135 Dr. Helut Quabek Finkeneg 39 6483 Babenausen Geran HQ-odellflugliteratur On te Longitudinal Stabilit of Gliders B Helut Quabek, bbreviations e 1, e, e 3 experiental sste of oordinates ord lengt ĉ ean aerodnai ord lengt C l L L L d D D i o o r lift-oeffiient of te profile lift-oeffiient lift-oeffiient of te ing lift-oeffiient of te elevator drag-oeffiient of te airfoil drag oeffiient of te ing indued drag-oeffiient of te ing oentu-oeffiient of te airfoil related to its aerodnai entre oentu-oeffiient of plane oentu-oeffiient of te lifting ing related to its aerodnai entre distane of te aerodnai entre of te elevator fro te.g.,y,z inertia-fores in te experiental-sste ing-area elevator-area L lift L lift of te ing L D D D i o N N lift of te elevator drag drag of te lifting ing indued drag piting-oent of te glider piting-oent of te ing related to te aerodnai entre position of te aerodnai entre of te glider position of te aerodnai entre of te lifting ing
.g. position of te entre of gravit veloit of te glider angle of attak ε ά ε Λ a donas-angle differene of te angles of inidene of elevator and ing d/dt rotational speed of te angle of attak angle of inlination, gliding-angle differene of angles of inidene aspet-ratio of te ing lift effiien-fator of te lifting ing a p airfoil effiien-fator of te lifting ing a x effiien-fator of te lifting ing, a x a @ a p Λ a aspet-ratio of te elevator sape influene lift effiien-fator of te elevator sape-influene ω rotational speed around te lateral -axis troug te.g. aerodnai pressure, ρ/ @ ρ σ i r i i Fn λ 1; λ 3;4 airdensit, 1.5 kg/ 3 at sea-level easure of te stati stabilit bod-ass ass of odel part i e.g. ing, fuselage, elevator, distane of te ass-entre of odel-part i fro te.g. ass-oent of inertia related to -axis.g. ass-oent of inertia of odel-part i related to -axis.g. arateristi euation it n solutions solutions of F4 for -disturbanes- solutions of F4 for --disturbanes ---
1. General Deterinations Generall in tis paper for all teoretial onsiderations te so-alled experiental sste e 1, e, e 3 of oordinates ill be osen, ere e 1 oinides it te gliding pat of te odel and as an angle of inlination against te geodeti orizon. e is osen orizontall and oinides it te lateral--axis of te glider in ing diretion, and e 3 is defined b e 3 e 1 x e. Te angle of attak of te ing tus is related to te axis e 1. Te angles and generall an var independent on ea oter and onseuentl also teir tie derivatives ά d/dt and ω d/dt. Conseuentl te diretion of te gliding speed points toards te negative diretion of te e 1 axis.. Fores at te Glider at longitudinal otion In te experiental sste of oordinates te euations of te fores ating at a glider, nael te fores due to ass inertia and te aerodnai lift and drag fores, are e 1 diretion: g sin D.1 e 3 diretion: Z Z g os + L. 3. Piting-oent of te Glider Te piting-oent aused b air fores at te glider in general depends on te angle of attak, te rotational speed of te angle of attak ά d/dt, and te rotational speed around te -axis lateral axes ω, ά, ω 3.1 Using a non-diensional oentu oeffiient, aording to aerodnai teor e get, ά, ω @ @ @ ĉ 3. Talor developent of te oentu-oeffiient provides ˆ,0,0 +, +, ω ω ˆ 3.3 Terein te derivatives ά /ά and ω / are dependent on. 3
Te seond ter signifies te oeffiient of a daping-oent i is proportional to te rotational speed ά. Essentiall it results fro te fat tat te angle t of te donas resulting fro te lifting ing at te elevator in ase of ά à 0 does not orrespond to te angle of attak t of te ing, but to te angle t + t. t r / and r is about te distane of te aerodnai entre of te elevator fro te entre of gravit,.g. E.g. for ά > 0 te donas angle gets saller and te resulting angle of attak at te elevator beoes t - t + t. Tus te lift at te elevator beoes larger ten for te stationar ase it ά 0, a negative piting-oent ill result, and in general e ave ά < 0. Te tird ter signifies te oeffiient of a dapening-oent i is proportional to te rotational speed ω around te lateral axis troug te.g. It results fro te fat tat te angle of attak and tus te lift at te elevator in ase of ω > 0 is inreased b te angle ω @ ĉ/ as opared to te stationar state it ω 0. gain a negative piting-oent results it ω < 0. Tese aerodnai piting-oents are ounterated b te ass-inertia of te glider-parts expressed b teir oents of inertia around te -axis of te glider. a i be te ass of a glider part and r i its distane fro te.g., ten te ass-oent of inertia of te glider an rougl be assessed to be r i Tus for te piting oent generall e ave i i + t t + t t 3.4 3.5 4. erodnai Centre of te Glider, N Te aerodnai entre of te glider is defined as te point N on its longitudinal axis about i te piting oent is onstant it respet to te angle of attak. Tus in ase of a ange of te angle of attak te resulting Lift L ill at troug N. Euall te aerodnai entre of te lifting ing is defined as te point N about i te piting oent of te ing, in generall dependent on te ing-sape and te properties of te osen airfoils, is onstant it respet to te angle of attak. Sine te lift- and oentu-arateristis, l and, of ost airfoils so a sligtl non-linear dependene on te angle of attak for lo Renoldnubers, Re < 1@10 6, te aerodnai entre of odel-ings an onl be onsidered to be stable for sall -ranges. Tis as to be taken into aount at te design of a odel-plane in order to aieve proper stati and dnai fligt beaviour under all fligt onditions. Te displaeent of te overall aerodnai entre of te plane N vs. te aerodnai entre of te ing N a be denoted b N N - N. Te oentu-euilibriu of te plane is ten given b N @ L r N @ L 4.1 Herein L is te ontribution of te elevator to te overall lift of te plane, and r N is te distane of te aerodnai entres of lifting ing and elevator. ording to aerodnai teories e get L l @ @ 4. l is te lift oeffiient of te elevator related to te area of te elevator and te aerodnai pressure at te loation of te elevator. For pratial reasons it is preferred to relate te lift oeffiient of te elevator to te ing area and its aerodnai pressure : L L @ @ 4.3 4
