Abstract:
A drag damper for use on a rotary-wing aircraft rotor comprises a body defining two variable volume chambers linked by a piston. The chambers filled with fluid in the damper are connected by a restrictor port in the piston or between the latter and the body, and via a channel of great length and small cross-section compared with the cross-section of the body, in which elastic means bear against and load the piston-rod assembly towards a neutral position, the anti-resonance frequency of the damper being matched substantially to the nominal rotation frequency of the rotor, and the restrictor port providing effective damping at the natural frequency (ωδ) of the blades in drag, differing by construction from the rotor frequency (Ω).

Description:
BACKGROUND OF THE INVENTION 
   The invention relates to a drag damper designed to be fitted between the hub of a rotary-wing aircraft rotor and a flapping mass, which comprises one of the blades of the rotor and a device connecting this blade to the hub so as to damp the angular drag movements of said flapping mass with respect to the hub, i.e. the angular deflections of the blade, and more globally of the corresponding flapping mass, about their drag axis which is substantially parallel to the axis of rotation of the rotor; 
   The rotor is more particularly a helicopter main rotor subject to the instability phenomena known as “ground resonance” and “air resonance”, although a conventional tail rotor may also be equipped with drag dampers according to the invention. 
   On rotors of the hinged type, the device connecting a blade to the hub may be arranged as a means of securing the blade and hinging it to the hub, when the blade is connected by its root, possibly in the form of a fork, to the hub, or as a device which is substantially radial (relative to the rotor axis) generally termed a cuff, and fitted with yokes at the ends to be connected to the blade root on the one hand and on the other to means of securing and hinging, such as a spherical laminated stop, itself connecting it to the hub, while on rotors of the semi-rigid type, this connecting device may be a flexible torsion arm, at the blade root, and surrounded by a torsionally rigid cuff integral with the blade root for controlling the blade in pitch, which is connected and hinged to the hub by this flexible torsion arm. 
   Numerous different embodiments of drag dampers are known, particularly dampers which are hydraulic, hydro-pneumatic, laminated with at least one layer of visco-elastic material stressed between two rigid fittings, or comprising combinations of these different means, these drag dampers comprising means of elastic return of defined stiffness and damping, when they are fitted to helicopter main rotors, to combat the resonance phenomena mentioned above. 
   It is a well-known practice to design into helicopter rotor blades and therefore into the corresponding flapping masses a natural drag frequency, also termed first drag mode or natural drag mode, which is different from the nominal rotation frequency at which the rotor is designed to be driven. 
   More generally, to avoid in particular fatigue problems resulting from the dynamic stresses in the blades and the fuselage, and problems of vibration levels in the fuselage, it is essential to position correctly the natural frequencies of the blades in flapping, torsion and drag relative to the nominal rotation speed of the rotor and its harmonics (multiples). 
   This results from the fact that a helicopter rotor constitutes a powerful vibration generator. Because of the variable angles of incidence and speeds of rotor blades and also of helicopters, alternating loads of aerodynamic origin are developed notably in the blades of rotors, and give rise in the latter to stresses as well as reactions on the attachments, particularly of the blades to the hubs. From this there result alternating loads and moments on the rotor heads, and the development of high vibration and stress levels in fuselages. The response of each blade, the stresses to which this blade is subjected and the loads which this blade transmits to the hub at the centre of the rotor are all the greater as at least one natural frequency of the blade (in drag, flapping and torsion) is close to the rotation frequency Ω of the rotor or one of its harmonics nΩ (where n is a whole number). 
   The dynamic characteristics of the rotor blades are therefore chosen to obtain suitable dynamic matching by ensuring that their natural vibration frequencies in flapping, drag and torsion are correctly positioned relative to the nominal rotation frequency Ω of the rotor and its multiples nΩ, which is why it is necessary to observe certain simple rules for positioning the frequencies, and in particular two essential rules, the first of which is to avoid positioning a natural vibration frequency in flapping, drag or torsion on or very close to a harmonic of the rotation speed nΩ (where n≧1), and the second to endeavour as far as possible to position only one of these three natural frequencies between two adjacent harmonics nΩ and (n+1) Ω of the rotation speed in order to avoid coupling. 
   In addition to these two essential rules, it is imperative to follow recommendations proper to each type of deformation in flapping, drag or torsion. 
   Concerning the recommendations relating particularly to the drag modes for hinged or semi-rigid (semi-hinged) rotors, the first drag mode (or natural drag frequency) is at the origin of “ground resonance” and “air resonance” problems due to coupling with modes of the helicopter structure. 
   On a rotor with blades hinged in drag, the angular frequency or pulsatance of the first drag mode is given by the expression: 
               ω   ⁢           ⁢   δ     =       Ω   ⁡     (         e   ·   m     ⁢           ⁢   δ       1   ⁢   δ       )         1   /   2             
 
where e is the drag eccentricity of each blade, mδ is the static moment of the flapping mass (blade+device connecting it to the hub) relative to the hinge (drag axis) and Iδ is the inertia of the flapping mass relative to this drag hinge.
 
