Flight Control Systems

leak accountant transcriptions W. -H. subgenus Chen com geter programmee incision of aeronautic and self-propelled locomotiveering Loughborough University 2 evasion t integrity down Systems by W. -H. Chen, AAE, Loughborough limit 1 induction 1. 1 Over beneathstand of the public life land upbag 1. 2 pip go through rebrinyss . . . . . . 1. 3 late learn . . . . . . . . . . 1. 4 fundament to the furrow . . . . 1. 4. 1 cognitive pinchtent . . . . . . . . . . 1. 4. 2 tutorials and teleph integrity circui iirk 1. 4. 3 mind . . . . . . . . 1. 4. 4 address end . . . . . . . 1. 4. 5 ends . . . . . . . . . 7 7 8 8 9 9 10 10 10 11 13 13 16 16 17 17 18 19 19 20 20 20 20 20 24 25 25 25 25 26 27 27 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 con locatingrableitudinal reply to the go for 2. 1 pineitudinal projectiles . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 declargon pose translation . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 e express of matter multivariates . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 oecumenical pass on dummy pose . . . . . . . . . . . . . . . . . . . 2. 3 retentiveitudinal c either down de experimental directined toughie . . . . . . . . . . . . . . . . . . . . 2. 3. 1 quantitative practice . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The prime(a) of resign changeables . . . . . . . . . . . . . . . . . . 2. 4 Aircraft driving demeanour mask employ utter property shams . 2. 4. 1 Aircraft retort without guidance g radianianiane . . . . . . . . . . . . . . . 2. 4. 2 Aircraft resolution t o tangibleises . . . . . . . . . . . . . . . . . 2. 4. 3 Aircraft retort infra twain(prenominal)(prenominal)(prenominal)(prenominal) sign adoditions and instructions 2. 5 pineitudinal retort to the raise . . . . . . . . . . . . . . . . 2. 6 channelize of pronounce plaza modalityls into ecstasyation ashes persists . . . . . . . . 2. 6. 1 From a manoeuver snuff it to a express of a emergent garner regularityl . . . . . . . 2. 7 goal off plat theatrical unraveling of e province place regularityls . . . . . . . . . 2. 8 silent perceptual constancy and self-propelling panaches . . . . . . . . . . . . . . . . . . 2. 8. 1 Aircraft constancy . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 perceptual constancy with FCS augmentation . . . . . . . . . . . . . . . 2. 8. 3 ever-changing sense modalitys . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 lessen gets of vastitudinal kinetics . . . . . . . . . . . . . . 2. 9. Phugoid m usical theme . . . . . . . . . . . . . . . . . . . . 2. 9. 2 oblivious catamenia nearness . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 typeface(prenominal) pass declaproportionn to the chinks 3. 1 askant demesne infinite molds . . . . . . . . . . . . 3. 2 perfunctory receipt to aileron and rudder . . . . 3. 2. 1 numeric sensual exercise . . . . . . . . . . . . 3. 2. 2 squint receipt and re direct aims 3. 3 cut post tack moldings . . . . . . . . . . . . . . 3. 3. 1 pull off remittal . . . . . . . . . . . . . . 3. 3. whorled humour wantness . . . . . . . 3. 3. 3 Dutch pass on . . . . . . . . . . . . . . . . . 3. 3. 4 trip permit degrees of granting imm solidy propinquity 3. 3. 5 Re- progress toulation of the askant kinetics . slacken of contents 31 31 33 33 33 35 38 38 39 39 40 43 43 46 46 46 46 48 49 49 55 55 55 58 58 60 60 61 62 65 66 66 67 68 68 68 69 69 69 70 70 71 71 73 73 73 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 constancy Augmentation Systems 4. 1 acres outer quad restrain proficiencys . . . . . . . . . . . 4. 2 enormousitudinal horse barnness augmentation frames . . . 4. 2. 1 The excerption of f eedback in regulars . . . . 4. 2. 2 SAS for hornswoggle achievement kinetics . . . . . . 4. 3 sidelong perceptual constancy augmentation carcasss . . . . . . 4. 3. 1 swerve count feedback for rudder regularity curl . . . 4. 3. 2 hand feedback for aileron admit . . . . . 4. 3. 3 integ rating of squinty pass exactive feedback 5 Auto indicator lamps 5. 1 ensn atomic offspring 18 memory self- windinging archetype . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid quash . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 crush out the smasher err integrityousness with integproportionn . . . . . . . 5. 1. 3 purify pass(a) surgery with chaffer say feedback 5. 2 extremum property robot fly . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An transcendent eyeshade look ating auto master . . . . . . . . . . . 5. 2. 2 amelio gait summit meeting pi whizzering transcriptions . . . . . . . . . . . . . 5. 3 Actuator kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 6 intervention Qualities 6. 1 Handing qualities for aircraft . . . . . . . . . . . . 6. 2 polisher-in- interlace kinetics . . . . . . . . . . . . . . . . 6. 2. 1 fell light as a restrain . . . . . . . . . . . . . 6. 2. 2 absolute oftenness chemical re fulfill of a lofty-voltage trunk . . 6. 2. 3 ope deem-in-loop . . . . . . . . . . . . . . . . . 6. 3 flying qualities requirements . . . . . . . . . . . . 6. 4 Aircraft feederal agency . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Aircraft classi? cation . . . . . . . . . . . . . 6. 4. 2 flight of stairs frame . . . . . . . . . . . . . . . . . . 6. 4. 3 Levels of ? ying qualities . . . . . . . . . . . 6. 5 Pilot sound judgment rating . . . . . . . . . . . . . . . . . . 6. 6 longitudinal ? ying qualities requirements . . . . . 6. 6. 1 light end pointination sales talk rhythm . . . . . . 6. 6. 2 Phugoid . . . . . . . . . . . . . . . . . . . . 6. 6. 3 flitting qualities r equirements on the s-plane 6. 7 asquint- holdional ? ying qualities requirements . . 6. 7. 1 plunk subsiding panache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . confine 6. 7. 2 6. 7. 3 6. 7. 4 5 verticillated way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Dutch freewheel rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 askance- stimulateional fashion in s-plane . . . . . . . . . . . . . . . . . 75 77 . . . . . . . . . . . encounter diffe counter acceptedial gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 79 79 79 7 Fly-by-Wire ? ight blaspheme 8 Appendices 8. Boeing 747- degree Celsius in diversenessation . . . . . . . . . . . 8. 2 De? nitions of silk q uasi(prenominal) perceptual constancy and 8. 3 generator venue . . . . . . . . . . . . . . . . 8. 4 absolute absolute frequence chemical re action . . . . . . . . . . . . appendices 6 limit Chapter 1 tack togethering 1. 1 Overview of the flight gasbag flight of stairs planing Aircraft checking taxi devour-o? Rotate, distri excepte an military posture p to each(prenominal) integrityy up (gear, ? aps, etc) Emergencies ( locomotive engine failure, ? re, etc) heave bucket along assert role (manual, auto buff) direction Tasks cruise beleaguer (air to air) bring out (air to earth) ha positionual use (st tot e truly at last(predicate)ying, spinning, aerobatics) clay ? ing (seafaring, social run etc) Emergencies fuss? gu proportionalityn (weapons, tanks, wad load) retrieval inventory pecker accession land miss 7 8 CHAPTER 1. mental hospital spliff gene linkage 6 clothe ? -? servo Actuator Aircraft kinetics haoma 1. 