Chapter 7 Dynamic stability analysis I Equations of motion and estimation of stability derivatives - 4 Lecture 25 Topics

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1 Chapt 7 Dynamic stability analysis I Equations of motion an stimation of stability ivativs - 4 ctu 5 opics 7.8 Expssions fo changs in aoynamic an populsiv focs an momnts Simplifi xpssions fo changs in aoynamic an populsiv focs an momnts 7.8. Stability ivativs 7.9 Final fom of small ptubation quations Small ptubation quations fo longituinal motion 7.9. Small ptubation quations fo latal motion Rmaks on solutions of small ptubation quations to obtain spons of aiplan to istubanc an to contol input 7.1 Estimation of stability ivativs 7.11 Divativs u to chang of u X/ u Z/ u M/ u 7.1 Divativs u to chang of w X/ w 7.8 Expssions fo changs in aoynamic an populsiv focs an momnts h aoynamic focs an momnts an th populsiv foc vay with Δu, Δv, Δw, Δp, Δq, Δ, Δδ a, Δδ an Δδ an thi ivativs. Accoing to Rf.1.1, chapt 3, Byan, who gav th basic fam wok of stability analysis in 1911, assum that ths focs an momnts can b xpss as functions of Dpt. of Aospac Engg., II Maas 1

2 th ptubation vaiabls. his can b xpss in th fom of a aylo sis as: ΔX (Δu, Δv, Δw, Δp, Δq, Δ, Δu, Δ v,..δδ a, Δδ, Δδ, Δδ ); X X... X... X X X X u v u δ δ δ δ u v u δ δ δ δ a a + high o tms (7.69) Not : Δδ is a paamt inicating ngin stting Simplifi xpssions fo changs in aoynamic an populsiv focs an momnts h small ptubation quations hav bn linaiz by ignoing th tms containing th pows of ptubation quantitis. Continuing th simplification, th high o tms in Eq.(7.69) a igno. Futh, to avoi unncssay complications, ΔX, ΔY, ΔZ, Δ, ΔM an ΔN a xpss in tms of only a fw quantitis which ictly affct thm. abl 7.4 psnts th quantitis an th ptubation vaiabls on which thy pn i.. X X X X ΔX = Δu+ Δw + Δδ + Δδ u w δ δ Z Z Z Z Z Z ΔZ = Δu+ Δw + Δw + Δq + Δδ + Δδ ; q = θ/t u w w q δ δ M M M M M M ΔM = Δu+ Δw+ Δw+ Δq+ Δδ + Δδ u w w q δ δ (7.7) (7.71) (7.7) Y Y Y Y ΔY = Δv+ Δp+ Δ + Δδ v p δ ' ' ' ' ' Δ' = Δv + Δp + Δ + Δδ + Δδ v p δ δa N N N N N Δ N = Δv+ Δp+ Δ + Δδ + Δδ v p δ δa a a (7.73) (7.74) (7.75) Dpt. of Aospac Engg., II Maas

3 Quantity ΔX ΔZ ΔM ΔY Δ ΔN Dpnnc on Δu, Δw, Δδ, Δδ Δu, Δw, Δw, Δq, Δδ, Δδ Δu, Δw, Δw, Δq, Δδ, Δδ Δv, Δp, Δ, Δδ Δu, Δp, Δ, Δδ, Δδ a Δu, Δp, Δ, Δδ, Δδ a abl 7.4 Changs in aoynamic focs an momnts an thi pnnc Rmaks: i) h simplification of xpssing ΔX, ΔZ ΔN, in tms of only a limit numb of vaiabls is possibl bcaus of th following asonabl assumptions. (a) ΔX, ΔW an ΔM a affct only by th vaiabls of longituinal motion i.. Δu, Δw, Δw, Δq, Δδ, Δδ ; th pnnc of ΔX on Δw is igno (Eq.7.7). (b) ΔY, Δ an ΔN a pnnt only on th vaiabls affcting latal an ictional motions viz. Δv, Δp, Δ an contol flctions Δδ, Δδ a. ii) hs assumptions a vali fo convntional aiplans with (a) plan of symmty, (b) high aspct atio wings (A>5) an (c) flying at moat angls of attack. Consult Rf.1.1 chapt 4 fo tatmnt of aiplans with low aspct atio wings an thos opating at high angls of attack Stability ivativs h quantitis X / u, X / w, N/ δ, N/ δ a in Eqs.(7.7) to (7.75) a call stability ivativs. 7.9 Final fom of small ptubation quations Substituting fo ΔX fom Eq.(7.7) in Eq.(7.63) yils: Δu X X X X m = Δu+ Δw+ Δδ + Δδ - mg Δθ cos θ t u w δ δ X X X X (m - )Δu - Δw + mg cosθ Δθ = Δδ + Δδ O t u w δ δ (7.76) (7.77) Dpt. of Aospac Engg., II Maas 3