   On a semi-rigid rotor, the first drag mode of a blade or flapping mass depends on the characteristics not only of the blade or flapping mass but also of the hub. The pulsatance of the first drag mode is then given by the expression: 
               ω   ⁢           ⁢   δ     =       Ω   ⁡     (           e   ·   m     ⁢           ⁢   δ       1   ⁢   δ       +       k   ⁢           ⁢   δ         Ω   2     ⁢   1   ⁢           ⁢   δ         )         1   /   2             
 
where kδ is the stiffness of the drag damper fitted between the blade or corresponding flapping mass and the hub of the rotor.
 
   The positioning of the first drag mode of a blade or of the corresponding flapping mass depends upon the modes of the helicopter structure (fuselage mass, inertia, stiffness of the landing gear and of any tyres which may be fitted to it), these modes of the structure being generally determined by specific tests, adjustment of the first drag mode being obtained by altering the term kδ representing the stiffness of the drag damper. 
   As a general rule, as the upper limit of the first drag mode ωδ, a value close to three-quarters of the nominal rotation frequency Ω of the rotor is taken, so as not to introduce excessively high stresses in the blades of the rotors. One of the other two natural vibration frequencies (in flapping and in torsion) of the blade or flapping mass is placed between Ω and 2Ω, and the other, as far as possible, between 2Ω and 3Ω. 
   For this reason, when the rotor is started up or stopped, and also at the end of a landing by the helicopter in autorotation, the instantaneous speed of rotation of the rotor intersects the resonance frequency in drag situated below the nominal speed. Because of this, and also because of the fairly large range of variations in rotor rotation speeds which are authorised for helicopters in flight, it is necessary to increase damping at the natural vibration frequencies of the blades in drag, and possibly to reduce this natural frequency by means of drag dampers, which is the reason why these dampers are also termed frequency adapters, the aim being that the blades should be sufficiently damped in drag to avoid going into resonance. 
   The invention relates more specifically to a drag damper of the general type comprising a tubular damper body in which a piston moving integrally with a damper rod and slidable axially is fitted, and in the damper body delimits two opposing variable volume working chambers, filled with a fluid of which volumes are transferred, by restriction of the fluid by at least one flow-restriction port arranged between the piston and the body and/or in the piston, between the working chambers when the piston moves in the damper body, upon which elastic means bear and load the piston and/or the rod so as to return the rod-piston assembly towards a neutral position in the body. 
   A drag damper of this kind, known in particular from FR 2 063 969, may be hinged on the one hand to a fixed point on the hub or on a bracket connected to the rotor hub, by means connecting the body or one end, external to said body, of the damper rod and on the other to a fixed point on the corresponding flapping mass, at the blade root of this flapping mass or on a device connecting this blade to the hub, by means connecting the end of the rod external to the body or the damper body respectively. 
   As the hinging point of one end of the drag damper on the hub or a bracket fixed to the hub is situated between the blade on which the drag damper is hinged at its other end and an adjacent blade, the stiffness of the damper introduces an equivalent angular stiffness, opposed to the angular deflections of the blade relative to the hub about its drag axis. It is thus possible to increase the natural frequency of the blades in drag to escape from the two resonance phenomena mentioned above, with additional damping at the natural drag frequency ωδ of the blade when the phenomena of air and ground resonance occur. 
   However, it is known these phenomena rarely appear during the life of a helicopter. Most of the time, the drag dampers are subject to forced excitation at the rotation frequency Ω of the rotor, on which the drag dampers dissipate energy to no purpose. 
   The mean power dissipated in a drag damper of a rotor can be expressed by the following relation: Pd=π·K″·f·Xe 2 , where K″ is the dissipative stiffness of the damper, f the frequency of the movement applied to the damper (axial movement of the rod-piston assembly in the cylinder) and Xe is the movement of said rod-piston assembly associated with frequency f. 
   For example, for drag dampers of the type presented above fitted to a four-bladed main rotor of a helicopter with a weight of about 8 to 10 tonnes, a comparison of the energy dissipated on the forced excitation at the rotation frequency Ω with that which is dissipated on the natural frequency in drag ωδ of the blades gives the following results. 
   For the same dissipative stiffness K″ of 400 daN/mm, the forced excitation at Ω with a frequency f of 4.5 Hz corresponds to an associated displacement Xe of 4 mm, that is to say a dissipated power of 900 W, whereas for the natural drag mode ωδ at a frequency f of 2 Hz, causing an associated displacement Xe of 1 mm, there corresponds a dissipated power of 30 W. 
   Each drag damper therefore dissipates 97% of its energy on forced excitation at Ω. Under these conditions of use, this energy is dissipated to no purpose, which entails substantial component fatigue, not only of the drag damper but of the means connecting it to the hub and to the flapping mass, and wasted weight due to oversizing of these parts. 
   On the helicopter with a weight of about 8 to 10 t considered, the forces applied to the drag dampers upon forced excitation at Ω are very high, which may cause incidents in service such as cracks in the yokes connecting the drag dampers to the flapping masses, damage to the drag damper spindles at the end where there are connected to the hub and also damage to the fittings connecting the drag dampers to the hub, and rapid deterioration of the ball joints used in these connection devices. 
