of the contact and the microscope observations are superimposed. The extension and location of these areas are confirmed. At the end of the running-in phase, the inner-ring and the most loaded roller present still signs of machining grooves.

Steady state (central phase)

During the first phase of steady state period from point 1 to point 2 (Figure 4) where the RT and RD curves are stable, the contact surfaces stay stable without significant modification on the macroscopic topography (Figure 7 and8). This explains the stability of the RT and RD curves. During that phase, at specific locations, the morphology of contact surfaces becomes smoother (Figure 7) which might imply the formation of a plasticized layer recovering machining grooves. During the first part of steady-state phase the asperities of the most loaded traces of the inner-ring interact with the roller and the grease, producing in some zones (Figure 7, Zone 1&2) a smoother topography. These smooth plasticized areas might result from the interaction of the inner ring, rollers and the grease. According to previous work on deep groove ball bearing [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF]and hybrid bearings [START_REF] Ghezzi | Damage evolution and contact surfaces analysis of high-loaded oscillating hybrid bearings[END_REF], this smoothed area could allow a lower rolling resistance torque and protect the contact surface from further degradation, leading to a long steady state phase.

The most loaded roller has an unchanged topography compared to the running-in phase.

Steady state (final phase)

During the second part of the steady state period, from point 2 to point 3 (Figure 4), the smoothed areas [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF] [25] are developed in a striking way on the areas between the traces, where the higher sliding ratio occurs with a local source and accumulation of wear particles (Figure 8, Zone 1).

On the contrary, the most loaded roller, where lower sliding ratio and lower particle accumulation occurs, has an unchanged topography compared to the running-in phase (Figure 8, Zone 2).

Damage initiation

At the end of the steady-state phase, after the point 4 (Figure 4), macroscopic damage wear process starts, and the RT increase rapidly. The observations of the inner-ring (Figure 9) shows the detachment of macroscopic slices of material (Zone 2), just after the increase of the RT (point 4, Figure 4). In this case, as shown in Figure 8, the detachment of material affects a large part of the raceway track. Traces 2 and 4, adjacent to track 1, do not show any damage, but only macroscopic plastic deformations of the surface (Zone 1, Figure 9), due to the macroscopic wear particles coming from the detachment on the track 1. They are those where the expansion of a smooth morphology is the most relevant too, maybe they are better protected.

The observation of the ring and roller surface topography leads to the statement that the inner ring is the first to be damaged. The increasing of the inner ring damage causes as well the initiation of the roller damage, which starts to show a very worked surface (Zone 3, Figure 9). This is due to the rolling of the roller on a highly irregular surface of the damaged raceway.

Second steady state

After the damage accumulation phase, two different scenarios are possible: of the roller and the raceway, high pressures seem help to develop a smooth area on the roller (Figure 11), with a similar topographical aspect of the one previously formed on the raceway surface (Figure 7).

Damage growth and bearing end-of-life

During the last part of the bearing life (between point 4 and 6 or 5 and 7), the irregularities on the damaged inner ring continue to evolve (Figure 12), and the RT and RD curves have an exponential trend up to the torque limit, corresponding to the bearing end of life.

At the end of the bearing life the contact surfaces and geometry are completely damaged (Figure 12), explaining the sudden increase of the RT and RD curves.

Accommodation mechanisms and bearing degradation scenario

The analysis of the contact surfaces during the different phases of the bearing life allowed for the different accommodation mechanisms at the interfaces to be reconstructed. The local contact stress and relative velocities at the interfaces are accommodated in different ways, as a function of the contact zones. In particular, the ratio between local sliding and sticking (Figure 6) can bring to either the detachment of wear particles from the initial topography (machining grooves) or the compression/plasticization on the superficial layer.

