Chapr 4 Longiudinal saic sabiliy and conrol Effc of acclraion (Lcur 15) Kywords : Elvaor rquird in pull-up; sick-fixd manuvr poin; sick forc gradin in pull-up; manuvr poin sick-fr; ovrall limis on c.g. ravl. Topics 4.1 Inroducion 4. Addiional lvaor dflcion in a pull-up 4.3 Elvaor angl pr g 4.4 Sick-fixd manuvr poin (x mp ) 4.5 Sick forc gradin in pull-up 4.6 Manuvr poin sick-fr (x mp ) 4.7 Limis on sick forc gradin pr g 4.8 saic sabiliy and conrol in a urn 4.9 Ovrall limis on c.g. ravl 4.10 Rmark on drminaion of x mp and x mp from fligh ss Exrciss Dp. of Arospac Engg., IIT Madras 1
Chapr 4 Lcur 15 Longiudinal saic sabiliy and conrol - Effc of acclraion-1 Topics 4.1 Inroducion 4. Addiional lvaor dflcion in a pull-up 4.3 Elvaor angl pr g 4.4 Sick-fixd manuvr poin (x mp ) Exampl 4.1 4.5 Sick forc gradin in pull-up 4.6 Manuvr poin sick-fr (x mp ) Exampl 4. 4.7 Limis on sick forc gradin pr g 4.8 saic sabiliy and conrol in a urn 4.9 Ovrall limis on c.g. ravl 4.10 Rmark on drminaion of x mp and x mp from fligh ss Exampl 4.3 4.1 Inroducion An acclrad fligh occurs whn an airplan (a) has acclraion or dclraion along a sraigh lin (acclrad lvl fligh or climb) or (b) prforms manuvrs lik loop and urn. In h cas of acclrad fligh along a sraigh lin, h sabiliy and conrol quaions ar h sam as hos for h unacclrad fligh. Howvr, h ngin hrus would b diffrn in acclrad and unacclrad flighs. This diffrnc in ngin hrus would rsul in slighly diffrn conribuion of powr o C mo and C mα. Significan changs in sabiliy and conrol ak plac whn an airplan gos hrough a manuvr. Rmark: In Europan books, h word manuvr is spl as manouvr. Dp. of Arospac Engg., IIT Madras
4. Addiional lvaor dflcion in pull-up Considr an airplan a h boom of a loop as shown in Fig.4.1. No : For an undisord viw of his figur, us h scrn rsoluion of 115 x 864 or 104 x 768 pixls. Fig. 4.1 Airplan in a loop L, h fligh vlociy, load facor and h radius of h loop b V, n and r rspcivly. Now, L = n W. Furhr, l ΔL b h xcss of lif ovr ha in lvl fligh. Thn, ΔL = (n-1) W. Th quaions of moion in h plan of symmry ar: T D = 0 (4.1) L W = (W/g)(V /r) = (W/g) Vω (4.) Dp. of Arospac Engg., IIT Madras 3
whr, ω is h angular vlociy. Equaion (4.) givs ω = (n - 1) g / V (4.3) Rmarks: Following hr imporan poins nd o b nod. i) As h airplan gos round h loop onc, i also gos onc around is c.g. To xplain his, Fig.4. shows h airplan a diffrn poins during h loop. I is vidn ha whil compling h loop h airplan has gon around islf onc. Thus q, h angular vlociy abou y-axis, is qual o ω i.. q = ω = (n-1)g / V (4.4) No : For an undisord viw of his figur, us h scrn rsoluion of 115 x 864 or 104 x 768 pixls. Fig. 4. Airplan aiud a diffrn poins in a loop ii) As h airplan roas wih angular vlociy q, h ail which is locad a a disanc of l from c.g., is subjcd o a downward vlociy Δv w = q l (Fig.4.3). Dp. of Arospac Engg., IIT Madras 4