   Moreover, FR 2 737 271 makes known a damper which senses acceleration and can be used in numerous applications in which it is necessary to damp elastic systems, and particularly as a damper for suspension systems, shock and yaw control, in transport and industry. 
   A damper of this kind comprises a tubular body in which a piston moving integrally with a damper rod and slidable axially is fitted, and in the body delimits two opposing variable volume working chambers connected to each other by at least one bypass channel, the length of which is very much greater than a main dimension of its cross-section, which is itself very much smaller than the cross-section of the body, a fluid filling at least the two working chambers in the body and the bypass channel, which can be made in the wall of the body, or in the piston, having the form of a spiral, helix or one or more concentric arcs of circle. When the piston is moved in the body, fluid is expelled and moves from one working chamber to the other using the bypass channel. By calculation, taking account of the law of conservation of fluid flows, the force to which the piston is subjected and accelerations of the piston on the one hand and of the column of fluid in the bypass channel on the other, it can be shown that the force to which the piston is subjected is proportional to the virtual fluid mass, similar to inertia, which can be maximised by adjusting the mass per unit volume of the fluid, the length of the bypass channel and above all the ratio between the cross-section of the body and the cross-section of the bypass channel. As viscous damping increases with the reduction in the cross-section and increase in the length of the bypass channel, choice of a low-viscosity fluid is indicated if the function of locking the rod-piston assembly in the cylinder by fluid inertia is to be favoured above the viscous damping function. 
   Moreover, EP 0 183 039 and GB 2 111 171 make known the practice of adjusting the damping force of a hydraulic damper of conventional type by filling its working chambers with an electro-rheological fluid circulating in a bypass channel connecting the two working chambers and subjected, in this bypass channel, to a variable electrostatic field enabling the viscosity of the fluid to be varied according to the signals from detectors such as velocity, acceleration, load detectors etc. 
   SUMMARY OF THE INVENTION 
   The idea underlying the invention is to propose a drag damper for helicopter rotor blades enabling the dynamic component at the rotation frequency Ω of the rotor to be eliminated in the damper by employing the principle of the fluid inertia damper according to FR 2 737 271, and to provide damping to the drag movement of the blades only at the natural frequency in drag ωδ, with the aim of improving the behaviour in service of all of the components constituting the damper and of the components connecting the damper to the hub on the one hand and on the other to the flapping mass, at the same time enabling the weight of the drag damper to be reduced. 
   To this end, the drag damper according to the invention is a drag damper as known from FR 2 063 969, and comprising: 
   a tubular damper body closed by two end faces, 
   a piston slidable axially fitted in the body and delimiting in and with said body two opposing variable volume working chambers, 
   a rod moving integrally with the piston and passing substantially axially through at least one end face of the body, 
   a spring bias acting on the rod and/or the piston and bearing on the body, and tending to return said rod-piston assembly to a neutral position in the body, 
   said body and said rod each comprising a connector for connecting respectively to one of the two components which are the rotor hub and said flapping mass, 
   a fluid filling at least the two working chambers in the body, and 
   at least one restriction port made in the piston and/or between the piston and the body, and restricting fluid passing from one working chamber to the other when the piston is moved in the body, 
   wherein the drag damper further comprises at least one bypass channel linking the two working chambers and filled with fluid, the length of said bypass channel being greater than a main dimension of its cross-section, which is itself smaller than the cross-section of the body, the body and the bypass channel having dimensional characteristics and said fluid having physical characteristics such that the damper has an anti-resonance frequency ωa substantially equal to the rotation frequency Ω of the rotor, so as to filter the Ω dynamic component in the loads applied to the damper, and wherein said at least one restriction port is calibrated so as to dampen appreciably the relative movements of the rod-piston assembly and of the damper body at a frequency which is substantially equal to the natural drag frequency ωδ of the flapping mass. 
   The drag damper according to the invention has the advantage of using the principle of the fluid inertia damper to filter or eliminate the Ω dynamic component in the loads it applies, by setting its anti-resonance frequency ωa to the rotation frequency Ω of the rotor, while providing damping in the natural drag mode of the blades at ωδ, to combat the problems of air and ground resonance, without degrading the filtering of the Ω component. By the anti-resonance frequency ωa of the drag damper is meant the frequency for which the force transmitted by a blade to the hub via the damper and the corresponding relative movements between the rod-piston assembly and the damper body are minimal. 
   In an advantageously simple mode of embodiment, when the bypass channel of length Lc has a constant cross-section sc, the damper body has a cross-section Sa, the spring bias has a stiffness k and the fluid has a mass per unit volume ρ, the anti-resonance frequency ωa of the damper, substantially matched to the frequency of rotation Ω of the rotor, is expressed as a function of the square root substantially of the ratio of the stiffness k of the spring bias to a virtual fluid mass Ma such that Ma=ρLc Sa 2 /sc. The relevance of such an embodiment is that its anti-resonance frequency, which is easily calculated, is independent of the mass of the blade (of the flapping mass) and of the hub, and is linked solely to the geometry of the drag damper, for a given mass per unit volume of the fluid, and therefore for a given fluid. 