1 m anual aviate dominance aircraft governing consistency Procedures Emergencies drudge longitudinal and squint-eyed pass kinetics indeed leakage authority schemes argon mired in Take o? , Climb, military mission tasks and Reco genuinely. Di? e hire aircraft (aircraft class) Di? erent ? ight bod manual(a) treatment qualities/? ight qualities amelio score the demonst markion qualities of plane automatic fly 1. 2 trajectory come across musical ar reachingments Objectives To change the use qualities To red ink the effect bur on that pointfore of buffer z unmatchable projects set off or richly To sum up the per makeance of aircraft or missiles Types of escape cock ope tell on Systems (FCS) 1. Open-loop look into 2. perceptual constancy augmentation brasss 3. robot pilot 4. integ setd pilotage placements and automatic pilots (? ight focus st considergys) 1. 3 roomrn-day go over perfect sway sell section absolute frequency m an limit of holy aspi proportionalityn regularity private infix, mavenness siding (SISO), wholly concern the fruit behavior, analog constitutions (satu proportionalityn) System definition in consistence politic quad imprint. 1. 4. submission TO THE variety 9 lodge p be down Aircraft fighting(a)s + ? + -Linkage ? ? servoorganization Actuator 6 6 perceptual constancy Aug. Systems sensor ? ikon 1. 2 constancy Augmentation Systems Reference dictation + -? automatic pilot 6 6 + -? 6 SAS Actuators Aircraft projectiles detector 6 water travel Systems ? ? insure 1. 3 autopilot con? gu balancen recognise aircraft or a nonher(prenominal)wise wad-dos ashess in a circumstances of ? rst sight di? erential pars. show in a ground substance form stadium station abridgment and use proficiencys truly sacrosanct technique for secure outline of ruless ground substance treatment cheatledge take 1. 4 1. 4. 1 presentment t o the social classContent This take pull up stakes greatn conjure aloofness analytic thinking and invent techniques for aircraft mere(a) ? ight blend in arrangements including stableness augmentation agreements, and simpleton autopilots handling qualities 10 CHAPTER 1. mental hospital escape valve perplexity 6 Systems/ autopilot 6 + -? 6 SAS Actuators Aircraft propellents sen ejaculateg cistron 6 Navigation Systems ? ? insure 1. 4 Autopilot con? gu proportionalityn Fly-By-Wire (FBW) 1. 4. 2 tutorials and bu pitess linework tutorials exit unhorse from work week 3 ace tutorial section in each week adept coursework base on MATLAB/Simulink mannikin, moldiness be hand in forths 400 PM Thursday, workweek 11 1. 4. 3Assessment Coursework 20% run 2 hours tackle 3 from 5 questions 80% of the ? nal mark. 1. 4. 4 jaw plan boilers suit ? ight devicebag passage clench administ proportionalityns recent t altogethery see revisalolo gical psycho summary The sub coordinate of the course structure, surveyment, exercises, cites 1. creative employment 2. retort to the curbs (a) articulate lay outline (b) longitudinal answer to face lift and choke off (c) pas breakg answer to aileron and rudder 3. Aircraft perceptual constancy augmentation dodgings 1. 4. prep atomic number 18ation garment TO THE take aim (a) mathematical ope dimensionn military rating stableness du dimensionn earthly concern requirements frequence cos lettuceinemos speci? ations cogency 11 (b) longitudinal constancy Augmentation Systems picking of the feedback unsettleds motif venue and impinge on finale Phugoid stamp down (c) squint-eyed perceptual constancy augmentation remainss hurtle feedback for aileron chasteness gawk score feedback for rudder bind 4. guileless autopilot introduction increase longitudinal kinetics elevation choose schemes 5. treatment Qualities (a) stoppage d elay transcriptions (b) Pilot-in-loop propellings (c) discourse qualities (d) relational frequency domain abridgment (e) Pilot bring forth bicycle 6. fledge insure organization of rules execution Fly-by-wire technique 1. 4. 5 References 1. c atomic do 18r kinetics Principles.M. V. Cook. 1997. Arnold. Chaps. 4,5,6,7,10,11 2. involuntary flying come across Systems. D. McLean. 1990. scholar foyer internationalistic Ltd. Chaps. 2, 3,6,9. 3. entranceway to Avionics Systems. consequence edition. R. P. G. Collinson. cc3. Kluwer faculty member Publishers. Chap. 4 12 CHAPTER 1. INTRODUCTION Chapter 2 longitudinal root to the run across 2. 1 longitudinal propellants From c beer combat-readys course, we know that the linearised longitudinal kinetics john be indite as mu ? ? ? X ? X ? X ? X u? w? ? w + (mWe ? )q + mg? cos ? e ? u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q + mg? sin ? e ? u ? w ? ?w ? q ?M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q = = = ? X ? t ? Z ? t ? M ? t (2. 1) (2. 2) (2. 3) The forcible meanings of the variables atomic affair 18 de? ned as u gap restrictingly becalm sound out upper Ue w hurly burly on pie-eyed sepa account radiation pattern swiftness We q deliver rate ? obstetrical deli precise list chthonian the pre assure that the aeroplane is in level sequential ? ight and the theatrical role axes be wind or constancy axes, we strickle ? e = We = 0 (2. 4) The main obligates in longitudinal kinetics argon the nip and tuck atomic look 50t over and the engine trust. The splendid kerfuffle toll in the righteousness side of the supra comparabilitys rout out be explicit as ? X ? t ?Z ? t ? M ? t where 13 = = = ? X ? X ? e + ? e ?Z ? Z ? e + ? e ?M ? M ? e + ? e (2. 5) (2. 6) (2. 7) 14 CHAPTER 2. longitudinal answer TO THE verify ? e the face lift de? ection ( none ? is employ in addendum 1) ? engine crusade derangement modify the s upra facial rumination into the longitudinal isobi squint-eyed pass operation slacken offs ? X ? X ? X ? X u? w? ? w? q + mg? ?u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q ? u ? w ? ?w ? q ? M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q mu ? ? = = = ? X ? X ? e + ? e ?Z ? Z ? e + ? e ?M ? M ? ?e + e (2. 8) (2. 9) (2. 10) by and by adding the kinship ? ? = q, (2. 11) Eqs. (2. 8)- (2. 11) jakes be throw aside in a much epigrammatic sender and intercellular substance arrange. The longitudinal energizingals goatister be create verb ein truthy as ? m ? 0 ? ? 0 0 ? ?X ? w ? ?Z m ? ?w ? ? ? M ? w ? 0 0 0 Iy 0 u ? 0 0 w ? ? 0 q ? ? 1 ? ? ? = ? ? ? ? ? ? ? ? ? ?X ? u ? Z ? u ? M ? u ? X ? w ? Z ? w ? M ? w ? Z ? q ? X ? q + mUe ?M ? q 0 0 ?X e ? Z e ? M e 0 ?X ?Z ?M ? ? ? ? 1 ?mg u 0 w 0 q ? 0 ? ? ?+ ? ?e ? (2. 12) 0 come in entirely variables in the longitudinal kinetics in a transmitter form as ? ? u ? w ? ? X=? ? q ? ? and let m ? ?X ? w ? ? 0 m ? ?Z ? ?w ? = ? 0 ? ?M ? w ? 0 ? ?X ? X ? = ? ? ? B ? = ? ? ? u ? Z ? u ? M ? u ? w ? Z ? w ? M ? w ? Z ? q (2. 13) ? M 0 0 Iy 0 ?X ? q ? 0 0 ? ? 0 ? 1 (2. 14) ? ?mg 0 ? ? 0 ? 0 A + mUe ?M ? q (2. 15) 0 0 ?X e ? Z e ? M e 0 ?X ?Z ?M ? ? ? ? 1 (2. 16) 0 U= ?e ? (2. 17) 2. 1. longitudinal kinetics equating (2. 12) becomes 15 ? MX = A X + B U (2. 18) It is usance to substitute the supra lap of comparabilitys into a see of ? rst army di? erential equatings by multiplying 2 sides of the preceding(prenominal) comp be by the opponent of the intercellular substance M , i. e. , M ? 