DYNAMIC PROPERTIES OF SOME GIMBAL AND TEETERING
TWO-BLADE HELICOPTER ROTOR HEADS
Alessandro Croce, Radek Possamai, Lorenzo Trainelli
Politecnico di Milano, Milano, Italy
Abstract
The primary aim of the paper is the characterization of the dynamic behaviour of an innovative two-blade main rotor design under cyclic perturbations. The consequent ‘wobbling’ motion, i.e. the 2/rev precession of the hub with respect to the mast, is known as a significant drawback of typical current two-blade rotor realisations, based on a teetering mount, as it induces considerable oscillating loads to the fuselage. As an alternative capable to alleviate this problem, the proposed design is based on a constant-speed gimbal mount. The performance of this solution is contrasted with a pure teetering one, as well as with intermediate architectures retaining only a part of the innovative elements of the new gimbal rotor, in order to appreciate their effects separately. All models are simulated within a high-fidelity finite element multibody framework. The results confirm the superior characteristics of the constant-speed architecture. The study is completed with a sensitivity analysis of the dynamic response of the proposed rotor model with respect to some important design parameters.
1. INTRODUCTION
The present paper aims to characterise the dynamic behaviour of an innovative two-blade main rotor design conceived as a convenient alternative to the standard teetering-type designs employed in current lightweight helicopters. It is well known that these rotors, with their remarkable design simplicity, suffer from some limitations and drawbacks. Among these, one of the most important is related to the vibratory loads transferred to the fuselage as a result of rotor cyclic flapping. This is the typical condition encountered in forward flight or while hovering under gust conditions, and involves a considerable impact on pilot workload, passenger comfort, vehicle handling qualities, and structural fatigue.
In an effort to alleviate this problem, Dr. Vladimiro Lidak (1944-2012), a missed Italian rotorcraft designer and inventor, conceived an innovative rotor head solution based on a homokinetic gimbal mount. This patented design [1] is currently developed by K4A S.p.A., a start-up helicopter manufacturer based in Naples, Italy, to be implemented in the KA-2HT two-seat helicopter.
In order to analyse the dynamic behaviour of this solution, in this paper four different two-blade rotor models fitting the same general requirements for a 650 kg MTOW class helicopter are considered and compared. The main characteristic of the dynamic response to cyclic perturbations, be it pilot control or wind gusts, is the ‘wobbling’ motion, i.e. the precessional 2/rev (two periods per rotor revolution) oscillations of the rotor head entailing analogous variations in the aerodynamic and inertial rotor forces. A thorough comparison of the wobbling behaviour is carried out contrasting Lidak’s design
with a basic teetering rotor head and variations thereof. Following this analysis, some parametric studies are carried out in order to determine the sensitivity of Lidak’s design to important design parameters. The present results are also commented in relation to those presented in a previous work for the same rotor model [2].
2. MULTIBODY SIMULATION
The present study has been conducted by the implementation of high-fidelity aero-servo-elastic models of the various rotor systems within the Cp-Lambda (Code for Performance, Loads and Aeroelasticity by Multi-Body Dynamic Analysis) software. This is based on a state-of-the-art finite-element multibody formulation [3] employing Cartesian coordinates for the description of all entities in the model, while all degrees of freedom are referred to a single inertial frame. The formulation handles arbitrarily large three-dimensional rotations and makes use non-conventional, unconditionally stable time-integration methods [4]. The software has been thoroughly employed in the aero-servo-elastic analysis of rotorcraft systems as well as wind turbine generator systems, e.g. [5, 6, 7].
Structural elements can be modelled as rigid bodies, beams and shells, and joint models. In particular, the blades, as well as other slender linkages, are modelled using geometrically exact, composite-ready beams of arbitrary geometry and accounts for axial, shear, bending and torsional stiffness. Joints are modelled through holonomic or nonholonomic constraints enforced by means of Lagrange multipliers and can account for backlash, free-play and friction.
