Majid Shahzad
email: majid.shahzad@isae.fr

Michel Chaussumier

Rémy Chieragatti

Catherine Mabru

Farhad Rezai-Aria

Surface characterization and influence of anodizing process on fatigue life of Al 7050 alloy

Keywords: C, Surface treatments E, Fatigue G, Scanning electron microscopy (SEM)

The present study investigates the influence of anodizing process on fatigue life of aluminium alloy 7050-T7451 by performing axial fatigue tests at stress ratio 'R' of 0.1. Effects of pre-treatments like degreasing and pickling employed prior to anodizing on fatigue life were studied. The post-exposure surface observations were made by scanning electron microscope (SEM) to characterize the effect of each treatment before fatigue testing. The surface observations have revealed that degreasing did not change the surface topography while pickling solution resulted in the formation of pits at the surface. Energy dispersive spectroscopy (EDS) was used to identify those constituent particles which were responsible for the pits formation. These pits are of primary concern with respect to accelerated fatigue crack initiation and subsequent anodic coating formation. The fatigue test results have shown that pickling process was detrimental in reducing the fatigue life significantly while less decrease has been observed for anodized specimens. Analyses of fracture surfaces of pickled specimens have revealed that the process completely changed the crack initiation mechanisms as compared to non-treated specimens and the crack initiation started at the pits. For most of the anodized specimens, fatigue cracks still initiated at the pits with very few cracks initiated from anodic coating. The decrease in fatigue life for pickled and anodized specimens as compared to bare condition has been attributed to decrease in initiation period and multi-site crack initiations. Multi-site crack initiation has resulted in rougher fractured surfaces for the pickled and anodized specimens as compare to bare specimens tested at same stress levels.

Introduction

High strength to weight ratios make aluminium alloys, specially 2000 and 7000 series, attractive to be used in aeronautical industry for different structural components. In aeronautical applications, an important aspect to take into consideration is the fatigue behaviour of the structures due to the influence of fluctuating loads [START_REF]Fatigue design of aluminium components and structures[END_REF]. Being subjected to different environmental conditions (for the aeronautical structures) corrosion resistance of the materials used is also an important aspect to be taken into account. To prevent the corrosion failure and enhance the corrosion resistance, anodizing is the most typical process used for aluminium alloys [START_REF][END_REF]. Despite the benefits obtained in terms of enhanced corrosion properties, the anodizing process has a damaging effect on the fatigue performance of the base material [START_REF] Lonyuk | The effect of oxide coatings on fatigue properties of 7475-T6 aluminium alloy[END_REF][START_REF] Sadeler | Effect of a commercial hard anodizing on the fatigue property of a 2014-T6 aluminium alloy[END_REF][START_REF] Rateick | Effect of hard anodize thickness on the fatigue of AA6061 and C355 aluminium[END_REF][START_REF] Cirik | Effect of anodic oxidation on fatigue performance of 7075-T6 alloy[END_REF][START_REF] Camargo | Influence of anodization on fatigue strength of 7050-T7451 aluminium alloy[END_REF] with the main effect being the encouragement of crack initiation. The anodization produces a brittle and hard oxide layer as compared to aluminium substrate with inherent pores and it easily cracks under cyclic stress [START_REF] Merati | Determination of fatigue related discontinuity state of 7000 series of aerospace aluminium alloys[END_REF].

Since oxide layer adheres extremely well to substrate, any crack that develops in it acts like stress raiser and propagates towards the substrate.

Prior to anodizing, the appropriate pretreatment of the surface is necessary which comprised of degreasing and pickling and the objective is to produce chemically clean surface ready to be anodized. Localized corrosion, in the form of pits, occurs during the pretreatment solution exposure and these pits have been identified as cause for accelerated crack nucleation during subsequent fatigue loading [START_REF] Monsalve | S-N-P curves in 7075-T7351 and 2024-T3 aluminium alloys subjected to surface treatments[END_REF][START_REF] Barter | Initiation and early growth of fatigue cracks in an aerospace aluminium alloy[END_REF]. Abramovici et al. [START_REF] Abramovici | The influence of etch pits on fatigue life of anodized aluminium alloys[END_REF] have shown that changing the pickling time had a great influence on the fatigue life of 7000 series. In a recent study for 7010 alloy, Shahzad et al. [START_REF] Shahzad | Influence of surface treatments on fatigue life of Al 7010 alloy[END_REF] have shown that pickling process resulted in the formations of pits by attacking the constituent particles at the surface with significant fatigue life implications.

In the presence of these surface defects, fatigue failure of a component can be greatly accelerated, so an inclusive understanding of these pre-treatments on localized corrosion mechanism is therefore of scientific interest. In this context, the objective of present paper is to quantify the effect of each surface treatment individually and their combined effects on fatigue life of the given alloy.

Moreover, constituent particles sizes and distributions can be different for longitudinal L, long transverse T L and short transverse S T directions, thus localized corrosion behaviour can be different for each treatment. However, in this article specimens have been machined in the long transverse 'T L ' direction and subsequent fatigue strength has been evaluated. Optical and scanning electron microscope coupled with energy dispersive spectroscope were used to study the effect of each treatment on surface topography and fracture surfaces of fatigue specimens to identify the crack origin sites and to understand the damage mechanism.

Experimental details

Material

The material investigated during this study is 7050-T7451 whose chemical composition, as determined by EDS technique, is given in Table 1. T7451 treatment consists of heat-treating, quenching and overaging [START_REF]Military Specification MIL-H6088G. Heat treatment of aluminium alloys[END_REF] for improved fracture toughness and minimal loss of tensile strength.

Metallographic analysis of the microstructure by optical microscope revealed that it is composed of unrecrystallized and recrystallized grains and latter are elongated in the rolling direction as shown in Fig. 1. Three types of constituent particles were found in this material: Al 7 Cu 2 Fe, Al 2 CuMg and Mg 2 Si as determined by EDS and their average size varied between 8 and 12 lm.

Mechanical properties of the alloy determined by tensile tests are; yield strength 448 MPa and 440 MPa, ultimate tensile strength 507 MPa and 504 MPa, elongation 12% and 11.4% in L and T L directions respectively.

Specimen preparation

Fatigue test specimens, as shown in Fig. 2, have been machined with initial surface roughness R a = 0.6 ± 0.1 lm by lathe turning, without using lubricant, with the loading axis along the long transverse (T L ) direction of the 7050 plate. Stress concentration factor 'K t ' of the specimen is 1.04 [START_REF] Peterson | Stress concentration factors[END_REF].

Surface treatments

Prior to anodizing, the specimens were initially degreased manually by diestone followed by degreasing in alkaline bath (pH $ 9) at 60 ± 5 °C during 3 min followed by water rinsing for 1-2 min. For pickling process, specimens were submerged in aqueous solution of H 2 SO 4 (10-15% by weight), HNO 3 (20-30% by weight) and ferric-sulphate Fe 2 (SO 4 ) 3 (20-30% by weight) for 3 min at 32 ± 2 °C followed by 1-2 min water rinsing. Anodizing was carried out in chromic acid CrO 3 solution (55 g/L) at 40 ± 2 °C for 50 min followed by water rinsing.

The thickness of anodic film was measured by optical microscopy and was also confirmed by SEM inspection. The average thickness of film produced by the process is measured to be about 3 lm.

Surface measurements

Besides the observations made by SEM, to characterize the effect of each surface treatment quantitatively, topography measurements were carried out with the help of a contact profilometer coupled with motorized table. To do so rectangular specimens of dimension 40 Â 40 mm 2 were machined with the help of a shaper to give same surface roughness 'R a ' values as generated for fatigue specimens machined by lathe. After machining, the specimens were subjected to different surface treatments. Several measurements adjacent to each other were done with zone size of 4 Â 4 mm 2 each with transverse and longitudinal resolutions of 5 lm in each direction. The characterization of pits was done with a specific program developed for this purpose. From surface topography data, a plane orientation correction was made using the least square method and pits were initially neglected in this step. Then, surface profile, calculated from machining condition, is subtracted from the real surface measured by profilometer to define the depth of pits.

Fatigue testing

Axial fatigue tests have been performed at 10 Hz in ambient conditions at stress ratio 'R' of 0.1 according to ASTM E 466 [START_REF]Standard practice for conducting force controlled constant amplitude axial fatigue tests of metallic materials[END_REF]. All tests were conducted under load controlled condition using a 100kN servo-hydraulic MTS machine. The nominal maximum cyclic stress was set at a value that was expected to result in a fatigue life of between 10 4 and 10 6 cycles and tests were stopped if the specimen did not fail at 1.2 Â 10 6 cycles.