B oparison one gets L l @ / @ / 4.4 If te airflo on te elevator ould not be influened b te ake fro te ing e ould get te derivative L, L l Hoever, sine donas is generated in te ake of te ing, te angle of attak at te elevator loation is redued b te donas angle /. Te differene of te angles of inidene beteen ing and elevator a be denoted b ε, ten te angle of attak at te elevator is given b 4.5 + + ε 4.6 Taking tis angle of attak into aount, te oeffiient of te elevator lift results to be L + + l, ε 4.7 t a disturbane of te longitudinal otion of te plane around te piting-axis it fixed rudder te ange of te elevator lift oeffiient it is given b te derivative L, : L, d l, 1 + d 4.8 Te overall lift-oeffiient is L L + L 4.9 and in ase of a disturbane around te piting axis te lift-slope L, of te ole plane in ase of fixed ontrols results to L, d L, + l, 1 + d 4.10 Using forulas 4.8 and 4.10 in forula 4.1, e ill reeive N L, L, r N 4.11 Related to te ean aerodnai ord of te ing finall results ˆ N L d l, 1 + d d, + l, 1 + d r ˆ N 4.1 In a larger distane beind te ing te free vorties of bot alves of te ing indue a donas angle of about L / π @ Λ. Tere fro results / L, / π @ Λ and for te aerodnai entre of te plane 5
L, l, 1 π Λ L, + l, 1 π Λ N ˆ, L r ˆ N 4.13 Tis forula is still rater oplex and for ost odellers ipossible to solve. a out of tis dilea is found for pratial ases en onsidering o te derivatives L, and l, depend on te lifting arateristis of te osen airfoils and on te sape of ing and elevator. Te slope of te lift-oeffiient of a lifting ing, nael L,, is losel related to te slope of te liftoeffiient of te applied airfoil, nael l, b an effiien fator denoted as a L, a @ l, 4.14 a takes into aount te influene of te ing-sape on te foration of te free vorties on te ingsurfaes. ording to te liited ing-span and te ing-sape te ideal lifting-effiien of te airfoil is redued. In an ideal non-visous environent te slope of an ideal airfoil ould be l, @π. Hoever, in a visous airflo for Re-nubers belo 1@10 6 non-linear deviations fro tis ideal slope a be experiened, in te pratiall non-ritial range of te angles of attak ostl an inrease up to 5% are experiened. Tis an be taken into aount b anoter effiien-fator a p, Tus e get L, a @ a p @ @π 4.15 l, a @ a p @ @π 4.16 Denoting a x a @ a p forula.13 after a fe rearrangeents an be reritten to a pratiall easier for: ˆ N a a r 1+ a a ˆ N 4.17 terein a and a denote te total lifting-effiienies of ing and elevator. Reark: Te lifting effiien fators a and a an easil be deterined b eans of te FF - progra of te autor. For an pratial ases it is suffiient to approxiatel ose a p.1, and if te aspet-ratio Λ $ 5 ten aording to te expanded lifting line teor te effiien-fators an be approxiated b a Λ + Λ + 4 4.18 ording to long tie experiene, using tis approa ver reliable result of fligt-stabilit ould be pratiall aieved, as ill be explained b an exaple later on. 5. Stati longitudinal stabilit, σ One of te ost iportant fligt eanial arateristis of a glider is te apabilit, to redress te balane of te original stationar longitudinal fligt state after disturbane of te angle of attak b itout using ontrols. disturbane of te angle of attak auses an inrease or derease of te lift b L of te plane, if tereb a oentu is aused tat fores te plane to rotate bak to te original state, te plane is featured statiall stable. Tus a glider beaves statiall stable if generall olds true tat 6
d σ @ dl 5.1 σ is a non-diensional positive onstant fator i is onsidered as a stabilit-easure for te stationar longitudinal fligt of a glider. Te larger it is, te larger te bak-leading oentu ill be. For σ 0 te stabilit-beaviour of te plane ill be indifferent and it ill no ore be ontrollable, for σ < 0 te longitudinal fligt of te plane ill beoe instable. Note: s ill be son later, besides σ also te ass-oents of inertia of te glider parts are to be taken into aount to opletel deterine te tie-dependent otion of a glider bak to fligt-balane after disturbane. Using non-diensional aerodnai oeffiients e get σ d / d L 5. Based on te explanations in apter te piting oent around te.g. is given b N.g. @ L + o r @ + o 5.3 Terein o is te piting-oent for te ing at te aerodnai entre, o is tat of te elevator, and r is te distane of te aerodnai entre of te elevator fro.g. oents aording to te vertial position of te fores an ostl be negleted for gliders. For te ange of te piting-oent around te.g. b ange of te lift ere fro results d / d L N.g. / ĉ 5.4 Ipleenting 3.4 in 3. ields σ N.g. / ĉ 5.5 Tereit e ave a ver useful, uantitative easure for te stati stabilit of te glider, nael te distane of te.g. fro te aerodnai entre of te plane related to te ean aerodnai ord of te lifting-ing. Beause of te reuireent σ > 0, te.g. ust be positioned in front of N in order to aieve longitudinal stati fligt stabilit. s ill be disussed in ore detail later on, usuall te position of te entre of gravit is to be osen su tat te lift oeffiient L at slo stationar gliding is eiter adapted to te optiu gliding or to te iniu sinkrate. One te.g. is deterined b evaluation of te profile- and ing-arateristis, b eans of te teoretial onsiderations in apter 4 te geoetrial paraeters of te glider an be osen su tat te reuired size of σ ill be aieved. One proble ere a be o it an be found out at te appropriate size of σ is. Te ost adeuate a is to deterine te stati stabilit of one or ore representative odels i are onsidered to ave good stabilit-beaviour. 6. Free Osillations of a Glider it Fixed Controls Te stati stabilit-easure σ in priniple just provides an anser to te uestion eter or not a glider ill beave stable on a stationar linear fligt pat. Hoever, it does not infor o fast te glider ill redress te original stationar balane after an disturbane. In ase of stati stabilit e an expet tat te glider perfors attenuated osillations. In te ost general ase a glider a ondut obined - and -osillations as ell as osillations of te.g along te gliding pat. s ill be outlined later on, in ost pratial ases te -osillations are u faster tan te.g.-osillations and b eans of appropriate oie of te glider-design-paraeters it an be aieved, tat tese osillations are daped to su a degree tat te glider returns to balane in a ver sort tie..g.-osillations take longer and annot so ell be daped, oever, in pratie te an easil be balaned out b proper RC-ontrolling of te pilot. In order to deterine te beaviour of a glider after disturbane of te angle of attak and/or te gliding angle tis oveents a be onsidered as sall disturbanes, und of a stationar linear 7