Thus, h ail is subjcd o a rlaiv wind in h upward dircion of magniud q l. This causs a chang in h angl of aack of h ail (Fig.4.3) by Δα givn by: ql V ql V Δ α = = 57.3 in dgrs (4.5) Fig.4.3 changs in α in pull-up iii) This chang in Δα rsuls in lif ΔL on ail and ngaiv ΔC mcg abou h c.g. To balanc his ΔC mcg, an addiional lvaor dflcion is ndd. Sinc, h ffc of going hrough a loop is o caus a rsising momn; his ffc is calld damping in loop. L, Δδ b h addiional lvaor dflcion ndd o balanc Δα. Thn -Δα (n-1)gl Δδ = = - 57.3 (4.6) V Th ohr componns of h airplan also xprinc changs in angl of aack du o h angular vlociy in loop. Th n ffc is approximaly accound for (Rf.1.7, chapr 7) by muliplying Eq.(4.6) by 1.1 i.. Δδ = 1.1x (- 57.3)(n-1)g l / ( V ) = - 63(n-1) g l / ( V τ) (4.7) Dp. of Arospac Engg., IIT Madras 5
Th pull-up fligh can b considrd as a par of a loop. Hnc, combining Eqs.(.84)and (4.7), h lvaor dflcion in pull-up (δ ) pullup is givn by: dc ( m ) dc sick-fixd L 63 (n-1) g l (δ ) pull-up = δocl - C L - C V In a pull-up wih load facor of n, L = n W. Hnc, C L = nw ρv S mδ dc ( m ) nw dc sick-fixd L 63 (n-1)g l Hnc, (δ ) pull-up = δocl - - ρv S C V 4.3 Elvaor angl pr g: mδ (4.8) (4.9) Th drivaiv of (δ ) pull-up wih n is calld lvaor angl pr g and from Eq.(4.9) i is givn by: dc ( m ) dδ sick-fixd 1 W dcl 63 gl ( ) pull-up = {- - } dn V ρs C Rmark: mδ (4.10) In lvl fligh, (dδ /dc L ) is zro whn (dc m /dc L ) sick-fixd is zro. From Eq.(4.10) i is sn ha (dδ /dn) is no zro whn (dc m /dc L ) sick-fixd is zro. This is bcaus h damping producd in a pull-ou maks h airplan apparnly mor sabl. From Eq.(4.10) (dδ /dn) is zro whn (dc m /dc L ) sick-fixd has h following valu: dcm 63 g l ρ C ( ) sick-fixd = - dcl ( W ) S 4.4 Sick-fixd manuvr poin (x mp ) mδ Th c.g. locaion for which (dδ /dn) pull-up is zro is calld sick-fixd manuvr poin and dnod by (x mp ). From Eq.(4.11) and noing ha (4.11) (dc m /dc L ) sick-fixd is zro whn c.g. is a x NP, h following xprssion is obaind for x mp. x x = - c 63 g l ρ C W S mp NP mδ c Using Eq(4.1) in (4.10) givs : (4.1) Dp. of Arospac Engg., IIT Madras 6
dδ C x L cg mp = - - dn pull-up Cmδ c c x (4.1a) Exampl 4.1 Considr an airplan wih W = 500 N, S = 15 m, aspc raio = 6, c =.50 m, l = 3c = 7.5m, C mδ = -0.01 dg -1 and τ = 0.5. Calcula h diffrnc bwn manuvr poin sick-fixd and nural poin sick-fixd. Soluion: Subsiuing h givn valus in Eq.(4.1) and assuming sa lvl condiions i.. ρ = 1.5 kg/m 3 givs: x x - 63 9.81 7.5 1.5 (-0.01) c c 0.5(500 /15) mp NP - = = 0.0378 4.5 Sick forc gradin in pull-up Th sick forc (F) is givn by : 1 F = G ρ V η c S {C hα α + C hδ δ + C hδ δ } Subsiuing for α and δ yilds: 1 F = G ρ V η c S [C hα (α 0Lw + i- i w ) + C hδ δ 0CL+ C hδ δ Hnc, n( W ) S dcm 57.3 g l (n-1) Chδ - C hδ( ) sick-fr + { C hα -1.1 }] ρv C dc V mδ L (4.13) W Gη c S ( ) C df hδ m hδ S dc ρ C ( ) sick-fr+ 57.3GηS cg l {Chα -1.1 } (4.14) dn C dc pull-up mδ L Th quaniy df/dn is calld h sick forc gradin pr g. 4.6 Manuvr poin sick-fr Th c.g. locaion for which (df/dn) quals zro is calld manuvr poin sick -fr. I is dnod by x mp. Rcalling ha h sick-fr nural poin is dnod by x NP, h following xprssion is obaind for x mp : Dp. of Arospac Engg., IIT Madras 7