Results and discussions

Characterizing the effects of pre-treatments

Microscopic examination was made after degreasing and pickling process to verify if these pre-treatments affect the surface topography of the specimen. By comparing micrographs Fig. 3a and b before and after degreasing process, we concluded that degreasing process did not change the surface topography of the specimens. Moreover, EDS analysis showed that the constituent particles; i.e. Al 7 Cu 2 Fe, Al 2 CuMg and Mg 2 Si were always present after the degreasing treatment. On the contrary, after the pickling solution exposure, numbers of pits were observed at the surface of the specimen. The pickling process was found to attack the constituent particles resulting in pits formations (Fig. 3c andd). During SEM examination of the pickled specimens, it was observed that some particles were completely dissolved while others were partially dissolved leaving behind a trace (Fig. 3d). This phenomenon has also been reported earlier [START_REF] Birbilis | Electrochemical characteristics of intermetallic phases in aluminium alloys[END_REF]; Birbilis et al. have shown that nature of the constituent particle had an influence on the pitting process. EDS analysis of some partially dissolved particles showed that these were particles rich in Cu and Fe; i.e. Al 7 Cu 2 Fe and Al 2 CuMg. Fig. 4 presents the results of topography measurements taken for specimen that has undergone the pickling treatment. Here we can observe that most of the pits have depth below 8 lm with very few exceeding this value. The possible reason for this phenomenon, as observed by SEM, that some particles were partially dissolved leaving behind the remnants. In their work for aluminium alloys, Ezuber et al. [START_REF] Ezuber | A study on the corrosion behaviour of aluminium alloys in sea water[END_REF] have shown that pit morphology depends on the type of constituent particles present and their reactivity with the exposed solution.

SEM observations of anodic coating

It is of scientific interest to study the effect of these pre-existing defects formed on the specimen surface during pretreatment solution exposure prior to the anodizing step. In Fig. 5, surface observa-tions made by SEM are shown for specimens undergone the anodization process. It can be clearly seen that anodization tend to form a smooth anodic coating with few pits still present at the surface (Fig. 5). These pits can act as stress raiser during fatigue loading and can significantly affect the fatigue life by reducing or even eliminating the initiation period. On the contrary to pickled specimens, no traces of partially dissolved particles were found for specimens that undergone complete anodization process. This phenomenon can be explained by the complete dissolution of remnant particles, left after pickling solution exposure, during anodizing process.

In Fig. 6 distribution of depth of pits measured by topography is given for the anodized specimens. Here we can observe that size of the largest pit observed has been increased as compared to the pits observed for pickled specimens which can be explained by the complete dissolution of constituent particles. In a recent investigation for aluminium alloy 7075, Savas and Earthman [START_REF] Savas | Surface characterization of 7075-T73 aluminium exposed to anodizing pretreatment solutions[END_REF] studied the effects of different anodizing pretreatment solutions on localized corrosion behaviour. They hypothesized that if pits, initiated during pretreatment solution exposure, were beyond a threshold size (10-20 lm), a higher current density existed at these locations during anodizing process, thus resulting in deeper pit structure. During this study, the maximum pit depth was determined to be 12 lm by topography measurements for anodized specimens.

S-N curves

To further validate the fact that degreasing has no effect on fatigue behaviour of this alloy, some specimens were tested after degreasing solution exposure and results showed no change in the fatigue life as compared to machined specimens. Fatigue test results for machined and pickled specimens are given in Fig. 7 and from the graph it is possible to evaluate the influence of pickling treatment by taking machined specimens as reference. It can be clearly seen that tendency of pickling process is to degrade the fatigue performance of the given alloy at all stress levels as compare to machined specimens. The presence of pits at the surface facilitates cracks initiation and as expected, decrease in the fatigue life of pickled specimens increases with the decrease of fatigue stress. The fatigue strength drops from 220 MPa for the machined specimens to 150 MPa for pickled specimens. The surface topography submitted to pickling treatment, shown in Fig. 3 before any fatigue loading, helps in understanding the strong effect of the process on fatigue strength for this alloy. In the Fig. 3, it is possible to observe many pits which resulted as dissolution of Al 7 Cu 2 Fe and Al 2 CuMg particles during pickling solution exposure. These pits can act like stress concentration and can accelerate the crack initiation mechanism in the material.

In case of anodized specimens, fatigue strength decreased from 150 MPa to 125 MPa as compared to pickled specimens for high cycle regime. By taking machined specimens as reference, 43% decrease has been observed for specimens after complete anodization cycle of which 32% decrease has been associated to pickling process (see Fig. 8). This small decrease of fatigue strength for anodizing as compared to pickling can be attributed to slight increase in pit size and brittle nature of oxide coating.

Fatigue fractographic analysis

Several specimens tested at different stress levels were examined after failure by SEM, using backscattered and secondary electrons, in order to understand the different fatigue damage mechanisms for machined, pickled and anodized conditions. Fatigue test results presented above indicate that the decrease in fatigue strength for pickled and anodized specimens was related to the crack initiation stage. Therefore, it is important to identify the origin of fatigue cracks for different specimens. For aluminium alloys, constituent particles and grain boundaries are the common sites for fatigue crack initiation [START_REF] Przystupa | Microstructure based fatigue life predictions for thick plate 7050-T7451 airframe alloys[END_REF]. For the specimens tested in as machined state, fatigue cracks have been observed to nucleate at constituent particles Mg 2 Si and Al 7 Cu 2 Fe particles as determined by EDS analysis while Al 2 CuMg particles were not effective in initiating cracks. It was found that for a given specimen if crack initiation originated at Mg 2 Si particle, it always started by the cracking of the particle and then crack propagated in matrix as shown in Fig. 9.

On the other hand, for Al 7 Cu 2 Fe particles, crack initiation always started at the interface of the particle and matrix as shown in Fig. 10. No cracking of the constituent particle or debonding was found at the crack initiation sites. The nucleation site was observed to quickly form a semi-elliptical surface crack approximately double the size of the constituent particle.

For pickled specimens, fatigue cracks were found to initiate at the pits formed by the dissolution of the Cu rich particles, i.e. Al 7 Cu 2 Fe and Al 2 CuMg as shown in Fig. 11. These pits may cause local stress concentration and may have a provocative effect on the subsequent cracks growth in the matrix. The fractographic examination of pickled specimens revealed that most of the pits which were found to initiate fatigue cracks was more than 8 lm deep (Fig. 11b). Although there were few pits which were beyond 8 lm depth, as determined by topography (Fig. 4) measurements, the fractography revealed that fatigue failure process exploits the deep pits existing on the specimen surface. In some cases, where two pits were close to each other, initiated cracks grew together to form single crack front as shown in Fig. 11a. Presence of the many pits on the surface also explains multi-crack initiation sites for the specimens which were treated with pickling solution. Some pickled specimens were observed at the surface to show that fatigue cracks are initiated from the pits (Fig. 11c).

Similar to pickled specimens, multiple crack nucleation sites were also found on the fractured surface of the anodized specimens. When analyzed thoroughly by SEM, it was found that almost all fatigue cracks for anodized specimens were initiated by pickling pits beneath the coating as shown in Fig. 12b while very few initiated in the coating and then propagated towards the substrate (Fig. 12c). Relatively small decrease in fatigue strength for anodized specimens as compared to pickled specimens, as shown by fatigue test results, can be explained in terms of fatigue crack initiation mechanism. Since for anodized specimens, crack initiation mechanism remains the same; i.e. fatigue initiation always started by the pickling pits. As compared to pickled specimens, the depth of pits which were found to initiate fatigue cracks in anodized specimens is 11 lm or more.

At same stress level, the numbers of cracks formed on fractured surface of the pickled and anodized specimens were larger than that of the machined specimens (Fig. 13). The fracture surface is not flat (Fig. 12a) which also reveals that number of fracture steps exists, indicating that fracture has occurred as a consequence of the propagation of several cracks that initiated from pits. This multicrack initiation is also responsible for rough fractured surface of the anodized specimen. Shiozawa et al. [START_REF] Shiozawa | Effect of anodized films on fatigue strength in aluminium alloy[END_REF] have also demonstrated this phenomenon that number of crack initiation sites increased after the anodizing treatment. For the pickled and anodized specimens, cracks at the multiple initiation sites propagated independently and coalesce with neighbouring cracks. Therefore, one can also conclude that decrease in fatigue strength for specimens after surface treatments is caused by the increase in number of crack initiation sites and crack propagation rates due to coalescence of cracks.

Conclusions

The fatigue strength of 7050 aluminium alloy has been evaluated for different surface treatments. The SEM micrographs of pickled and anodized specimens have revealed the presence of pit like defects at surface before any fatigue loading. These pits were formed as a result of dissolution of constituent particles (Al 7 Cu 2 Fe and Al 2 CuMg) during different solution exposure. Topographical measurements were made to determine the size of these pits. The fatigue test results indicated that pickling and anodizing processes reduce the fatigue strength of the base material and this could be attributed to surface pits which can facilitate an early fatigue crack initiation and accelerate subsequent crack growth.