8 fligt pat. Ten te euations for te fores at te glider, given in apter, a be developed into Talor-progressions ereb iger poer eleents are negleted: 6.1 and 6. Euall te oentu euation of apter 3 is developed to 6.3 Rearrangeent of te fore euations 6.1 and 6. and of te oentu euation 6.3 provides 6.4 B eans of an exponential desription of te disturbanes aording to o @ e λt, o @ e λt, o @ e λt te arateristi euation of te sste F 4 λ beoes: 6.5 + + D os g D + + Z Z Z L Z sin g Z L Z + + + + dt d dt d,0,0,0,0 ω, +,, ω 0 + + + 0 0 dt d dt d dt d Z Z dt d Z dt d λ λ λ λ λ λ λ Z Z Z F 4
In fligt-eanial teories tis euation is usuall ritten in subseuent for λ 4 + B @ λ 3 + C @ λ + D @ λ + E 0 6.6 ording to te stabilit riteria of Huritz for an osillating sste like te one onsidered E > 0! 6.7 is a neessar reuireent for te longitudinal stabilit of te glider. Taking euations 6.1 to 6.3 into aount, in detail e get D L E g sin g os L D + g os g sin 6.8 Tis an be reritten to E L g os D g sin + L g os D g sin L g os D g sin Here fro results L g os D g sin E + δ δ 6.9 Under noral angles of attak L/@@g@os - D/@@g@sin > 0, tus te reuireent 6.7 is idential it te reuireent + δ 0 δ 6.10 δ/δ denotes te deviation of te speed b, terefore te differential-uotient of te overall pitingoent derived b and taken along te speed polar ust be negative in order to aieve stati longitudinal stabilit: δ/δ > 0 6.11 Tis result is ell in orrespondene it tose of apter 5. 6.1 Fast piting-osillations t stationar free fligt gliding under ondition 6.11 for stati longitudinal stabilit, beause of te reuireents 0 te arateristi euation 6.5 ill be redued to λ λ 0 6.1.1 Tis is te arateristi euation of an attenuated pit-osillation around te.g. Sine te attenuation fator ά is alas positive b nature, tis euation provides real roots in ase of stabilit it >0. B eans of euations 6.3 e get 9
λ, +, ˆ λ, ω ˆ 0 6.1. In order to solve tis euation, next te derivatives erein ave to be deterined. 6.1.a Te -derivative Dependent on te rotation of te glider around te lateral axis it te angular speed ω, te so alled -derivatives, ill pla a roll. Te result fro te distint air as i eerges at te various parts of te glider b interferene of te general airflo it speed and of te loal vertial air flo it speed @ r ω @ r of te rotation, and ere r is te distane of te glider-part fro te.g. Te ange of te flo-diretion tereupon ten orresponds to an inreental angle of attak, also alled dnai angle of attak dn, given b dn atan @ r / @ r / 6.1.3 Tereb at te elevator an inreental lift results i is given b L, l ω r 6.1.4 Terein r is te distane of te aerodnai entre of te elevator fro.g., for te inreental lift oeffiient follos L l, ω r 6.1.5 Taking into aount tat / ω ˆ / L, ω L 6.1.6 te -derivative of te elevator beoes r ˆ l, ω l, 6.1.7 and for te overall derivative ill result r ˆ L, ω L, ω ing + fuselage + l, 6.1.8 Te elevator ontribution of te pit attenuation oents,ω tus is r L l, ω r 6.1.9 Wit,ω @ ω @ ĉ/ @ @ @ ĉ e get r ˆ, ω l, 6.1.10 For te overall pit-attenuation-oent it follos 10
, ω, ω ing + fuselage, ω 6.1.11 t onventional glider-onfigurations, lo seep of te lifting ing, and proper elevator distane fro.g., usuall te ontributions of ing and fuselage are less tan 1/10 of te elevator ontribution. For ost pratial ases in odel fling e an assue tat te aerodnai pressure at ing and elevator are about eual, / 1, and tus e finall get: r ˆ, ω l, 6.1.1 Using forula 4.16, tis attenuation derivative an finall be ritten in te for, ω r π a a p ˆ 6.1.13 Tis euation is of ajor iportane for te design of a glider. Te effiien fator a pas regard to te geoetri sape and to te aspet ratio Λ of te elevator, aording to te extended lifting-line-teor it good approxiation a Λ /+Λ +4 1/, e.g. for an elevator it Λ 6 e rougl get a 0.7. s desribed in apter 4 te fator a p pas regard to te visous flo-effets at te elevator-airfoil on its slope of te lift-oeffiient it. Usuall te.g of a glider is osen for optiu gliding or iniu sinkrate, ten te lift at te elevator is lose to zero and aordingl also te angle of attak. In order to inrease te veloit of te glider, an inrease of te angle of attak is reuired at te elevator. Rougl, ost oonl used airfoils of elevators ave setrial sape and teir visosit-fator at loer Re-nubers a deviate onsiderabl fro te ideal value a p 1 for ig speed. Tus, in order to guarantee a distint pit-attenuation-derivative,ω at an possible fligt-veloit, for a p te iniu a p 1 sould be osen and te paraeters a, and r of te elevator aordingl be adapted. Tis eans, for ost pratial purposes it is suffiient to use te euation, ω π a r ˆ 6.1.13a In an ases te value of,ω an be adopted fro odels knon to provide good attenuation beaviour. 6.1.b Te ά - derivative of of Te attenuation-derivative b ά d/dt on te one side takes te retarded ne foration of te airflo fro te lifting ing into aount i results fro an aelerated oveent of te angle of attak, on te oter side it pas regard to te donas-fration arriving at te elevator it dela after an nonstationar airflo-anges at te ing. Wit ange of te angle of attak b in te first instane tere ill appear eual -anges at ing and elevator. But onl after a tie dela of t te donas-ange of te lifting ing beoes effetive at te elevator at ten leads to an inreental ange of te angle of attak of te elevator. Under stationar fligt-onditions /@ orresponds to te relation of donas and angle of attak. In a larger distane beind te lifting ing it uasi-elliptial ingsape it an be assued tat π Λ d L d 6.1.15 s son in apter 4, L, a @a p @@π, erein a p takes are of te visous airflo-effets in te boundar-laer of te airfoil used at te lifting ing. Tus e an rite 4 a a p Λ 4 a Λ 6.1.16 11
1 For te ase of non-stationar airflo La Plae-transforation of te donas-anges at te elevator ields it p1/t 6.1.17 and for te effetive inreent of te angle of attak at te elevator follos 6.1.18 Developent into a progression for sall piting-freuenies, p <<1/t, ill suppl 6.1.19 Hereit for te ange of te lift at te elevator results to 6.1.19a 6.1.19b Te first ter in euation 6.1.19b orresponds to te stationar ange, te seond ter orresponds to an ά derivative, and taking into aount te definition e ill get 6.1.0 t an be alulated b eans of te speed and te pat r * i te anged ake as to travel fro te lifting ing to te elevator, nael t r */. Hereit for te ά-derivative turns to be 6.1.1 Te oentu-derivative b eans of te euation 6.1. an be ritten in te for 6.1.3 Te position of te aerodnai entre of te ing, N, depends on te lift- and oentuderivatives of te osen airfoils aording to 6.1.4 erein L, a @ a p @ @π. t p e p p 1 t p e p p + p p p 1 l L, + t l l 1, ˆ / /, L L t l L,, l L r ˆ,, ˆ / /,.. N g r L r L l N g r r 1 ˆ,..,,, 4 1 L N
