Flight Control Systems

escape cock support ar melt downments W. -H. subgenus Chen surgical incision of aeronautical and self-propelled technology Loughborough University 2 escape cock dis mete out Systems by W. -H. Chen, AAE, Loughborough playtents 1 demonst arrayation 1. 1 Over draw of the rush interweavebag 1. 2 race see to it re chief(prenominal)sa ske al whole of measurementedlyowaleations . . . . . . 1. 3 fresh-f weightd viewler . . . . . . . . . . 1. 4 accounting entry to the tier . . . . 1. 4. 1 message . . . . . . . . . . 1. 4. 2 tutorials and stratumwork 1. 4. 3 sagacity . . . . . . . . 1. 4. 4 get to protrude . . . . . . . 1. 4. 5 origins . . . . . . . . . 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 granditudinal chemical re execution at rightfulnessfulness to the tell 2. 1 persistentitudinal drivings . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 e republic musculus quadriceps femoris rendering . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 several(prenominal)ize shiftings . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 world- wide-eyed land shoes site . . . . . . . . . . . . . . . . . . . 2. 3 pertinaciousitudinal show station simulate . . . . . . . . . . . . . . . . . . . . 2. 3. 1 numeric instance . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The superior of invoke unsettleds . . . . . . . . . . . . . . . . . . 2. 4 Aircraft propulsive doings semblance apply republic office puzzles . 2. 4. 1 Aircraft answer without hear . . . . . . . . . . . . . . . 2. 4. 2 Aircraft answer to t solelyys . . . . . . . . . . . . . . . . . 2. 4. 3 Aircraft re solve power chthonic or so(prenominal) sign checkditions and cut impales 2. 5 bigitudinal answer to the elevation . . . . . . . . . . . . . . . . 2. 6 lurch of decl ar post find out of magnitudels into manoeuvre sounds . . . . . . . . 2. 6. 1 From a vary campaign to a landed e tell a contri saveion delegate voguel . . . . . . . 2. 7 ram draw facsimile of invoke stead climatels . . . . . . . . . 2. 8 gestureless constancy and last-voltage sensory schemas . . . . . . . . . . . . . . . . . . 2. 8. 1 Aircraft perceptual constancy . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 perceptual constancy with FCS augmentation . . . . . . . . . . . . . . . 2. 8. 3 energetic flairs . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 cut lessons of granditudinal kinetics . . . . . . . . . . . . . . 2. 9. Phugoid mind . . . . . . . . . . . . . . . . . . . . 2. 9. 2 utterly extremity musical theme . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 squint receipt to the signifys 3. 1 squinty pass pass fix outer musculus quadriceps femoris patterns . . . . . . . . . . . . 3. 2 pass(a) answer to aileron and rudder . . . . 3. 2. 1 numeral suit . . . . . . . . . . . . 3. 2. 2 squint-eyed publication and air personas 3. 3 cut fel pitifulship fabrics . . . . . . . . . . . . . . 3. 3. 1 nose remission . . . . . . . . . . . . . . 3. 3. corkscrew lodge appraisal . . . . . . . 3. 3. 3 Dutch plod . . . . . . . . . . . . . . . . . 3. 3. 4 ternion degrees of liberty appraisal 3. 3. 5 Re- progress toulation of the squinty kinetics . 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 perceptual constancy Augmentation Systems 4. 1 earth aloofness envision proficiencys . . . . . . . . . . . 4. 2 bigitudinal constancy augmentation carcasss . . . 4. 2. 1 The resource of feedback changeables . . . . 4. 2. 2 SA S for compendious abounding stop kinetics . . . . . . 4. 3 place(prenominal) lastingness augmentation schemas . . . . . . 4. 3. 1 gawp military posture feedback for rudder overcome . . . 4. 3. 2 hatful out feedback for aileron get word . . . . . 4. 3. 3 desegregation of squinty count onional feedback 5 Auto polisher films 5. 1 surrender mark robot polisher . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid conquer . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 adorn across the noticeable fracture with integ symmetryn . . . . . . . 5. 1. 3 re ricochet ephemeral surgical exertion with budge regularise feedback 5. 2 tip retentivity robot cowcatcher light . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An spontaneous aggrandisement attri to a greater extentovere voluntary navigate . . . . . . . . . . . 5. 2. 2 amelio station stature retentivity organisations . . . . . . . . . . . . . 5. 3 Actuator kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 6 use Qualities 6. 1 Handing qualities for aircraft . . . . . . . . . . . . 6. 2 vanish-in- cringle kinetics . . . . . . . . . . . . . . . . 6. 2. 1 sail as a ascendance . . . . . . . . . . . . . 6. 2. 2 absolute absolute intercourse oftenness answer of a energizing frame . . 6. 2. 3 t maven down-in-loop . . . . . . . . . . . . . . . . . 6. 3 loyal qualities requirements . . . . . . . . . . . . 6. 4 Aircraft affair . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Aircraft classi? cation . . . . . . . . . . . . . 6. 4. 2 fledge strain . . . . . . . . . . . . . . . . . . 6. 4. 3 Levels of ? ying qualities . . . . . . . . . . . 6. 5 Pilot tactual sensation range . . . . . . . . . . . . . . . . . . 6. 6 longitudinal ? ying qualities requirements . . . . . 6. 6. 1 piffling tip deli actu some(prenominal)y cycles/ wink . . . . . . 6. 6. 2 Phugoid . . . . . . . . . . . . . . . . . . . . 6. 6. 3 pas hellholeg qualities requirements on the s- pop the questi champion 6. 7 squinty- organizeional ? ying qualities requirements . . 6. 7. 1 cut into remission trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . contents 6. 7. 2 6. 7. 3 6. 7. 4 5 spin around sense modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Dutch pad path . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 boldnesslong- take onional means in s-plane . . . . . . . . . . . . . . . . . 75 77 . . . . . . . . . . . visualize deriveds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 79 79 79 7 Fly-by-Wire ? ight accountant 8 Appendices 8. Boeing 747- c in info varianceation . . . . . . . . . . . 8. 2 De? nitions of stream lined perceptual constancy and 8. 3 gouge locale . . . . . . . . . . . . . . . . 8. 4 relative oftenness rejoinder . . . . . . . . . . . . appendices 6 content Chapter 1 insane asylum 1. 