1. Given the longitudinal equations of motion of an aircraft in the following format,
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1 Cht 4. Gin th lnitdinl tins f tin f n icft in th fllin ft, U cs W (4. n th in dinlss f fd t ind s. Discss th lti its f th tins f tin in dinl, dinlss nd cncis fs. Ans f it is ssd tht th icft is in ll fliht, thn th tins f tin cn b cntd t ind s b sttin tid itch ttitd nd tid nl lcit W t. Thf (4. bcs, U (4. Th tins f tin ndd dinlss b diidin ch tin b n it t. Th il nd nl fc tins nlisd b diidin thh b th dnic fc t, hil th itchin nt tin is nlisd th dnic nt t c. Oth ts sd in th dlnt f th -nlisd tins dfind s flls: Dinlss ti: σ t tˆ h σ (4.3 nitdinl lti dnsit fct: c (4.4 Dinlss lcitis: û,, ˆ σ (4.5
2 t is ls ntd tht bcs ll fliht hs bn ssd, C (4.6 Th il fc tin ill b cnsidd fist. Hin diidd thh b th dnic fc t, th nt st is t ltil thh b th t, c c (4.7 This lls th il fc tin t b ittn s, c c c c (4.8 U th ss (4.3 t (4.6, tin (4.8 dcs t, σ σ C ˆ c c û (4.9 Th il ccltin cnnt is -nlisd s flls, ( û dtˆ dû t d d dt d σ σ σ (4. nd th nl ccltin is dlt ith in siil nn. Etin (4.9 cn thn b ittn in th cnnint f, C ˆ û û (4. h,,, nd th dinlss ditis, hs dfinitins fll f tin (4.9. iill, th nn-dinl nl fc nd itchin nt tins cn b dld, ˆ ˆ i û ˆ ˆ û (4. h i is th nn-dinl itch inti in b,
3 i (4.3 c. Th iht hndd thnl is sst ( shn in th Fi (4. bl is fid in iid if sch tht is cincidnt ith th cnt f it. U,, c, (,,,,,,, W, Fi 4. Gnlisd bd s sst nd tin ibls Th cnnts f lcit nd fc ln,, nd U,, W, nd,, sctil. Th cnnts f nl lcit bt,, nd,, sctil. Th int (,, in th if hs lcl lcit nd ccltin cnnts,,, nd,, sctil. h tht b si th tin f th s ( n t th tin f th int (,,, th bslt ccltin cnnts f (,, in b, U W W U W U ( ( ( ( ( ( ( ( ( (4.4 Fth, s th ss f th icft t b nifl distibtd sh tht th ttl bd fc cnnts in b, U W h is th ss f th icft. Ans ( W ( W U ( U Th lcit cnnts t int (,, lti t th iin in b, (4.5 (4.6 Ech lcit cnnt ciss lin t nd t ttinl ts. Th iin f th ts d t ttinl tin in th cnnt, f l, is illsttd in
4 Fi 4. bl. Th ttinl ts in th ss f nd btind in siil. - kin in t s sst ln is kin in t s sst ln is Fi 4. lcit ts d t ttinl tin n Fi 4., bth nd snt tnntil lcit cnnts ctin ln lin thh (,, lll t th is., c th is sst shn in Fi 4. is fid in iid if thn th lin ts, nd nd tins (4.6 dc t, (4.7 t, th csndin cnnts f ccltin t (,, lti t in b, (4.8 ik th lcit cnnts t (,,, th ccltin cnnts ch cis lin t nd t ttinl ts. As n l, th iin f th ts d t ttinl tin in is illsttd in Fi 4.3. Bth nd snt tnntil ccltin cnnts ctin ln lin thh (,, lll t th is. Th ccltins is f th tl intctin f th lin cnnts f lcit ith th cnnts f nl lcit. B si th tin f th s ( n t th tin f th int (,,, th bslt lcit cnnts (,, f (,, btind, U U W W (4.9 h th ss f (,, sbstittd f (4.7.
5 - kin in t s sst ln is kin in t s sst ln is Fi 4.3 Accltin ts d t ttinl tin iill, th cnnts f intil ccltin (,, t th int (,, in b, (4. Etins (4.9 thn diffntitd ith sct t ti nd if iid bd is ssd, U W (4., b sbstittin (4. nd (4.9 int (4., th bslt ccltin cnnts f (,, cn b ittn s, U W W U W U ( ( ( ( ( ( ( ( ( (4. Cnsid n incntl ss lcd t int (,,. Alin tn s scnd l t th incntl ss, th incntl fc cnnts ctin n th ss in b (,,. Ths, th ttl fc cnnts (,, ctin n th bd btind b th fc incnts th hl bd, Σ Σ Σ (4.3 bstittin th ss f th intil ccltin cnnts (4. int (4.3 lds t,
6 Σ U Σ Σ W ( W ( ( ( ( W U ( ( ( ( U ( ( ( (4.4 tin tht c th iin f th s sst nd th cnt f it cincidnt, Σ Σ Σ (4.5 thn th sltnt cnnts f ttl fc ctin n th bd in b, U W ( W ( W U ( U (4.6 h is th ttl ss f th bd. 3. Th linisd lnitdinl tins f tin f n icft dscibin sll tbtins bt std tid ctilin fliht cnditin in b, ( ( t ( t W ( t ( ( t ( t U ( t ( t ( t (4.7 Dl ss f (t, (t nd (t nd hnc clt th tins f tin fd t nlisd icft bd s. Wht silifictins b d if ind s fnc nd ll fliht ssd? Ans F fliht in stbl tsh, th distbin fcs nd nts ci (t, (t nd (t cn b ssd t b d t dnic ffcts, ittinl ffcts, nt f cntl sfcs nd ffcts. Hnc, tins (4.7 cn b ittn s, ( ( t ( t W ( ( t ( t U ( t c c c (4.8 F sll distbncs bt ti, ss cn b dld t lc th ts n th iht hnd sid f tins (4.8. Gittinl Ts: Th ittinl cntibtin t tins (4.8 is btind b slin th icft s iht int th distbd bd s.