Lifting lines blade ele aerodynami unsteady co elements. T terms of t number of s the aerody correspondi up tables, w drag, and m local angle An inflow e lines so as code imple wake mode momentum stream-tube models are 3. ROTOR As mention response to rotor is ca reference t rotor head d on a configu mount, allo relative mo as teetering both the m (i.e. rotation implement which only freedom. In suspended teetering hi configuratio popular Rob the T1 mod Lidak’s des illustrated in type flybar realizing a ‘ This consis ‘direct’ com swashplate ‘mixer’. This by the flyba As a conse combination delivered by the blade ro given by θ swashplate is the prima In order to joint and th s based on ement theo ic centre orrections, c Their geome hree-dimens span-wise st ynamic cha ing to each l which store th moment coe of attack, Re lement can s to model t ements the el [8], as well (BEM) mo e theory with also conside R HEAD MO ned above, o cyclic pertu arried out b two-blade a designs are uration involv owing two d tion between g (i.e. rotatio mast and the
n about the a more clas permits the n this confi on the ma nge. This is on in lightwe binson R-22 el. sign, referre n Figure 1. fixed to the mixed contro sts in the co mmand prov and transm s element als ar, the ‘secon equence, the n of the tw
y the pitch h oot. The resu = τp θsw+ ( tilt, η is the ary command appreciate t he mixed p n classical ory, accou offset, twis can be assoc etric descrip sional twiste tations along racteristics ocation are he values of efficients as eynolds and be associate the rotor inf
Peters–He as a classic odel based wake swirl. ered. DELS the study urbations of L y a compa architectures investigated ving a consta degrees of n hub and m on about an e blade axis blade axis) ssical, much e teetering s guration, th ast by a re by far the m ight helicopt [9], and will ed to as th It includes a hub and a ol’ strategy fo ombination o ided by the mitted by pitc so receives ndary’ or ‘ind e blades are wo control a horns connec ulting cyclic p (1 – τp) η, w hub feather d ratio. he contribut pitch control two-dimens nting for st, sweep, ciated with b ption is give ed curves. A g each lifting of the ae given using f the sectiona functions of Mach numbe ed with the l flow effects. dynamic in cal blade-ele on the an Tip and hub
of the dyn Lidak’s gimb rison with o . Four diffe d. Two are b ant-speed gi freedom to mast, referre n axis norm s) and feathe ). The other h simpler mo single degre e rotor hea evolute joint, most represe ters, such as be referred t e H1 mode also a Bell-H complex lin or blade pitc of a ‘primary pilot throug ch links up mechanical direct’ comm e feathered actions, whic cting the mix pitch comma where θsw is ring angle, a ion of the gi to the dyn sional the and beam en in At a line, rofoil look-al lift, f the ers. lifting The nflow ment nular b loss namic balled other erent ased mbal o the ed to mal to ering r two ount, ee of ad is , the ented s the to as el, is Hiller-kage ching. y’, or gh a to a input mand. by a ch is xer to and is s the nd τp mbal namic resp of th featu cont Sim flyba cons desi tilt a with cont T2 m Figu four with cons roto desc Fi Fig ponse of Lida his design is ures a traditi tribution from ilarly, in an ar upon th sidered an a ign by addin about an axis the teeter trol is consid model. ure 2 summa architecture the same s sidered for t r models an cripted. igure 1. CATIA considered i
gure 2. The fou
ak’s rotor he s also inves onal direct p m the flybar d effort to asc he dynamic augmentation ng a Bell-type s fixed to the ing axis. N dered in this arizes the re es. All of the et of blades, the KA-2HT. nd their mu
model of the s n this work (co
ur rotor head m work. ead, a ‘degra stigated. Thi pitch control, dynamics. certain the in c response n of the bas e stabilizing mast and at No contributi case, referr elationships em have bee , representa . In the follo lti-body idea shaft-rotor hea ourtesy of K4A models investig . aded’ version s H2 model without any nfluence of a e, we also sic teetering bar, free to t right angles on to pitch red to as the between the en equipped tive of those owing, these alization are ad assembly A S.p.A.). gated in this n l y a o g o s h e e d e e e
3.1. Mode The H1 ro companion prominent convenienc original hom hub to the m revolute pa further con scissor-like are crucial t the transmi attempt to p in spite of th This comp constant ra internal ele bisector pin the mast an thus app transmissio Figures 3 implementa in the rotor chain subs resulting m elements, a degrees of f 3.2. Mode The H2 ro eliminating fact, the pit without any to Figure 4, connecting (starting at to the pitch way, the fly