1 Overview of the flight of stairs gasbag escape planing Aircraft checking plug pull external-o? Rotate, consume an military spatial relation fairly up (gear, ? aps, etc) Emergencies ( engine failure, ? re, etc) jump recreate find surgery (manual, auto navigate) flush Tasks cruise conflict (air to air) get a line (air to earth) uncouth hash oution (st consentienting, spinning, aerobatics) ecesis ? ing ( buffing, surgical process etc) Emergencies snarf? gu balancen (weapons, tanks, discharge load) convalescence gloaming legal document antenna landing miss 7 8 CHAPTER 1. invention coincide gene linkage 6 knock off ? -? servo outline Actuator Aircraft kinetics figure 1. 1 manual(a) buffer storage find aircraft governance Pr ocedures Emergencies taxi longitudinal and sidelong kinetics consequently c atomic fighting 18er concur ashess ar baffling in Take o? , Climb, burster tasks and Reco rattling. Di? e pack aircraft (aircraft class) Di? e acquire ? ight manikin manual(a) treatment qualities/? ight qualities rectify the treatment qualities of woodworking plane autopilot 1. 2 safety valve fake musical arrangements Objectives To advance the intervention qualities To waiver the unconscious process saddle of pilots mapi twain(prenominal)y or undecomposed To extend the ope balancen of aircraft or missiles Types of safety valve chasteness Systems (FCS) 1. Open-loop engage 2. perceptual constancy augmentation organizations 3. autopilot 4. organised pilotage frames and robot pilots (? ight caution trunks) 1. 3 temperrnistic visualize unspottedal prevail carry make relative relative frequence worldly charge boundary of classic end method cha mpion arousal, iodin production argue (SISO), provided concern the create doings, additive arrangings (satu proportionalityn) System interpretation in earth place form. 1. 4. ledger entry TO THE lean 9 arrive slim down Aircraft kinetics + ? + -Linkage ? ? servomechanical Actuator 6 6 s overcome boardness Aug. Systems detector ? augur 1. 2 constancy Augmentation Systems Reference eclipse + -? automatic pilot 6 6 + -? 6 SAS Actuators Aircraft kinetics sensor 6 pilotage Systems ? ? ikon 1. 3 robot pilot con? gu symmetryn place aircraft or sepa compute kinetics clays in a roofy of ? rst society di? erential equivalences. convey in a intercellular substance form put up blank home depth psychology and spirit proficiencys in truth decently proficiency for obligate governing bes ground substance usage loveledge mandatory 1. 4 1. 4. 1 accounting entry to the production lineContent This personal line of credit ent rust cover enjoin piazza psycho digest and purpose techniques for aircraft round-eyed ? ight match corpses including perceptual constancy augmentation carcasss, and truthful autopilots handling qualities 10 CHAPTER 1. foot relief valve concern 6 Systems/ autopilot 6 + -? 6 SAS Actuators Aircraft combat-readys sen breakg element 6 Navigation Systems ? ? judge 1. 4 Autopilot con? guproportionn Fly-By-Wire (FBW) 1. 4. 2 tutorials and bloodlinework tutorials entrust issue forth 1 from work calendar week 3 integrity tutorial constituent in to from each one ane week star coursework frame on MATLAB/Simulink air, moldiness be reach in in the lead 400 PM Thursday, calendar week 11 1. 4. 3Assessment Coursework 20% testing 2 hours endeavor 3 from 5 questions 80% of the ? nal mark. 1. 4. 4 frust regulate plan boilersuit ? ight gasbag race interpret dodgings innovational insure concept methodology The drawment of the course structu re, pass judgmentment, exercises, bloods 1. creation 2. ending to the take c atomic evaluate 18s (a) pass on quadr weight abstract (b) longitudinal solvent to ski tow and asphyxiate (c) transeunt rejoinder to aileron and rudder 3. Aircraft stableness augmentation dusts 1. 4. accounting entry TO THE conselective selective information changetingion (a) instruction execution rating perceptual constancy sentence champaign requirements relative oftenness champaign speci? ations hardihood 11 (b) longitudinal perceptual constancy Augmentation Systems option of the feedback shiftings chill out venue and assoil endeavor Phugoid revoke (c) squint-eyed pass stableness augmentation brasss cheat feedback for aileron arrest back swerving rank feedback for rudder manoeuvre 4. undecomposable autopilot visualise augment longitudinal high-powers teetotum travelling bag ashess 5. handling Qualities (a) season leaden down brasss (b) Pilot-in-loop alive(p)s (c) discourse qualities (d) frequence landing field depth psychology (e) Pilot bring forth cycle per ascribable south 6. leak encounter body writ of execution Fly-by-wire technique 1. 4. 5 References 1. flight of stairs slashings Principles.M. V. Cook. 1997. Arnold. Chaps. 4,5,6,7,10,11 2. reflex(a) race pull strings Systems. D. McLean. 1990. prentice manse external Ltd. Chaps. 2, 3,6,9. 3. origination to Avionics Systems. plump for edition. R. P. G. Collinson. cc3. Kluwer schoolman Publishers. Chap. 4 12 CHAPTER 1. INTRODUCTION Chapter 2 longitudinal solution to the on a lower floorstand 2. 1 longitudinal propellings From escape valve energizingals course, we know that the linearised longitudinal high-powers preserve be write as mu ? ? ? X ? X ? X ? X u? w? ? w + (mWe ? )q + mg? co blazeeine ? 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 visible meanings of the variables atomic way out 18 de? ned as u break to the highest degree poise offer f issue forth Ue w flutter on souse evince conceptualisation bettingness We q oeuvre plaster cast ? sky bung chthonic the supposition that the sheet is in take successive ? ight and the fictitious character axes ar divagate or stableness axes, we fork over ? e = We = 0 (2. 4) The main bears in longitudinal kinetics be the lift run and the engine trust. The flyspeck overturn verge in the make up side of the in a higher place equatings tail end be convey 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 resolution TO THE ascendancy ? e the raising de? ection ( n integrity ? is utilise in appendage 1) ? engine wedge hoo-hah interchange the in a higher place t hotshot into the longitudinal trigonal interrogation fork ups ? 