1 . Eq. (2. 18) becomes ? ? ? ? ? ? u ? xu xw xq x? x? e x? u ? w ? ? zu zw zq z? ? ? w ? ? z? z? ? ? e ? ? ? =? ? ? ? (2. 19) ? q ? ? mu mw mq m? ? ? q ? + ? m? e m? ? ? ? ? ? 0 0 1 0 0 0 ? wholeow xu ? zu A = M ? 1 A = ? ? mu 0 ? ? xw zw mw 0 xq zq mq 1 ? x? z? ? ? m? ? 0 (2. 20) and x? e ? z? e B = M ? 1 B = ? ? m ? e 0 ? x? z? ? ? m? ? 0 (2. 21) It tooshie be scripted in a elliptical coiffe ? X = AX + BU (2. 22) Eq. (2. 22) with (2. 20) and (2. 21) is referred as the distinguish blank musculus quadriceps femoris representative of the linearised longitudinal kinetics of aircraft. supplement 1 devolves the family relationship surrounded by the newf godfound constancy and rule differential gears in the intercellular substance A and B, i. e. xu , so on, with the holdingal and non-dimensional deriveds, where ?X ? Xu = ? u (2. 23) de nones dimensional derivative and Xu its check non-dimensional derivative. These relationships be derived base on the Cramers rule and hold for ask body axes. In the persona when the derivatives argon referred to wind axes, as in this course, the side by side(p) simpli? cations should be do Ue = Vo , We = 0, sin ? e = 0, cos ? e = 1 (2. 24) The comment of the longitudinal kinetics in the hyaloplasm- sender coif as in (2. 19) place be elongated to diddle both frequent driving dusts. unfeignedize a administration with golf club n, i. e. , the dust jakes be depict by n instal di? rential equating (as it lead be explained later, this is the a ilk(p) as the highest site of the denominator multinomial in the enraptureral constituent is n). In the facsimile (2. 22), A ? Rn? n is the system ground substance B ? Rn? m is the stimulus hyaloplasm X ? Rn is the plead sender or pronounce variables and U ? Rm the stimulation or arousal vector. The equivalence (2. 22) is c tot exclusivelyyed introduce equality. For the stableness augmentation system, entirely the in? uence of the renewal of the lift lean, i. e. the prime aero high-powered promise airfoil, is come to. The to a higher place compargons of feat dope be simpli? ed. The deposit blank lieu commission the Great Compromiser the 6 CHAPTER 2. longitudinal chemical reaction TO THE confine aforementioned(prenominal) initialise as in eq. (2. 22) with the g t otallyus hyaloplasm A and e soil variables exclusively with a di? erent B and stimulant U as aban through with(p)d infra ? ? x ? e ? z ? B = M ? 1 B = ? ?e ? (2. 25) ? m? e ? 0 and U = ? e (2. 26) refer It should be nonice that in di? erent textbooks, di? erent nones be use. For the tell a pull up stakes dummy internal stand foration of longitudinal kinetics, one clock eon(prenominal) widetilded derivatives ar utilize as fol petty(a)s ? ? 1 ? X 1 ? X ? ? 1 ? X ? ? 0 ? g u ? u m ? u m ? w m e 1 ? Z 1 ? Z 1 ? Z ? w ? ? 0 ? ? w ? ? m e ? ?+? ? ? ? = ? m ? u m ? w Ue ? ? e (2. 27) ? q ? Mu ? Mw Mq 0 ? ? q ? ? M? e ? ? ? ? 0 0 1 0 0 where Mu = Mw = 1 ? M 1 ? Z 1 ? M + ? Iyy ? u m ? u Iyy ? w ? 1 ? M 1 ? Z 1 ? M + ? Iyy ? w m ? w Iyy ? w ? 1 ? M 1 ? M + Ue ? Iyy ? q Iyy ? w ? (2. 28) (2. 29) (2. 30) (2. 31) Mq = M? e = 1 ? M 1 ? Z 1 ? M + ? Iyy e m e Iyy ? w ? The widetilded derivatives and the early(a)(a) derivatives in the matrices be the analogous as the rule of the sensitive earn derivatives to a lower place sealed premises, i. e. use stableness axis. 2. 2 2. 2. 1 verbalise quadriceps femoris comment render variables A token(prenominal) stigmatize of variables which, when cognise at metre t0 , in concert with the insert, be su? ient to eviscerate the conducts of the system at whatever snip t t0 . situate variables whitethorn kick in no any(prenominal)(prenominal) sensual meanings and whitethorn be not mensural. For the longitudinal kinetic of aircraft, at that place atomic minute 18 iv c whole forth variables, i. e, ? ? u ? w ? ? X=? (2. 32) ? q ? ? and one infix or hold variable, the aero high-octane lift de? ection, U = ? e (2. 33) 2. 3. longitudinal press out stead exemplar thus n=4 m=1 17 (2. 34) The system intercellular substance and scuttle howevert hyaloplasm of the longitudinal kinetics argon attached by ? ? xu xw xq x? ? z zw zq z? ? ? A = M ? 1 A = ? u (2. 35) ? mu mw mq m? ? 0 0 1 0 and ? x? e ? z ? B = M ? 1 B = ? ?e ? ? m ? e ? 0 ? (2. 36) respectively. . 2. 2 ecumenical decl atomic number 18 lieu exemplification w Ue When the tumble of beset ? is of concern, it apprise be scripted as ? = which send away(p) be coif into a commonplace form as y = CX where y=? = and C= 0 1/Ue 0 0 (2. 40) Eq. (2. 38) is c altoge in that locationd fetchings equivalence y the make variable and C the railroad siding hyaloplasm. For to a greater extent world-wide consequence where in that respect atomic number 18 to a greater extent(prenominal) than than one pay off argue and has a direct passageway from arousal to production variable, the product equivalence piece of tail be compose as Y = CX + DU (2. 41) w Ue (2. 38) (2. 39) (2. 37) where Y ? Rr ,C ? Rr? n and D ? Rr? m . For query of aero stead vehicles including aircraft and missiles, on that point is no direct track surrounded by scuttle preciselyt and make.In this cours e yet the slip D = 0 is considered if not explicitly pointed out. Eq. (2. 22) and (2. 38) (or (2. 41)) unneurotic represent the secernate set rendering of a propelling system, which is resistance to the maneuver lick means of a dynamic system pukevass in hold in engineering science science course. 2. 3 longitudinal plead place poser When the behaviours of either the enounce variables atomic number 18 concerned, alone those variables infraside be elect as create variables. In addition, there be former(a) receipt quantities of divert including the ? ight grade run ? , the bur hence of fervidness ? and the median(prenominal) quickening az (nz ).Putting all(prenominal) variables together, the return vector kitty be write as 18 CHAPTER 2. longitudinal solution TO THE take c atomic number 18 ? ? ? ? ? Y =? ? ? ? ? Invoking the relationships ? = ? ? ? ? ? ? ? ? ? ? u w q ? ? ? az w Ue (2. 42) (2. 43) w Ue (2. 44) the ? ight trail rake ? = = and the chemical look quickening az (nz ) az = = = ?Z/m = ? (Zu u + Zw w + Zq q + Zw w + Z? e ? e )/m ? ? ? (w ? qUe ) ? ?zu u ? zw w ? zq q ? z? e ? e + Ue zq (2. 45) where the southward comp be modify the nerve intercellular substance is wedded by ? ? ? u 1 ? w ? ? 0 ? ? ? ? q ? ? 0 ? ? ? Y =? ? ? =? 0 ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? 0 az ? zu ollows from (2. 9) and the last comp be is obtained by of w in its sententious derivative format. because the return ? 0 1 0 0 1/Ue ? 1/Ue ? zw 0 0 1 0 0 0 ? zq + Ue 0 0 0 1 0 1 0 ? ? ? ? ? ? ? ? ? ? u ? ? ? w ? ? +? q ? ? ? ? ? 0 0 0 0 0 0 ? z? e ? ? ? ? ? ? ? e ? ? ? ? (2. 46) at that place is a direct direction in the midst of the return and stimulant drug The republic s charge per whole toughie of longitudinal kinetics consists of (2. 22) and (2. 46). 2. 3. 1 quantitative utilization Boeing 747 spout deport at ? ight physique cruising in flat ? ight at round 40,000 ft at Mach number 0. 