   In order to match the performance of the drag damper to helicopters on which the main rotors have a variable speed of rotation, it is also advantageous that the cross-section sc of at least part of the bypass channel should be variable and controlled by a control device according to at least one rotor rotation speed signal received by said control device, which controls the variation in said cross-section sc so as substantially to match the anti-resonance frequency ωa of the damper to the rotation frequency Ω of the rotor. In the case of a bypass channel of circular cross-section, its diameter and thus its cross-section may be controlled by such an active control device. 
   According to the invention, the bypass channel may be inside the damper body. In this case, this bypass channel can be made at least partially inside the piston, and/or said spring bias may comprise at least one spring in which at least a part of said bypass channel is made. Different spring structures may be used in order to constitute both the spring bias providing stiffness, and means in which parts of the bypass channel are made, for example metal helical or coil springs, conical or cylindrical springs. 
   As a variant, the bypass channel may be at least partially external to said damper body. This variant lends itself more favourably to controlling and monitoring the cross-section of at least part of said bypass channel, when this cross-section is variable and adjustable, as indicated above. 
   In order to ensure optimum restriction of the fluid at the natural drag frequency ωδ, the piston may be drilled with several restriction ports calibrated at different cross-sections. 
   Finally, to reduce the dissipation forces linked to pressure losses and to viscous damping forces, which are unwanted forces limiting total compensation of the stiffness forces (elastic forces transmitted by the springs) by the hydraulic force at the anti-resonance frequency, it is also advantageous to be able to vary the viscosity of fluid and, to this end, and as known from EP 0 183 039 and GB 2 111 171 quoted above, the fluid used in the drag damper may be an electro-rheological fluid, the variable viscosity of which is controlled by the control of an electric and/or magnetic field to which at least part of said fluid is subjected. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other characteristics and advantages of the invention will emerge from the description given below of examples, though not limited to these, of embodiments described with reference to the attached drawings in which: 
       FIG. 1  is a schematic view in axial half-section on the left-hand half-view and in offset section on the right-hand half-view, for a four-bladed helicopter main rotor head, 
       FIG. 2  is a partial schematic view partly in section and partly in plan of the rotor head in  FIG. 1 , with a drag damper fitted between each of the four rotor blades and the hub, 
       FIG. 3  is a schematic view of the assembly of a blade, a drag damper and the rotor hub, to explain the operation of the damper, 
       FIGS. 4 and 5  are schematic views respectively in axial and radial cross-section of a first example of drag damper operating as explained with reference to  FIG. 3 , 
       FIG. 6  is a schematic view in axial section of another example of a drag damper operating according to  FIG. 3 , and 
       FIG. 7  shows two curves of the change in equivalent stiffness of the drag damper in  FIG. 3  as a function of excitation frequency, respectively without pressure loss and with pressure loss. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIGS. 1 and 2  show schematically the head of a four-bladed helicopter main rotor substantially as described in EP 0 213 016 with reference to FIGS.  4  and  5  in that document, to which reference may advantageously be made for further information. 
   It will be recalled that this rotor head comprises a one-piece tubular mast-hub  1 , the lower substantially cylindrical part of which forms a mast  2  with a base  4  designed to be connected in rotation to a main gearbox of the helicopter H (shown schematically in phantom outline) to drive the rotor in rotation about the axis A of the mast-hub  1 . The latter also comprises an upper part forming a hub  3 , which is an extension of the mast  2  and has the general external shape substantially of a truncated cone hollowed out at the end opposite to the mast  2  as far as a reinforcement ring  5  which constitutes the upper free end, thickened locally on its periphery, of the hub  3 . Radial (relative to axis A) openings  6 , identical and equal in number to the number of rotor blades, are made in the portion of hub  3  which is directly adjacent the ring  5  and are regularly distributed in a circumferential direction over the periphery of this portion of hub  3 . Each opening  6  has a substantially half-moon or greatly rounded bean shape, the general concavity of which is facing the ring  5 , which surrounds the opening in the upper end of the hub  3 . This ring  5 , the shape of which in plan (see  FIG. 2 ) is that of a pseudo-regular polygon, with small sides  5   a  equal, rounded and convex alternating with large sides  5   b  equal, rectilinear and equal in number to the number of rotor blades, is reinforced by a composite belt  5   c  in a peripheral groove in the ring  5 . Each of the openings  6  is made in the hub  3  respectively directly under one of the small sides  5   a  of the ring  5 , these convex small sides  5   a  corresponding to the portions of the ring  5  to which are secured spherical laminated stops  8  housed in the opening in the upper end of the hub  3  and constituting the devices for securing and hinging the blades  7  to the mast-hub  1 . 