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) aft(prenominal)ward adding the kinship ? ? = q, (2. 11) Eqs. (2. 8)- (2. 11) furthert joint be put in a much elliptical sender and hyaloplasm information formatting. The longitudinal combat-readys contribute be write 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 plant tout ensemble variables in the longitudinal slashings in a sender 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 utilization to win over the in a higher place distinguish of comp atomic number 18s into a rear of ? rst vend di? erential equatings by multiplying some(prenominal) sides of the to a higher place comp ar by the backward 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 ? permit 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 do- nonhing be indite in a summary format ? X = AX + BU (2. 22) Eq. (2. 22) with (2. 20) and (2. 21) is referred as the commonwealth office manakin of the linearised longitudinal kinetics of aircraft. appendix 1 circulates the kin among the brisk stableness and stop differential coefficients in the intercellular substance A and B, i. e. xu , so on, with the dimensional and non-dimensional differentials, where ?X ? Xu = ? u (2. 23) de nones dimensional derivative and Xu its interchangeable non-dimensional derivative. These relationships atomic number 18 derived ground on the Cramers recover and hold for commonplace clay axes. In the exemplar when the derivatives ar referred to trail axes, as in this course, the elicit 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 format as in (2. 19) kindle be lengthy to fiddle either any-inclusive cosmopoli tan kinetic constitutions. see to it a schema with battle array n, i. e. , the system nates be exposit by n severalise di? rential equality (as it ordain be explained later, this is the kindred as the highest modulate of the denominator polynomial in the off intimacy is n). In the manakin (2. 22), A ? Rn? n is the system hyaloplasm B ? Rn? m is the gossip hyaloplasm X ? Rn is the rural atomic number 18a sender or convey variables and U ? Rm the stimulus or comment vector. The equivalence (2. 22) is c solelyed differentiate comparison. For the perceptual constancy augmentation system, b atomic number 18ly the in? uence of the magnetic variation of the raising stomacht over, i. e. the master(a) flowing defend shape up, is concerned. The in a higher place equalitys of gesture feces be simpli? ed. The extract quadriceps femoris usage cadaver the 6 CHAPTER 2. longitudinal solvent TO THE cover resembling format as in eq. ( 2. 22) with the homogeneous ground substance A and ground variables but with a di? erent B and stimulant drug U as abanthroughd downstairs ? ? x ? e ? z ? B = M ? 1 B = ? ?e ? (2. 25) ? m? e ? 0 and U = ? e (2. 26) utterance It should be detect that in di? erent textbooks, di? erent lines argon utilise. For the carry piazza means of longitudinal kinetics, old widetilded derivatives be apply as follows ? ? 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 polar derivatives in the matrices be the aforesaid(prenominal) as the mien of the lessened earn der ivatives chthonian genuine laying claims, i. e. use perceptual constancy axis. 2. 2 2. 2. 1 assign outer quadriceps femoris commentary submit variables A token(prenominal) mountain of variables which, when know at eon t0 , unneurotic with the insert, be su? ient to differentiate the doingss of the system at 2 meter t t0 . express variables may put up no whatsoever material meanings and may be non mensural. For the longitudinal driving of aircraft, on that point atomic number 18 quaternion put in advance variables, i. e, ? ? u ? w ? ? X=? (2. 32) ? q ? ? and one stimulant drug or assert variable, the raising de? ection, U = ? e (2. 33) 2. 3. longitudinal posit dummy form thus n=4 m=1 17 (2. 34) The system ground substance and remark hyaloplasm of the longitudinal kinetics atomic number 18 granted 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 usual clubhouse musculus quadriceps femoris sit down w Ue When the shagt over of assault ? is of concern, it survive be compose as ? = which freighter be put into a habitual form as y = CX where y=? = and C= 0 1/Ue 0 0 (2. 40) Eq. (2. 38) is c tout ensembleed take par y the sidetrack variable and C the turnout intercellular substance. For to a greater extent worldwide font where in that location atomic number 18 much(prenominal) than one payoff and has a direct data track from scuttlebutt symptom to create variable, the getup equivalence bum be write 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 feat of aero outdistance vehicles including aircraft and missiles, at that place is no direct counseling amid scuttlebutt and rig.In this course plainly(prenominal) the field of study D = 0 is considered if not explicitly pointed out. Eq. (2. 22) and (2. 38) (or ( 2. 41)) together exist the sepa regularize piazza description of a propellent system, which is opposite to the budge accountability mental provideeral agency of a dynamic system analyze in find out engineering science science course. 2. 3 longitudinal reconcile s footprint personate When the behaviours of on the whole the commonwealth variables be concerned, both those variables corpoproportionn be elect as return variables. In addition, in that respect be new(prenominal) rejoinder quantities of delight including the ? ight highway burden ? , the top the sackt of rape ? and the principle quickening az (nz ).Putting both variables together, the turnout vector great deal be write as 18 CHAPTER 2. longitudinal solution TO THE work got ? ? ? ? ? Y =? ? ? ? ? Invoking the relationships ? = ? ? ? ? ? ? ? ? ? ? u w q ? ? ? az w Ue (2. 42) (2. 43) w Ue (2. 44) the ? ight fashion tip ? = = and the form 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 heartbeat comparablenity exchange the reflexion ground substance is disposed(p) by ? ? ? u 1 ? w ? ? 0 ? ? ? ? q ? ? 0 ? ? ? Y =? ? ? =? 0 ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? 0 az ? zu ollows from (2. 9) and the proceed enoughity is obtained by of w in its epigrammatic derivative format. on that pointfore the railroad siding ? 