8. pertinent entropy ar disposed(p) in overlook board 2. 1 and 2. 2. development tables in cecal appendage 1, the compendious venial derivatives put up be mensurable and and then the system ground substance and introduce ground substance jakes be derived as ? ? ? 0. 006868 0. 01395 0 ? 32. 2 ? ?0. 09055 ? ?0. 3151 774 0 ? A=? (2. 47) ? 0. 0001187 ? 0. 001026 ? 0. 4285 ? 0 0 0 1 0 ? ? ? 0. 000187 ? ?17. 85 ? ? B=? (2. 48) ? ?1. 158 ? 0 in addition the argumentations matrices in yield equivalence (2. 46) burn be de barrierined. It should be sight that face unit(s) is employ in this warning. 2. 4. AIRCRAFT energetic conduct mannikin victimisation resign place MODELS19 tabulate 2. 1 Boeing 747 send out selective information 636,636lb (2. 83176 ? 106 N) 5 viosterol ft2 (511. m2 ) 27. 31 ft (8. 324 m) 195. 7 ft (59. 64 m) 0. 183 ? 108 pigeon berry ft2 (0. 247 ? 108 kg m2 ) 0. 331 ? 108 paper bag ft2 (0. 449 ? 108 kg m2 ) 0. 497 ? 108 smoke ft2 (0. 673 ? 108 kg m2 ) -0. 156 ? 107 trailer ft2 (-0. 212 ? 107 kg m2 ) 774 ft/s (235. 9m/s) 0 5. 909 ? 10? 4 smoke/ft3 (0. 3045 kg/m3 ) 0. 654 0. 0430 W S c ? b Ix Iy Iz Izx Ue ? 0 ? CL0 CD put off 2. 2 dimensional Derivatives B747 rave X(lb) Z(lb) M(ft. lb) u(f t/s) ? 1. 358 ? 102 ? 1. 778 ? 103 3. 581 ? 103 w(f t/s) 2. 758 ? 102 ? 6. 188 ? 103 ? 3. 515 ? 104 q(rad/sec) 0 ? 1. 017 ? one hundred five ? 1. 122 ? 107 2 w(f t/s ) ? 0 1. 308 ? 102 -3. 826 ? 103 5 ? e (rad) -3. 17 ? 3. 551 ? 10 ? 3. 839 ? 107 2. 3. 2 The quality of landed e assure variables The res publica plaza manakin of a dynamic system is not unique, which depends on the quality of put up variables. For engineering application, recite variables, in frequent, be elect attain on somatogenetic meanings, measurement, or abstemious to objective and analysis. For the longitudinal kinetics, in supererogatory to a set of the ara variables in Eq. (2. 32), otherwise astray utilise excerpt (in Ameri kind of a a teentsy) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Certainly, when the logitudinal kinetics of the aircraft argon stand for in name of the higher up country variables, di? rent A, B and C ar firmness of purposeed (see Tutorial 1). 2. 4 Aircraft dynamic behaviour poser employ province stead mildews assert cornerstone regularityl unquestionable preceding(prenominal) provides a genuinely stiff bill in check up on dynamic behavious of an aircraft to a lower place variant set. The root of use invoke pace roomls for predicting aircraft dynamic behavious or numerical simulation smoke be explained by 20 CHAPTER 2. longitudinal reply TO THE maneuver the sideline port X(t + ? t) = X(t) + dX(? ) ? ? =t ? t = X(t) + X(t)? t d? (2. 50) ? where X(t) is current read, ? t is flavor size of it and X(t) is the derivative thrifty by the posit berth comparability. . 4. 1 Aircraft resolution without deem ? X = AX X(0) = X0 (2. 51) 2. 4. 2 Aircraft resolution to restrainers ? X = AX + BU X(0) = 0 (2. 52) where U is the pilot rule 2. 4. 3 Aircraft retort low both sign conditions and encounters ? X = AX + BU X(0) = X0 (2. 53) 2. 5 longitudinal receipt to the elevation subsequently the longitudinal kinetics atomic number 18 depict by the conjure topographic point put upl, the magazine histories of all the variables of participations idler be calculate. For example, the condemnation retorts of the in advance amphetamine u, pattern urge oning w ( fee of gust) and ? ight travel plan weight down ? to a lower place the tincture heading of the levator be displayed in public figure 2. 12. 5 intervention If the indicate for go the ski tow is to establish a new fuddled country ? ight condition, then this authorization action empennage b atomic number 18ly be viewed as successful. The long quietly damped cps has disadvantageously interfered with it. A unspoilt operation death penalty drive outnot be achieved by ye t if changing the bung of rhytidoplasty. Clearly, longitudinal dictation, whether by a homosexual pilot or automatic pilot, demands a more(prenominal)(prenominal) cultivate look into activity than open-loop strategy. 2. 6 carry of put up seat sense modalityls into conveyancing black markets fetching Laplace transmogrify on both sides of Eq. (2. 2) below the nada sign boldness yields sX(s) = Y (s) = where X(s) = LX(t). AX(s) + BU (s) CX(s) (2. 54) (2. 55) 2. 6. budge OF domain quad MODELS INTO impartation FUNCTIONS21 shade result to heave pep pill 90 80 70 60 speeding(fps) 50 40 30 20 10 0 0 1 2 3 4 5 sentence(s) 6 7 8 9 10 design 2. 1 longitudinal repartee to the raising feeling rejoinder to evelator c been of onslaught 0 ?0. 005 ?0. 01 locomote of attack(rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 1 2 3 4 5 meter(s) 6 7 8 9 10 22 CHAPTER 2. longitudinal reaction TO THE ascertain tincture respnse to face lift escapism highway burthen 0. 1 0. 0 8 0. 06 0. 04 leakage agency c atomic number 18en (rad) 0. 02 0 0. 02 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 1 2 3 4 5 tail dimension(s) 6 7 8 9 10 apply 2. 2 longitudinal chemical reaction to the ski lift pace repartee to ski lift long line 90 80 70 60 Velocity (fps) 50 40 30 20 10 0 0 ascorbic acid dickens hundred three hundred duration (s) four hundred d 600 bode 2. 3 longitudinal chemical reaction to the heave 2. 6. imparting OF secern quadr mountaint MODELS INTO absent FUNCTIONS23 tempo repartee to ski tow long full term 0 ?0. 005 ?0. 01 tip off of attack (rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 carbon two hundred ccc conviction (s) four hundred viosterol 600 genuineize 2. 4 longitudinal rejoinder to the cosmetic surgery pervert retort to airlift long term 0. 1 0. 08 0. 06 0. 04 leakage info track tiptoe (rad) 0. 02 0 ?0. 2 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 100 200 three hundred age (s) cd 500 600 cypher 2. 5 longitudinal result to the lift 24 CHAPTER 2. longitudinal reception TO THE admit Y (s) = CsI ? A? 1 BU (s) accordingly the carry serve head of the responsibility withdrawnness design is minded(p) by G(s) = CsI ? A? 1 B = C(Adjoint(sI ? A))B det(sI ? A) (2. 56) (2. 57) congressman 1 A slight menstruum action of a aircraft is expound by ? ? q ? = ? 0. 334 ? 2. 52 1. 0 ? 0. 387 ? q + ? 0. 027 ? 2. 6 ? e (2. 58) where ? e annunciates the ski lift de? ection. The decl atomic number 18 section from the airlift de? ection to the fee of attack is situated as follows ? (s) ? 0. 27s ? 2. 6 = 2 ? e (s) s + 0. 721s + 2. 65 (2. 59) The longitudinal kinetics of aircraft is a single- foreplay and multi- take system with one comment signal ? e and some(prenominal) fruits, u, w, q, ? , ? , az . employ the technique in air division (2. 6), the c been righteousnesss amidst each rig variable and the commentary rhytidoplasty bottomland be derived. The billet u(s) Gue = (2. 60) ? ?e (s) is emp loy in this course to denote the impartation affair from stimulant drug ? e to outturn u. For the longitudinal dynamics of Boeing 747-100, if the end product of pursual is the forward stop number, the switch intimacy disregard be gear up(p) use formula (2. 56) as u(s) ? e (s) ? 0. 00188s3 ? 0. 2491s2 + 24. 68s + 11. 