For standard gliders r * r and. and tus it suffiient aura e an rite r. g. N, 1 l, ˆ r 6.1.4 ording to apter 4, l, a @ a p @ @π, a x a @ a p, and finall te attenuation derivative due to ά turns out to be r. g. N, π a 1 ˆ r 6.1.5 Even at ajor anges of N due to visous airfoil-effets for ost standard-gliders.g.! N << r for te non-ritial -region of te ing-setions terefore it suffiient aura e an assue tat r, π a ˆ 6.1.5 ltogeter te attenuation-derivatives ill suppl, ω +, r ˆ 1 + π a 6.1.6 ording to forula 6.1.16 / is of te order of agnitude of 4@a * /Λ, and onseuentl for gliders it iger aspet ratios of te lifting ing te donas-derivative / a be negleted itout ajor error. Tus, finall it an be stated tat te ajor ontribution to te attenuation of te rotational oveent of a glider results fro te -derivative. 6.1. Te - derivative of Te lift-dependene of a glider on te angle of attak itin te non-ritial -range of te osen airfoils in a first approa is oposed of sares fro te lifting ing and te elevator. Te influene of te fuselage sall ere be negleted. Sine effets resulting fro drag are also of seondar iportane, fro apter and it os 1 and D @ sin << L @ os te total lift of te glider is given b L L + L, 6.1.7 and using te anner of riting it lift oeffiients We get L L @ @, L L @ @, L l @ @ L L + l 6.1.8 Taking into aount te donas fator /, te overall lift slope of te glider results to L, + 1 l, L, 6.1.9 B use of forula 6.1.9 te -derivative of te piting-oent turns out to be. g. N L, + 1 l, ˆ, r ˆ 6.1.30 13
Tis derivative essentiall depends on te size of te elevator and its distane fro.g. Te position of te aerodnai entre of te ing is influened b te visous effets of te osen airfoils and given b euation 6.1.4. B use of euations 4.15 and 4.16 and under te assuption tat it finall follos. g. N r, π a + 1 π a ˆ ˆ 6.1.31 6.1.d Conseuenes for te Fast Piting ing-osillations 1,0 0,8 0,5 ttenuated Osillation Generall te arateristi euation of an attenuated osillation is ritten in te for λ + @ δ @ λ + ω o 0 6.1.3 Zt 0,3 0,0 0 1 3 4 5 6 7-0,3-0,5-0,8 tere in ω o [s -1 ] is alled irular eigen - freuen and δ [s -1 ] is alled attenuation onstant. ttenuated osillation is given for te ase en δ < ω o. -1,0 t In tis ase euation 6.1.3 as to onjugated oplex solutions λ 1;! δ ± j ω it ω ω o δ j denotes te iaginar unit. disturbane Zt ten results fro te general solution Z jωt t C1 e + C e jωt e δ t 6.1.33a t t 0, Z Z o Zt o for te undeterined oeffiients C 1 and C results Z o C 1 +C, and for our purposes Zt an finall be transfored into euation δ δ t Z t Z t os ω t + ϕ o 6.1.33 bove grapi illustrates te attenuation of a disturbane Zt it tie, te to enveloping urves desribe te tie dependent daping of te osillator otion after te disturbane. Te larger te attenuation onstant δ, te ore rapidl te envelope Zt o @exp-δ@t approaes 0. Te usual easure for it is D δ/ω o. B oparison of euation 6.1. it euation 6.1.3, for te attenuated osillations of te piting oveent of te glider after disturbane of te stationar gliding e get δ 1 ˆ, +, ˆ ω 6.1.34 Taking into aount euation 6.1.6 e ill finall get π δ a r 1 + ρ 6.1.35 Here ρ is te densit of te air. Tis euation tells us tat after disturbane of te angle of attak and/or te gliding angle, te daping of piting-osillations ainl depends on sape and geoetr of te elevator and in partiular ost strongl on te distane of te elevator fro.g. sine tis ats it te seond poer. 14
In te non-ritial -range usuall te visosit fator of te elevator airfoil a p 1, in partiular at ver lo Re-nubers ere visosit-effets in te boundar laer of te airfoil pla a onsiderable role for te airflo. In order to ake sure tat a glider ill provide desired attenuation, te loer liit a p 1 sould be osen in euation 6.1.35, and orrespondingl te oter paraeters for te reuired δ. s entioned earlier, regard to te donas is paid b / 4@a /Λ. In priniple it ill beoe saller it inreasing aspet-ratio of te lifting ing, and as ill still be disussed later, due to deteriorating visous-effets it inreasing fligt veloit it ill derease it inrease of te veloit. t loer veloit, for gliders it sall aspet-ratio Λ 10 te donas fator a beoe / 0.5, at iger speed for gliders it iger aspet ratio Λ 5 te loer border ill be in te range of / 0.15. Tis eans tat te attenuation of te piting-osillation ill beoe saller it inreasing aspet-ratio of te lifting ing i as to be taken into aount for te size and te oentu-ar of te elevator. We also learn fro euation 6.1.35 tat te attenuation of te piting osillation inreases it te speed and it te ean aerodnai ord C of te glider. fator to i ost often not suffiient attention is dran is te ass-oent of inertia around te lateral axis of te glider. ording to euation 3.4. te asses of te tail and te nose of te glider ontribute ost to tis oent, tus in order to aieve proper attenuation and to keep te elevator diensions sall, aording to euation 6.1.35 a onstrution goal sould be to keep te elevator ass as lo as possible orrespondingl te ass in te front part of te fuselage an be redued. Te attenuation of te piting osillation, oever, ust not be osen too strong beause on te oter side te response to te elevator ontrol-panel a beoe too slo for te neessar anoeuvrabilit. Wen designing a ne glider ostl it an be ver elpful to deterine te values of te arateristi paraeters for te attenuation fro gliders knon to provide te reuired δ-easure. For te irular eigen -freuen ω o of te orresponding non-attenuated osillation it atters 1 ωo, ˆ 6.1.36 and taking into aount euation 6.1.31 it ill turn out to beoe ωo π a 1. g. ˆ N 1 π a r ˆ ˆ 6.1.37 Sine te aerodnai entre of te lifting ing is deterined b te ing-design and te osen airfoils and te position of te.g in priniple results fro te reuireents for optiu gliding and/or iniu sinkrate, and sine all oter paraeters are deterined b te reuireent for suffiient stati stabilit and attenuation, tere is no ore possibilit to affet ω o. 6. Slo Osillations of te Centre of Gravit t instationar longitudinal otion of a glider it onstant angle of attak, 0, aording to F.W. Lanaster a so alled pit-pugoid develops after disturbane in and/or. Tis usuall is a longperiod ode in i te.g. arries out a ligtl daped osillation about its stationar fligt pat. It involves a slo piting-osillation over an seonds in i energ is exanged beteen vertial and forard veloit. Te euations of otion no just provide inforation on te angle of te elevator ontrol neessar to aintain a onstant angle of attak. Te relations beteen te eigt G @g, te lift L, te drag D of te plane, and te gliding angle are to be derived fro euations 6.1 and 6. of te fores in te diretions of te x- and z-axes. Wen setting 0 e reeive 15