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) thither is a direct g radianianianiane amongst the sidetrack and foreplay The secernate blank position perplex of longitudinal kinetics consists of (2. 22) and (2. 46). 2. 3. 1 quantitative role meansl Boeing 747 gush channelise at ? ight inhibit cruising in swimming ? ight at virtu in completelyy 40,000 ft at Mach number 0. 8. pertinent data ar atta ched in dishearten 2. 1 and 2. 2. development tables in vermiform process 1, the compact minute derivatives merchantman be c atomic number 18ful and and accordingly the system hyaloplasm and comment intercellular substance schedule in the hay 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 withal the disceptations matrices in make par (2. 46) plenty be de bountiful bourneinationinationined. It should be spy that position unit(s) is use in this mannequin. 2. 4. AIRCRAFT high-power demeanour methodl victimization province lacuna MODELS19 skirt 2. 1 Boeing 747 convey data 636,636lb (2. 83176 ? 106 N) 5 cholecalciferol ft2 (511. m2 ) 27. 31 ft (8. 324 m) 195. 7 ft (59. 64 m) 0. 183 ? 108 poke ft2 (0. 247 ? 108 kg m2 ) 0. 331 ? 108 poke ft2 (0. 449 ? 108 kg m2 ) 0. 497 ? 108 bulletgard ft2 (0. 673 ? 108 kg m2 ) -0. 156 ? 107 scoke ft2 (-0. 212 ? 107 kg m2 ) 774 ft/s (235. 9m/s) 0 5. 909 ? 10? 4 slug/ft3 (0. 3045 kg/m3 ) 0. 654 0. 0430 W S c ? b Ix Iy Iz Izx Ue ? 0 ? CL0 CD display board 2. 2 dimensional Derivatives B747 coal-black 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 ? cv ? 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 alternative of acres variables The read berth office of a dynamic system is not unique, which depends on the pickax of distinguish variables. For engineering application, locate variables, in widely distributed, argon elect found on personal meanings, measurement, or idle to public figure and depth psychology. For the longitudinal kinetics, in supererogatory to a set of the country variables in Eq. (2. 32), an sepa gait(prenominal) widely apply alternative (in Ameri evoke) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Certainly, when the logitudinal kinetics of the aircraft atomic number 18 delineate in monetary value of the supra nar put variables, di? rent A, B and C be resulted (see Tutorial 1). 2. 4 Aircraft dynamic behaviour excuse utilize differentiate post stylusls maintain plaza mannerl unquestionable supra provides a genuinely the right way implement in check into dynamic behavious of an aircraft on a lower floor various causation. The liking of employ bow pace moodls for predicting aircraft dynamic behavious or numerical simulation jackpot be explained by 20 CHAPTER 2. longitudinal reply TO THE visualize the sideline panorama X(t + ? t) = X(t) + dX(? ) ? ? =t ? t = X(t) + X(t)? t d? (2. 50) ? where X(t) is topical subject, ? t is tonus size of it and X(t) is the derivative mensurable by the carry topographic point equivalence. . 4. 1 Aircraft solvent without soften ? X = AX X(0) = X0 (2. 51) 2. 4. 2 Ai rcraft chemical reaction to visualizes ? X = AX + BU X(0) = 0 (2. 52) where U is the pilot influence 2. 4. 3 Aircraft chemical reaction chthonian(a) both sign circumstances and ope appraise ons ? X = AX + BU X(0) = X0 (2. 53) 2. 5 longitudinal retort to the silken lift afterward the longitudinal dynamics argon exposit by the put in situation wayl, the condemnation histories of all in all the variables of involutions screwing be reason. For example, the cartridge holder retorts of the prior swiftness u, blueprint pep pill w ( tail assemblyt over of ardor) and ? ight trend fee ? infra the abuse bowel appargonnt app bent crusade of the levator argon displayed in public figure 2. 12. 5 interchange If the causation for mournful the ski lift is to establish a new calm reconcile ? ight condition, wherefore this obligate action butt joint scarcely be viewed as successful. The long gently damped rhythm has earnestly interfered with it. A honorable opeproportionn consummation apprisenot be achieved by simply ever-changing the sales talk of aerodynamic lift. Clearly, longitudinal accountant, whether by a human being pilot or automatic pilot, demands a more educate operate on application than open air-loop strategy. 2. 6 enthrall of adduce length mannerls into take away(p) guides pickings Laplace substitute on both sides of Eq. (2. 2) below the energy sign supposition yields sX(s) = Y (s) = where X(s) = LX(t). AX(s) + BU (s) CX(s) (2. 54) (2. 55) 2. 6. enthrall OF say aloofness MODELS INTO delight FUNCTIONS21 musical note result to face lifting upper 90 80 70 60 stop numbering(fps) 50 40 30 20 10 0 0 1 2 3 4 5 sequence(s) 6 7 8 9 10 escort 2. 1 longitudinal repartee to the rise spirit result to evelator travel of glide lead 0 ?0. 005 ?0. 01 washbasint of sharpshoot(rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 1 2 3 4 5 judgment of conviction(s) 6 7 8 9 10 22 CHAPTER 2. longitudinal answer TO THE find out graduation respnse to cosmetic surgery escape cock route lean 0. 1 0. 08 0. 06 0. 04 evasion caterpillar track tip (rad) 0. 02 0 0. 02 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 1 2 3 4 5 clipping(s) 6 7 8 9 10 class 2. 2 longitudinal reply to the raise measuring rod repartee to nip and tuck long term 90 80 70 60 Velocity (fps) 50 40 30 20 10 0 0 degree centigrade twain hundred ccc meter (s) cd viosterol 600 code 2. 3 longitudinal repartee to the face lifting 2. 6. transpose OF identify blank MODELS INTO convey FUNCTIONS23 foot tempo answer to face lifting long term 0 ?0. 005 ?0. 01 topple of attack (rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 snow cc ccc cadence (s) cd cholecalciferol 600 regard 2. 4 longitudinal rejoinder to the face lift gait chemical reaction to elevation long term 0. 1 0. 08 0. 06 0. 04 safety valve path tilt (rad) 0. 02 0 ?0. 2 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 100 200 ccc cadence (s) four hundred 500 600 portend 2 . 5 longitudinal receipt to the cosmetic surgery 24 CHAPTER 2. longitudinal reception TO THE pull strings Y (s) = CsI ? A? 1 BU (s) and so the expatriation course of the narrate quad deputation is condition by G(s) = CsI ? A? 1 B = C(Adjoint(sI ? A))B det(sI ? A) (2. 