x' mp x' NP 57.3 g l ρ Cmδ Chδ = + {C hα - 1.1 } c c W ( )C hδ S Using Eq(4.15) in Eq(4.14) givs df dn pull-up df C xcg x' hδ mp = - G c S η W/S - dn pull-up Cmδ c c sick-forc gradin pr g as : (4.15) (4.16) Exampl 4. For h airplan in xampl 4.1 assum furhr ha C hα = - 0.003 dg -1 and C hδ = - 0.005 dg -1. Obain h diffrnc bwn sick-fr manuvr poin and sick-fr nural poin. Subsiuing various quaniis in Eq.(4.15) givs: x' x' 57.3 9.81 7.5 1.5 (-0.01) 1.1 (-0.005) c c 1500 (-0.005) 0.5 x' x' mp NP - = {(- 0.003)- } = 0.075 mp c NP = + 0.075 c Rmark: x mp lis bhind x NP bcaus h airplan has acquird apparn incras in sabiliy in a pull up. 4.7 Limis on sick forc gradin pr g Th sick forc gradin pr g or (df/dn) indicas as or difficuly in carrying ou a manuvr. Hnc i should li wihin crain limis. Rfrnc 1.7 givs h limis as 3 lbs/g o 8 lbs/g or 14 N/g o 36 N/g for fighrs and h uppr limi of 156 N/g for bombrs and cargo airplans. As (df/dn) dpnds on c.g. locaion, hs limis impos rsricions on c.g. ravl (s scion 4.9). 4.8 Saic sabiliy and conrol in a urn Rfrncs 1.7 and 1.1 considr, in addiion o pull up, h sabiliy and conrol in a urning fligh. Howvr, h rquirmns in his fligh ar lss criical han hos in a pull up. Dp. of Arospac Engg., IIT Madras 8
4.9 Ovrall limis on c.g. ravl Taking ino accoun various considraions discussd in chaprs, 3 and 4, h limis on c.g. ravl ar shown in h Fig.4.4. I is o b nod ha gnrally, c.g. ravl should b limid o abou 8% of m.a.c. for a gnral aviaion airplan and abou 15% of m.a.c. for a passngr airplan. Th sringn naur of limiaions on c.g. ravl can b gaugd from h following cas. For an airplan of lngh =10 m, b = 10 m and aspc raio = 10, h valu of c would b around on mr. Thus, a prmissibl c.g. ravl of 8% implis jus 8 cm of c.g movmn for a fuslag of 10 m lngh. No: Th symbols A,B, I indica limiaions on c.g. movmn du o h following considraions. Limis on af c.g. movmn: A: (x NP ) powr off ; B: (df/dn) = 0; C: (x NP ) powron ; D: (df/dn) minimum ; E: (x NP ) powron. Limis on forward c.g. movmn: F: δ for C Lmax in fr fligh wih n=1; G:δ for C Lmax in fr fligh wih n = n max ; H: δ for C Lmax wih ground ffc; I: (df/dn) max. Fig.4.4 Summary of limis on c.g. ravl du o various considraions- schmaic 4.10 Rmark on drminaion of x mp and x mp from fligh ss In scions.13 and 3.5 h fligh s chniqus for drminaion of x NP and x NP ar mniond. Th procdur o drmin x mp and x mp can b brifly dscribd as follows. (i) Choos a c.g. locaion, (ii) Choos a fligh spd, Dp. of Arospac Engg., IIT Madras 9