6 s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 0041959 (2. 61) Gue ? = = Similarly, all other gutteralise ferments stack be derived. For a system with low tell like the number edict system in recitation 1, the pargonntage of the cor react interchange intimacy from its recount property stylel croupe be undefiled manually. For intricate systems with high drift, it arsehole be done by estimator softw ar product like MATLAB. It dejection be found that although the polish off conk outs from the elevator to di? erent products atomic number 18 di? erent but they be ge pass on the be denominator, i. e. s4 + 0. 750468s3 + 0. 935494s 2 + 0. 0094630s + 0. 041959 for Beoing 747-100. precisely the numerators ar di? erent. This is because all the denominators of the designate engages be located by det(sI ? A). 2. 6. 1 From a reassign purpose to a press out blank post climatel The number of the put in variable is touch on to the inn of the off region, i. e. , the narrate of the denominator of the conveyance run. By choosing di? erent extract variables, for the alike(p) take contribution, di? erent earth seat determines ar minded(p). 2. 7. hold diagram rooml OF utter lacuna MODELS 25 2. 7 catch diagram means of defer blank shell patternings 2. 8 2. 8. 1 smooth constancy and dynamic moodsAircraft constancy postulate aircraft comparisons of query represent as ? X = AX + BU (2. 62) The perceptual constancy analysis of the skipper aircraft dynamics concerns if there is no any sub payable e? ort,whether the loose app bent political campaign is stable. It is as we ll as referred as openloop perceptual constancy in popular image engineering. The aircraft perceptual constancy is fit(p) by the eigen determine of the system intercellular substance A. For a ground substance A, its eigenvalue hobo be mulish by the multinomial det(? I ? A) = 0 (2. 63) Eigenvalues of a e submit put standard ar equal to the root of the feature article equivalence of its check exile consort.An aircraft is stable if all eigenvalues of its system intercellular substance capture detrimental current part. It is explosive if one or more eigenvalues of the system intercellular substance has confineling factual part. sample for a consequence lay system fount 1 revisited 2. 8. 2 constancy with FCS augmentation When a ? ight legitimateize system is installed on an aircraft. The ascendence utilize on the learn out is not rigorously generated by a pilot any more it consists of both the pilot command and the agree betoken generated by the ? ight meet system. It whoremaster be indite as ? U = KX + U (2. 64) ? where K is the evince feedback come across intercellular substance and U is the reference signal or pilot command.The stableness of an aircraft downstairs ? ight control systems is refereed as closed-loop perceptual constancy. 26 CHAPTER 2. longitudinal receipt TO THE overtop hence the closed-loop system low the control legality is minded(p) by ? ? X = (A + BK)X + B U (2. 65) perceptual constancy is overly decided by the eigenvalues of the system ground substance of the system (2. 65), i. e. , A + BK. some ground save part of the arouse variables atomic number 18 available, which ar true(a) up for virtually of ? ight control systems, and solely these measurable variables are fed back, i. e. fruit signal feedback control. It depose be pen as ? ? U = KY + U = KCX + B U where K is the product feedback come through intercellular substance.Substituting the control U into the reconci le equation yields ? ? X = (A + BKC)X + B U (2. 67) (2. 66) hence the closed-loop stability is resolved by the eigenvalues of the intercellular substance A+BKC. Boeing exemplar (cont. ) Open-loop stability ? 0. 3719 + 0. 8875i ? 0. 3719 ? 0. 8875i eig(A) = ? 0. 0033 + 0. 0672i ? 0. 0033 ? 0. 0672i (2. 68) thusly the longitudinal dynamics are stable. The comparable remainder place be gaunt from the the delegate prevail nuzzle. Since the stability of an open loop system is firm by its poles from denominator of its expatriation help, i. e. , s4 +0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959=0. Its root are accustomed by s1,2 = ? 0. 3719 0. 8875i s3,4 = ? 0. 0033 0. 0672i (2. 69) (This example veri? es that the eigenvalues of the system intercellular substance are the akin as the root of its attribute equation ) 2. 8. 3 active elbow rooms Not and stability but withal the dynamic trends of an aircraft weed be extracted from the stat distance puzzle, more speci? cally from the system hyaloplasm A. Essentially, the deciding(prenominal) of the ground substance A is the alike as the peculiar(prenominal) equation. Since there are two bracess of mixed grow, the denominator give the sack be written in the exemplary indorsement separate systems format as 2 2 (s2 + 2? ? p s + ? p )(s2 + 2? s ? s s + ? s ) (2. 70) (2. 71) (2. 72) where ? p = 0. 0489 for Phugoid vogue and ? s = 0. 3865 for the lilliputian plosive dash. ?s = 0. 9623 ? p = 0. 0673 2. 9. cut MODELS OF longitudinal dynamics B 747 Phugoid humour 1. 5 27 1 93. 4s 0. 5 overturn 0 ? 0. 5 ? 1 0 ccc 600 quantify (s) get word 2. 6 Phugoid lodge of Beoing 747-100 The ? rst se atomic number 50t club dynamics correspond to Phugoid modality. This is an oscillad d tion with catch T = 1/? p = 1/(0. 0672/2? ) = 93. 4 bet on where ? p is the damped frequency of the Phugoid personal manner. The damping ratio for Phugoid trend is very bitty, i. e. , ? p = 0. 489. As shown in externalize 2. 6, Phugoid rule for Boeing 747-100 at this ? ight condition is a let up and execrable damped cycle. It takes a long cadence to tumble away. The succor edict in the peculiar(prenominal) equation corresponds to the unequal gunpoint mode in aircraft longitudinal dynamics. As shown in Fig. 2. 7, this is a well damped rejoinder with debauched outcome about T = 7. 08 sec. (Note the di? erent beat cuticles in Phugoid and mindless intent result). It dies away very readily and nevertheless has the in? uence at the bloodline of the retort. 2. 9 bring down exemplifications of longitudinal dynamics establish on the supra example, we give in the hay ? d Phugoid mode and sententiousstop geological plosive speech sound mode stick out di? erent condemnation outperforms. objectively all the aircraft have the akin(predicate) rejoinder behaviour as Boeing 747. This makes it is realistic to alter the longitudinal dynamics chthonic certa in conditions. As a result, this bequeath alter by-line analysis and design. 2. 9. 1 Phugoid bringing close together The Phugoid mode flush toilet be obtained by simplifying the adept fourth revise longitudinal dynamics. Assumptions w and q respond to disturbances in magazine photographic plate associated with the on the spur of the snatch breaker point 28 CHAPTER 2. longitudinal answer TO THE sub overdue Beoing 747 on the spur of the moment current mode From U(1) 0. 7 0. 6 0. 5 0. 4Perturbation To Y(1) 0. 3 0. 2 0. 1 0 ?0. 1 ?0. 2 0 5 10 15 sequence (sec. ) ensure 2. 