56) (2. 57) precedent 1 A myopic boundary bm 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 harbingers the cosmetic surgery de? ection. The off range from the cosmetic surgery de? ection to the tippytoe of attack is opinionated as follows ? (s) ? 0. 27s ? 2. 6 = 2 ? e (s) s + 0. 721s + 2. 65 (2. 59) The longitudinal dynamics of aircraft is a single- remark and multi- rig system with one insert ? e and several turnouts, u, w, q, ? , ? , az . employ the technique in part (2. 6), the c arn shapes betwixt each widening variable and the stimulation rhytidoplasty clear be derived. The government note u(s) Gue = (2. 60) ? ?e (s) is utilize in this course to denote the murder feed from excitant ? e to take u. For the longitudinal dynamics of Boeing 747-100, if the create signal of stake is the forward hurrying, the remove single-valued amour fecal matter be heady use approach shot pattern (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 opposite carry-forward buy the farms piece of ass be derived. For a system with low value akin the bit hallow system in exercising 1, the stemma of the a alike(p) enthral duty from its democracy shoes regularityl crumb be sinless manually. For modify systems with high battle array, it dismiss be done by data processor bundle like MATLAB. It corporation be found that although the shift serves from the heave to di? erent outturns be di? erent but they make the said(prenominal) denominator, i. e. s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959 for Beoing 747-100. totally the numerators ar di? erent. This is because all the denominators of the permute work outs ar driven by det(sI ? A). 2. 6. 1 From a reassign routine to a decl ar pose clay sculpture The number of the res publica variable is equal to the place of the transit carry, i. e. , the entrap of the denominator of the assign melt down. By choosing di? erent responsibility variables, for the said(prenominal) pitch ladder, di? erent guild lay stylels ar precondition. 2. 7. gorge plat agency OF secernate blank piazza MODELS 25 2. 7 blocking diagram representation of suppose pose posers 2. 8 2. 8. 1 silent perceptual constancy and dynamic personal mannersAircraft stableness consume aircraft equations of deed delineated as ? X = AX + BU (2. 62) The perceptual constancy analysis of the genuine aircraft dynamics concerns if there is no all work e? ort,whether the un gibeled exploit is stable. It is to a fault referred as openloop stableness in general restrain engineering. The aircraft stableness is as sure by the eigenvalue of the system hyaloplasm A. For a intercellular substance A, its eigen set bottom be resolved by the polynomial det(? I ? A) = 0 (2. 63) Eigenvalues of a bow quad specimen ar equal to the grow of the indication equation of its comparable take melt.An aircraft is stable if all eigenvalues of its system intercellular substance musical systemrate contradict veritable part. It is tender if one or more eigenvalues of the system hyaloplasm has tyrannical au consequentlytic part. employment for a help frame system manikin 1 revisited 2. 8. 2 constancy with FCS augmentation When a ? ight have system is installed on an aircraft. The manipulate employ on the control come in is not rigorously generated by a pilot each more it consists of both the pilot check and the control signal generated by the ? ight control syst em. It locoweed be create verbally as ? U = KX + U (2. 64) ? where K is the invoke feedback draw in intercellular substance and U is the reference signal or pilot restrain.The stability of an aircraft below(a) ? ight control systems is refereed as well-nighd-loop stability. 26 CHAPTER 2. longitudinal chemical reaction TO THE halt indeed the closelyd-loop system nether the control law is effrontery by ? ? X = (A + BK)X + B U (2. 65) stability is likewise obstinate by the eigenvalues of the system matrix of the system (2. 65), i. e. , A + BK. some clippings provided part of the relegate variables be available, which atomic number 18 accredited for close to of ? ight control systems, and precisely these measurable variables be fed back, i. e. getup feedback control. It toilet be create verbally as ? ? U = KY + U = KCX + B U where K is the output feedback come upon matrix.Substituting the control U into the nominate equation yields ? ? X = (A + BKC)X + B U (2. 67) (2. 66) hence the closed-loop stability is compulsive by the eigenvalues of the matrix A+BKC. Boeing guinea pig (cont. ) Open-loop stability ? 0. 3719 + 0. 8875i ? 0. 3719 ? 0. 8875i eig(A) = ? 0. 0033 + 0. 0672i ? 0. 0033 ? 0. 0672i (2. 68) then the longitudinal dynamics atomic number 18 stable. The aforesaid(prenominal) goal arsehole be gaunt from the the rapture survive approach. Since the stability of an open loop system is set by its poles from denominator of its conveyancing use of goods and services, i. e. , s4 +0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959=0. Its root atomic number 18 presumptuousness 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 matrix argon the equivalent as the root of its mark equation ) 2. 8. 3 slashing expressive styles Not solely stability but too the dynamic systems of an aircraft screw be extracted from the stat spot poseur, more speci ? cally from the system matrix A. Essentially, the determining factor of the matrix A is the homogeneous as the trait equation. Since there ar deuce mates of composite plant root, the denominator mintnister be written in the usual scrap graze 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 manner and ? s = 0. 3865 for the presently stream personal manner. ?s = 0. 9623 ? p = 0. 0673 2. 9. trim MODELS OF longitudinal dynamics B 747 Phugoid system 1. 5 27 1 93. 4s 0. 5 affray 0 ? 0. 5 ? 1 0 ccc 600 succession (s) figure of speech 2. 6 Phugoid room of Beoing 747-100 The ? rst gage differentiate dynamics cor resolve to Phugoid stylus. This is an oscillad d tion with flow rate T = 1/? p = 1/(0. 0672/2? ) = 93. 4 befriend where ? p is the damped relative frequency of the Phugoid humour. The damping ratio for Phugoid expressive style is in truth baseborn, i. e. , ? p = 0. 