Figure 3 el H1 otor head is paper [10] features are e. The rotor mokinetic gim mast. This is airs mutually nstrained in mechanism to the quasi-ssion from t preserve the he relative ti plex arrang atio of the re ments of the axis actually nd the hub a proximating n. 3 and 4 ation of this in head transm systems for odel include and 44 joints freedom. el H2 otor head is the mixed tch links dire contribution the H2 mod elements fro point C) and h link (desce ybar motion ( 3. Topological transmiss s described ]. Here, so e recalled f head is cha mbal joint con s composed y arranged n their rela s called bise -constant-sp the mast to value of the lt between th ement enfo elative rotatio e gimbal mo y bisects the xis normal to an idea represent nnovative so mission and the sake es 59 rigid b s, for a total s derived fr pitch contro ectly actuate n from the fly del is obtaine om the flybar d connecting ending from (i.e. the hub
sketch of the H sion subsystem
in detail in me of its for the sak aracterized b nnecting the of a sequenc at right an ative motion ectors. The l eed behavio the hub, i.e e angular vel he two elem orces a n ons between ount. In fact e angle forme o the rotor p al homoki the multi olution, separ the pitch co of clarity. bodies, 21 b number of rom the H1 l mechanism the pitch ho bar. With res ed by erasing r to the pitch
the latter dir point A). In feathering) H1 rotor-head m. n the most ke of by an rotor ce of ngles, n by latter our of e. the locity ents. early n the t, the ed by lane, inetic body rated ontrol The beam 1756 1, by m. In orns, spect g the horn rectly n this does not impo actu the p 3.3. The cons The teete resp flyba cont and mod 3.4. The T1 m latte mea mov mas F influence the ortant functi uate the hub proper behav Model T1 T1 rotor sidered for tw rotor head i ering hinge pect to the m ar, and nat trol. The T1 aerodynam dels. Model T2 T2 rotor he model, with t er is a stabiliz ans of a re vement arou st and the tee
Figure 4. Topo Figure 5. To e blade pitch ion of deve feathering m viour of the g head is th wo-blade hel s provided w allowing rig mast. The m urally involv model retain mic characte ead consists he addition o zing device c evolute joint nd an axis etering axis. logical sketch chain subsy pological sketc h. However, eloping the motion need gimbal joint. he simplest licopter rotor with a centra gid blade fl model does n ves only a ns all geom eristics of H in a modific of a Bell-type connected to t that allow at right ang In the prese of the H1 mod ystem. ch of the T1 m it retains the forces that ed to insure t realization rs (Figure 5). al single axis lapping with not include a direct pitch etric, inertial H1 and H2 cation of the e flybar. The the mast by ws its tilting gles with the ent case, the
del control odel. e t e n . s h a h l 2 e e y g e e
flybar is the inertial and mechanical that the flyb 4. COMPA As a first a the conside analysis w performed. is 504 rpm. start from correspondi helicopter’s given cyclic 4.1. Cycli In the first hover is rea command w and finally suitable tim reach period Figure 6 sh conditions longitudinal are drawn in represent th with respect It can be ob teetering m trajectories axis). The r of model T2 the presenc constant-sp characterise Figure 6. Hu swashplate H e same used aerodynami linkage con bar acts also ARATIVE WO assessment ered four rot with the Cp
The angular All the time a steady ing to the sa weight, 64 c perturbation ic command set of simul alised as ste which tilt the to 10°. Aft me lapse is dic condition
hows the hu correspond tilt, for the f n the plane o he hub longi t to fuselage bserved that odels, T1 an passing thro reduction in t 2 with respec ce of the fly peed mode ed by much ub wobbling in e longitudinal t H1 (red) and H2 d in the H1 ic properties ntributing to as a dampin OBBLING A of the dynam tor head mo p-Lambda c r velocity for e-marching d y-state hov ame rotor thr 400 N) and n. d perturbatio ations, the p p inputs of lo e swashplate ter comman provided to ns. ub wobbling ing to a four models of the angles itudinal and e-fixed axes. the hub mot nd T2, descr ough the orig the amplitude ct to model T ybar. On the els, H1 smaller wob n a periodic co tilt for the T1 (b 2 (magenta) mo model, inclu (but without pitch control ng device. ANALYSIS mic behavio odels, a wob code has b r all rotor mo ynamic anal vering cond rust (equal to proceed w on perturbation ongitudinal c e from zero t nd applicatio o allow the
for the per 10° swash . The traject s u1 and u2, w
lateral tilt an tions for both ribe large cir