4 h following notations a commonly us to simplify th small ptubation quations. 1 X 1 X 1 X 1 X X =, X =, X =, X = ; m u m w m δ m δ u w δ δ Using ths, Eq.(7.77) can b wittn as: ( - X )Δu - X Δw + g cosθ Δθ = X Δδ + X Δδ t u w δ δ (7.78) (7.79) In a simila mann, th following notations a us to simplify th xpssions fo ΔZ, ΔM, ΔY, Δ an ΔN. 1 Z 1 Z 1 Z 1 Z 1 Z Z =, Z =, Z =, Z =, Z =, m u m w m w m δ m δ u w w δ δ (7.8) 1 M 1 M 1 M 1 M M =, M =,...,M =, M = I u I w I δ I u w δ δ yy yy yy yy 1 Y 1 Y 1 Y Y v =, Y =, Y δ = m v m m δ 1 ' 1 ' 1 ' 1 ' 1 ' ' =, ' =,' =,' =, ' = I v I p I I δ I δ v p δa δ xx xx xx xx a xx (7.81) (7.8) (7.83) 1 N 1 N 1 N 1 N 1 N N =, N =,N =, N =,N = I v I p I I δ I δ v p δa δ zz zz zz zz a zz Small ptubation quations fo longituinal motion h small ptubation quations fo longituinal motion a: ( - X )Δu - X Δw + g cosθ Δθ = X Δδ + X Δδ t u w δ δ (7.84) (7.85) -ZuΔu + [(1- Z w ) - Z w ] Δw - [(u +Z q) - g sinθ ]Δθ = ZδΔδ + Z δ Δδ (7.86) t t u w w q - M Δu - (M +M )Δw + ( - M )Δθ t t t = M δ Δδ + M δ Δδ (7.87) Dpt. of Aospac Engg., II Maas 4

5 7.9. Small ptubation quations fo latal motion h small ptubation quation fo latal motion a: ( - Y ) Δv - (u - Y ) Δ - gcos θ Δφ = Y Δδ t v δ I -' Δv + ( -' ) Δp - [ +' ] Δ = ' Δδ + ' Δδ xz v p δa a δ t Ixx t I -N Δv - ( + N ) Δp - [ - N ]Δ = N Δδ + N Δδ xz v p δa a δ Izz t t (7.88) (7.89) (7.9) Rmaks on solutions of small ptubation quations to obtain spons of aiplan to a istubanc an to th contol input h Eqs.(7.85) to (7.9) constitut th linaiz small ptubation quations fo longituinal an latal motions. hi solution woul yil answs to two typs of poblms namly (a) spons to a istubanc an (b) spons to a contol input. In th cas of spons to a istubanc, with th contol fix, it is assum that Δδ, Δδ, Δδ a, an Δδ a zo. o stuy th ffct of istubanc, it is assum that on of th paamts fom among u, w,...., is givn a small valu at tim t =. hn, th quations a solv with this initial conition. h changs, with tim, in th valus of th chosn paamts woul giv infomation about th ynamic stability. Howv, it will b point out in sction 8. that it is not ncssay to solv th abov iffntial quations to know whth th aiplan is stabl o not. h is a simpl appoach to inf about th stability of th aiplan. In th cas of spons to th contol input, it is assum that th contol flction is givn as a function of tim an th solution of th abov quations is obtain. Fo xampl, it may b pscib that th lvato flction Δδ changs fom zo to a valu Δδ 1, in a small intval of tim an thn mains constant (Fig 7.4a). h solution of th quations woul giv th infomation about chang in angl of attack an th tim takn to achiv th final valu. Figu 7.4b shows a spons whin th aiplan attains th final angl of attack aft ov-shooting it (final valu). In som cass, th final valu may b Dpt. of Aospac Engg., II Maas 5

6 achiv monotonically. Howv, calculation of spons is an involv task. Som inication about this is givn in chapt 1. (a) Elvato flction (b) Possibl spons of chang in angl of attack. Fig.7.4 Rspons to lvato flction 7.1 Estimation of stability ivativs h solution of small ptubation quations fo longituinal motion woul b takn up in chapt 8 an fo th latal motion in chapt 9. Howv, to solv ths quations th stability ivativs a qui. h following subsctions al with thi stimation Divativs u to chang of Δu hs ivativs inclu X / u, Z / u an M / u X / u h changs in ΔX a caus by changs in th ag an th thust i.. Δ X = - ΔD + Δ (7.91) Continuing with th linaiz tatmnt of th poblm, th vaiation of ΔX with Δu is xpss as: D X u u (7.9) u u X D 1 1 CD O ( ρ u S CD ) ρs ( u ucd ) u u u u u u u (7.93) Rcall that X u = (1 / m) ( X / u) an lt C Du = C D / (u / u ) Dpt. of Aospac Engg., II Maas 6