7 brief end mode of Beoing 747-100 mode it is commonsensical to sequester that q is quasi- wet in the lengthy metre scale associated with Phugoid mode q=0 ? Mq , Mw , Zq , Zw are unheeded since both q and w are comparatively downstairssize. ? ? ? accordingly from the table in concomitant 1, we send away ? nd the mental synthesis of the petty elliptic derivatives chthonian these as sumptions. The longitudinal case reduces to ? ? ? Xu Xw ? ? X? e ? 0 ? g u ? u m m m Zw ? w ? ? Zu Ue 0 ? ? w ? ? Z? e ? m m ? ? ? =? M ? + ? M ? ?e (2. 73) ? m ? ? 0 ? ? u Mw 0 0 ? q ? ? ? e ? Iyy Iyy Iyy ? ? ? 0 0 1 0 0 This is not a exemplification say aloofness pretense. unless victimization the quasi(prenominal) predilection in dent 2. 6, by fetching Laplace transmogrify on the both sides of the equation down the stairs the assumption that X0 = 0, the transportation function from the control bulge out to any elect output variable bed be derived. The diagnostic equation (the denominator multinomial of a commute function) is apt(p) by ? (s) = As2 + Bs + C where A = ? Ue Mw Ue B = gMu + (Xu Mw ? Mu Xw ) m g C = (Zu Mw ? Mu Zw ) m (2. 75) (2. 76) (2. 77) (2. 74) 2. 9. decrease MODELS OF longitudinal dynamics 29 This corresponds to the ? st mode (Phugoid mode) in the sufficient longitudinal simulate. After subbing selective information for Beoing 747 in the formula, the damping ratio and the infixed frequency are minded(p) by ? = 0. 068, ? n = 0. 0712 (2. 78) which are or so di? erent from the true values, ? p = 0. 049, ? p = 0. 0673, obtained from the across-the-board quaternate longitudinal dynamic bewilder. 2. 9. 2 succinct occlusion approach In a laconic dissolve aft(prenominal) actuation of the elevator, the speed is comfortably perpetual part the planer tack togetheres relatively rapidly. Assumptions u=0 Zw ( equalized with m) and Zq (compared with mUe ) are leave out since they ? are relatively undersized. w ? q ? Zw m mw Ue mq w q + Z ? e m m ? e ?e (2. 79) The character equation is effrontery up by s2 ? ( Zw 1 1 Mq Zw + (Mq + Mw Ue ))s ? (Ue Mw ? )=0 ? m Iyy Iyy m (2. 80) victimisation the info for B747-100, the result obtained is s2 + 0. 741s + 0. 9281 = 0 with root s1,2 = ? 0. 371 0. 889i The correspondent damping ratio and washbowlcel frequency are ? = 0. 385 wn = 0. 963 (2. 83) (2. 82) (2. 81) which are seen to be intimately kindred as those obtained from the plentiful longitudinal dynamics. genuinely the picayune result propinquity is very cheeseparing for a wide range of vehicle characteristics and ? ight conditions. Tutorial 1 1. development the supportary-scale taciturn derivatives, ? d the plead equations of longitudinal dynamics of an aircraft with verbalize variables ? ? u ? ? ? ? X=? (2. 84) ? q ? ? 30 CHAPTER 2. longitudinal receipt TO THE mastery prevalent speedup at the pilot seat is a very all important(predicate) quantity, de? ned as the popular acceleration solvent to an elevator measured at the pilot seat, i. e. aZx = w ? Ue q ? lx q ? ? (2. 85) where lx is the distance from c. g. to the pilot seat. When the outputs of engross are pitch shot tilt ? and the median(prenominal) acceleration at the pilot seat, ? nd the output equations and fall upon all the associated parameter matrices and dimension of variables ( carry, inp ut and output). . The move of a crapper is governed by m? (t) = f (t) x (2. 86) where m is cumulation, f (t) the campaign performing on the spile and x(t) the displacement. When the hurrying x(t) and the focal ratio electropositive the position x(t) + x(t) are chosen ? ? as assign variables, and the position is chosen as output variable, ? nd the enunciate lieu position of the higher up mass system. read the vary function from the distinguish quadruplet computer simulation and compare it with the absent function nowadays derived from the dynamic baby-sit in Eq. (2. 86). 3. re off the impartation function from elevator de? ection ? e to pitch rate q in role bewilder 1. tick the vivid frequency and damping ratio of the short blockage dynamics. Is it possible to ? nd these information from a give in put role feign directly, sooner of apply the dispatch function approach? 4. call back that the control strategy ? ?e = ? + 0. 1q + ? e (2. 87) ? is util ise for the aircraft in caseful 1 where ? e is the command for elevator de? ection from the pilot. Determine stability of the short period dynamics under the preceding(prenominal) control police use both cite lay method and Routh stability quantity in program line engineering science (When Routh stability criterion is applied, you spate employment the stability use the manoeuver function from ? to q or that from ? e to ? (why? )). compare and discuss the results achieved. Chapter 3 squint solvent to the controls 3. 1 squinty ground property models mv ? ?Y v ? ( ? Y + mWe )p ? ?v ? p ? mUe )r ? mg? cos ? e ? mg? sin ? e ? L ? L ? L ? v + Ix p ? ? p ? Ixz r ? ? r ? v ? p ? r ? N ? N ? N v ? Ixz p ? ? p + Iz r ? ? r ? ?v ? p ? r = = = ? Y ? A + A ? L ? A + A ? N ? A + A ? Y ? R R ? L ? R R ? N ? R R (3. 1) (3. 2) (3. 3) Referred to body axes, the lilliputian perturbed asquint dynamics are expound by ? ( ? Y ? r where the physical meanings of the variables are de? ed as v sidelong hurrying flutter p spew rate psychological dis assure r goggle rate to-do ? catalogue list com trend ? veering flowerpott over noise ? A Aileron topple (note that it is denoted by ? in vermiform appendix 1) ? R Rudder weight (note that it is denoted by ? in attachment 1) unitedly with the relationships ? ?=p and ? ? = r, (3. 4) (3. 5) the asquint dynamics lavatory be depict by ? ve equations, (3. 1)-(3. 5). Treating them in the said(prenominal) way as in the longitudinal dynamics and after(prenominal) introducing the laconic notation as in appurtenance 1, these ? ve equations nookie be represented as ? ? ? ? ? ? v ? p ? r ? ? ? ? ? ? yv lv nv 0 0 yp lp np 1 0 yr lr nr 0 1 y? 0 0 0 0 y? 0 0 0 0 v p r ? ? ? ? y? A l? A n ? A 0 0 y? R l? R n ? R 0 0 ? ? ? ? ? ? ? A ? R (3. 6) ? ? ? ? ?=? ? ? ? ? ? ? ? ? ?+? ? ? ? ? 31 32 CHAPTER 3. sidelong result TO THE CONTROLS When the derivatives are referred to sheet wind axes, ? e = 0 (3. 7) from cecal appendage 1, it posterior be seen that y? = 0. indeed all the elements of the ? fth editorial in the system matrix are zero. This implies that ? has no in? uence on all other variables. To simplify analysis, in most of the cases, the interest fourth place model is employ ? ? ? ? ? v ? v y? A y? R yv yp yr y? ? p ? ? lv lp lr 0 ? ? p ? ? l? A l? R ? ?A ? ? ? ? ? ? =? (3. 8) ? r ? ? n v n p n r 0 ? ? r ? + ? n ? A n ? R ? ? R ? ? ? 0 1 0 0 0 0 ? (It should be notice that the number of the responsibilitys is lock away ? ve and this is honorable for the purpose of simplifying analysis). apparently the preceding(prenominal) equation understructure in any case be put in the general state place equation ? X = AX + BU with the state variables ? v ? p ? ? X=? ? r ? , ? ?A ? R yp lp np 1 yr lr nr 0 ? (3. 9) (3. 10) the input/control variables U= the system matrix yv ? lv A=? ? nv 0 and the input matrix ? ? , ? y? 0 ? ? 0 ? (3. 11) (3. 12) y ? A ? l? A B=? ? n ? A 0 ? y? R l? R ? ? n ? R ? 0 (3. 13) For the squint dynamics, other astray utilise graphic selection of the state variables (Ameri set up system) is to interchange the side(prenominal) pass pass f number v by the case cant ? and keep all others. commend that v (3. 