   The spherical laminated stops  8  are well-known components, each hinging a blade  7  to the hub  3  about the three axes of flapping, drag and pitch change of the blade, which intersect at the pivoting point determined by the centre of rotation of the corresponding spherical laminated stop  8 . Each stop  8  mainly comprises a central laminated part  10  constituted by an alternating stack of rigid layers, of metal for example, and of a visco-elastic material, such as a synthetic rubber, and in the form of segments of a sphere, this central laminated part being vulcanised between two rigid fittings  9 ,  11 , of metal for example. 
   Each spherical laminated stop  8  is placed against the inner face of the ring  5  and is secured against the latter by its external radial fitting  9  shaped as a fork straddling the ring  5  at an opening  6  while its inner radial fitting  11 , integral with the laminated central part  10  on the side opposite the outer fitting  9  is secured by bolting as spacers between the two branches  13  and  14  of an inner radial yoke of a connecting device  12 , termed a cuff, connecting a blade  7  to the hub  3  via the corresponding spherical laminated stop  8 . In this method of assembly, the lower branch  13  of the inner radial yoke  13 - 14  of the cuff  12  passes through the corresponding opening  6 . 
   The blade  7  has its root  7   a  secured between the two branches  15  and  16  of an external radial yoke of the cuff  12  by two tubular pins  17 , parallel with each other and substantially with the rotor axis A, and passing through aligned bores in the blade root  7   a  and the branches  15  and  16  of this outer yoke. 
   In the simple form of embodiment in  FIGS. 1 and 2 , the radial connecting cuff  12  comprises two radial plates arranged substantially one above the other, the lower plate embodying the lower branches  13  and  15  and the upper plate embodying the upper branches  14  and  16  respectively of the inner and outer yokes of the cuff  12 , and these two plates have as a spacer between them the inner fitting  11  of the stop  8  and, outside the ring  5 , by a spacer  18  onto which the two plates are bolted, and which supports a laterally projecting pitch change lever  19  for controlling the pitch of the corresponding blade  7 . 
   Drag damping of each flapping mass constituted by a blade  7  and its cuff  12  connecting it to the hub  3 , and the elastic return of this flapping mass to its neutral axis are provided by a device external to the mast-hub  1  and arranged laterally between the ring  5  of the hub  3  and this flapping mass. This device, termed a drag damper, an elastic drag return strut with built-in damping or even a frequency adapter is a device  20  arranged as a particular hydraulic damper the structure and mode of operation of which are described below with reference to  FIGS. 3  to  6 . 
   The drag damper  20  is coupled by a ball joint  21  at its inner end in an attachment yoke  22  fitted onto an outward radial projection from the ring  5 , approximately in the middle of the large side  5   b  of this ring  5  which extends between the cuff  12  of the blade  7  considered and the cuff  12  of an adjacent blade  7  of the rotor. 
   At its outer end, the damper  20  is coupled by a ball joint  23  in a yoke formed between two fittings  24  which each form a single part with one of the two lower and upper plates of the connecting cuff  12 , each fitting  24  extending both radially outwards and towards the trailing edge of the corresponding blade  7 , starting from the outer radial end of the corresponding lower or upper plate. 
   The drag damper  20  in  FIG. 2  is represented schematically in  FIG. 3 , coupled between the hub  3 , of mass Mm, by the end fitted with the ball joint  21  in  FIG. 2 , and the flapping mass constituted by the blade  7  and the cuff  12 , and of mass Mp, by the ball joint  23  at its other end in FIG.  2 . 
   The drag damper  20  comprises a cylindrical tubular body  25  of circular cross-section Sa, closed at its axial ends by two end faces  26  and  27 , a piston  28  fitted and slidable axially in the body  25  and delimiting in this body  25 , and with the latter and the end faces  26  and  27 , two opposing variable volume working chambers  29  and  30 , the piston  28  moving integrally with an axial rod  31  passing in a sealed manner through not only the end face  26 , to be coupled by its outer free end to the cylinder  20 , to the flapping mass  7 - 12 , but also in this example, the other end face  27  coupled to the hub  3 . The damper  20  also comprises two identical springs  32  housed in the chambers  29  and  30 , each of which bears on the one hand against one of the end faces  26  and  27  respectively and on the other respectively against one of the two opposite faces of the piston  28 , so that it loads the rod  31 -piston  28  assembly so as to return this assembly to a neutral position in the body  25 , by the springs  32  bearing on this closed body, and a bypass channel  33  permanently connects the two chambers  29  and  30 , this channel having a length Lc which is very much greater than a main dimension, for example the diameter, of its constant cross-section sc, the latter being itself very much smaller (by about one order of magnitude) than the constant cross-section Sa of the body  25 , and a fluid, which is a hydraulic oil, fills the two chambers  29  and  30  and also the channel  33 , and is restricted by an annular calibrated restriction port  34  between the body  25  and the piston  28 , when this piston  28  is moved axially in the body  25  by relative movements in drag of the flapping mass  7 - 12 , connected to the rod  31 , and of the hub  3 , connected to the body  25 . 
   In the case of an external excitation being applied to the flapping mass  7 - 12 , and represented by the force Fe applied to this flapping mass in  FIG. 3 , if Fm is the force transmitted to the hub  3 , via the drag damper  20 , and if xp and xm represent the axial movements relative to a common origin on the axis of the damper  20 , respectively of the flapping mass  7 - 12  and the hub  3 , as indicated schematically in  FIG. 3 , application of the general theorems of mechanics gives the following relations (1) and (2) for the two masses Mp and Mm in translation:
         for the blade
 