(iii) Fly h airplan in a pull-up or a sady urn a chosn load facor (n). During h s, h valus of avrag aliud, fligh spd, lvaor dflcion, conrol forc and load facor ar nod. Corrcions o h radings ar applid for any rrors. Carry ou h ss a diffrn fligh spds and load facors. (iv) Rpa ss a diffrn c.g. locaions. (v) From h fligh ss for a chosn c.g. locaion bu a diffrn valus of n, plo δ vs n and F vs n. Calcula (vi) Obain dδ /dn and df/dn. dδ /dn and df/dn a diffrn c.g. locaions. (vii) Plo ( dδ /dn ) vs c.g. locaion. Exrapola h curv. Th c.g. locaion for which ( dδ /dn ) quals zro is h manuvr poin sick-fixd (x mp ). (viii) Plo df/dn vs c.g. locaion. Exrapola h curv. Th c.g. locaion for which (df/dn) quals zro is h manuvr poin sick-fr (x mp ). For furhr dails, s chapr 4 of Rf..5. Exampl 4.3 Givn an airplan wih h following gomric and arodynamic characrisics: S = 19.8 m, b = 10.5 m, C Lαw = 0.078 dg -1, c =. m; l = 5.0 m, i w = 0, dε / dα = 0.48, S = 3.6 m, span of ail plan (b ) = 4.0 m, C Lα = 0.058 dg -1, i = 1 0, S = 1.08 m, c = 0.8 m; η = 0.9; C hα = -0.004 dg -1, C hδ = -0.009 dg -1, G = 1.6 m -1, W = 40,000 N, (x NP ) = 0.35 c. Calcula sick forc gradin in pull-up a sa lvl for c.g. locaion of 0.0 c, 0.6 c and 0.37 c. Plo hs valus vrsus c.g. posiion and by graphical inrpolaion or xrapolaion drmin sick-fr manuvr poin. If i is rquird ha h airplan has sick forc pr g bwn 36 N/g and 14 N/g. Wha would b h c.g. limis? Assum ha h ab is always usd o rim ou sick forc a n =1. Soluion: i) Calculaion of (df/dn) pull-up. Dp. of Arospac Engg., IIT Madras 10
V H = = = 0.413 df m hδ ( ) pull-up = - C hδ ( ) sick-fr + 57.3G η S c g l (Chα- ) dn Cmδ dcl S /S = 1.08 / 3.6 = 0.3; hnc, = 0.5 (Fig..3) S l 3.6 5.0 S c 19.8. G ηs c (W/S) dc ρ 1.1C C = - V η C = -0.413 0.9 0.058 0.5 = -0.0108 dg mδ H Lα W/S = 40000/19.8 = 00N/m x = 0.35c NP -1 CLα dε C x' NP = xnp- VH η (1- ) C dα C Lαw hα hδ x' NP 0.058 (-0.004) = 0.35-0.413 0.9 (1-0.48) 0.5 = 0.35-0.03 = 0.318 c 0.078 (- 0.009) G ηsc (W/S)Chδ 1.6 0.9 1.08 0.8 00 (- 0.009) = = - 733.0 C (- 0.0108) mδ ρ 1.1Chδ 57.3 G η S c g l (Chα- ) 1.5 (- 0.009) = 57.3 1.6 0.9 1.08 0.8 9.81 5.0 {- 0.004-1.1 } = 11.85 0.5 df ( ) pull-up = - 733.0 (dc m/dc L) sick-fr +11.85 dn (E 4.3.1) (dc m /dc L ) sick-fr for diffrn locaion of c.g. ar givn in abl E4.3. Furhr using Eq. (E4.3.1), h valus of (df/dn) pull-up ar also givn in h sam abl. c.g. (dc m /dc L ) sick-fr (df/dn) pull-up 0. c -0.118 98.34 0.6 c -0.058 48.66 0.37 c 0.05-6.5 Tabl E4.3 (dc m /dc L ) sick-fr and (df/dn) pull-up for diffrn locaions of c.g. (ii) Manuvr poin sick-fr : This is h c.g. locaion for which (df/dn) pull-up is zro. From Eq.(E4.3.1): Dp. of Arospac Engg., IIT Madras 11
0 = -733.0(dC / dc ) + 11.85 m L sick-fr m L sick-fr or (dc /dc ) a manuvr poin sick - fr = 11.85 / 733.0 = 0.016 Hnc, manuvr poin sick fr is: = (0.318 + 0.016) c = 0.334 c (iii) c.g. limis for df/dn bwn 14 o 36 N/g can b obaining using corrsponding (dc m /dc L ) sick-fr. 14 = -733.0 (dc /dc ) + 11.85 m L sick-fr1 m L sick-fr1 (dc /dc ) = -.15 / 733.0 = - 0.0093 36 = - 733.0(dC /dc ) + 11.85 m L sick-fr - (36-11.85) (dc m/dc L ) sick-fr = = - 0.039 733.0 Hnc, c.g. limis ar (0.318-0.039) c & (0.318-0.009)c i.. 0.85 c & 0.315 c. Rmark: In his paricular xampl, h c.g. limis for propr df/dn sm o b oo narrow; h limis on (df/dn) prhaps corrspond o a fighr airplan. Dp. of Arospac Engg., IIT Madras 1