+ 6..1 Z + Z 6.. 0 λ Z λ Z 6..3 Tereupon te arateristi euation of te pit-pugoid an be ritten in te for D D L λ + G sin + λ + G sin + G os 0 6..4 Sine L L and D D, it follos L/ @L/ and D/ @D/ and it L G@os, D G@sin e get λ + G sin + G sin λ + G sin + G os 0 6..5 Tus finall te arateristi euation an be ritten in te for g sin g λ + λ + os sin 0 6..6 For saller gliding angles sin 0. 6..a Conseuenes for te slo.g.-osillations Like for te fast piting osillations, te general arateristi euation of te daped.g.-osillation is to be ritten in te for λ + @ δ @ λ + ω o 0 6..7 Terein ω o [t -1 ] is alled irular eigen -freuen and δ [t -1 ] is te daping onstant. ttenuated osillation is given for te ase en δ < ω o. Like for te arateristi euation of te fast pit osillation, ten tere ill exist to solutions λ 3 and λ 4 λ 3;4! δ ± j ω it ω ω o δ Tis again leads to a daped osillating disturbane. Coparison of euation 6..7 it euation 6..6 ields: g sin δ Daping onstant: 6..8 g ωo os sin g os Eigen -freuen: 6..9 16
Tus, daping and eigen -freuen onl depend on te veloit and on te gliding angle of te orresponding stationar fligt-state. Te do not depend on te arateristis of a given glider. ttenuated C.G.-Osillation Zt 1,0 0,8 0,5 0,3 0,0 0 1 3 4 5 6 7-0,3.g.: Zt.g.: e-dt -0,5.g.: edt pit: ZT -0,8 pit: e-dt pit: -e-dt -1,0 t Te left grapi provides a roug idea of te differene beteen te fast attenuated pitingosillations and te slo, daped.g. osillations. Te subseuent exaples ill provide te relations as te are observed in fling pratie. 6.3 Coupled Pit- and C.g.-Osillations In soe ases it a be desired to onsider te euations of otion for a onurrent disturbane in veloit, gliding angle and angle of attak. In tis ase te arateristi euation F 4 λ euation 6.5 and 6.6 as to be solved. Te oupling of fast pit- and slo.g.-osillations ill ause a ertain sift of te roots λ 1 to λ 4. s before λ 1; a be te roots of te fast piting-otion and λ 3:4 tose of te slo pugoid-otion. Using te designation of te oeffiients of F 4 as in euation 6.6 te ateati evaluation provides folloing euations for te four roots: λ 1 + λ!b λ 3 + λ 4 λ 1 @λ C λ 3 @λ 4 λ 1 + λ @λ 3 + λ 4 λ 3 @λ 4 E / λ 1 @λ λ 3 + λ 4! D+ λ 1 + λ @λ 3 @λ 4 / λ 1 @λ 6.3.1a 6..3.1b 6.3.1 6.3.1d For te initial approxiation λ 3 + λ 4 0 0 and λ 3 @λ 4 0 0 as a first solution is ielded: λ 1 + λ 1!B 6.3.a λ 1 @λ 1 C 6.3.b λ 3 @λ 4 1 E / C 6.3.. λ 3 + λ 4 1!D@C + B@E / C 6.3..d In a seond step ten te roots λ 1 to λ 4 an be deterined and te orresponding otion investigated. Tis ill not furter be folloed up in tis ontext. For RC-ontrolled planes it is rater iportant tat te fast piting-osillation is suffiientl daped sine oterise te pilot ill not be able to orret tese disturbanes. On te oter and, to orret slo pugoidal osillations does usuall not ause an probles. ording to experiene for ost standard gliders te daping of te pit-disturbanes is su tat te fligt beaviour after disturbanes an be predited b eans of te separated arateristi euations of otion for fast piting-osillations and slo pugoidal oveent. It a be iportant to solve te oupled arateristi euation for ing onl -gliders it inor degree of longitudinal attenuation in order to predit teir fligt stabilit. 17
7. Exaples of Proven Gliders 7.1 ssessent of te ass oent of Inertia nertia, ording to apter 3 te ass oent of inertia is given b te euation r i i i Inertia fores derive fro te attribute of te ass to resist aelerations. Te ass of rotational aelerations is represented b ass oent of inertia ters. Te total ass oent of inertia related to te rotation of a glider around te lateral -axis troug te.g.,, results fro te various parts of te glider: te lifting ing, te fuselage, and te tail-parts fin and elevator, or -tail. n approxiate value of an be assessed for ost gliders aording to te approa @ r + f,f @ r f,l + f,r @ r f,r + t @ r t 7.1.1 Terein denotes te ass of te lifting ing, r te distane of te.g. fro te ass entre of te ing, f,f te sare of te fuselage-ass in front of te.g., r f,f te distane of te.g. fro te assentre in te front of te fuselage, f,r te ass of te rear-tube of te fuselage beind te.g., r f,r te distane of te.g. fro te ass-entre of te rear-fuselage part, t te ass of te tail and r t te distane of te.g. fro te ass-entre of te tail. Later on to exaples ill be given. One of te ill be tat of an F3-glider it 3.7 eter ingspan and a ass of approxiatel.3 kg. For tis odel it as teoretiall estiated tat 1.30 kg, f,f 0.68 kg, f,r 0.8 kg, t 0.1 kg, r 0.03 eter r f,f 0.4 eter r f,r 0.7 eter r t 1.15 eter Hereit te ass-oent of inertia as expeted to beoe 1.30 @ 0.03 + 0.68 @ 0.4 + 0. @ 0.7 + 0.1 @ 1.15 kg @ 0.001 + 0.109 + 0.098 + 0.159 kg @ 0.367 kg @ We see tat te sallest ontribution results fro te lifting ing beause its ass-entre is rater lose to te.g., ilst te largest ontribution results fro te tail-part i as te loest ass, but its distane fro te.g. is te largest. Generall, in order to keep te ass-oent of inertia sall as desired b te daping-reuireents, at standard-gliders te eigt of te tail sould be kept as lo as possible. Ea gra saved at te tail also redues te balaning-ballast in te nose of te fuselage about fator to 3 and orrespondingl also te ass-oent of inertia of te fuselage-front. s as son in apter 6, plas an iportant roll for te attenuation of te fast piting osillations, see euation 6.1.35. Wit inreasing value of in general te size or te oentu ar r of te elevator ave to be inreased to opensate for. If also a ertain stati stabilit is reuired aording to apter 4, euation 4.1.7., te rigt balane beteen and r as to be found. 18