In as machined condition, fractographic examination showed that crack initiation has been associated with constituent particles Mg 2 Si and Al 7 Cu 2 Fe. While for pickled specimens damage mechanism was associated with pit initiation and most of the pits found to initiate cracks were more than 8 lm deep. The microscopic aspect of fracture surface for anodized specimen has shown a dual micro-mechanism of failure, with almost all crack initiation sites started from pits while very few were found to start from anodic coating. The decrease in fatigue life could also be attributed to the increase number of crack initiation sites. The fracture surface was not flat, revealing the presence of a number of fracture steps, which points out that fracture has occurred as a consequence of the propagation of several cracks that initiated from surface of the specimens.

Fig. 1 .

 1 Fig. 1. Optical micrograph of 7050-T7451 alloy microstructure.

Fig. 2 .

 2 Fig. 2. Fatigue specimen geometry (dimensions in mm).

Fig. 3 .

 3 Fig. 3. (a) SEM micrograph of the surface of machined specimen, (b) SEM micrograph of the specimen undergone degreasing only, (c) formation of pits at the surface of specimen subjected to pickling process and (d) dissolution of constituent particles to various extents.

FrequencyFig. 4 .

 4 Fig. 4. Distribution of depth of pits formed after pickling solution exposure (topography measurements for a zone of 4 Â 4 mm 2 ).

Fig. 5 .FrequencyFig. 6 .

 56 Fig. 5. (a) Surface appearance of specimen after complete anodization cycle and (b) larger pits resulted as complete dissolution of particles.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Fatigue test results for machined and pickled specimens.

Fig. 9 .

 9 Fig. 9. (a) SEM image for machined specimen at low magnification showing crack initiation site and (b) EDS analysis of (a) showing cracking of Mg 2 Si particle and subsequent crack growth.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Fracture surfaces of machined specimen, at low and high magnification showing fatigue crack initiated by Al 7 Cu 2 Fe particle and subsequent crack growth.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. (a) Multi-site crack initiation for anodized specimen, (b) SEM image showing crack initiation from pickling pit and (c) cracking from anodic coating and subsequent crack growth to substrate.

Table 1

 1 Chemical composition of 7050-T7451 alloy.

	Element	Si	Fe	Cu	Mn	Mg	Ti	Zn	Zr	Al
	wt.%	0.28	0.19	1.76	0.07	2.42	0.06	6.15	0.14	Bal.


Matthieu Bizeul

Christophe Bouvet
email: christophe.bouvet@isae.fr

Jean-Jacques J Barrau

Rémy Cuenca

Fatigue crack growth in thin notched woven glass composites under tensile loading. Part I: Experimental

Keywords: A, Structural composites A, Glass fibres B, Fatigue B, Fracture

teaching and research institutions in France or abroad, or from public or private research centers.

Fatigue crack growth in thin notched woven glass composites under tensile loading.

Introduction

Woven composites are advanced materials that are commonly used in aerospace applications. Their use is interesting owing to their excellent drapability over complex geometries, their effective manufacturing cost and their good damage tolerance properties.

For example, the skin of helicopter blades is often made of a few woven plies (Fig. 1). In these thin structure parts, a through-thethickness crack might appear in service; such a crack could initiate and propagate from existing defects caused by manufacturing processes, stress concentration or low-energy impact [START_REF] Hansen | Damage development in woven fabric composites during tensiontension fatigue[END_REF]. In order to anticipate such a scenario which could slightly modify the rotor dynamic behaviour, it seems useful to study the crack propagation in these skins in fatigue. Blade loads are mainly defined by centrifugal forces and flap and drag bending moments due to cyclic aerodynamic loads. These structures are therefore subjected to fatigue stresses where the primary load is tension due to centrifugal effects.

Few studies are available related to fatigue crack propagation in woven laminates. Mandell [START_REF] Mandell | Fatigue crack propagation rates in woven and non woven fiber glass laminates[END_REF] studied the propagation of a through-the-thickness crack in notched specimens loaded in tension-tension fatigue (R = 0.1, f = 4-7 Hz). Laminates were made of polyester resin and glass woven plies lined up with the load direction. The thickness was about 2.5 mm and the initial notch was cut with a 0.63 mm thickness diamond saw. The main characteristic underlined was that the crack growth was linked to fibre tow width. Moreover, difficulty was encountered in measuring the crack length because of the stepwise nature of the crack growth and the extensive damage region associated with the crack tip. More recently, Shindo et al. [START_REF] Shindo | Characterization of Mode I fatigue crack growth in GFRP woven laminates at low temperatures[END_REF] examined the mode I fatigue behaviour of notched plain woven glass laminates. For all specimens, the weft fibre bundles were aligned with the load axis. Load control fatigue tests were performed on CT specimens of thickness 25 mm. Crack lengths were calculated from the compliance data obtained during test using finite element analysis. Three stages of fatigue crack growth were identified: crack initiation, stable crack growth and unstable crack propagation. Optical micrograph of fatigue crack path taken at approximately N/N f = 90% showed an amount of damage near the crack tip. The damage zone consisted especially of matrix cracks in the fibre bundle undulation region. Furthermore, the size scale of damage that occurred around the crack tip was not negligible compared to the other significant dimensions. Pegoretti and Ricco [START_REF] Pegoretti | Fatigue crack propagation in polypropylene reinforced with short glass fibres[END_REF] investigated fatigue crack growth in polypropylene composites reinforced with short glass fibres. Experiments were conducted on a single-edge notched tension specimen of width 27 mm and of thickness 2.7 mm at room temperature: the notch length measured was 3 mm. The crack growth rate level was found to decrease as the fibre weight fraction increased. A further analysis of the data indicated that crack propagation was also governed by viscoelastic creep. Other works merely focused on the experimental determination of the fracture toughness of thin quasi-isotropic laminates made of glass woven plies in single edge notch (SEN) of center notch (CNT) specimens loaded in tension [START_REF] Belmonte | Characterisation and modelling of the notched tensile fracture of woven quasi-isotropic GFRP laminates[END_REF][START_REF] Marissen | Fracture of quasi-isotropic composite sheets with sharp notches[END_REF]. Constant values of fracture toughness are found for different crack lengths.

The objective of this present paper is to study the tension-tension fatigue propagation of a through-the-thickness crack in notched laminates made of few woven glass plies. Several laminate stacking sequences are used for skin blades but some directions are commonly used. Thus, three simple stacking sequences are considered:

-An orthotropic laminate [0] 2 with two glass woven plies lined up with the tensile load. The influence of the tows nature (warp or weft) of the woven ply on the crack growth is supplemented.

The fibre fatigue behaviour influence on the crack growth is then evaluated. In this paper, the laminate with the warp yarns of woven plies in the tensile direction is noted [0°] 2 and actually presents 50% of fibres in 0°and in 90°directions; similarly, the laminate with the weft yarns in the tensile direction will be noted [90°] 2 and presents 50% of fibres in 0°and in 90°d irections. -A bias laminate [45] 2 . This laminate exhibits an important nonlinear behaviour in tension. In this regard, the influence of matrix fatigue behaviour on the crack growth is assessed. It can be noted that this laminate presents the warp yarns in 45°direction and the weft ones in À45°direction, and in fact presents 50% of fibres in 45°and in À45°directions. -A ''quasi-isotropic'' laminate [45; 0; 45] with a middle woven ply having warp fibres in the load direction. This one may present coupled phenomena of the two previous laminates. Basically, this laminate is not quasi-isotropic, because quasi-isotropic needs the same number of plies in each direction, but this term is still used in the text below in order to simplify notations and is in quotes to avoid confusion.

Fatigue test specimen

The crack propagation is studied in glass/epoxy woven composite. These laminates are made of 8-harness satin balanced woven fabric (8-HS) pre-pegs plies with a fibre volume fraction of 50%. The yarn size is 0.5 mm-width and 0.1 mm-thickness. The elastic modulus are 21.5 and 20.5 GPa, the limit strength 385 and 280 MPa, respectively in the warp and weft directions, the shear modulus is 3.5 GPa and the shear strength 65 MPa. The samples are always manufactured with the warp woven face on the top of the thickness, so the top face is a warp side and the underside face a weft one.