gin (i.e. the e of the wob T1 is the res other hand and H2, bling oscillat ndition with 10 blue), T2 (cyan) odels. uding t any l), so our of bbling been odels lyses dition o the ith a from cyclic to 5° on, a rotor riodic plate tories which ngles h the rcular mast bbling ult of d, the are tions. The cond mea appr with diffe limit and is su mixi later long wob is st from the c By resu diffe In f assu the a As wob dyna a s long lowe man later appe in av In a have mod resu load sam mod 0° ), Fi per precession ditions in the ans that the
roximately th a residual erences betw ted, and invo
u2, while the
ubstantially t ng mechanis ral tilt and a gitudinal tilt. bbling betwee trikingly evid m 5° to less t constant-spe comparing ults of [2], erences relat fact, the us umptions of absence of w a further c bbling motion amic transien second app gitudinal pitc er wobblin noeuvre, invo ral tilt. As ap ears much s verage longit addition to h e been anal dels. Figure 8 ultant interna ding imposed me conditions dels present igure 7. Hub w rturbations of 5 (bl n in this c e vicinity of th e average he same as amount of a ween models olve only the e amount of the same. T sm induces a a slight highe We remark en model T1 dent, with am han 1°, show eed gimbal so the model we can ted to the d e of a sim an ideal ho wobbling in a confirmation, n of models H nt from null c lication of ch command ng amplitu olving a cer pparent, the simpler, with tudinal tilt, at ub motion, lysed to cha 8 refers to th al force vecto d by the roto s as Figure larger amplit obbling evolut 5° swashplate l ue) and H1 (red
case reach he value u1 = hub longitu that of the average late s H1 and H e average v f the wobblin The absence a lower value er value of k that the d 1 and model mplitude val wing the effe
olution. H1 respons appreciate issimilarity i mplified mod omokinetic jo analogous co , Figure 7 H1 and T1 d cyclic to 5° , a variation d. Model H ude throu rtain amount behaviour o h a progress t null lateral t several othe aracterise th he wobbling or, which rep or to the airfr
6. Again, t tudes with re
tion for two su longitudinal til d) models. es periodic = 10°, which udinal tilt is swashplate, eral tilt. The H2 are very values for u1 ng amplitude of the pitch e of average the average difference in l H1 and H2 ues passing ectiveness of se with the significant n modelling. del plus the oint leads to onditions.
shows the uring the full , followed by n of 5° in H1 shows a ghout the t of average of model T1 ive increase tilt. er quantities he four rotor of the mast presents the rame, for the he teetering espect to the bsequent t for the T1 c h s , e y e h e e n 2 g f e t . e o e l y n a e e e s r t e e g e
constant-sp complex co effects cont the force os amplitude w addition, the higher ave models. Figures 9 a internal forc with respec axis) within Again, the between the the superi solutions. Figure 11 behaviour w the 5° and Figure periodic c for the T1 Figure 9. T internal forc longitudina peed models oupling of ro tributing to th scillations fo with respect e force resp erage latera nd 10 show ce and mom t to fuselage n one revolu wide differe e different m or behavio allows a within the co 10° pitch co 8. Wobbling of condition with 1 (blue), T2 (cya m Time histories o ce in a periodic al tilt for the T1 H2 (mag . The elliptic otor aerodyna he mast load or model T1 t to the hub onse of mod al tilt than
the time hist ment compo e-fixed axes ( ution in per nce in the a models is app ur of the appreciating omplete man ommand app
f the mast inte 10° swashplate an), H1 (red) an models. of the compone c condition wit (blue), T2 (cya enta) models. orbits reflec amic and in ing. Interesti show a dou b oscillations del T1 involv constant-s tories of the nents, evalu (z being the riodic condit amplitude va parent, confir constant-s the tran noeuvre invo plications. A rnal force in a e longitudinal t nd H2 (magent
ents of the mas h 10° swashpla an), H1 (red) an ct the ertial ingly, ubled s. In ves a peed mast uated mast tions. alues rming peed nsient olving Again, tilt ta) Fi i sw Fi p fo Fi s st ate nd gure 11. Time internal force f washplate long H1 (re gure 12. Wobb eriodic conditi or the T1 (blue gure 10. Time internal mom washplate long H1 (re histories of the for two subseq gitudinal tilt fo ed) and H2 (ma
bling of the roto ion with 10° sw ), T2 (cyan), H1 models histories of the ments in a perio gitudinal tilt fo ed) and H2 (ma
e components quent perturbat or the T1 (blue) agenta) models or aerodynami washplate long 1 (red) and H2 s. e components odic condition or the T1 (blue) agenta) models of the mast tions of 5° , T2 (cyan), s. c force in a itudinal tilt (magenta) of the mast with 10° , T2 (cyan), s.