7 Distbd Cs Tid Cs Hin Fi 4.4 Aicft iht cnnts in th lnitdinl ln f it is ssd tht th iin f th bd s is cincidnt ith th c, th is n iht nt bt n f th s s. inc th icft is initill flin ins ll, f th ti cnditin th cnnts f iht nl in th lnitdinl ln f st, (4.9 cs Fllin distbnc f ti, th ttitd tbtin is in b (φ,, ψ nd th cnnts f iht in th distbd bd s did ith th tin, h D is th dictin c ti dfind s, D (4.3 cs csψ D φ csψ csφ ψ csφ csψ φ ψ cs ψ φ ψ csφ csψ csφ ψ φ csψ φ cs (4.3 csφ cs F sll tbtins, csφ cs csψ nd φ φ,, ψ ψ, hnc th s f D cn b dcd t, ψ D ψ φ (4.3 φ
8 h dcts f sll tbtins h bn nlctd, s tin (4.3 bcs cs φ φ ψ ψ (4.33 Thf, cs cs (4.34 Adnic Ts: t is ssd tht th dnic fc nd nt ts dndnt n th tbtin tin ibls (,, nd (,,, ls thi ditis. U Tl sis snttin in hich,. nl th fist ts in ch sis fnctin is cnsidd ificnt b. hih d ditis, ct ths inlin, nlctd th dnic t f il fc, f l, cn b ssd s, (4.35 h is cnstnt. Etin (4.35 cn b cnnintl ittn s, (4.36 h, tc. dnic stbilit ditis. iill, th nl fc nd itchin nt ts in b, (4.37 (4.38 Cntl Ts: As th fcs nd nts ctd b th dflctin f th lt, ilns nd dd is f chns in th dnic cnditins, it is sl t ntif thi ffct in siil nn t th dnic ts., f l, th il fc d t th dnic cntl ints cn b ittn s, c (4.39
9 h, nd lt, iln nd dd nls, sd lti t ti sttins, nd sctil, nd, nd dnic cntl ditis. iill, th nl fc nd itchin nt ts in b, c (4.4 c (4.4 P Ts: P, nd hnc thst, is sll cntlld b thttl l. nt f th l css thst chn hich in tn slts in chn in th cnnts f fc nd nt ctin n th icft. P ffcts ssd in siil nn t th dnic stbilit nd cntl ditis. Ths, (4.4 (4.43 (4.44 h, nd nin thst ditis. t tht, lik th dnic cntl ts, thst is sd lti t ti sttin. Hin dld ss f th ts n th iht hnd sid f tins (4.8, th tins f tin cn b ittn s, ( t cs U ( t ( t ( cs W ( t ( ( t (4.45, t tid fliht cnditins ll tbtin ibls nd thi ditis, b dfinitin,. Ths, th std stt dnic ts cn b fnd t b, cs (4.46 Fth, if it cn b ssd tht lnitdinl nd ltl-dictinl tin cn b dcld nd th distbnc is cnstind t th lnitdinl ln f tin, thn
10 th ltl tin ibls, nd ll nd th css-clin ditis nliibl sll i.. Thf, tins (4.45 dc t, U cs W (4.47 F th cs f ll fliht nd if ind s fnc is ssd thn, W (4.48 nd tins (4.47 dc fth t, U ( tt th sstins d in diin th sll tbtin lnitdinl tins f tin f n icft. F ch sstin i listic l f n icft t, cnfitin, hich k th sstin inlid. Ans Th sstins d in diin th lnitdinl tins f tin f sll tbtins s flls: i Distbncs f ti sll this is n inlid sstin if th ints t th cntls f th ilt cs l ditins f th ti cnditin ii Qsi-std ifl th icft - this is n inlid sstin t hih nls f ttck nd in th st-stll in iii tic ss distibtin this is n inlid sstin f icft ith n stic sts cnfitin i tin is cnstind t th lnitdinl ln f st this is n inlid sstin if cntl int css tbd tin in bth th lnitdinl nd ltl-dictinl lns Aif is iid ith n lstic ffcts this is n inlid sstin f l ciil tnst icft nd hih sd fliht i Aicft is initill tid ith ll, nd sidsli this sstin is inlid if th icft is ndin n sch s std ll cdintd tn (nn- ll nl
11 5. h tht hn th dct f inti is ch sll thn th nts f inti in ll nd, nd sctil, th ltl-dictinl ditis in difid Aicn nlisd f b itd b th Aicn nlisd f. Ans Th dcld ltl-dictinl tins f tin cn b ssd in ts f Aicn nlisd ditis s flls, ( ( / / cs U W ψ φ (4.5 Th nbl difid Aicn nlisd f f tins is, ( ( cs U W ψ φ (4.5 h, f l, th difid nlisd ditis in b ss sch s, / / (4.5 n th cs f th dct f inti bin ch sll thn th nts f intis nd, / (4.53 U this itin, th difid nlisd tins f tin cn b ittn s,
12 ( ( cs U W ψ φ (4.54 t cn b sn tht if th itin in tin (4.53 is ls lid t th Aicn nlisd tins in (4.5, th sltin ss ill b idnticl t ths in (4.54. Thf, hn is ch sll thn nd, th ltldictinl ditis in difid Aicn nlisd f b itd b th Aicn nlisd f.
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