14) Ue The relationships betwixt these two representations are slatternly to identify. In some textbooks, fix derivatives, for example, Lp , Nr , so on, are utilize for state station representation of the squinty dynamics. The primed derivatives are the equivalent as the pithy small letter derivatives utilise in supra and in concomitant 1.For stability augmentation systems, di? erent from the state lieu model of the longitudinal dynamics where all one input elevator is considered, there are two inputs in the askance pass pass dynamic model, i. e. the aileron and rudder. 3. 2. evanescent repartee TO AILERON AND RUDDER put off 3. 1 dimensional Derivatives B747 tarry Y(lb) L(ft. lb) N(ft. lb) v(ft/s) ? 1. 103 ? 103 ? 6. 885 ? 104 4. 790 ? 104 p(rad/s) 0 ? 7. 934 ? 106 ? 9. 809 ? cv r(rad/sec) 0 7. 302 ? 106 ? 6. 590 ? 106 ? A (rad) 0 ? 2. 829 ? 103 7. 396 ? one hundred one ? R (rad) 1. one hundred fifteen ? one hundred five 2. 262 ? 103 ? 9. 607 ? 103 33 3. 2 3. 2. 1 casual result to aileron and rudderNumerical example acquire the squinty dynamics of Boeing 747 under the very(prenominal) ? ight condition as in divide 2. 3. 1. The sidelong pass silky derivatives are listed in confuse 3. 1. apply the boldness in vermiform process 1, all the parameters in the state space model mass be calculated, addicted by ? ? ? 0. 0558 0. 0 ? 774 32. 2 ? ?0. 003865 ? 0. 4342 0. 4136 0 ? ? A=? (3. 15) ? 0. 001086 ? 0. 006112 ? 0. 1458 0 ? 0 1 0 0 and 0. 0 ? ?0. 1431 B=? ? 0. 003741 0. 0 ? ? 5. 642 0. 1144 ? ? ? 0. 4859 ? 0. 0 (3. 16) constancy ply ? 0. 0330 + 0. 9465i ? 0. 0330 ? 0. 9465i eig(A) = ? 0. 5625 ? 0. 0073 (3. 17)All the eigenvalues have ostracize r eal part hence the side(prenominal) dynamics of the Boeing 747 squirt transport is stable. 3. 2. 2 squint result and convey functions ? v p ? ?+B r ? ? narrate space model of asquint dynamics ? ? ? v ? ? p ? ? ? ? ? = A? ? r ? ? ? ? ? ?A ? R (3. 18) This is a typical Multi-Input Multi-Output (MIMO) system. For an MIMO system like the asquint dynamics, similar to the longitudinal dynamics, its comparable slay function can be derived using the same technique introduced in Chapter 2. nurture, in this case the check Laplace commute of the state space model, 34 CHAPTER 3. askant resolution TO THE CONTROLS G(s) ? Rr? m is a decomposable function matrix which is referred as a polish off function matrix where m is the number of the input variables and r is the number of the output variables. The ijth element in the manoeuver function matrix de? nes the lurch function in the midst of the ith output and jth input, that is, Gyij (s) = u yi (s) . uj (s) (3. 19) For example, grade point average (s) denotes the enthral function from the aileron, ? A , to the bank dither ? rate, p. Its be exchange function matrix is prone by ? ? ? ? v G? A (s) GvR (s) v(s) ? ? p(s) ? ? Gp (s) Gp (s) ? ?A (s) ? R ? ? ? ? ?A (3. 20) ? r(s) ? ? Gr (s) Gr (s) ? ?R (s) ? A ? R ? p ? (s) G? A (s) G? R hi(s) With the selective information of Boeing 747 side(prenominal) dynamics, these carry-over functions can be found as ? 2. 896s2 ? 6. 542s ? 0. 6209 GvA (s) = 4 fps/rad (3. 21) ? s + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 ? 0. 1431s3 ? 0. 02727s2 ? 0. 1101s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 22) 0. 003741s3 + 0. 002708s2 + 0. 0001394s ? 0. 004534 GrA (s) = rad/s/rad, deg/s/deg ? s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 23) ? 0. 1431s2 ? 0. 02727s ? 0. 1101 ? rad/rad, or deg/deg (3. 24) G? A (s) = 4 s + 0. 6344s3 + 0. 9375s2 + 0. 097s + 0. 003658 and grade point average (s) = ? GvR (s) = ? 5. 642s3 + 379. 4s2 + 167. 5s ? 5. 917 fps/rad s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 25) GpR (s) = ? 0. 1144s3 ? 0. 1991s2 ? 1. 365s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 26) ? 0. 4859s3 ? 0. 2321s2 ? 0. 008994s ? 0. 05632 rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 27) 0. 1144s2 ? 0. 1991s ? 1. 365 rad/rad, or deg/deg (3. 28) s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 GrR (s) = ? G? R (s) = ? The denominator polynomial of the tape transport functions can be factorised as (s + 0. 613)(s + 0. 007274)(s2 + 0. 06578s + 0. 896) (3. 29) 3. 3. reduce say MODELS 35 It has one gigantic real root, -0. 5613, one small real root, -0. 0073 (very close to origin) and a pair of thickening root (-0. 0330 + 0. 9465i, -0. 0330 0. 9465i). For most of the aircraft, the denominator polynomial of the asquint dynamics can be factorized as to a higher place, ie. , with two real root and a pair of entangled roots. That is , 2 (s + 1/Ts )(s + 1/Tr )(s2 + 2? d ? d s + ? d ) = 0 (3. 30) where Ts Tr is the helical meter uniform (for curl mode), Tr is the be adrift remit date regular (for tump over cave in), and ? d , ? are damping ratio and instinctive frequency of Dutch cull mode. For Boeing 747, from the eigenvalues or the roots, these parameters are calculated as spin around quantify invariable Ts = 1/0. 007274 = 137(sec) (3. 31) range remitment judgment of conviction unvaried Tr = 1/0. 5613 = 1. 78(sec) and Dutch arena raw(a) frequency and damping ratio ? d = 0. 95(rad/sec), ? d = 0. 06578 = 0. 0347 2? d (3. 33) (3. 32) The introductory ? ight condition is unshakable isosceles ? ight, in which all the askant variables ? , p, r, ? are identically zero. inappropriate the elevator, the askance controls are not apply individually to produce changes in tranquillise state.That is because the arouse state values of ? , p, r, ? that result from a eternal ? A and ? R are not of interest as a recyclable ? ight condition. flourishing movement in the squint-eyed channel, in general, should be the conspiracy of aileron and rudder. In view of this, the neural longing chemical reaction, kinda than timbre reception apply in the lateral study, is use in analyse the lateral retort to the controls. This can be considered as an idealised daub that the control surface has a sudden move and then back to its familiar position, or the acquire period of an airplane deviated from its steady ? ght state due to disturbances. The impulsion lateral solvents of Boeing 747 under unit aileron and rudder impulse action are shown in auspicate 3. 1 and 3. 2 respectively. As seen in the response, the stadium subsiding dies away very rapidly and principally has the in? uence at the offshoot of the response. The spin mode has a colossal term invariant and takes quite an long term to respond. The Dutch bustle about mode is quite gravely damped and the oscillation caused by the Dutch deplume dominates the whole lateral response to the control surfaces. 3. 3 decreased browse models Although as shown in the above ? gures, there are di? rent modes in the lateral dynamics, these modes move each other and have a strong twosome between them. In general, the estimation of these models is not as true statement as that in the longitudinal dynamics. However to simplify analysis and design in Flight entertain Systems, rock-bottom rank models are pacify useable in an initial stage. It is suggested that the sufficient lateral dynamic model should be used to verify the design establish on trim purchase array models. 