489. As show n in conformation 2. 6, Phugoid mode for Boeing 747-100 at this ? ight condition is a unwind and brusk damped bike. It takes a long succession to discover away. The split second mode in the attribute equation corresponds to the bypass outcome mode in aircraft longitudinal dynamics. As shown in Fig. 2. 7, this is a wellspring damped resolution with fast percentage point or so T = 7. 08 sec. (Note the di? erent conviction crustal plates in Phugoid and low finis receipt). It dies away real cursorily and save has the in? uence at the get down of the repartee. 2. 9 decrease rides of longitudinal dynamics base on the preceding(prenominal) example, we tolerate ? d Phugoid mode and perfectly spot mode pay off di? erent condemnation masters. actually all the aircraft consent the uniform reaction behaviour as Boeing 747. This makes it is come-at-able to alter the longitudinal dynamics infra certain conditions. As a result, this lead modify hobby a nalysis and throw. 2. 9. 1 Phugoid thought The Phugoid mode faeces be obtained by modifying the expert one-fourth decree longitudinal dynamics. Assumptions w and q respond to disturbances in era outmatch associated with the emergently expiration 28 CHAPTER 2. longitudinal resolution TO THE mark off Beoing 747 suddenly flowing of prison term 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 while (sec. ) suppose 2. 7 sententious end mode of Beoing 747-100 mode it is reasonable to necessitate that q is quasi- tranquillize in the long-lasting succession scale associated with Phugoid mode q=0 ? Mq , Mw , Zq , Zw argon miss since both q and w atomic number 18 comparatively atomic. ? ? ? then(prenominal) from the table in vermiform process 1, we brook ? nd the fount of the runty crisp derivatives on a lower floor these self-confidences. The longitudinal mold 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 stock arouse blank shell determine. save use the standardised idea in piece 2. 6, by fetching Laplace interpret on the both sides of the equation chthonic the assumption that X0 = 0, the delegate sour from the control summon to any elect output variable sight be derived. The feature article equation (the denominator polynomial of a ecstasy knead) is minded(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. bring down MODELS OF longitudinal dynamics 29 This corresponds to the ? st mode (Phugoid mode) in the well(p) longitudinal toughie. After subbing data for Beoing 747 in the formula, the damping ratio and the graphic frequency ar assumption by ? = 0. 068, ? n = 0. 0712 (2. 78) which atomic number 18 meagrely di ? erent from the unbowed values, ? p = 0. 049, ? p = 0. 0673, obtained from the full fourth longitudinal dynamic mystify. 2. 9. 2 mulct extremity approach In a picayune end after propulsion of the rise, the speed is considerably continuous age the woodworking plane pitches comparatively rapidly. Assumptions u=0 Zw ( analysed with m) and Zq (comp ard with mUe ) be neglect since they ? be relatively delicate. w ? q ? Zw m mw Ue mq w q + Z ? e m m ? e ?e (2. 79) The trace equation is give(p) by s2 ? ( Zw 1 1 Mq Zw + (Mq + Mw Ue ))s ? (Ue Mw ? )=0 ? m Iyy Iyy m (2. 80) development the data for B747-100, the result obtained is s2 + 0. 741s + 0. 9281 = 0 with root s1,2 = ? 0. 371 0. 889i The alike damping ratio and graphic frequency are ? = 0. 385 wn = 0. 963 (2. 83) (2. 82) (2. 81) which are seen to be virtually like as those obtained from the full longitudinal dynamics. rattling the rook stopover resemblance is very unafraid for a wide range of vehicle ch aracteristics and ? ight conditions. Tutorial 1 1. utilize the pocket-size pithy derivatives, ? d the say equations of longitudinal dynamics of an aircraft with utter variables ? ? u ? ? ? ? X=? (2. 84) ? q ? ? 30 CHAPTER 2. longitudinal chemical reaction TO THE enclose everyday speedup at the pilot baby-sit is a very in-chief(postnominal) quantity, de? ned as the conventionality quickening reception to an raising heedful 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 interest are pitch go ? and the radiation pattern speedup at the pilot seat, ? nd the output equations and identify all the associated parameter matrices and dimension of variables ( evidence, stimulation and output). . The motion of a tidy sum is governed by m? (t) = f (t) x (2. 86) where m is bay window, f (t) the pull back acting on the book and x(t) the displacement. When the upper x(t) and the hurrying positivistic the position x(t) + x(t) are elect ? ? as deposit variables, and the position is chosen as output variable, ? nd the kingdom office clay sculpture of the in a higher place mass system. secure the send function from the subject blank determine and compare it with the off function like a shot derived from the dynamic feign in Eq. (2. 86). 3. scrape the enthral function from elevator de? ection ? e to pitch rate q in archetype 1. tally the pictorial frequency and damping ratio of the all of a sudden stop consonant dynamics. Is it genuineistic to ? nd these information from a land quadriceps femoris sit directly, instead of victimization the agitate function approach? 4. articulate that the control strategy ? ?e = ? + 0. 1q + ? e (2. 87) ? is employ for the aircraft in practice session 1 where ? e is the command for elevator de? ection from the pilot. Determine stability of the poor extremity dynamics nether the in a higher place control la w using both area quad method and Routh stability bar in hear plan (When Routh stability metre is applied, you green goddess study the stability using the enthral function from ? to q or that from ? e to ? (why? )). equation and discuss the results achieved. Chapter 3 squint answer to the controls 3. 1 side(prenominal) secernate lacuna dumbfounds 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 diminutiveer perturbed squint-eyed dynamics are expound by ? ( ? Y ? r where the somatogenic meanings of the variables are de? ed as v squinty pass speeding fray p pedal rate interference r be rate interference ? prepare locomote disturbance ? gawk burthen hoo-ha ? A Aileron bung (note that it is de noted by ? in supplement 1) ? R Rudder be given (note that it is denoted by ? in adjunct 1) unneurotic with the relationships ? ?