 Fe−Fm=Mp{umlaut over (x)}p,   (1)
   and for the hub
 
 Fm=Mm{umlaut over (x)}m,   (2)
 
where {umlaut over (x)} represents the second derivative of the movement with respect to time, i.e. acceleration, so that {umlaut over (x)}p is the acceleration undergone by the flapping mass  7 - 12  and {umlaut over (x)}m is the acceleration undergone by the hub  3 .
       

   Additional equations are given by linking the internal forces to the degrees of freedom of the system. By the constitution of the damper  20 , the force Fm transmitted to the hub  3  is the sum of the elastic return force of the springs  32  (each of which has a stiffness k/2), the forces of pressure transmitted by the fluid present in the body  25  of the damper  20  and of the viscous friction force Fh. Hence the relation (3):
 
 Fm=−k ( xm−xp )+ Fh−c ( {dot over (x)}m−{dot over (x)}p )  (3)
 
where c represents the coefficient of viscous damping and {dot over (x)} the first derivative of the movement with respect to time, i.e. velocity. Moreover, we have the relation (4):
 
 Fh =−( P   1 − P   2 )  Sa,   (4)
 
where P 1  et P 2  are the pressures obtained respectively in chambers  29  and  30 . Considering the mass of fluid in motion in the bypass channel  33  and acceleration of this fluid (the oil), we also have the relation (5):
 
( P   1 − P   2 ) sc=−Mfc {umlaut over (x)}fc,   (5)
 
where Mfc represents the mass of fluid in motion in the channel  33 , i.e. Mfc=ρLc sc, where ρ is the mass per unit volume of the fluid and {umlaut over (x)}fc is the acceleration of the fluid moved in the channel  33 .
 
   The fluid inertia effect is introduced by the hydraulic force term Fh, since the relation of velocity to the passing of the fluid between the channel  33  and the body  25  leads to the following relation (6):
 
 sc{dot over (x)}fc=Sa ( {dot over (x)}m−{dot over (x)}p )  (6)
 
whence the relation (7): 
                   Fh   =       ⁢       -     (     P1   -   P2     )       ⁢           ⁢   Sa                 =       ⁢       -       ρ   ⁢           ⁢     LcS   2     ⁢   a     sc       ⁢     (         x   ¨     ⁢   m     -       x   ¨     ⁢   p       )                   =       ⁢       Ma   ⁡     (         x   ¨     ⁢   p     -       x   ¨     ⁢   m       )       ⁢           ⁢   taking   ⁢     :                   Ma   =       ⁢     ρ   ⁢           ⁢   Lc   ⁢         S   2     ⁢   a     sc                     (   7   )             
 
which represents an apparent mass (or fluid inertia) which may be very much greater than the actual mass of the fluid Mf in motion.
 
   As the movement xp of the flapping mass  7 - 12  is of the sinusoidal type of pulsatance ω, its acceleration {umlaut over (x)}p can be expressed in the form {umlaut over (x)}p=−ω 2 xp, which, by combining the expressions (1), (3) and (7), allows the movement equation to be obtained for the degree of freedom xp:
 
 Mp{umlaut over (x)}p+c ( {dot over (x)}p−{dot over (x)}m )+( k−Maω   2 )( xp−xm )= Fe (ω)  (8)
 
where Fe(ω) is the force applied to the flapping mass  7 - 12  at the frequency of the external excitation.
 
   Similarly, as the movement xm of the hub  3  is of the sinusoidal type of pulsatance ω, its acceleration {umlaut over (x)}m can be expressed in the form {umlaut over (x)}m=−ω 2 xm which, by combining expressions (2), (3) and (7) allows the movement equation to be obtained for the degree of freedom xm:
 
 Mm{umlaut over (x)}m+c ( {dot over (x)}m−{dot over (x)}p )+( k−Maω   2 )( xm−xp )=0  (9)
 
   The effect of the fluid inertia is therefore expressed by the addition of a negative fluid stiffness Kf=−Ma ω 2 , a function of the excitation pulsatance ω. The equivalent stiffness Keq of this system is therefore the sum of the two stiffness values combined in parallel and given by:
 
 Keq=k−Ma ω   2   (10)
 
   The anti-resonance frequency of such a drag damper  20 , which is the frequency for which minimisation of the force Fm and of the movement xm at the hub  3  is obtained, is the frequency ωa for which the equivalent stiffness Keq is zero, and is therefore given by the following relation
 
 ωa =( k/Ma )½.  (11)
 