the constan throughout t The resulta rotor was al this quantity periodic c component forces, but a a doubled o single rotor This charac where the aerodynami evaluated a Figure 14 s of the time force magn Figure 13. T the magnitu a periodic c for the T1 Figure 14. F force in longitudin nt-speed mo the simulatio ant aerodyna lso investiga y a different onditions n as seen for also a 1/rev orbit in the ( revolution. cteristic can e time hi ic force after reaching shows the F e histories o nitude for m Time histories o ude (below) of condition with (blue), T2 (cya m FFT of the mag n a periodic co nal tilt for the T
odels show l on. amic force g ted. As seen result is obt not only in hub motion a component. u1,u2) plane be apprecia istories of components g periodic co Fast Fourier of the resulta models T1 a of the compon the rotor aerod 10° swashplate n), H1 (red) an models. gnitude of the r ondition with 10 T1 (blue) and H ower oscilla generated by n in Figure 12 ained, in tha nduce a 2 and mast inte This appea , as a result ted in Figure the resu s are sh nditions. Transform ( ant aerodyn nd H1. The ents (above) a dynamic force e longitudinal t d H2 (magenta rotor aerodyna 0° swashplate H1 (red) models ations y the 2, for at the 2/rev ernal rs as t of a e 13, ultant hown, (FFT) namic e two sign with anal whic To c orde traje vect simi almo som mod 4.2. The pert (hov nd in tilt a) amic s. Fig l s H nals display a the 2/rev lysing the h ch show only complete the er to grasp ectories of th tor. It is see lar, so that i ost perpend mewhat lowe dels. Advance second se urbation in t ver) to 20 m/ gure 15. FFT of force in a pe longitudinal tilt Figure 16. W aerodynamic swashplate lon H1 (red) and H2 represented b thi a significant one. This ub wobbling y the 2/rev ha e illustration, the shift e hub and o en that the m n all cases t dicular to er variation ratio pertur et of simula he asymptot /s, which cor f the magnitude riodic conditio t for the T1 (blu
Wobbling of the force in a peri gitudinal tilt fo (magenta) mo by thicker lines nner lines with
1/rev comp is not the g, as seen in armonic com Figure 16 is between th of the aerody motion is ap the aerodyna the hub ns for con rbation ations consid
tic wind spe rresponds to
e of the rotor a on with 10° swa ue) and H1 (red
e hub and of th iodic condition or the T1 (blue) odels. Hub traje s, aerodynamic h ‘x’ markers. ponent along case when n Figure 15, mponent. s included, in he wobbling ynamic force pproximately amic force is plane, with nstant-speed ders a step ed from null an advance aerodynamic ashplate d) models. he rotor n with 10° ), T2 (cyan), ectories are c force by g n , n g e y s h d p l e
ratio μ = perturbation allow the ro Figure 17 s conditions models. As the hub mo and T2, des the origin, and H2, oscillations. cyclic pertu and T2 is e and H2 is m amplitude r model H1 o which is the origin in the Figure 17. H advance rat Figure 18. periodic c (blue), T2 ( 0.1, in the n, a suitable otor reach pe shows the h correspondin seen in the otions for bo scribe larger while both feature sig . It is observ rbation case extremely sim markedly dif reduction for oscillates abo e farther awa e (u1,u2) plane Hub wobbling io for the T1 (b (magen . Wobbling of t ondition with 0 (cyan), H1 (red) longitudinal e time lapse riodic condit ub wobbling ng to μ = 0 e case of cy oth the teete r trajectories constant-sp gnificantly l ved that, at v e, the respon milar, while th fferent, with r model H2. out an avera ay from the in e. in a periodic co blue), T2 (cyan) nta) models.