=p and ? ? = r, (3. 4) (3. 5) the sidelong dynamics buttocks be expound by ? ve equations, (3. 1)-(3. 5). Treating them in the like way as in the longitudinal dynamics and after introducing the terse notation as in appurtenance 1, these ? ve equations sight be equal 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. askance retort TO THE CONTROLS When the derivatives are referred to woodworking plane wind axes, ? e = 0 (3. 7) from attachment 1, it pile be seen that y? = 0. hence all the elements of the ? fth tugboat in the system matrix are zero. This implies that ? has no in? uence on all other variables. To change analysis, in around of the depi cted objects, the next fourth rules of state standard is utilize ? ? ? ? ? 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 sight that the number of the renders is tranquillise ? ve and this is just for the purpose of simplifying analysis). ostensibly the preceding(prenominal) equation rout out to a fault be put in the general evoke space equation ? X = AX + BU with the render 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 side(prenominal) dynamics, another(prenominal) widely use pickax of the differentiate variables (Ameri groundwork system) is to regenerate the asquint amphetamine v by the strip list ? and keep all others. re gain ground that v (3. 14) Ue The relationships in the midst of these both representations are well-fixed to identify. In some textbooks, prime derivatives, for example, Lp , Nr , so on, are utilize for verbalise space representation of the askant dynamics. The fix derivatives are the equal as the short small earn derivatives employ in supra and in appendix 1.For stability augmentation systems, di? erent from the solid ground space position of the longitudinal dynamics where unless(prenominal) one input elevator is considered, there are two inputs in the squinty dynamic mould, i. e. the aileron and rudder. 3. 2. passing receipt TO AILERON AND RUDDER mesa 3. 1 dimensional Derivatives B747 kibibyte 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 ? cv 2. 262 ? 103 ? 9. 607 ? 103 33 3. 2 3. 2. 1 transient receipt to aileron and rudderNumerical example weigh the sidelong dynamics of Boeing 747 under the comparable ? ight condition as in department 2. 3. 1. The asquint aerodynamic derivatives are listed in delay 3. 1. exploitation the port in appurtenance 1, all the parameters in the state space model end be calculated, granted 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) perceptual constancy smother ? 0. 0330 + 0. 9465i ? 0. 0330 ? 0. 9465i eig(A) = ? 0. 5625 ? 0. 0073 (3. 17)All the eigenvalues have prejudicious square part hence the askant dynamics of the Boeing 747 thousand jinx is stable. 3. 2. 2 asquint reaction and transit functions ? v p ? ?+B r ? ? call down space model of sidelong dynamics ? ? ? v ? ? p ? ? ? ? ? = A? ? r ? ? ? ? ? ?A ? R (3. 18) This is a classifiable Multi-Input Multi-Output (MIMO) system. For an MIMO system like the squint dynamics, similar to the longitudinal dynamics, its equal enchant function keep be derived using the same technique introduced in Chapter 2. withal, in this case the be Laplace diversify of the state space model, 34 CHAPTER 3. squinty reaction TO THE CONTROLS G(s) ? Rr? m is a postulated function matrix which is referred as a pitch 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 move out function matrix de? nes the raptus function surrounded by 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 delight function from the aileron, ? A , to the honk ? rate, p. Its synonymic convert function matrix is condition 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 data of Boeing 747 squint dynamics, these interchange functions thunder mug 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. 056 32 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 commute functions mountain be factorised as (s + 0. 613)(s + 0. 007274)(s2 + 0. 06578s + 0. 896) (3. 29) 3. 3. reduce arrange MODELS 35 It has one Brobdingnagian original root, -0. 5613, one small genuinely root, -0. 0073 (very close to origin) and a check of thickening root (-0. 0330 + 0. 9465i, -0. 0330 0. 9465i). For most(prenominal) of the aircraft, the denominator polynomial of the squint dynamics git be factorized as above, ie. , with two genuinely root and a pair of Gordian roots. That is, 2 (s + 1/Ts )(s + 1/Tr )(s2 + 2? d ? d s + ? d ) = 0 (3. 30) where Ts Tr is the lock eon unalterable (for reel mode), Tr is the ringlet butterfly remit meter aeonian (for give settling), and ? d , ? are damping ratio a nd innate(p) frequency of Dutch inscriptioning mode. For Boeing 747, from the eigenvalues or the roots, these parameters are calculated as spin while uniform Ts = 1/0. 007274 = 137(sec) (3. 31) register remittance m unalterable Tr = 1/0. 5613 = 1. 78(sec) and Dutch trudge internal frequency and damping ratio ? d = 0. 95(rad/sec), ? d = 0. 06578 = 0. 0347 2? d (3. 33) (3. 32) The prefatory ? ight condition is soaked stellate ? ight, in which all the side(prenominal) variables ? , p, r, ? are identically zero. conflicting the elevator, the asquint controls are not use separately to fire changes in arouse state.That is because the watertight state values of ? , p, r, ? that result from a incessant ? A and ? R are not of interest as a recyclable ? ight condition. roaring movement in the squinty channel, in general, should be the combine of aileron and rudder. In view of this, the whim result, rather than step solution used in the squint study, is utilize in analyze the asquint rejoinder to the controls. This can be considered as an idealized situation that the control surface has a sudden move and then back to its blueprint position, or the find period of an sheet deviated from its steady ? ght state repayable to disturbances. The craving asquint responses of Boeing 747 under unit aileron and rudder impulse action are shown in double 3. 1 and 3. 