   We thus obtain a drag damper  20  having an anti-resonance frequency ωa independent of the masses of the flapping mass  7 - 12  and of the hub  3 , and linked solely to the geometry of the damper  20 . 
   In the latter, the relevance of the bypass channel  33  is to create a high difference in dynamic pressure by causing a small fluid mass (of oil) to pass into this channel  33  of cross-section sc far smaller than the cross-section Sa of the body  25 . This fluid inertia effect results from conservation of the fluid flow at the point of convergence, when fluid enters the channel  33  coming from the one of the two chambers  29  and  30  which is compressed by movement of the piston  28 , and at the point of divergence where the fluid leaves the channel  33  to enter the one of the two chambers  30  and  29  where expansion is occurring. At the anti-resonance frequency ωa, the hydraulic force developed is used to compensate the elastic force transmitted by the springs  32 . 
   Simultaneously, the dissipation forces connected with the pressure losses and with the viscous damping forces are unwanted forces which limit total compensation of the forces of stiffness by the hydraulic force at the anti-resonance frequency. 
   The drag damper  20  is therefore dimensioned, as regards the stiffness k of the two springs  32 , the length Lc and the cross-section sc of the channel  33 , and the cross-section Sa of the body  25 , and the fluid (the oil) used has physical characteristics, particularly a mass per unit volume p, so that the anti-resonance frequency ωa of the damper  20  is set substantially at the nominal rotation frequency Ω of the rotor, of which it is known that it is different by design from the natural frequency in drag ωδ of the flapping mass  7 - 12 . 
   At the same time, the cross-section through the restrictor port  32  between the piston  28  and the body  25  is calibrated so as to optimise the dissipative phenomena, i.e. to damp the drag mode sufficiently at the natural drag frequency ωδ of the flapping mass  7 - 12 , without degrading the filtering of the Ω component. 
   In these conditions, the damper  20  filters the Ω dynamic component in the loads it applies, so that damping of the forced excitation at Ω is zero or practically zero, but on the other hand the restrictor port  34  provides substantial damping of the relative movements of the rod  31 -piston  28  assembly and of the body  25  at a frequency which is substantially equal to the natural drag frequency ωδ of the flapping mass  7 - 12 . 
   An example of dimensioning of the dampers  20  for the main rotor of a helicopter with a weight of the order of eight to ten tonnes leads to the following geometry: diameter of the body  25 : Da=0.15 m, diameter of the channel  33 : Dc=0.014 m, length of the channel  33 : Lc=1.45 m, mass per unit volume and viscosity of the oil ρ=850 kg/m 3  and ν=20·10 −6 m     2   /s, coefficient of pressure loss of the channel  33  equal to 3.4 for a desired stiffness of the damper  20 , at the natural drag frequency ωδ, of 200 daN/mm, the anti-resonance frequency ωa and the rotor rotation frequency Ω being 30.6 rad/s. 
   Disregarding pressure losses, the drag damper  20  defined above provides optimum performance, the Ω forced excitation is not damped, there is maximum gain, and the fluid undergoes no restriction. The equivalent stiffness Keq of such a fluid inertia drag damper, as a function of frequency, is shown in  FIG. 7 , in which the curve  35  corresponds to an ideal device without pressure losses. The curve  35  shows that the damper eliminates the damping loads at the rotation frequency of the rotor Ω, equal to the anti-resonance frequency ωa, and works only on the damping of the drag mode of the blades at ωδ, which is the natural drag frequency of the blades. At this frequency ωδ, the damper  20  ensures an equivalent stiffness practically equal to the stiffness of the damper  20 , i.e. of the order of 200 daN/mm. Damping at ωδ is obtained by restriction of the fluid in the body  25 , in this example via the annular restrictor port between the piston  28  and this body  25 . 
   In fact, the pressure losses reduce the effectiveness of the system at frequency Ω, and when all of the pressure losses and the turbulent nature of the flow are taken into account, the performance of the drag damper  20  corresponds to the curve  36  in  FIG. 7 , for which the equivalent stiffness Keq is minimal (but not zero) for an anti-resonance frequency ωa slightly less than Ω, damping at the natural drag frequency of the blades ωδ remaining effective. 
   Despite the turbulent nature of the flow and allowance for all of the pressure losses, the fluid inertia drag damper  20  provides a reduction of about 40% to about 50% in the dynamic stresses at Ω compared with a conventional drag damper, which is still highly advantageous. 
     FIGS. 4 and 5  show schematically a first example of embodiment of such a fluid inertia drag damper  40 , comprising a cylindrical tubular body  45  closed by two end faces  46  and  47 , a piston  48  slidable axially and, in this example, with sealing in the body  45  and being integral with an axial rod  51  also slidable with sealing in the end faces  46  and  47 , and one end of which external to the body  45  comprises a ball end  51   a  for hinging to a rotor blade or to a device connecting this blade to the rotor hub, while the end face  47  is integral with a ball end  47   a  for hinging to the body  45  on the rotor hub, two springs  52 , identical and of a helical or coil type, each being housed respectively in one of the two working chambers  49  and  50  delimited by the piston  48  in the body  45 , each spring  52  being guided by the body  45  and bearing respectively against one of the two end faces  46  and  47  at one end, and at the other, against the piston  48  to return the rod  51 -piston  48  assembly to a neutral position in the body  45 , the two chambers  49  and  50 , filled with a fluid such as hydraulic oil, being in permanent communication with each other via a restrictor port  54  drilled in the base of the piston  48  and calibrated to obtain substantial damping at the natural drag frequency ωδ of the blade considered, this