In analyzing a typical blade section (Fig. 1), it can be stated that the skin has to follow the longitudinal strain of the spar mainly loaded by centrifugal effect. This remark leads to the development of a technological fatigue test specimen (Fig. 2) having a width 50 mm. A severe notch is performed on one edge with a 0.2 mm diameter diamond thread to limit the through-the-thickness crack initiation duration. On the other edge, a strip of T300/914 unidirectional carbon fibres having a stacking sequence of [0] 2 is placed to get closer to the real loading conditions and plays the role of spar. It also reduces the likelihood of global inelastic strains occuring under fatigue loading. The fatigue tests are conducted under strain control at room temperature (about 20 °C) on a 100 kN closed-loop servo-hydraulic Instron tension machine, with an extensometer placed on the carbon strip, at a frequency of 20 Hz. The imposed strain levels are e min = 10 À3 and e max = 3 Â 10 À3 . This maximum fatigue strain level corresponds to the infinite life asymptote in S-N curve [START_REF] Shindo | Cryogenic fatigue behavior of plain weave glass/epoxy composite laminates under tension-tension cycling[END_REF][START_REF] Pandita | Tensile fatigue behaviour of glass plain-weave fabric composites in on -and off-axis directions[END_REF][START_REF] Xiao | Damage and fracture of notched non-woven and woven composites laminates[END_REF][START_REF] Fujii | Microscopic fatigue processes in a plain-weave glass-fibre composite[END_REF][START_REF] Talreja | Fatigue of composite materials: damage mechanisms and fatigue life diagrams[END_REF] and is widely used as design fatigue strain for structures made of glass plies. The specimen is cooled down to room temperature by a fan and its temperature is saved (the thermocouple sensor is located 10 mm under the notch). Furthermore, no increase in temperature was recorded.

The crack growth is monitored with the help of a CCD camera (Fig. 3). The different devices are driven with the help of a Labview program. Due to the relative transparency of the glass fibres reinforced plastics, the propagation of a white damage zone by back-illumination is observed. This kind of phenomenon was also described by [START_REF] Belmonte | Characterisation and modelling of the notched tensile fracture of woven quasi-isotropic GFRP laminates[END_REF][START_REF] Xiao | Damage and fracture of notched non-woven and woven composites laminates[END_REF]. In a previous work [START_REF] Bizeul | Influence of woven ply degradation on fatigue crack growth in thin notched composites under tensile loading[END_REF], this damage zone was identified as a through-the-thickness crack all along by microscopic investigations. Since the beginning of the crack growth, this damage zone has presented tows breakage even in the tip. This damage zone also contains transverse yarn cracking and metadelaminations [START_REF] Patel | Durability of a graphite/epoxy woven composite under combined hygrothermal conditions[END_REF].

Fatigue crack growth results

Warp and weft direction

Earlier experiments [START_REF] Bizeul | Influence of woven ply degradation on fatigue crack growth in thin notched composites under tensile loading[END_REF] showed a different behaviour between warp and weft directions of the studied balanced woven ply under static loading as well as in fatigue, especially in crack propagation tests in samples of 30 mm width (Fig. 4). The main fact underlined by these results is the large dispersion between samples in both cases, either for the laminates with warp tows coinciding with load direction, or for the weft ones. It can be noted on this figure, the six warp and weft samples are exactly the same, only the initial crack tip position relative to the yarns should be different (this manufacturing parameter is difficult to control). These differences have been connected to the crack initiation time [START_REF] Bizeul | Influence of woven ply degradation on fatigue crack growth in thin notched composites under tensile loading[END_REF] which depends on the sollicitation direction and on the initial crack tip position relative to the yarns position. In fact, more the initiation time is long, more the crack growth decreases because of the extended of the damage zone. This fact may be due to the small number of plies [START_REF] Boniface | Damage development in woven glass fibre/epoxy laminates under tensile loading[END_REF] and perhaps because the experiments are led in strain controlled. Moreover, the crack growth initiation duration in warp laminates was lower than in weft laminates. As the weft direction undulation is more severe, the matrix damage in this direction appears sooner. So, the initiation phase lasts more and the crack growth rate decreases because of the extended damage zone. In order to study the crack growth in a larger domain and to reduce the mean initiation duration, the geometric characteristics of the specimen have been modified (Fig. 2): The ratio a/w has been increased in order to limit the initiation duration and the specimen width is larger.

The crack growth in 50 mm width samples exhibits qualitative similarity (Fig. 5). It can be noted, on this figure, the three warp and the two weft samples are exactly the same, only the initial crack position relative to the yarns should be different. Three stages are easily noticed: the first two are commonly observed [START_REF] Shindo | Characterization of Mode I fatigue crack growth in GFRP woven laminates at low temperatures[END_REF][START_REF] Pegoretti | Fatigue crack propagation in polypropylene reinforced with short glass fibres[END_REF]: the ''initiation'' phase where crack length does not exceed 3 mm and the ''stable propagation'' phase. The last one corresponds to the crack arrest due to the carbon strip influence.

Actually, two main observations are noteworthy. First, the different behaviour between warp and weft directions in crack propagation does not subsist, as the crack growths are similar and even overlap for two warp and weft samples. The second point is about the crack propagation test duration. This period of time is divided by around 10 compared to the previous geometry where the specimens' width was equal to 30 mm. Moreover, as the crack initiation duration does not vary as much as the one corresponding to 30 mm width specimens, the crack growth rates are of the same order (Table 1). Table 1 recalls the extreme results of crack growth in 30 mm width samples of the previous experimental campaign and specifies key results of crack growth in the 50 mm width samples. The initiation duration is determined owing to the typical decrease of crack growth rate at the beginning of the propagation (Fig. 6). This phenomenon is associated with the microstructure influence on crack growth [START_REF] Shindo | Characterization of Mode I fatigue crack growth in GFRP woven laminates at low temperatures[END_REF][START_REF] Pegoretti | Fatigue crack propagation in polypropylene reinforced with short glass fibres[END_REF]. Bizeul et al. [START_REF] Bizeul | Influence of woven ply degradation on fatigue crack growth in thin notched composites under tensile loading[END_REF] stated that crack growth evolves regularly when three to five tows are broken. In other words, as the yarn width is equal to 0.5 mm, microstructure influence seems to exist when the crack length is less than 3 mm. The microstructure of the studied woven ply acts not only on the initiation duration but also on the crack path. It is interesting to examine the local crack path in relation to the woven scheme (Fig. 7). The warp side of the woven ply is observed on the photograph; the warp tows are then visible on the surface and lined up with the fatigue tensile load. The theoretical basic 8-harness satin weave pattern is shown next to the photograph. The crack path is overall straight but in focusing on the local crack path, it can be seen that the cross-over points of the woven plies are ideal places for fibre tow failure. The photograph also highlights that thin longitudinal matrix cracks are localized in the interstice between two consecutive tows and their lengths are limited by cross-over points.

These observations are in accordance with [START_REF] Belmonte | Characterisation and modelling of the notched tensile fracture of woven quasi-isotropic GFRP laminates[END_REF][START_REF] Pandita | Tensile fatigue behaviour of glass plain-weave fabric composites in on -and off-axis directions[END_REF][START_REF] Fujii | Microscopic fatigue processes in a plain-weave glass-fibre composite[END_REF][START_REF] Osada | Initial fracture behaviour of satin woven fabric composites[END_REF][START_REF] Woo | Analysis of fracture process in single-edge-notched laminated composites based on the high amplitude acoustic emission events[END_REF] where fatigue damage in the on-axis directions is initiated by cracks in transverse fibre yarn, and are deflected along the longitudinal yarn (meta-delamination) and finally leads to 0°tow fracture at crimp regions.

Additionally, from examination of the crack, it seems realistic to say that a tow of fibre always breaks entirely. This remark is consistent with the work of [START_REF] Schulte | Fatigue behaviour and damage development in woven fabric and hybrid fabric composites[END_REF]. The crack extends in a discrete manner and is linked to tow width [START_REF] Mandell | Fatigue crack propagation rates in woven and non woven fiber glass laminates[END_REF].

Bias direction

Usually, helicopter blade skins are mostly made of ±45°woven plies in order to ensure the torsional stiffness of the profile (Fig. 1). Thus, fatigue tests at controlled strain ratio are carried out on specific specimens (Fig. 2) with carbon strip on one edge in order to represent the blade spar and ±45°woven laminate elsewhere. The propagation of a white damage zone is monitored in particular test configurations detailed below. With the initial specimen (Fig. 2), the crack growth is very slow and even stops even in the case when the notch length is increased. A damage zone appears in the tip of the notch in a V-form (Fig. 8) and stays stationary. Owing to back-illumination, this zone appears in dark on the photograph taken at the maximum fatigue strain level (5 Â 10 6 cycles), which results in a through-the-thickness crack of less than 2 mm. It should be pointed out that matrix damage extends over the crack tip in the form of thin and straight matrix cracks between aligned tows.