Among the accommodation mechanisms that seem to allow for a stability of the bearing interface at high contact pressures, the formation of smoothed area has been identified. From the observation of the sites where this takes place, two main factors can be assumed: the combined presence of local sliding & rolling and the presence of wear particles in situ. The bearing working condition seems to be at the origin of this superficial evolution [START_REF] Eleöd | Conception of numerical and experimental tools for study of the Tribological Transformation of Surface (TTS)[END_REF], which accommodates the high contact stresses and allow for the establishment a long steady state in the operative life of the bearing. The smoothed areas are formed on both the raceway and roller surfaces. However, the damage of the bearing is initiated at the raceway, and more precisely at the Trace 1. This seems to be a consequence of the less smoothed surface on this trace, with respect to the adjacent ones. Figure 13 shows the distribution of the phenomena observed on the overall raceway. This difference in the forming of the smoothed area is due to both the lower amount of sliding, bringing to lower source of wear particles in situ, and the expulsion of the particles toward the adjacent traces. Figure 13 shows how the more smoothed surface corresponds to the areas on the borders of Traces 2 and 4, where the wear particles agglomerate.

On the other side, the lower sliding at the Trace 1 is demonstrated by the observation of the lateral surfaces of the roller (Figure 14). The Figure shows scuffing marks at one side of the rollers. Nevertheless, for the Trace 1 the marks are localized only along the oscillating angle, showing the absence of overall sliding of the roller. On the contrary, the scuffing marks cover all the circumference of the roller R2 and R4,

showing an overall sliding of these rollers the respect to the corresponding contact traces.

Moreover, the observation of the roller sides allows for highlighting an asymmetry in the contact, due to the bending of the shaft under the radial load. This observation can justify as well the large scattering of the endurance tests (Figure 3), due to the bearing misalignment and the corresponding distribution of the contact stresses. When focusing on the damaged trace (Trace 1), the initiation of the damage on the raceway, instead of the roller, is in agreement with the results from the numerical simulations.

Roller bearings versus deep-groove ball bearings

The results obtained in this study on roller bearings are here compared with those from previous studies, focused on deep-groove ball bearings subjected to the same boundary conditions [START_REF] Tonazzi | Numerical analysis of contact stress and strain distributions for greased and ungreased high loaded oscillating bearings[END_REF] For the two cases, the tests have been conceived to guarantee the same contact pressure on the most loaded contact pairs and the same oscillatory motion. For the both analyses, the imposed oscillation amplitude is small enough to avoid overlapping of the rolling traces. The aim of the comparison is the investigation effect of the change in the geometry on the accommodation mechanisms and on the degradation scenario.

Figure 16 presents the models of the bearing life and the evolution of the surface topographies reconstructed for the roller bearing and the deep-grove ball bearing. Observation which can be made thanks to that comparison are:

1. In both cases it is possible to recognize five phases that characterize the bearings life; these have different duration, but in both cases, they represent precise and defined changes in both the contact surfaces and evolution of the signals.

2. Some recurring accommodation mechanisms take place. In particular, the creation of a smoothed topography in particular areas on the inner-ring raceway is observed (Figure 17). The smoothed areas are created in both cases during the steady state phase; however, while in ball bearings they are developed at the beginning of steady state (just after running-in), in roller bearing they appear clearly during the central phase. In the roller bearings, also the roller developed an evident zone with smoothed topography, which covers its surface after that the first damage of the raceway occurs, when superficial stresses and wear flows increase.

3. In the case of deep-groove ball bearing, the smoothed areas spread more uniformly at the center of the inner ring raceway. While in the case of roller bearing, traces 2 and 4 are preferential sites for such superficial transformations. In fact, in these two traces a higher sliding rate occurs with respect to trace 1. The two main aspects that could justify a different development of the smoothed topographies on different sites are:

i.

The geometry of deep-groove ball bearing. In fact, the inner-ring raceway is characterized by a deep-groove that allows to better accumulate the wear particles near the contact. The roller bearing raceway is more open and moreover there are grooves on the sides of the raceway, in which grease and wear particles can be accumulated, decelerating in this way the formation of the smoothed area.

ii.