natural frequency ωδ being different by design from the nominal rotation frequency Ω of the rotor. This damper  40  also comprises a bypass channel of substantial length and small cross-section connecting the two chambers  49  and  50 , the main difference compared with the schematic mode of embodiment in  FIG. 3  being that this bypass channel  53  is internal to the body  45  and more precisely made inside the piston  48 . This channel  53  is arranged in two radial adjacent spirals in the piston  48 , one of which  53   a  diverges from a central end communicating via an inlet port  53   b  with the chamber  49  as far as its outer radial end communicating via an axial passage  53   c  with the outer radial end of the second spiral  53   d  which converges as far as its inner radial end which runs via an opening  53   e  into the other chamber  50  of the damper  40 . The geometry of this damper  40  and the physical characteristics of the fluid used, in particular the lengths and cross-section of the channel  53 , the cross-section of the body  45  and the mass per unit volume and viscosity of the oil used are such that the anti-resonance frequency of the system is close to the rotor rotation frequency Ω, as explained with reference to  FIGS. 3 and 7 . 
   The example of embodiment in  FIG. 6  differs from that in  FIGS. 4 and 5  in that the fluid inertia drag damper  60  comprises a bypass channel  73  which, as in the schematic example in  FIG. 3 , is external to the damper body  65  closed with sealing by the end faces  66  and  67  added on and secured with screws, the end faces themselves having running through them, with sealing, the axial rod  71  moving integrally with the piston  68  slidable with sealing in the body  65 . Another difference is that the two helical springs  72  are wound around the rod  71 , the end external to the body  65  of which has a bore  71   a  for attachment of a ball end for hinging to the blade, while the end  67  is extended axially outwards by an end piece  67   a  having a bore for attaching a ball end for hinging to a hub. The permanent communication between the two working chambers  69  and  70  of the damper  60  is provided not only by the bypass channel  73  but also by one or more restrictor ports  74  drilled through the piston  68  and calibrated to produce substantial damping at the natural drag frequency ωδ of the blade, while the dimensional characteristics of the damper  60  and the physical characteristics of the oil which it contains are selected, particularly as regards the length and cross-section of the channel  73  and also the cross-section of the body  65 , so that the anti-resonance frequency ωa of the assembly is as close as possible to the nominal rotation frequency Ω of the rotor. 
   As a variant, the piston  68  is drilled with several restrictor ports  74  calibrated at different cross-sections, in order to ensure optimum restriction of the oil at the natural drag frequency ωδ. 
   Also as a variant, the springs used to obtain the necessary stiffness may have different structures, particularly helical or coil springs, or springs which are cylindrical or conical, and in general of metal. In certain damper architectures, such as the one in  FIGS. 4 and 5 , the springs can be used to fulfil not only the stiffness function but also that of at least part of the bypass channel, by using hollow springs communicating with the channel which runs through the piston. 
   Also as a variant, and as shown schematically in  FIG. 6 , a part  75  of the bypass channel  73  may be of variable cross-section and controlled, in the form of an adjustable restriction, by a control device  76  which receives at  77  a rotor rotation speed signal, in order to control the variation in the cross-section of this portion  75  of the channel  73  so as to give real-time adjustment of the anti-resonance frequency o of the damper  60  at a variable speed of rotation Ω of the rotor. In this way, the performance of the drag damper is adapted to variable speed rotors. 
   In general, the geometry of the bypass channel may have complex forms, particularly of cross-section and variation of cross-section, in order to optimise pressure losses. 
   Moreover, to reduce the viscosity of the fluid used and thus reduce the effect of pressure losses, in order to optimise the operation of the drag damper, the fluid filling this damper may be an electro-rheological fluid, the variable viscosity of which is controlled by the control of an electrical and/or magnetic field to which at least part of the fluid is subjected. This can be embodied in the manner imparted in patents GB 2 111 171 and EP 0 183 039, to which reference should be made for further information on the matter, for example by arranging in a portion  78  of the channel  73 , which is raised to a certain electrical potential by connection to a source generator  79 , a cuff or plate  80  raised to a different electrical potential by connection to the same variable electrical field generator  79 . It is thus possible to subject the fluid circulating in this portion  78  of the bypass channel  73  to an electrical field controlled by the control of the source generator  79 , so that the viscosity of the fluid can be varied in an appropriate manner. 
   Of course, the fitting of such a fluid inertia drag damper is not limited to the type of main rotor according to  FIGS. 1 and 2 , and it can be fitted to other types of main rotors, particularly such as those described in FR 2 427 251, in which the hub is a plate or radial ring with axial recesses running through it, each of them housing means of connecting and hinging a blade to the hub, or FR 2 456 034, or FR 2 529 860, in which the hub body is a central sleeve which supports two radial plates spaced apart, between which are secured the means of connecting and hinging the blades to the hub, and the root of each blade may comprise a loop which surrounds the means of securing and hinging, themselves secured between the two plates of the hub. 
   The invention has now been described in detail for purposes of clarity of understanding. However, it will be appreciated that certain changes and modifications may be practised within the scope of the appended claims.