The accumulation of matrix damage specific to [±45] n laminates [START_REF] Tong | Characteristics of fatigue crack growth in GFRP laminates[END_REF], tends to limit the stress concentration in the vicinity of the crack tip and basically the crack growth. Hence, changes are made on the specific specimen to improve the initiation and the propagation of a through-the-thickness crack in [45] 2 laminates while maintaining the fatigue strain levels previously fixed. In order to get closer to the complex service loading conditions of blade skins, the trailing edge strip influence has been taken into account (Fig. 1). This rear strip has many functions such as closing the torsion box of the profile, dragging a part of centrifugal load and adding trailing flexural stiffness to the blade. The fibres of this structural part are then aligned with the blade longitudinal axis like leading edge spar fibres. Yet, the initial specific specimen lacks longitudinal stiffness on the notch side and the main part of the tensile load goes on the other side in the carbon strip. A small part of the load is thus drained in the notch tip, slowing even the crack initiation. In order to observe a fatigue crack growth in these bias woven laminates, the technological specimen has been modified accordingly (Fig. 9). A similar UD carbon strip has been placed on the other edge of the sample, acting as a trailing edge spar. After polymerization and cutting a notch with a diamond thread, whose length is narrowly greater than the strip width.

The crack initiation is instantaneous with such a test configuration (Fig. 10), however the crack growth tends to slow down gradually. The fatigue tests show very similar results and have a duration of about 5 Â 10 6 cycles. This may be attributed to the fact that in the beginning of the test, crack growth rate is high, the crack initiation duration remains thus low and relatively close to a specimen to another. Unlike oriented laminates, the initiation duration has negligible influence on crack growth.

After measuring the crack growth in fatigue, the crack growth rate is deduced and plotted versus crack growth for each sample (Fig. 11). In the beginning of the crack growth, its rate decreases and then reaches a steady state when crack length is between 10 and 20 mm. Finally, the crack growth tends to ''stop'' when the crack length measures 25 mm, corresponding to a distance of 8 mm from the carbon strip acting as a leading edge spar. Thus, inelastic strains are likely to occur in fatigue in [45] 2 laminates as reported by [START_REF] Pandita | Tensile fatigue behaviour of glass plain-weave fabric composites in on -and off-axis directions[END_REF][START_REF] Hochard | A progressive first ply failure model for woven ply CFRP laminates under static and fatigue loads[END_REF], especially in the vicinity of the crack tip. This phenomenon is due to the damage of the resin under shear sollicitations and to the fibres rotations. As the fatigue tests are led in a strain control environment, the stress concentration in front of the crack is reduced with the appearance and development of inelastic strains. It seems that there is a threshold below which propagation cannot occur, as underlined by the fatigue tests on specimens without UD reinforcement corresponding to blade rear strip. The visible white damage zone which propagates corresponds to a through-the-thickness crack (Fig. 12). Tow breakage are easy to recognize in microscopic investigations if observations are carried out perpendicular to the fibres. Tow fracture mainly appear in the fibre undulation areas. Typical damages in woven ply also exist i.e., meta-delaminations, transverse microcracks, etc.

Again, optical analysis reveals that the local crack path in [45] 2 laminates is influenced by the weave pattern and the tow width (Fig. 12). As observed by Woo et al. [START_REF] Woo | Analysis of fracture process in single-edge-notched laminated composites based on the high amplitude acoustic emission events[END_REF], the through-the-thickness crack propagates perpendicular to yarn fibres and thus shifts alternately between ±45°directions. The entire crack path remains overall straight and perpendicular to the load direction. In addition, it is noticeable that the crack extends according to the fibre tow width and the tows always seem to break completely.

Once more the crack path has to go through the nearest crossover points of the woven plies; yarn failures mainly occur in these crimp areas. Finally, it is interesting to note that matrix damage surround the crack path. These consist especially of thin and long matrix cracks parallel to the fibres. It seems appropriate to associate these observations with the matrix degradation in the gap that exists between two consecutive yarns. Only the thin cracks parallel to the fibres closest to the surface are visible. They form a halo around the crack which extends over a length around 2-3 mm in the direction of fibres on both sides. Damage in V-form is visible in the crack tip.

''Quasi-isotropic'' laminate

Stacking sequences of blades skins are mostly a combination of those simple ones previously studied. In accordance with this remark, the crack growth in [45; 0; 45] ''quasi-isotropic'' laminates is investigated. The warp direction of the 0°woven ply corresponds to the loading direction. The specimen is similar to that used for the warp and weft campaign (Fig. 2) and thus presents a carbon strip on one edge. The other test characteristics are identical.

The crack growth in ''quasi-isotropic'' laminates follows the same three stages of the warp laminates i.e., the crack growth is very slow when the crack length remains less than 3 mm, then the crack growth rate increases until it reaches a maximum corresponding to a crack length of 20 mm and eventually decreases quickly (Fig. 13). The maximum crack growth rate levels are very close to those recorded in warp laminates. Nevertheless, it should be noted that the first stage of crack growth, usually called 'initiation phase', lasts a long time in comparison with the warp laminates one. The test duration is between 0.6 Â 10 6 and 2.1 Â 10 6 cycles and the 'initiation phase' corresponds to 70% of this period, while the test duration for warp laminates is four times lower and the first related stage is equal to 40% of this time.

These comparisons seem to show that the fatigue crack propagation in ''quasi-isotropic'' laminates amounts to crack growth in warp laminates whose initiation duration lasts a long time. Therefore, the layers at ±45°tend to slow the through-the-thickness crack initiation and its growth, at least until reaching a certain length. Then after this time period, the fatigue crack growth in ''quasi-isotropic'' laminates is similar to that found in the warp laminates. Marissen et al. [START_REF] Marissen | Fracture of quasi-isotropic composite sheets with sharp notches[END_REF] mentioned the same type of influence for ±45°plies in notched quasi-isotropic laminates.

Conclusions

Fatigue crack growth in thin woven glass laminates has been investigated in this research. A technological test specimen has been developed to get closer to the real loading conditions acting on helicopter blade skins. From the present experimental results, following conclusions can be drawn:

-in warp and weft laminates, the maximum crack growth rate level seems to be related to the crack initiation duration, namely when this duration increases the maximum rate decreases. As this time period does not vary for several samples, the crack growth rate seems to be independent of the tow type (warp or weft). Nevertheless, weft laminates present longer crack initiation duration due to their earlier ability to matrix damage; -in bias laminates, an amount of matrix damage avoids the crack initiation in front of the notch tip. This phenomenon leads to a modification of the technological test specimen in order to observe a crack growth; -in ''quasi-isotropic'' laminates, the crack growth is similar to warp laminates qualitatively and quantitatively except the 'initiation phase' which lasts more. 45°woven plies seem to slow down the crack growth for crack lengths less than 3 mm; -in each previous laminate, the microstructure influence is emphasized. The cross-over points of the woven ply are ideal places for fibres tow failure. Hence, the weave pattern influences the local crack path. Moreover, the crack propagates perpendicular to yarn fibres as the global crack path remains straight. It also seems that tow fibre always fails entirely. The through-the-thickness crack extends in a discrete manner and is linked to tow width. Typical matrix damage of woven ply also surrounds the crack path. On one hand it consists of meta-delaminations, while on the other hand, thin and long matrix cracks parallel to the fibres are located in the interstice between two consecutive tows, where their length is limited by cross-over points.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Typical helicopter blade section.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. General and schematic views of the experiment setup.

Fig. 6 .Fig. 7 .Fig. 8 . 5 Fig. 9 .

 67859 Fig. 6. Crack growth rates versus crack length in warp [0] 2 and weft [90] 2 laminates in samples of 50 mm width.

Fig. 10 . 6 Fig. 11 .

 10611 Fig. 10. Fatigue crack growth in [45] 2 laminates with two carbon strips.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Crack path in relation to woven fabric scheme in [45] 2 notched laminate loaded in fatigue.

Table 1

 1 Key results for fatigue crack growth in warp [0] 2 and weft [90] 2 laminates.

		Notch length (mm)	Initiation duration (cycles)	Maximum recorded crack growth rate (mm/cy.)
	Sample width 30 mm
	Warp 5	2.9	1.3 Â 10 6	2.6 Â 10 À5
	Warp 4	2.4	2.0 Â 10 5	2.0 Â 10 À4
	Weft 4	2.8	11 Â 10 6	4.0 Â 10 À6
	Weft 2	2.9	6 Â 10 5	9.7 Â 10 À5
	Sample width 50 mm
	Warp 1	7.1	2.0 Â 10 5	3.1 Â 10 À4
	Warp 2	6.9	6.5 Â 10 4	5.0 Â 10 À4
	Warp 3	7.0	5.0 Â 10 4	3.9 Â 10 À4
	Weft 1	6.9	2.4 Â 10 5	2.6 Â 10 À4
	Weft 2	6.9	1.8 Â 10 5	3.7 Â 10 À4


Leonardo Sanches
email: leonardo.sanches@isae.fr

Guilhem Michon
email: guilhem.michon@isae.fr

Alain Berlioz
email: berlioz@cict.fr

Daniel Alazard
email: daniel.alazard@isae.fr

Instability zones for isotropic and anisotropic multibladed rotor configurations

Keywords:

Helicopter ground resonance is an unstable dynamic phenomenon which can lead to the total destruction of the aircraft during take-off or landing phases. The earliest research in this domain was carried out by Coleman and Feingold during the decade of 60s. The instability was predicted by using classical procedures once the rotor was considered as isotropic, consequently, the periodic equations of motion could be simplified to a system with constant coefficients by introducing a change of variables, known as the Coleman Variable Transformation. The goal of the present work is to further comprehend the phenomenon and the influence of the anisotropic properties of rotors by analyzing the periodic set of equations of motion. For this, Floquet's Theory (Floquet's Method -FM) is used. The analysis for predicting the ground resonance phenomenon in isotropic and anisotropic rotor configurations is explored. The conclusions lead to verify the appearance of bifurcation points depending on the anisotropic characteristic present in the rotor. The temporal response analysis in the motion of helicopter with one asymmetric blade at unstable regions highlighted the presence of non symmetric rotor deformation shapes.