A greater sliding motion of the most loaded rolling element in deep-groove ball bearing than in roller bearing. This seems fundamental for formation of the smoothed areas. In fact, a greater sliding implies a greater plasticization and can accelerate in this way the superficial transformation, as showed by the attitude of forming a smoothed area at the border of the traces 2 and 4. 4. a subsurface plastic strain accumulation, in the case of deep-groove ball bearings, is located on the most loaded ball (Figure 17) and brings the ball to damage first. On the contrary, in the case of roller bearings the roller is not subjected to such failure mechanism, and the surface of the raceway is damaged first.

5. rollers can slide also on the flanges of the inner-ring raceway so there are other contacts in the roller bearing, which are not present in the deep-groove ball bearing. The scuffing marks analysis on the roller-ends allowed some hypothesis to understand the degradation scenario of the roller bearing:

i. the presence of scuffing on one side of the roller highlights the misalignment of the bearing, which involves a change in the distribution of stresses.

ii.

The scuffing marks allows us to have information about the status of the investigated contacts. It was possible to notice how the roller 1 is characterized by pure overall rolling within the oscillation angle, while the adjacent traces showed overall sliding. A higher sliding/rolling ratio results in a higher local particle detachment, linked with the formation of the smoothed areas.

Conclusions

The evolution of greased spherical roller bearings under high load and oscillating motion has been investigated, by the analysis of the resistance torque and wear signals, together with the observations of the contact surfaces at the different life phases.

The reconstructed degradation model revealed five main phases, which characterize the greased roller bearings lifetime, under the tested loading conditions

The observation of the contact topographies, at the different phases of the bearing life, allows for providing a hypothesis on one of the main accommodation mechanisms. The generation of a smoothed area plays a key role in the bearing life by accommodating contact stresses and allowing for a long steady state in the bearing lifetime.

When comparing the roller and ball bearings, during the evolution of the bearing life, two main accommodation mechanisms can be identified: i) the strain accommodation within the TTS on the inner ring raceway surface; ii) the strain accumulation in subsurface, leading to fatigue cycles of the rolling element. The competition between these two accommodation mechanisms leads to the damage initiation on either the inner ring (in the case of roller bearing) or the rolling element (in the case of deep-groove ball bearing). The different geometry allowed as well for correlating the different accommodation mechanisms with the distribution of contact stresses and local sliding/rolling ratio.

Figure 1

 1 Figure 1-b.

Figure 1 :

 1 Figure 1: tested bearing and scheme of the test bench "R2".

Figure 2 :

 2 Figure 2: internal load distribution and definition of the most loaded traces and rollers.

Figure 3 :

 3 Figure 3: RT and RD normalized curves of lifetime tests. TEST CYCLES HLTA33-1 HLTA33-2 HLTA33-3 HLTA33-4 46 053 116 759 209 925 148 499

  The tested bearings have shown a very different life duration. Nevertheless, the tested bearings show common trends in the curve of the RT and RD curves, corresponding to a reproducible evolution of the bearing degradation scenario. The schemes of the two possible scenarios of the evolution of the RT and RD curves, during the overall bearing lifetime, are presented in Figure4. The main common points of lifetime tests are:

Figure 4 :

 4 Figure 4: Wear model of RT (up) and RD (down) for a greased roller bearing.

Figure 5 :

 5 Figure 5: Initial topography of a roller bearing. Inner-ring (left) and roller (right) SEM analysis. Mode: TOPO.

Figure 6 :

 6 Figure 6: a) End of running-in: SEM analysis of the inner ring (Mode: SE), with zooms on different zones in the in the contact area (Mode: TOPO). b) Roller contact status (Mode: SE).

Figure 7 :

 7 Figure 7: Central part of steady-state phase: SEM analysis of the inner ring (Mode: SE), with zooms on different zones in the contact area (Mode: TOPO).