7. Experiental Deterination of te ass -oent of Inertia nertia, siple pratial etod to deterine te ass-oent of inertia of an given bod is as follos. If a bod like tat in te left grapi is suspended it an axis troug it an be stiulated to sing around tis axis and te tie T for one full period of osillations is given b T π g z 7..1 Herein is te ass-oent of inertia related to te axis troug, z is te distane fro te.g. ording to psial eanis an also be desribed in te for.g. + @ z 7.. ere.g. is te ass-oent of inertia for te bod related to te axis troug te entre of gravit, parallel to te axis troug. Cobining te to euations e get. g. T π g z z 7..3 ω π/t is te osillation-freuen of tis sing of pendulu. B eans of tis pendulu-etod for a given odel-plane te ass oent of inertia around te lateral -axis troug te.g. an easil be deterined. For exaple, en te F3-odel given in setion 7. as ung up it nose don at te end of te fuselage it sung it a period-tie T.3 s. Wit a distane of te singing-axis fro te.g., z 1. eters, b eans of forula 7..3 tis ields.3/π @.3 @ 9.81@ 1.!.3 @ 1. 0.38 kg@ Te above teoretial estiate of 0.367 kg@ differs not u fro te pratial result. In order to deterine te appropriate values of te geoetri paraeters of te odel for proper stati stabilit and attenuation of te fast piting osillations it as a good guide. 7.3 Exaple of an F3-odel Te left grapi sos te 3 side-draft for a ne F3-odel planned b te autor. Fligt-eanial arateristis of te odel as given belo ave been deterined b eans of te FF - progra Fligt-eanis for Fligt- odels i is desribed in ore detail on te oepage.-odellflug.de. ajor goals for te odel ere superior sinkrates and gliding-perforane at all fligt onditions, as ell as proper fligtstabilit and anoeuvrabilit as reuired in F3-ontests. a. Sine te lifting ing is ainl responsible for te perforane of a glider-odel, ajor attention as been turned to its geoetri outla and its aerodnai arateristis su as lift-effiien, airfoil- 19
and indued drag. Te airfoils finall osen are te HQ/W-.5/8.5 strait troug for te lifting ing, and te HQ/W-0/9 for elevator and fin. Te distribution of te ing ord as osen su tat te liftdistribution of te odel is lose to ideal. ording to good pratial experiene, stall probles at te lifting ing an be andled b appropriate ingtips. Cl HQ/W-,5/8,5 dnai Cl-Cd-Polars of F3-odel 1,5 dn Polare, 30 g/d Re 100000l Re 00000 Re 400000l 1,0 Re 800000 0,5 0,0 0,000 0,005 0,010 0,015 0,00 0,05 0,030 Cd b. Usuall te seletion of distint airfoils for a lifting ing is done b a oparison of te perforane of potential airfoils over te possible speed range given b te eigt/unit area. For anned gliders tis at te end is given in te for of a uasi-stationar veloit polar. In te first instane it reuires tat for all possible stationar veloities of te glider te orresponding lift, te airfoiland te indued drag ust be deterined for te lifting ing. In te left polargrapi of te HQ/W-,5/8,5 -profil tis partiular uasi-stationar polar is indiated b te red polar urve. Oter l - d -values ten tose on te red uasi-stationar polar a be reaed under instationar fligt onditions, su as given at a fast turn or a loop, oever, tis is of inor iportane for te perforane-onsiderations. Te stationar gliding- and sink-veloities of a glider are given b g os 4 ρ os L L 7.3.1 z g ρ W 3 L 3 os 4 W 3 L 3 os 7.3. Te orresponding stationar gliding nuber is given b G.N. 1/ tan L / D L / D 7.3.3 In order to deterine te potential perforane of a given lifting ing it osen ing-setions te autor usuall asertains te funtional dependene of te ing onl sinkrates and gliding nubers given b S 3. R. D L G. N. L D 7.3.4 7.3.5 If onl one airfoil is osen itout tist, like for te F3-odel, ten L a @ l, ere a denotes te lift-effiien-fator of te ing i an easil be deterined b te FF-progra and l is te lift oeffiient of te osen airfoil. In oter ases L ust be deterined b integration i ill not furter be explained ere e.g. su a etod is inluded in te FF-progra. Te drag related to te osen lifting-oeffiient results fro te properties of te airfoil and fro te free vorties, in total e ave D Dp + Di Dp + L /πλ. Correspondingl for te F3-odel under onsideration te grapi belo reflets te dependene of te gliding nuber G.N. and of te sinkrate S.R. on te veloit, indiretl given b te l of te airfoil. In tis art are also inluded te G.N.- and S.R.-urves for flap defletion. B eans of su a art te optiu L - D -orking point an easil be identified eiter for iniu sinking or optiu gliding as reuired at slo stationar fling in F3-ontests. 0
Gleitzalen und Sinkraten der Tragfläe für Profil HQ/W-,5/8,5 35 0,5 30 0,4 5 Gleitzal GZ 0 15 10 5 0,3 0, 0,1 Sinkrate SR GZHQW-,5/8,5 GZ4 GZ-3 dn GZ SRHQW-,5/8,5 SR4 SR-3 dn SR 0 0,0 0,0 0,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1, 1,3 Profilauftriebsbeierte a For te planned F3-odel te optiu orking point is around l 0.9. Next e ill see ere te.g. ust be loated in order to aieve tis orking point ilst fling.. Having deterined te optiu L - D -orking point of te lifting ing for slo perforane gliding, next te position of te entre of gravit.g. i enables te glider to aieve tese optiu fligt onditions at soaring as to be found. ording to fligt dnais, it f denoting te oentu-oeffiient of te fuselage, and assuing tat te oentu oeffiient of te elevator an be negleted, te longitudinal oentu euation for te entre of gravit generall ields. 7.3.6.a For te tinner F3-fuselages f an be negleted, tus for zero lift at te elevator it turns out. g. ˆ. g. ˆ N 1 r + L o f ˆ ˆ ˆ N L o L 7.3.6.b In ases like te one being onsidered ere te sae profile is used in all ing setions, te oentu oeffiient o orresponds to tat of te profile and te lift oeffiient L is given b L a @ l, ere l is te lift oeffiient of te profile. For exaple te oeffiients o and l are taken for te HQ/W-,5/8.5 -airfoil fro te orresponding -FOIL-polar-diagras son belo. HQ/W-,5/8,5; dnai Cl-,C-Polars, 30 g/d Cl 1,5 1,0 0,5 0,0-5,0 -,5 0,0,5 5,0 7,5 10,0 1,5 lpa 0,00-0,05 C -0,10-0,15 dn Cl-polar Cl, Re100000 Cl, Re400000 dn Co-polar C, Re100000 C, Re400000 In tis grapi in partiular te uasistationar polars of te oeffiients for lift tik red urve and te airfoil oentu tik dark blue line are given for a ing load of 3.0 kg/. B eans of te FF-Progra te lift effiien-fator of te ing as alulated to be a 0.897. 1