Introduction

The stability of the helicopter dynamics on ground was first studied by Coleman and Feingold [START_REF] Coleman | Theory of self-excited mechanical oscillations of helicopter rotors with hinged blades[END_REF]. They observed that the ground resonance phenomenon consists of a self-excited oscillation caused by the interaction between the lagging motion of the rotor blades and other modes of helicopter motion. The prediction of the unstable regions with isotropic rotor configurations was easily achieved once the periodic characteristics of the governing equations of motion disappeared by a change of variables. This process, applicable only to isotropic rotors, is identified as the Coleman Transformation and more generally the multi blade coordinate transformation.

Later on, research works were carried out to verify the influence of the air resonance on the stability of helicopters on ground [START_REF] Donham | Ground and air resonance characteristics of a soft inplane rigid rotor system[END_REF][START_REF] Lytwyn | Airborne and ground resonance of hingeless rotors[END_REF]. Major contributions in understanding the ground resonance phenomenon in hingeless and bearingless rotors were made by Army researchers, for example, Dawson [4] and Hodges [START_REF] Hodges | An aeromechanical stability analysis for bearingless rotor helicopters[END_REF].

Recently, the influence of nonlinear springs and dampers on the instability zones was analyzed by Kunz [START_REF] Kunz | Nonlinear analysis of helicopter ground resonance[END_REF]. Byers and Gandhi [START_REF] Byers | Rotor blade with radial absorber[END_REF] explored the passive control of the problem. Also, Duchemin [START_REF] Duchemin | Dynamic behaviour and stability of a rotor under base excitations[END_REF] studied the dynamic behavior of flexible rotors in onboard systems.

In all the above mentioned works, only symmetric rotors have been examined. Nevertheless, the study of anisotropic rotors is very interesting from a practical point of view. Indeed, due to aging, the properties of stiffness and damping can vary from one blade to another.

With the objective of further understanding the ground resonance phenomenon, the analysis is extended to verify the influence of dissimilarities in blades on the stability of the phenomenon. For this, Coleman Transformation is no more valid, therefore, the treatment of the periodic equations of motion is envisioned.

By keeping the periodic terms in the equations, Floquet's Theory [START_REF] Nayfeh | Nonlinear Oscillations[END_REF] (Floquet's Method -FM) was employed to carry out the stability analysis of helicopters with isotropic rotor and the results were compared with those obtained by applying the Coleman Transformation [START_REF] Sanches | Modélisation Dynamique d'un Rotor sur Base Flexible (Dynamical Modelling of a Rotor over a Flexible Base)[END_REF]. Also, Chopra and Wang [START_REF] Wang | Dynamics of helicopters in ground resonance with and without blade dissimilarities[END_REF] have studied the effects of blade dissimilarities. However they considered few examples and small dissimilarities on the analysis.

The same theory was applied to the vibration analysis of asymmetric rotating machines [START_REF] Lazarus | A 3D finite element model for the vibration analysis of asymmetric rotating machines[END_REF][START_REF] Genta | Vibration dynamics and control[END_REF] and on the dynamic system with parametric excitations [START_REF] Dufour | Parametric instability of a beam due to axial excitation and to boundary conditions[END_REF].

Moreover, new routs of research have been introduced recently by an extension of Floquet's Theory to treat time-delayed models of dynamical systems with parametric excitations [START_REF] Garg | Stability of a time-delayed system with parametric excitation[END_REF][START_REF] Insperger | Stability chart for the delayed Mathieu equation[END_REF].

The present paper uses the Floquet Theory to predict and study the ground resonance phenomenon through illustrations of stability maps for isotropic and anisotropic rotors. A wide range of anisotropic rotors were considered and high level of blade asymmetries.

A mechanical model which is used to develop the dynamic equations of motion is presented in Section 2. In Section 3, the applied methodology -Floquet's Methodis briefly described. In Sections 4 and 5, the predicted critical rotor speeds are presented for an isotropic and anisotropic rotor and compared.

1.1. Nomenclatures a [m] Rotor eccentricity b [m] Blade length F ext External force vector G, K, M
Damping, stiffness and mass matrix of the dynamical system

Iz b k [kg m 2 ]
Lag rotational inertia of the k th blade around its center of gravity

K b k [N m] k th blade lead-lag stiffness K f_X , K f_Y [N m -1 ]
Longitudinal and transversal stiffness of fuselage m f , m b k

[kg] Fuselage mass and mass of k th blade r a k ffiffiffiffiffiffiffiffi ffi

ar bk p r m k [m]
Ratio between the static moment of the k th blade over the total mass of the helicopter

r b k [m -1 ]
Ratio between the static moment over the total lead-lag rotational inertia of the k th blade S, v

State matrix and vector of Eq. (4) u General variables x(t), y(t)

[m] Longitudinal and transversal displacement of the fuselage x b_x , y b_x

[m] Blade's position in x and y directions (x, y, z)

Mobile coordinate system attached to the rotor hub (X 0 , Y 0 , Z 0 )

Mechanical model

Similar to that proposed by Coleman and Feingold, the present mechanical model is developed to characterize the dynamic behavior of a helicopter with a hinged rotor. In other words, it consists of figuring out the relation between the longitudinalx(t)and lateral displacementy(t)of the fuselage, and the k th blade lag angle -φ k (t)in terms of the rotor speed Ω and time t. Fig. 1 illustrates a general schema of the system.

The fuselage is considered to be a rigid body with its center of gravity at point O. At the initial time, the origin of an inertial coordinate system (X 0 ,Y 0 ,Z 0 ) is coincident at this point. The body is connected to springs which represent the flexibility of the landing skid.

The rotor head system consists of an assembly of one rigid rotor hub with Nb blades. Each blade is represented by a concentrated mass located at a distance b from the lag articulation (point B) and, on each articulation, a torsional spring is present. The origin of a mobile coordinate system (x, y, z), parallel to the inertial one, is located at the geometric center of the rotor hub (point A). Both, body and rotor head, are joined by a rigid shaft, the aerodynamical forces on the blades are neglected and any viscous damping is taken into account.

The governing equations of motion

In order to obtain the governing equations of motion, following assumptions are made:

• The fuselage has a mass m f and the spring stiffness connected to it K f -X and K f -Y through x and y directions, respectively; • The rotor is composed of Nb = 4 blades and each blade k has an azimuth angle of θ k = 2π(k -1)/Nb with the x-axis;

• Each blade has the same mass m b k and moment of inertia around the vertical axis is Iz b k ;

• The angular spring stiffness for each k blade is K b k ;

• The k th blade position projected in the inertial coordinate system is:

x b k = a cos Ω t + θ k ð Þ+ b cos Ωt + θ k + φ k t ð Þ ð Þ + x t ð Þ ð1Þ
y b k = a sin Ω t + θ k ð Þ+ b sin Ωt + θ k + φ k t ð Þ ð Þ + y t ð Þ ð2Þ
where a is the hinge offset. The general variable u, which corresponds to the degrees of freedom of the system, is considered as follows,

u 1::6 t ð Þ f g= x t ð Þ y t ð Þ φ 1 t ð Þ φ 2 t ð Þ φ 3 t ð Þ φ 4 t ð Þ f g ð3Þ
The equations of motion are then obtained by applying Lagrange equation to the kinetic and potential energy expressions of the system, given in Appendix A.

These equations are obtained by following a first order Taylor's series expansion of the trigonometric blade lead-lag angle terms and then only the linear terms are considered. In a matrix form, these equations are presented as below,

M ü+ G u + Ku = F ext ð4Þ
where M, G and K are the linear periodic mass, damping and stiffness matrices, respectively. These are non-symmetric and nondiagonal matrices due to the presence of parametric terms. Moreover, F ext is the external force vector which is equal to zero for an isotropic rotor configuration. All of these matrices are given in Eqs. ( 5)-( 8), in the following.