Figure 8 :

 8 Figure 8: Final part of steady-state phase: SEM analysis of the inner ring (Mode: SE), with zooms on different zones in the contact area (Mode: TOPO).

Figure 9 :

 9 Figure 9: Damage initiation: SEM analysis of the inner ring and roller (Mode: SE), with zooms on different zones in the contact area (Mode: TOPO).

  1st scenario: either the most loaded rollers and raceway tracks begin to highly damage very quickly, bringing to a fast end of the bearing life 2nd scenario: or the modification of contact geometry continues until a new contact equilibrium state is established. In that second scenario, the second steady state may occupy a relatively short number of cycles because of the bad contact conformity. Then, the bearing degradation increases exponentially again, up to the bearing end of life. The second scenario lead to the point 7 of the

Figure 4 .

 4 Figure 4.

Figure 10

 10 Figure 10 shows how, in this 2nd degradation scenario, the damage on the inner ring track is smaller than the one in the previous case. The extent of detachment of material on Track 1 can affect the appearance (or not) of a second stationary phase. In this case, considering the smaller extent of the damage, the roller can easily plasticize the irregularities on Track 1 and a new equilibrium state is reached.

Figure 10 :

 10 Figure 10: comparison of material detachment on trace 1 between the two different scenarios.While the roller showed slight changes up to this stage, the beginning of the damage of the track with the consequent detachment of material involves a modification of its topography. Due to a bad conformation

Figure 11 :

 11 Figure 11: Second steady-state: SEM analysis of the most loaded roller (Mode: SE), with zoom in the contact area (Mode: TOPO).

Figure 12 :

 12 Figure 12: End of life: Optical microscope analysis of the inner ring (left) and roller (right).

Figure 13 :

 13 Figure 13: scheme of zones where smoothed plasticized areas developed with a qualitative scale of its formation.

Figure 14 :

 14 Figure 14: scuffing marks on each face on the most loaded rollers.

Figure 15

 15 shows the observation of the damage initiation, when the test is stopped at the second steady state the first damage occurs (Figure15-right), and the numerical results on the residual plastic strain distribution (Figure15-left), on both the roller and the raceway. The numerical results show a higher plastic deformation on the surface of the inner ring raceway with respect to the roller surface. Moreover, the contact zone showing the highest plasticization, due to higher shear stress, is the one close to the borders of the contact race (green and yellow areas in the Figure15, top left). This is the same zone where the detachment of the superficial material first occurs (Figure15, top right), showing how the numerical simulation predicts correctly the areas where the concentration of stresses and strains brings to the initiation of the damage.

Figure 15 :

 15 Figure 15: Comparison between numerical results (Equivalent Plastic Deformation) and experimental results (Topographical analysis).

Figure 16 :

 16 Figure 16: comparison of degradation scenario models.

Figure 17 :

 17 Figure 17: comparison of accommodation mechanisms between ball bearings (left) and roller bearings (right).

Table 1 :

 1 radial load distribution in function of the angle .

	Lubricant	Aeroshell grease 33
	Level of lubrication	100% filled
	Radial static load applied	10 785 N
	Motion type	Oscillatory motion
	Amplitude of the oscillatory motion	40°
	Oscillation frequency	5 Hz

Table 2 :

 2 test conditions.

Table 3 :

 3 lifetime tests.

  [START_REF] Houra Komba | Experimental damage analysis in high loaded oscillating bearings[END_REF] [8][START_REF] Berthier | Wear mechanism in oscillating bearings[END_REF] [START_REF] Ghezzi | Damage evolution and contact surfaces analysis of high-loaded oscillating hybrid bearings[END_REF]. The two bearing types have different geometrical dimensions, a different kind of rolling element, but are made of the same material (steel 100Cr6). The grease type and lubrication level applied in the endurance tests are the same. For the both, the experimental tests have been performed on the same SKF test bench.