the mast intern 0.1 advance rat ) and H2 (mage
direction. e is provide
ions. g for the per
0.1, for the yclic perturba ering models passing thro eed models ower ampl variance with nse of model hat of model an advantag In this insta age tilted pos nitial trim, i.e
ondition with 0 ), H1 (red) and
nal force in a tio for the T1 enta) models. After ed to riodic four ation, s, T1 ough s, H1 itude h the ls T1 ls H1 ge in ance, sition e. the Figu resu is c very to t mod 5. In a beha been som the flyba stiffn 5.1. The varia diffe para dyna afb S afb t flyba padd cond 3, 4, Figu cond it ca inert and the v This valu aver wob 0.1 H2 Fi s ure 18 refe ultant interna confirmed th y effective in he mast, w del. PARAMETR an effort to aviour of m n carried out me design p inertial and ar, the pitc ness. Flybar ine first param ation of the erent values ameter has amics throug Sfb (Rfb)3/Jfb, the lift-curve ar paddle re dle referenc ducted by us , 5, where Jre ure 19 show ditions with a an be seen, t tia induces a a trend for values of the s is clearly se e for u1 tend rage value o bbling amplitu gure 19. Hub w washplate long va ers to the l force vecto at the cons reducing the with a super RIC STUDIES further cha model H1, p t to assess th arameters. aerodynam ch mixing ertia metric study wobbling re s for the f a significant gh the flybar where ρ rep slope of th eference surf ce radius. A sing the value
ef denotes its ws the hub a 10° swash the result of a reduction o the average e swashplate een in Figure ds to increas of u2 tends
ude. This phe
wobbling in a p gitudinal tilt fo alues of the fly
wobbling o or. In the pre stant-speed e 2/rev loads rior position S aracterise t parametric s he impact of These stud ic character ratio, and y was focus esponse with flybar inerti t role on th r Lock numb presents the he flybar pad rface, and R A sensitivity e Jfb = nJref w s reference d b wobbling hplate longitu an increase of the wobblin e hub tilt ang e tilt angles. e 20, where se towards 1 to vanish, a enomenon s periodic condit or model H1 wit ybar inertia. of the mast esent case, it models are s transferred for the H1 he dynamic tudies have variations in ies involved ristics of the the gimbal sed on the h respect to a Jfb. This e feathering ber γfb = 2 ρ e air density, ddle, Sfb the Rfb the flybar study was with n = 1, 2, design value. in periodic udinal tilt. As in the flybar ng amplitude gles towards the average 0°, while the as does the eems to hint ion with 10° th different t t e d c e n d e l e o s g ρ , e r s , . c s r e s e e e t
that a desig featuring fo flybar, may accurate co 5.2. Flyba In this sec slope afb is the flybar considered reference manoeuvre obtaining Fi As appare relatively li increase in displaceme values of versa. The by these va Figure 20. S amp Figure 21. H swashplat v gn employing our blades, i y be conce onstant-speed ar aerodyna ction, a vari considered, Lock num variations design va with 10° igure 21. ent, the im imited for t the flybar l nt of the w longitudinal wobbling am riations. Sensitivity of av plitudes with re Hub wobbling te longitudinal values of the fl g the same g nstead of tw eived to ac d transmissio amics ation in the as another mber. For of ±5% an alue and swashplate pact of thi the values ift-curve slo wobbling axis and lateral mplitude app
verage tilt ang espect to flyba
in a periodic c tilt for model H lybar lift-curve gimbal mount wo blades a chieve extre on performan e flybar lift-c term include this study, nd ±10% of simulated longitudinal s paramete considered. pe determin s towards l tilts, and pears unaffe
les and wobbli ar inertia.. condition with 1 H1 with differe slope. t, but and a emely nce. curve ed in we f the the l tilt, er is . An nes a ower vice-ected Figu aver amp 5.3. An i repr prim of t cons refe lead long Figu cons the limit the blad seco ing 10° nt w F s ure 22 repor rage values plitude as fun Control m important de resented by t mary comman he wobbling sidered vari rence design ding to perio gitudinal tilt. ure 23 illustra sidered varia parameter ted. An incre influence o de pitch, at t ondary comm Figure 22. Se wobbling amp igure 23. Hub w swashplate lon valu rts the slowly of u1 and nctions of flyb mixing esign parame
the pitch mix nd ratio τp. To
g response ations of ± n value and r odic conditio
ates the wob ations. Here, variation ag ase in the m f the prima the expense mand) determ ensitivity of ave litudes with res
wobbling in a p ngitudinal tilt fo ues of the pitch
y varying tre u2, and th bar lift-curve eter of Lidak xing, represe o evaluate th to this par ±5% and ±1 repeated the ons at 10° bbling traject the relative gain appea mixing ratio (i ary comman e of the influ mines a disp
erage tilt angle spect to flybar
periodic condit or model H1 wi h mixing ratio.