Fro te above art te oentu oeffiient orresponding to te lift oeffiient for optiu perforane, l 0.9, as taken to be Tus e get o!0,059. 0.059 0.897 0.9 ˆ 0. 073 ˆ ˆ. g. N N + If no visous effets ould influene te airstrea at te profile te aerodnai entre of te ing ould be at 5 % and te.g. ould ten be at 3.3 % of te C. s an be seen in te polar art, aording to te -Foil-analses te lift- and oentu-oeffiients for a given Renold-nuber, e.g. Re100000 and Re400000, do not beave linear, even not in te non-ritial -range, but approa linear perforane it inreasing Re-nubers. s alread pointed out in apter 6, due to te visous airstrea at te airfoil te aerodnai entre of te ing sifts aa fro te ideal 5 % position aording to forula 6.1.4 ˆ N 1 4, L, Te uasi-stationar developent of te profile derivatives, and l, as funtions of te angle of attak is son in te folloing grapi. Te urves are traed bak to te orresponding urves in te previous grapi. Te tinner lines represent te dependene of te derivatives on te angle of attak and te tik red line teir ratio. Cl ; C /Cl HQ/W-,5/8,5 dnai Derivatives, 30 g/d 0,0 0,00 0,15 0,015 0,10 0,010 0,05 0,005 0,00 0,000-0,05-0,005-5,0 -,5 0,0,5 5,0 7,5 10,0 lpa C Cl dn dc/dc l dn C dn s entioned earlier, L, a @ l, in ase of a lifting ing it unifor airfoil. Tus it te derivatives taken fro te left art for l 0.9, for te planned F3odel te teoretial aerodnai entre turns out to be 1 4 N, ˆ a l, 0.5 0.065 / 0.897 0,178 nd onseuentl te position of te entre of gravit sould be osen at g.. ˆ 0.178 + 0.073 0.51 Rearks and oparison to fligt experiene Tis.g.-position appears to be rater far in front of te lifting ing; oever, to aspets ave to be onsidered: First, te osen orking point l 0.9 is rater ig but te orresponding.g. ill allo to aieve all stationar fligt onditions for all saller l -values. Seondl, for exeplar reasons te aerodnai oeffiients and teir derivatives for tis partiular F3-ase ave been developed b eans of te -Foil-progra of ark Drela i to soe degree sees to overepasise te visous airstrea effets. Te Profile-progra of
Riard Eppler on te oter and appears to underestiate te and ould ave ielded a position for te aerodnai entre loser to te uarter-point of te C. s te long ter pratial experiene of te autor it an different odels as son, in ost ases te aerodnai entre of te lifting ings is loser to te 5 % position ten to tat resulting fro te -Foil-progra! But, fortunatel, in pratie a bad position of te.g. ill soon be found out and orreted. s as been son in previous apters, te.g.-position is te point of referene for te stati and dnai stabilit of te glider. If te aerodnai entre of te ing is assued too far in front of te ing and onseuentl also te.g. it a appen tat te stabilit-arateristis su as te elevator size and te oentu ar a be osen too sall. Tis ill furter still be disussed in te next setion. E.g. for a orking point l 0.8 te position of te aerodnai entre ould turn out to be at 19.4 % and tat of te.g. at 8. %, te soaring perforane ould not u differ. d. Next step in te design-routine of a plane ill usuall be to deterine te diensions and aerodnai features of te eleents for longitudinal fligt ontrol and stabilit. For a noral glider tese eleents are te rear fuselage-part beind te.g. and te elevator. For ing onl odels S-saped profiles, seep, and negative tist of te lifting ing ill take over te longitudinal stabilit funtions, but tis ase ill be disussed in anoter paper. Te stati stabilit for sure is ost iportant for te longitudinal fligt-stabilit of a plane, as laid out in apter 5: σ N.g. / ĉ Fro an glider onstrutions te autor got te feedbak, tat te longitudinal stabilit of F3-gliders sould at least be 0.1: σ 0.1! Conseuentl e ave te reuireent for te elevator and its oentu ar related to te.g. tat aording to apter 4 te ave to be sized su tat te aerodnai entre of te total plane fulfils te reuireent N / ĉ 0.1 +.g. / ĉ In apter 4 te final euation for te dependene of te aerodnai entre on te ing- and elevatorarateristis as ˆ N a a r 1+ a a ˆ N it a x a @ a p and a x a @ a p. Fro te last grapi it an be taken, tat te -derivative of te profile-lift-oeffiient for te lifting ing varies beteen l, 0.08 grad -1 for te osen orking point l 0.9 and l, 0,13 grad -1 for lo speed, ile at ediu angles of attak 1-4 it sos values about l, 0.104 grad -1. In te alulations ere ust be transfored to rad-units, ergo: 4.58 l, 7.44 rad -1 Sine a p l, /π, orrespondingl e reeive 0.73 a p 1.19, 3
For te range of ore dnai soaring l, 6 grad -1 i is not far fro te ideal π-value and a p 1. s e ave seen earlier in tis apter, te lift effiien-fator of te osen ingfor is a 0.897, and aording to experiene te lift-effiien of te elevator is assued to be about a 0.75. Te ajor orking-onditions of te elevator are about zero lift. Te -Foil-analses of suitable airfoils for te elevator tell us, tat for te lift range of te elevator a p 1,. Based on tese data and on a first rude estiate / 0.1 for te ratio of te areas of elevator and lifting ing, and te siplifiation tat approxiatel r N r e reeive 0.056 @ r / ĉ N / ĉ 0.088 @ r / ĉ Taking te.g. as found for te orking point l 0.9, nael.g. /ĉ 0.51, on te oter and side e ave N /ĉ 0.15+0.51 0.4. Conseuentl te range for te oentu-ar r is found to be 6.3 r / ĉ 4.0, e. In setion 6.1.d it as derived tat te attenuation-onstant δ of fast piting-osillations is affeted b various paraeters as finall given in euations 6.1.34/35, ritten in a ore pratial for e get ˆ δ ρ ˆ 4, 1 +, ω π a r ˆ 1 + First e see tat te daping of te osillations inreases it te soaring veloit. Tus, inor influenes due to invisous airfoil effets i are refleted b te fators a p and a p are generall overeled it inreasing veloit. Seondl, for a given ing te ajor paraeters b i attenuation an be influened are te assoent of inertia and size, sape fator a, and oentu ar r of te elevator. For a lifting ing te -derivative of te donas far beind te ing is given b / 4@a /Λ. Tus, it dereases it te aspet-ratio Λ of te lifting ing. Wit data given earlier te -derivative for te planned F3-Glider ranges aording to 0.15 4@a H / Λ 0.5 and annot be influened b te elevator arateristis. Conseuentl te onl reaining design-eleent for proper daping of disturbanes is te ratio of te - derivative and te ass-oent of Inertia, ω 1 π a r ˆ Fro pratial experiene it various F3-odels and analses of suessful oter F3-odels te autor as found tat appropriate dnai daping is aieved en tis ratio ranges itin -30 to -40. s as laid out in setion 7.1, for a loer-eigt F3-odel te ass-oent of inertia is around 0.4 kg@. Hoever, en a odel is being build it a easil appen tat te eigt of te tail gets iger tan desired and, sine te ass of te tail ontributes ost to, a inor daping tan planned ill appear. Tus, in order to be on te safe side onerning dnai longitudinal daping, it a often be better to assue tat 0.5. Sine daping inreases it fligt-veloit, te visous airstreaeffets at te elevator are in fat onl iportant for fligt-onditions near te orking point l 0.9. ssuing for te planned F3-odel as before tat a 0.75, a p 1., / 0.1, and 0.367 e ield te reuireent 3.8 ˆ 5.1 r 4