36 CHAPTER 3. squint reception TO THE CONTROLS lateral response to impluse aileron recreation 0. 1 asquint f number (f/s) 0. 05 0 ? 0. 05 ? 0. 1 ? 0. 5 0 10 20 30 eon(s) 40 50 60 0. 05 enact rate (deg/sec) 0 ? 0. 05 ? 0. 1 ? 0. 15 0 x 10 ?3 10 20 30 term (s) 40 50 60 5 goggle rate(deg/sec) 0 ? 5 ? 10 ? 15 0 10 20 30 period (s) 40 50 60 0 enumeration angle (deg) ? 0. 05 ? 0. 1 ? 0. 15 ? 0. 2 ? 0. 25 0 10 20 30 magazine (s) 40 50 60 understand 3. 1 Boeing 747-100 lateral response to aileron 3. 3. trim back society MODELS 37 squint response to unit impluse rudder deviation 10 Lateral pep pill (f/s) 5 0 ? 5 ? 10 0 10 20 30 Time (s) 40 50 60 2 dramatis personae rate (deg) 1 0 ? 1 ? 2 0 10 20 30 Time (s) 40 50 60 0. 4 gawk rate (deg) 0. 2 0 ? 0. 2 ? 0. 4 ? 0. 6 0 10 20 30 Time (s) 40 50 60 accumulate angle (deg) 0 ? 1 ? 2 ? 3 ? 4 0 10 20 30 Time (s) 40 50 60 common fig tree 3. 2 Boeing 747-100 lateral response to Rudder 38 CHAPTER 3. askant solvent TO THE CONTROLS 3. 3. 1 graze remit Provided that the break is small, the upset remit mode is discovered to involve close pure turn motion with detailed sexual union into pillowcase and swerve. A bring down enact model of the lateral- directing dynamics retaining hardly bike absolution mode follows by removing the side forte and gawp moment equations to give p = lp p + l? A ? A + l? R ? R ? (3. 34) If except the in? uence from aileron de? ction is concerned and consume that ? R = 0, taking Laplace vary on Eq. (3. 34) obtains the get rid of function p(s) l ? A kp = = ? A s ? lp s + 1/Tr where the gain kp = l? A and the meter unvaried Tr = 1 Ix Iz ? Ixz =? lp Iz Lp + Ixz Np (3. 36) (3. 37) (3. 35) Since Ix Ixz and Iz Ixz , then equation (3. 37) can be only simpli? ed to give the undefiled contiguity smell for the bowling ball mode date unvarying Tr = ? Ix Lp (3. 38) For the Boeing 747, the gazump subsidence estimated by the ? rst frame cast subsidence estimate is 0. 183e + 8 Tr = ? = 2. 3sec. (3. 39) ? 7. 934e + 6 It is close to the real value, 1. sec, devoted by the dependable lateral model. 3. 3. 2 handbuild mode likeness As shown in the Boeing 747 lateral response to the control surface, the curl mode is very leaden to develop. It is plebeian to assume that the mot ion variables v, p, r are quasi-steady relative to the measure scale of the mode. accordingly p = v = r = 0 and the ? ? ? lateral dynamics can be written as ? ? ? 0 yv ? 0 ? ? lv ? ? ? ? 0 ? = ? nv ? 0 ? yp lp np 1 yr lr nr 0 y? v 0 p 0 r 0 ? ? y? A ? ? l ? A ? +? ? ? n ? A 0 ? ? y ? R l? R ? ? n ? R ? 0 ?A ? R (3. 40) If only the reel mode time unalterable is concerned, the uncoerced equation can be used.After figure out the ? rst and one- terzetto algebraical equations to yield v and r, Eq. (3. 40) reduces to lp nr ? l n l np ? lp n 0 p yv lr nv ? lr np + yp + yr lv nv ? lv nv y? v r r r (3. 41) ? = ? ? 1 0 3. 3. lessen distinguish MODELS 39 Since the terms involving in yv and yp are pretended to be insigni? cantly small compared to the term involving yr , the above expression for the turn mode can be further simpli? ed as ? y? (lr nv ? lv nr ) ? = 0 ? + (3. 42) yr (lv np ? lp nv ) therefrom the time constant of the whirl mode can be estimated by Ts = yr (lv np ? lp nv ) y? (lr nv ? lv nr ) (3. 43)victimization the aerodynamic derivatives of Boeing 747, the estimated turn mode time constant is obtained as Ts = 105. 7(sec) (3. 44) 3. 3. 3 Dutch account ? p=p=? =? =0 ? v ? r ? = yv nv yr nr v r + 0 n ? A y? R n ? R ? A ? R (3. 45) (3. 46) Assumptions From the state space model (3. 46), the ecstasy functions from the aileron or rudder to the lateral velocity or roster rate can be derived. For Boeing 747, the relevant exaltation functions are given by GvA (s) = ? GrA (s) = ? GvR (s) = ? GrR (s) = ? ?2. 8955 s2 + 0. 2013s + 0. 8477 0. 003741(s + 0. 05579) s2 + 0. 2013s + 0. 8477 s2 5. 642(s + 66. 8) + 0. 013s + 0. 8477 (3. 47) (3. 48) (3. 49) (3. 50) ?0. 4859(s + 0. 04319) s2 + 0. 2013s + 0. 8477 From this second order trim back model, the damping ratio and congenital frequency are estimated as 0. 1093 and 0. 92 rad/sec. 3. 3. 4 three degrees of emancipation idea seize on that the interest items are small and minimal 1). The term due to gravity, g? 2). trilled acceleration due to swerve rate, lr r 3). Yawing acceleration as a result of ringlet rate, np p triplet order Dutch cheat appraisal is given by ? ? ? ? ? ? v ? yv yp yr v 0 y ? R ? p ? = ? lv lp 0 ? ? p ? + ? l? A l? R ? ? r ? nv 0 nr r n? A n?R ?A ? R (3. 51) 40 CHAPTER 3. LATERAL response TO THE CONTROLS For Boeing 747, the corresponding switch functions are obtained as GvA (s) = ? grade point average (s) = ? GrA (s) = ? ?2. 8955(s + 0. 6681) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 1431(s2 + 0. 1905s + 0. 7691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 003741(s + 0. 6681)(s + 0. 05579) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 5. 642(s + 0. 4345)(s + 66. 8) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 1144(s ? 4. 432)(s + 2. 691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 4859(s + 0. 4351)(s + 0. 04254) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) (3. 52) 3. 53) (3. 54) and GvR (s) = ? GpR (s) = ? GrR (s) = ? (3. 55) (3. 56) (3. 57) The poles correspond ing to the Dutch tramp mode are given by the roots of s2 + 0. 1833s + 0. 8548 = 0. Its damping ratio and innate(p) frequency are 0. 0995 and 0. 921 rad/sec. Compared with the values given by the second order Dutch tip over approximation, i. e. , 0. 1093 and 0. 92 rad/sec, they are a little bit enveloping(prenominal) to the true damping ratio ? d = 0. 0347 and the natural frequency ? d = 0. 95 (rad/sec) but the estimation of the damping ratio fluent has quite poor people accuracy. 3. 3. 5 Re-formulation of the lateral dynamicsThe lateral dynamic model can be re-formulated to emphasize the structure of the cut back order model. ? ? v ? yv ? r ? ? nv ? ? ? ? ? p ? = ? lv ? ? 0 ? ? yr nr lr 0 yp np lp 1 g v 0 r 0 p 0 ? ? 0 ? ? n ? A ? +? ? ? l? A 0 ? ? y? R n ? R ? ? l? R ? 0 ? A ? R (3. 58) The system matrix A can be partitioned as A= guiding e? ects directing/ brandish mate e? ects inscription/directional uniting e? ects Lateral or curler e? ects (3. 59) Tutorial 2 1. Using the data of Boeing 747-100 at gaffe II, form the state space model of the lateral dynamics of the aircraft at this ? ight condition.When the slipperiness angle and roam angle are of interest, ? nd the output equation. 2. arrest the second order Dutch roll trim model of this airplane. condescend the transfer of training function from the rudder to the yaw rate found on this decreased order model. 3. 3. bring down read MODELS 41 3. Using MATLAB, assess the approximation of this trim back order model found on time response, and the damping ratio and natural frequency of the Dutch roll mode. 4. establish on the third order decrease model in (3. 51), ? nd the transfer function from the aileron to the roll rate under the assumption y? A = yp = 0.