ends for the he wobbling slope. k’s design is ented by the he sensitivity rameter, we 10% of the e simulations swashplate
tories for the influence of rs relatively i.e. enlarging d upon the uence of the placement of es and r inertia. tion with 10° ith different e g s e y e e s e e f y g e e f
the wobbl longitudinal vice-versa. unaffected b Figure 24 r average va amplitude a 5.4. Gimb The last p stiffness. Th comparative of simplicit teetering r conceived torsional sti improve con torsional sp feathering a The referen Nm/rad. W simulations 100% of K ratio betwee Figure 24. S ampli Figure 25. H swashplate ing axis tilt and high Again, the by these var reports the s alues of u1 as functions o bal stiffness arametric st his paramete e studies dis ty in contr rotor mode for the K ffness within ntrol power i prings have axis and Kt = nce design We perform with values Kref with 10% en Kf and Kt. Sensitivity of a itudes with res
Hub wobbling i e longitudinal t values of the towards lo her values o wobbling am iations. slowly varyin and u2, an of pitch mixin s tudy consid er was not c scussed abo rasting cons els. Howeve KA-2HT hel n the gimbal j n low-g fligh different valu 2 Kf about t value for K med the s of Kf in a % increments
average tilt ang spect to pitch m
n a periodic co tilt for model H e gimbal stiffne wer values of lateral tilt, mplitude app ng trends fo nd the wob ng ratio. ered the gi considered in ove, for the
stant-speed er, the de icopter incl joint, as a w ht conditions. ues Kf abou he teetering Kf is Kref = 3 usual dyn range from s, preserving
gles and wobbl mixing ratio. ondition with 1 H1 with differen ess. s of , and pears r the bbling mbal n the sake and esign udes way to . The ut the axis. 3610 namic 0 to g the As s sign both incre osci disp later Figu aver amp 6. In th four blad type The in d with inno mas pert of t simu mult simu subj pert traje resu supe Moti cons asse flyba and inert wob valu ling 0° nt Fig seen in Figu nificantly on h average easing the llations abo placed beyon ral displacem ure 26 show rage values plitude as fun CONCLUDIN his paper, a different m de lightweigh e and two of gimbal mod etail in [10] out pitch c ovative cons st and hub urbations of he hub wit ulated by me tibody formu ulate the com jected to cy urbations. T ectory, intern ultant rotor a eriority of the ivated by this stant-speed essing the im ar lift-curve gimbal stiff tia and gimb bbling respo es and ampl gure 26. Sensit amplitude ure 25, the g the wobblin values and gimbal st out an axis nd the swas ment become ws the rapidly of u1 and nctions of the NG REMAR study of th ain rotor mo ht helicopter a gimbal typ dels are the and a variat ommand m stant-speed . The typic all these ro h respect to eans of a hi ulation. This mplexity in th yclic pitch i he analysis nal mast for aerodynamic e constant-s s result, a se design h mpact of va slope, pitch ness. It is o al stiffness c nse, both in litude. tivity of averag s with respect gimbal stiffn ng response d amplitud tiffness indu which is lo shplate tilt es very impo y varying tre u2, and th e gimbal stiffn RKS e dynamic b odels design rs, two of t pe, has been
novel desig tion of the s mixing. Both transmissio cal respons otors, a 2/rev to the mast igh-fidelity fin s allows to he rotor resp input or ad has been ba rces and mo c force. In al speed types ensitivity ana has been riations in fl h command observed tha can significa n terms of ge tilt angles a t to gimbal stiff ess impacts in terms of e. In fact, uces larger ongitudinally value. Also, rtant.
ends for the he wobbling
ness.
behaviour of ned for two-he teetering n carried out. gn described same design feature an on between se to cyclic v precession t, has been nite element o accurately ponses when dvance ratio ased on hub oments, and ll cases, the is apparent. alysis for the performed, lybar inertia, mixing ratio at the flybar ntly vary the average tilt nd wobbling fness. s f , r y , e g f -g . d n n n c n n t y n o b d e . e , , o r e t
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