f. For te F3-odel son above te folloing odel-paraeters ere osen: ean ord ĉ 09.54 Lifting-area of te ing 0.704 spet ratio of te ing Λ 17.41 Lift-effiien of te ing a 0.897 irfoil of te ing HQ/W-,5/8,5 ean ord of te elevator ĉ 101.5 Lifting-area of te elevator 0.065 Lift-effiien of te elevator a 0.76 irfoil of te elevator HQ/W-0/9 For te ost iportant fligt state around te osen orking point l 0.9 e assue irfoil effiien-fator a p 1. at zero elevator lift Furter ass oent of inertia 0.367 kg @ Tere fro dnai stabilit is alulated to be: Dnai stabilit easure,ω / -33.9! Tis is ell itin te reuired stabilit range. Taking into aount te visous effets of te airstrea around te lifting ing -FOIL-analsis, above e ad found tat in order to aieve te orking point onditions around l 0.9 te position of te entre of gravit sould be osen at Centre of gravit.g. /ĉ 0.51 Ten te lengt of te oentu ar beteen.g. and te aerodnai entre of te elevator beoes Lengt of oentu ar r 1.05 Suffiient stati longitudinal fligt-stabilit ust be given. s son before, te overall aerodnai entre of te odel is deterined b te forula ˆ N a a r 1+ a a ˆ N Wit a p 1., a p 1., and r r N e get erodnai entre of glider N /ĉ 0.51 + 0.159 0.41 nd finall te stati longitudinal stabilit-easure for te osen position of te.g. orresponding to te orking point l 0.9 turns out to be 5
Stati stabilit σ N -.g. /ĉ 0.16! ording to experiene tis ould be uite a good stabilit-value for an F3-odel. Folloing also te results of te usual onventional non-visous alulations of te stabilit values are presented for oparison: erodnai entre of te ing N /ĉ 0.5 Zero oentu oeffiient of te ing o!0.08 Centre of gravit.g. /ĉ! o / L opt 0.08/0.897@ 0.9 0.0991.g. /ĉ 0.5 + 0.099 0.349 Lengt of oentu ar r 1.004 erodnai entre of te glider N /ĉ 0.5 + 0.84 0.534 Stati stabilit σ N - S /ĉ 0.19! oparison sos tat te alulation of te stati stabilit-easure found b onsideration of visous airstrea effets as te result fro te -FOIL-profile-analsis ields a value lose to tat of te nonvisous stabilit-onsideration. In priniple tis is due to te nearl eual sift of N and.g. toards te leading edge of te lifting ing as a result of te visous airstrea-effets on te l - and - derivatives as predited b te -Foil-progra. But: fter all te experiene te autor as gained in designing and RC-fling of an different glider-odels in over 30 ears, it as never found tat for a odel like te one under onsideration it a abered airfoil like te HQ/W-.5/8.5 te.g. sould be tat lose at te uarter-point of te C for te optiu orking-point l 0.9 as it turns out b eans of te -Foil profileanalsis. On te ontrar, in fligt pratie te.g. as alas found to be lose to te one alulated for te optiu orking-point b eans of te non-visous approa as given above. Tis is te autor prefers te PROFILE-progra of Prof. Riard Eppler over -FOIL, at least for alulations of fligt stabilit. Calulation of te.g. and stabilit-easures never failed en based on te PROFILE-analsis ile -FOIL alas predits a.g. too far toards te front of te planes. One ajor onlusion to be dran a be tat te profile-analses as onduted b te -FOILroutines obviousl overepasize te visous effets and tus predit rater strong deviations of te - derivatives for te lift and oentu-oeffiients fro te ideal non-visous slopes, in partiular for lo Re-nubers. profound revision of te parts of te progra it respet of te influene of visous effets ould for sure be ost appreiated b all odellers! g. Finall it ill be of interest to i degree fast piting-osillations of te planned F3-glider ill be attenuated b te fligt-eanial arateristis alulated before. Having in ind at as said before about te visous airstrea-effets, te furter alulations ill use te non-visous approa. In apter 6.1.d e ad son tat a disturbane of te angle of attak is desribed b euation 6.1.33 δ δ t Z t Z t os ω t + ϕ o Werein ω ω o δ an be alulated aording to 1. g N ωo π a. 1 π a ˆ π δ a r 1 + ρ r ˆ ˆ 6
ssuing tat a x a 0.76, a a 0.897, and / 4@a /Λ 0., ρ 1.5 kg/ 3 e get Eigen -freuen of pit-osillations ω o 0.53 @ [s -1 ] Daping onstant of pit-osillations δ 0.7 @ [s -1 ] Freuen of pit-osillations ω ω o! δ 1/ 0.46 @ [s -1 ] Osillation-freuen and attenuation inrease it inreasing speed, tus te orst fligt-state onerning daping of pit-disturbanes is given for te state related to te orking-point ere te fligt-veloit is at its iniu eloit at optiu orking-point 4 @ // a @ a 0.9 1/ 7.7 /s Here e ave ω o 4.08 s -1, δ.08 s -1, ω 3.54 s -1. For te slo.g.-osillations it as son in apter 6..a tat Zt ttenuated Osillation of an F3-odel for 7.7 /s 1,0 0,8 0,5 0,3 0,0 0 1 3 4 5 6 7-0,3.g.: Zt.g.: e-dt -0,5.g.: edt pit: ZT -0,8 pit: e-dt pit: -e-dt -1,0 t ω δ o,. g. g. g os. g sin Wit.7 and 7.7 /s ω o,.g. 1.80 s -1, δ.g. 0.030 s -1 ω.g. 1.80 s -1 Te attaed grapi sos, tat alread after one period te piting-osillations oe to rest and te glider finds bak to te stationar fligt state. t lo fligt-veloit te slo.g.-osillations of te F3- odel are onl oderatel daped. Hoever, in fligt pratie tese long.g.-osillations are usuall not a proble; ost pilots intuitivel orret te it te elevator-ontrol of te RC-transitter. Zt ttenuated Osillation of an F3-odel for 15 /s 1,0 0,8 0,5 0,3 0,0 0 1 3 4 5 6 7-0,3.g.: Zt.g.: e-dt -0,5.g.: edt pit: ZT -0,8 pit: e-dt pit: -e-dt -1,0 t Wit inreasing fligt-veloit also te gliding-angle dereases. s an easil be taken fro te above forulae, te dapening onstant δ.g dereases sligtl it inreasing and, ile te osillation freuen dereases. For exaple in te left and side grapi te pit and.g.-osillations are given for 15 /s and 3.8. eloit-polar und Gliding-Nnubers of an F3-odel 0 5 10 15 0 0,0 0-0,5 10 z [/s] CL/CD Finall te left grapi provides te teoretial perforane paraeters of te oplete F3-odel for its expeted operational fligt range under inlusion of all sorts of drag related to te odel. -1,0 0 z GZ -1,5 x [/s] 30 7