M = 1 0 -r m 1 sin Ω t + θ 1 ð Þ -r m 2 sin Ω t + θ 2 ð Þ -r m 3 sin Ω t + θ 3 ð Þ -r m 4 sin Ω t + θ 4 ð Þ 0 1 r m 1 cos Ω t + θ 1 ð Þ r m 2 cos Ω t + θ 2 ð Þ r m 3 cos Ω t + θ 3 ð Þ r m 4 cos Ω t + θ 4 ð Þ -r b 1 sin Ω t + θ 1 ð Þ r b 1 cos Ω t + θ 1 ð Þ 1 0 0 0 -r b 2 sin Ω t + θ 2 ð Þ r b 2 cos Ω t + θ 2 ð Þ 0 1 0 0 -r b 3 sin Ω t + θ 3 ð Þ r b 3 cos Ω t + θ 3 ð Þ 0 0 1 0 -r b 4 sin Ω t + θ 4 ð Þ r b 4 cos Ω t + θ 4 ð Þ 0 0 0 1 2 6 6 6 6 6 6 4 ð5Þ
G = 0 0 -2 Ω r m 1 cos Ω t + θ 1 ð Þ -2 Ω r m 2 cos Ω t + θ 2 ð Þ -2 Ω r m 3 cos Ω t + θ 3 ð Þ -2 Ω r m 4 cos Ω t + θ 4 ð Þ 0 0 -2Ω r m 1 sin Ω t + θ 1 ð Þ -2Ω r m 2 sin Ω t + θ 2 ð Þ -2Ω r m 3 sin Ω t + θ 3 ð Þ -2Ω r m 4 sin Ω t + θ 4 ð Þ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2
K = ω 2 x 0 Ω 2 r m 1 sin Ω t + θ 1 ð Þ Ω 2 r m 2 sin Ω t + θ 2 ð Þ Ω 2 r m 3 sin Ω t + θ 3 ð Þ Ω 2 r m 4 sin Ω t + θ 4 ð Þ 0 ω 2 y -Ω 2 r m 1 cos Ω t + θ 1 ð Þ -Ω 2 r m 2 cos Ω t + θ 2 ð Þ -Ω 2 r m 3 cos Ω t + θ 3 ð Þ -Ω 2 r m 4 cos Ω t + θ 4 ð Þ 0 0 ω 2 b 1 + Ω 2 r 2 a 1 0 0 0 0 0 0 ω 2 b 2 + Ω 2 r 2 a 2 0 0 0 0 0 0 ω 2 b 3 + Ω 2 r 2 a 3 0 0 0 0 0 0 ω 2 b 4 + Ω 2 r ð6Þ
F ext = ∑ Np k = 1 Ω 2 r m k a + b b cos Ω t + θ k ð Þ ∑ Np k = 1 Ω 2 r m k a + b b sin Ω t + θ k ð Þ 0 0 0 0 ð7Þ
where

r m k = b m b k m f + ∑ Np k = 1 m b k ; r b k = b m b k b 2 m b k + Iz b k ; r 2 a k = a r b k ω 2 x = K f X m f + ∑ Np k = 1 m b k ; ω 2 y = K f Y m f + ∑ Np k = 1 m b k ; ω 2 b k = K b k b 2 m b k + Iz b k k = 1::Np:
It is noted that r m k and r b k factors, which represent the ratio between the blade static moment over the total inertial mass of the helicopter and the total rotational inertia of the blade, respectively, are smaller when compare to unity.

The terms ω x and ω y are the resonance frequencies of the fuselage in the x and y directions, respectively. Moreover, ω b 3..6 are the lead-lag resonance frequencies of blades 1 to 4.

Floquet's method

In this section, a stability analysis of the periodic equations of motion is carried out in order to determine the critical rotor speeds at which the unstable oscillations in helicopter may appear. The Floquet's Theory is used.

Representing the dynamical system in a state-space format and considering S(t) the state-space matrix with period T of Eq. (4), the system becomes,

v t ð Þ = S t ð Þv t ð Þ; t N t 0 v t 0 ð Þ = v 0 ; v t ð Þ = u t ð Þ u t ð Þ Â Ã T ð9Þ
where v(t) is the state variable. By Floquet's Theory, a transition matrix Φ, which relates v(t) with v(t 0 ), is defined as,

Φ t; t 0 ð Þ= P t; t 0 ð Þe t-t 0 ð ÞQ : ð10Þ
The monodromy matrix R or the Floquet Transition Matrix (FTM), defined as R = Φ(t 0 + T, t 0 ) is computed by admitting the periodic system in Eq. ( 4) as a switched periodical system. As reported Berlioz et al. [START_REF] Dufour | Parametric instability of a beam due to axial excitation and to boundary conditions[END_REF], the matrix S(t) is approximated by a series of p step functions over the period T and the monodromy matrix is then obtained as,

R = ∏ p k = 1 exp S k Δt ð Þ ð11Þ
where S k , within the interval Δt = t kt k -1 , is constant and defined at S(t k -1 ).

The dynamical system Eq. ( 4) is exponentially stable if R is Schurz. It means that the norm of the eigenvalues (λ) of R, known as characteristic multipliers, is smaller than unity.

Numerical results

This section aims at predicting the boundaries of critical regions at which the system is unstable for one isotropic and a wide range of anisotropic rotor configurations. For this the Floquet's Method is used.

Isotropic rotor configuration

The numerical values used for the system's inputs are defined in Table 1. It is important to notice that the subscript k in the terms is eliminated once the rotor is isotropic.

Table 1

Numerical values of the fuselage and rotor head inputs.

Fuselage

Rotor

m f = 2902.9 [kg] m b = 31.9 [kg] a = 0.2 [m] b = 2.5 [m] ω x = 3 [Hz] ω y = 3 [Hz] ω b = 1.5 [Hz] I zb = 259 [kg m 2 ]
Fig. 2. Evolution of the characteristic multipliers as function of the rotor speed.

For every rotating speed defined between 0 and 10 Hz by a step of 0.02 Hz, the FM is applied and the characteristic multipliers are obtained. Fig. 2 presents the evolution of biggest norm value of the characteristic multipliers with respect to the speed of rotation. The period T is divided in p = 64 parts.

Once the instability zone is determined through Fig. 2, the boundary limits (beginning and ending points) of the region are then collected and presented in Table 2.

Considering the Coleman Transformation expressions described in Ref. [START_REF] Sanches | Modélisation Dynamique d'un Rotor sur Base Flexible (Dynamical Modelling of a Rotor over a Flexible Base)[END_REF] and applying them on Eq. ( 4), the final equations of motions and the criteria of stability are presented in Appendix B.

Using the data defined in Table 1 and analyzing the stability of the dynamical system for every rotating speed varying from 0 to 10 Hz by a step of 0.02 Hz, the instability zone is identified and the boundary limits found are joined in Table 2 with those obtained by using FM.

Both methods give the same instability zone.

Anisotropic rotor configuration

The interest of this subsection is to determine the instability zones where the dynamical system becomes unstable for anisotropic rotors by using the method described in Section 3.

The numerical values used for the system's inputs are the same as those defined in Table 1, except for the lead-lag resonance frequencies of blades ω b k . In Case study 1, only one blade is modified. Finally, for Cases 2 and 3, two blades are altered by considering them in adjacent and opposite locations, respectively.

• Case 1:

Three blades are considered having the same lead-lag resonance frequency equal to 1.5 Hz, while the last one is varied from -100% to + 100% by a step of 9% of the blade lead-lag frequency indicated previously. Similar to the procedure applied for an isotropic rotor and for each specific case of anisotropy, FM is applied. The evolution of the characteristic multipliers along with the rotor speed is obtained, the instability zones are determined and only the boundary speeds are collected. The same value for p = 64 parts is considered. Fig. 3 shows the evolution of unstable regions as a function of the rotor speed and the asymmetry parameter (Δω 4 ). • Case 2:

In this case, two adjacent blades (3rd and 4th blades) have been altered from -100 to +100% by a step of 9% of the reference value of 1.5 Hz which is the blade lead-lag resonance frequency considered for the other blades. By repeating the procedure used in Case study 1, Fig. 4 illustrates tri-dimensionally the limits of the instability zones as function of the adjusting parameters and the rotor speed. The gray color represents the unstable regions.

Introducing different constant values for ω b 3 (-100%, -52.381%, 52.381% and 100%), Fig. 5 shows the evolution of the instabilities zones as function of Δω b 4 for each rotor configuration, similar to the procedure used in Case 1.

Legend:

Starting point Ending point Fig. 3. Helicopter stability analysis by considering asymmetries on the 4th blade.

• Case 3:

In this case, the same procedure is carried out as discussed previously. However, the blades which have been modified are in opposite positions (2nd and 4th blades).

Discussion

Regarding the stability analysis for isotropic helicopter as in Fig. 2, the results highlight the presence of unstable oscillations at 4.35 ≤ Ω ≤ 5.18 Hz. Chopra in his work [START_REF] Wang | Dynamics of helicopters in ground resonance with and without blade dissimilarities[END_REF] explained that this instability is due to the coalescence frequencies between fuselage and an asymmetric rotor mode shape (forward whirling mode) at a specific range of the rotor speed.