7 As gas th tm, / u th following may b not. a) Fo gliing flight, = an hnc, / u =. b) Fo a jt aiplan, is almost constant ov small intvals of u an hnc, / u =. c) Fo a piston ngin aiplan with vaiabl pitch popll, th HP is naly constant ov a small ang of u. Hnc, = HP/u an consquntly / u = - HP/u = - D/u As gas C Du th following facts may b not. a) Fo subsonic flights with Mach numb lss than th citical Mach numb, th ag cofficint mains constant with Mach numb an hnc C Du =. b) Whn C D is a function of Mach numb (M 1 ), C Du is wittn as: C Du = C D / (u / u ) = u C D / (a M 1 ) = (u / a ) C D / M 1 =M 1 C D / M 1 ; wh, a = sp of soun un conitions of unistub flight. h symbol M 1 is us fo Mach numb to avoi confusion with pitching momnt (M). With th abov consiations, X u can b wittn as: 1 X ρ u S 1 X u = = - (C Du + C D) + (7.94) m u m m u ρ u S -D X = - (C +C ) + o ( ) u Du D mu mu (7.95) QS -C QS 1 - (C +C ) o ( ); Q = ρu (7.96) mu mu D Du D QS = - {(C Du+C D)+ o (- C D)} mu Following Rf.1.1, chapt 3, two nw quantitis C X an C Xu a intouc : X CX 1 X C X= ; C Xu= = QS (u/u ) (1/)ρ Su u Consquntly, 1 X ρsu X u = = - C m u mu Xu (7.97) (7.98) (7.99) C Xu = -{(C Du+C D)+ o (-C D)} (7.1) Dpt. of Aospac Engg., II Maas 7

8 7.11. Z/ u h foc in Z-iction is u to th wight an th lift. Z = W - 1 ρu S C Ignoing th chang in wight uing th istubanc, givs ( Z / u ) as: Z 1 C - ρ S u (C u+c ); Cu u ( u/u ) (7.11) (7.1) t, 1 Z C Zu = ; hn, C zu = -(C u + C ) (1/)ρ S u u (7.13) O 1 Z ρsu Z = (C + C ) u m u mu u (7.14) QS = - (C u + C ) mu (7.15) It may b a that: a) At low subsonic Mach numb C u can b nglct. b) At sub-citical Mach numbs Rf 1.1, chapt 3 stats that C / M 1 can b calculat using Pantl -Glaut ul applicabl to aifoils. Howv, Rf.1.1, chapt 4 suggsts that: C C = M = M α C α u 1 1 M1 M1 h tm C α as a function of Mach numb is givn as: πa C α = in a A β K (1+ tan Λ c/) β1-1 (7.16) (7.17) wh, A = Aspct atio of wing, Λ c/ = swp of th mi-cho lin, β 1 = (1- M 1 ) 1/ an K = (lift cuv slop of aifoil) / π. Rmak: h stability ivativ C u fo Boing-747 is calculat at M 1 =.8 in Appnix C. Fo this pupos C α is valuat at M 1 =.8 an M 1 =.78. Using ths two valus C α / M is calculat at M 1 =.8. Dpt. of Aospac Engg., II Maas 8

9 M/ u Noting that 1 M= ρu S c C mcg M ( 1 ρs u c C mcg) u u 1 C mcg = ρ S c (u + uc mcg) u But, C mcg = in quilibium flight. Consquntly, M 1 = ρ S c u C wh, C = M u mu mu (u/u ) givs: (7.18) C Analogous to Eq.(7.15), C mu can b xpss as: C C = M m mu 1 M 1 C (7.19) mα =M1 α wh, M 1 is Mach numb (7.11) M 1 Hnc, 1 M Q S c M u = = C I u u I yy yy mu (7.111) Changs in C mα with M 1 a foun out by obtaining C mα at naby Mach numbs (s Appnix C ). h valu is also affct by lastic bning of th wing an fuslag. 7.1 Divativs u to chang of Δw hs inclu X / w, Z / w an M / w. h iscussion in this subsction is bas on Rf.7., chapt 4. It may b not that in th unistub flight, X s - axis is along th flight iction. i.. V = u i, w =, v =. Aft th istubanc, th aiplan has Δw (Fig.7.5). hus, th lativ win maks an angl Δα = Δw / u. h lift () an ag (D) a now ppnicula an paalll spctivly to th lativ vlocity (V R ) as shown in Fig.7.5. Dpt. of Aospac Engg., II Maas 9

10 Fig.7.5 Stability ivativs u changs of w X / w Fom Fig.7.5, X = sin Δα - Dcos Δα (7.11) Futh, X w 1 u X (7.113) Hnc, X 1 D = {cos Δα + sinδα + Dsin Δα- cos Δα} w u α α Sinc, Δα is small, cos Δα = 1 an tms involving sin Δα a igno. Hnc, 1 X 1 D X w ( ) m w mu (7.114) (7.115) { ρu SC ( ρu SC D)} mu ρu S QS (C -C ) (C -C ) (7.116) mu mu D D Xw mu t, C Xα = hn, C Xα = C - CDα QS, (7.117) Not: Fo Mach numb lss than th citical Mach numb th ag pola is givn by: C C C = C +. Hnc, C = C πa πa D D Dα α. Dpt. of Aospac Engg., II Maas 1