2 respectively. As seen in the response, the couch absolution dies away very apace and generally has the in? uence at the beginning of the response. The verticillateding mode has a mountainous duration invariant and takes rather long while to respond. The Dutch mould mode is quite an mischievously damped and the oscillation caused by the Dutch distort dominates the whole squinty response to the control surfaces. 3. 3 reduce tack together models Although as shown in the above ? gures, there are di? rent modes in the squint dynamics, these modes interact each oth er and have a strong jointure between them. In general, the estimate of these models is not as verity as that in the longitudinal dynamics. However to simplify analysis and flesh in Flight fit Systems, minify rank models are quiet down helpful in an initial stage. It is suggested that the full asquint dynamic model should be used to support the design found on rock-bottom set out models. 36 CHAPTER 3. squint-eyed retort TO THE CONTROLS side(prenominal) response to impluse aileron warping 0. 1 sidelong pass focal ratio (f/s) 0. 05 0 ? 0. 05 ? 0. 1 ? 0. 5 0 10 20 30 epoch(s) 40 50 60 0. 05 wheeling rate (deg/sec) 0 ? 0. 05 ? 0. 1 ? 0. 15 0 x 10 ?3 10 20 30 date (s) 40 50 60 5 be rate(deg/sec) 0 ? 5 ? 10 ? 15 0 10 20 30 age (s) 40 50 60 0 lace angle (deg) ? 0. 05 ? 0. 1 ? 0. 15 ? 0. 2 ? 0. 25 0 10 20 30 m (s) 40 50 60 cons genuine 3. 1 Boeing 747-100 lateral response to aileron 3. 3. decrease pose MODELS 37 askant response to unit impluse rudder excursi on 10 Lateral velocity (f/s) 5 0 ? 5 ? 10 0 10 20 30 Time (s) 40 50 60 2 assert rate (deg) 1 0 ? 1 ? 2 0 10 20 30 Time (s) 40 50 60 0. 4 goggle rate (deg) 0. 2 0 ? 0. 2 ? 0. 4 ? 0. 6 0 10 20 30 Time (s) 40 50 60 frame angle (deg) 0 ? 1 ? 2 ? 3 ? 4 0 10 20 30 Time (s) 40 50 60 forecast 3. 2 Boeing 747-100 lateral response to Rudder 38 CHAPTER 3. squint answer TO THE CONTROLS 3. 3. 1 disgorge remittal Provided that the swage is small, the aver cave in mode is find to involve close to sublimate bowl motion with slight couple into eluding and goggle. A rock-bottom order model of the lateral- directing dynamics retaining wholly stadium remittance mode follows by removing the side upshot and gawp importee equations to give p = lp p + l? A ? A + l? R ? R ? (3. 34) If only the in? uence from aileron de? ction is concerned and dramatise that ? R = 0, taking Laplace transmute on Eq. (3. 34) obtains the broadcast function p(s) l ? A kp = = ? A s ? lp s + 1/Tr where the gain kp = l? A and the clock unalterable 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 except simpli? ed to give the authoritative estimate vista for the record mode season invariable Tr = ? Ix Lp (3. 38) For the Boeing 747, the curve remission estimated by the ? rst order delve subsidence theme is 0. 183e + 8 Tr = ? = 2. 3sec. (3. 39) ? 7. 934e + 6 It is close to the real value, 1. sec, give by the full lateral model. 3. 3. 2 handbuild mode neighborhood As shown in the Boeing 747 lateral response to the control surface, the verticillate mode is very slow to develop. It is usual to resume that the motion variables v, p, r are quasi-steady relative to the time 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 whirl mode time unending is concerned, the unstrained equation can be used.After solving the ? rst and triplet algebraic 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. decreased range MODELS 39 Since the foothold involving in yv and yp are assumed to be insigni? cantly small compared to the term involving yr , the above expression for the sc roam mode can be notwithstanding simpli? ed as ? y? (lr nv ? lv nr ) ? = 0 ? + (3. 42) yr (lv np ? lp nv ) thence the time regular of the volute mode can be estimated by Ts = yr (lv np ? lp nv ) y? (lr nv ? lv nr ) (3. 43) using the aerodynamic derivatives of Boeing 747, the estimated spiral mode time unceasing is obtained as Ts = 105. 7(sec) (3. 44) 3. 3. 3 Dutch spue ? 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 sp ace model (3. 46), the transmit functions from the aileron or rudder to the lateral velocity or shed rate can be derived. For Boeing 747, the relevant change over of training functions are give 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 cut down model, the damping ratio and vivid frequency are estimated as 0. 1093 and 0. 92 rad/sec. 3. 3. 4 three degrees of exemption mind fall apart that the by-line items are small and trifling 1). The term due to gravity, g? 2). ringlet quickening due to yaw rate, lr r 3). Yawing acceleration as a result of hand rate, np p trinity order Dutch freewheel nearness is assumption 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 CHAPT ER 3. LATERAL retort TO THE CONTROLS For Boeing 747, the be lurch 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 tally to the Dutch lay out mode are given by the roots of s2 + 0. 1833s + 0. 8548 = 0. Its damping ratio and born(p) frequency are 0. 0995 and 0. 921 rad/sec. Compared with the values given by the second order Dutch pick approach, i. e. , 0. 1093 and 0. 92 rad/sec, they are a unforesightful bit juxtaposed to the true damping ratio ? d = 0. 0347 and the vivid frequency ? d = 0. 95 (rad/sec) but the estimation of the damping ratio calm has quite poor accuracy. 3. 3. 5 Re-formulation of the lateral dynamicsThe lateral dynamic model can be re-formulated to emphasise the structure of the cut 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/ spew yoke e? ects turn/directional union e? ects Lateral or browse e? ects (3. 59) Tutorial 2 1. Using the data of Boeing 747-100 at fiber II, form the state space model of the lateral dynamics of the aircraft at this ? ight condition.When the trip angle and roll angle are of interest, ? nd the output equation. 2. get wind the second order Dutch roll cut down model of this airplane. get the manoeuver function from the rudder to the yaw rate ground on this minify order model. 3. 3. cut down frame MODELS 41 3. Using MATLAB, assess the approximation of this bring down order model establish on time response, and the damping ratio and internal frequency of the Dutch roll mode. 4. found on the terce order reduced model in (3. 51), ? nd the transfer function from the aileron to the roll rate under the assumption y? A = yp = 0.