Legend:

Starting point Ending point Considering the influence of asymmetries in one blade in Case 1, Fig. 3 illustrates the presence of a main instability region at 4.4 ≤ Ω ≤ 5.2 Hz which is not altered for almost all values of Δω b 4 . It corresponds exactly to the same critical region found for an isotropic rotor.

Also, another unstable region is observed at low rotor speeds for a wide range of asymmetries blade. The critical rotating speed is around 3 Hz which corresponds to the natural frequency of the fuselage.

Nevertheless, the two braches come out from the main region defining the evolution of instability zones as function of blade dissimilarity (Δω b 4 ). This is shown in the figure like an "x" shape with the main instability region.

Referring to the specific case where Δω b 4 = -60%, Fig. 8 illustrates the deformation shapes of helicopter at the instants of maximal amplitudes of longitudinal and lateral displacement of the fuselage for each unstable regions. These deformation shapes are obtained from the temporal responses by using a numerical step-by-step integration method (see Appendix C).

Legend:

Starting point Ending point In order to help the visualization of the figure, a gain of 40 in case a) and 20 in remainder cases are applied in all deformation amplitudes. The intersection point between two dashed red lines corresponds to the rest point of the helicopter and the dashed black lines indicate the level zero of blade lead-lag oscillations.

Moreover, from the analysis of deformation shapes and temporal responses, one verifies a non symmetric rotor deformation in all cases a) to c). For Ω = 2.929 and 3.945 Hz, high oscillations of the dissymmetric blade are observed when compared to other blades. Nevertheless, the sine signals which describe the motion of blades present mean values different from zero in the first case; whereas, in the second case, are zero for all blades.

In case c) where Ω = 4.797 Hz, the amplitude of motion of the dissymmetric blade is smaller than the others. One remarks also the highest amplitude of oscillations for the blade situated in opposition with respect to the asymmetric blade.

Regarding the influence of other blades dissimilarities (Cases 2 and 3), examining the obtained results and comparing to the first case, the same characteristics for the instability regions were obtained in both cases: a main region at 4.4 ≤ Ω ≤ 5.2 Hz, another at low speeds for almost all values of Δω b 4 and two branches coming out of the main region.

Nevertheless, due to the changes in two blades, the appearance of a new main region is also detected which moves depending on the values attributed to Δω b 3 or Δω b 2 . As they increase, the new region moves to high values of the rotor speed.

Small differences between Cases 2 and 3 can be found in the boundary values of the new main unstable region and in the instability zone at a low rotor speed.

Conclusions

For the comprehension and analysis of the ground resonance phenomenon in helicopter with lead-lag articulated bladed rotors, this paper focused on predicting the critical rotor speeds at which unstable oscillations may occur for an isotropic and wide range of anisotropic rotor configurations. For this, Floquet Theory was used.

For the isotropic rotor configuration, the numerical results showed one well defined instability zone at rotor speed values between 4.35 and 5.18 Hz. However, in order to determine the influence of blade dissimilarities on the stability of ground resonance phenomenon in helicopters, three case studies were performed. For the first case only one blade was altered and, for the second and third cases, two blades in adjacent and in opposite positions, respectively, were modified.

The investigated asymmetries were achieved by varying the blade lead-lag resonance frequency in percentage from -100 to +100 related to the reference values of an isotropic rotor.

Maps of stability were determined for these types of rotors and a complex evolution of instability zones, beyond the appearance of new critical regions, was observed.

Nevertheless, similarities in the stability maps of helicopters with isotropic and anisotropic rotors were found. In all cases of anisotropic rotor configurations, the results verified the presence of the same instability zone found in the isotropic rotors.

Analysis on the deformation shapes on each unstable region of a helicopter with one asymmetric blade was carried out. The deformations, obtained from the temporal responses, highlighted the influence of the dissymmetric blade.

Regarding the similarities between dissymmetric rotors, a common critical region is observed at low values of the rotor speed. Moreover, small differences were noted between Case studies 2 and 3.

Appendix A

In the present section, the kinetic and potential energies expressions of the dynamical system are presented.

• Kinetic energies:

The total kinetic energy T Tot expression of the dynamical system are given as

T Tot = T Fus + T RH ðA: 1Þ
where T Fus and T RH are the kinetic energy expressions of the fuselage and rotor head system, respectively. These energy expressions are presented below as

T Fus = m f 2 ẋ2 + ẏ2 ðA:2Þ T RH = 1 2 ∑ Np k = 1 m b k ẋ2 bPk + ẏ2 bPk = 1 2 m b k … ∑ Np k = 1 ẋ2 + ẏ2 + b 2 φ2 k + 2b 2 Ω φk + b 2 Ω 2 a 2 + b 2 2 a bcos φ k ð Þ Ω 2 + Ω φk h i + 2a Ω -ẋ sin Ω t + θ k ð Þ+ ẏ cos Ω t + θ k ð Þ ½ 2 b Ω ẏ + ẏ φk ð Þ cos Ω t + θ k + φ k ð Þ -2 b Ω ẋ + ẋ φk ð Þ sin Ω t + θ k + φ k ð Þ 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; :
ðA:3Þ

• Potential energies:

The total potential energy U Tot expression of the dynamical system are given as

U Tot = U Fus + U RH ðA: 4Þ
where U Fus and U RH are the potential energy expressions of the fuselage and rotor head system, respectively. These energy expressions are presented below as ðB:2Þ

U Fus = 1 2 K f -X x 2 + K f -Y y 2 ðA:5Þ U RH = 1 2 ∑ Np k = 1 K b k φ
Note that the subscript k is eliminate from Eq. ( 4) once it is isotropic rotor. The dynamical system in Eq. (B.1) is stable at a rotating speed Ω if the real part of all eigenvalues is defined negative [START_REF] Coleman | Theory of self-excited mechanical oscillations of helicopter rotors with hinged blades[END_REF][START_REF] Sanches | Modélisation Dynamique d'un Rotor sur Base Flexible (Dynamical Modelling of a Rotor over a Flexible Base)[END_REF].

Appendix C

Through a numerical step-by-step integration method ("Adams113") and by using MATLAB®, this section presents the temporal responses of a helicopter with one dissymmetric blade Δω b 4 = -60%.

The rotating speeds values considered for the analyses correspond to the mean rotor speeds values at each unstable region observed in Fig. 3. They are, in Hz, equal to 2.929, 3.945 and 4.797.

The initial conditions imposed are 0.1°for the lead-lag oscillation of 4th blade, and zero for all remainder amplitudes and speeds.

Fig. 1 .

 1 Fig. 1. General schema of the mechanical model.

Fig. 6

 6 illustrates, tri-dimensionally, the limits of the instability zones as a function of Δω b 2 and Δω b 4 . Moreover, assuming constant values for ω b 2 (-100%, -52.381%, 52.381% and 100%), as in previous case, Fig. 7 illustrates the evolution of the instabilities zones as function a of Δω b 4 .

Fig. 4 .

 4 Fig. 4. Helicopter stability analysis by considering asymmetries on the 3rd and 4th blades.

Fig. 5 .

 5 Fig. 5. Anisotropic rotor analysis for the 4th blades considering different values of ω b 3 .

Fig. 6 .

 6 Fig.6. Helicopter stability analysis by considering asymmetries on the 2nd and 4th blades.

Fig. 7 .

 7 Fig. 7. Anisotropic rotor analysis for the 4th blades considering different values of ω b 2 .

Fig. 8 .

 8 Fig. 8. Deformation shapes of an (Δω b 4 = -60%) for Ω, in Hz, equal to a) 2.929 Hz, b) 3.945 and c) 4.797 at the maximum 1) longitudinal and 2) lateral displacement.

  In this appendix, the periodic equations of motion are transformed by using Coleman Transformation. The final governing equations are obtained and the criteria of stability are presented.Once the Coleman Transformation[START_REF] Sanches | Modélisation Dynamique d'un Rotor sur Base Flexible (Dynamical Modelling of a Rotor over a Flexible Base)[END_REF], defined as:η = -2 Np ∑φ k sin Ωt + θ k ð Þ and ζ = -2 Np ∑φ k cos Ωt + θ k ð Þ ð B:1Þis applied on Eq. (4), the below a matrix differential equation is obtained.

  Fig. C.1 illustrates the results obtained at each unstable region.

Fig. C. 1 .

 1 Fig. C.1. Temporal Responses at Ω, in Hz, equal to a) 2.929 Hz, b) 3.945 and c) 4.797 for a one blade dissymmetric helicopter (Δω b 4 = -60%).

Table 2

 2 Limits of instability zones predicted by the Floquet's Method in Hz.

	Critical rotor speed	Limits of instability zone [Hz]
	Floquet	4.35	5.18
	Coleman	4.357	5.187