Fligh dynamics II Sabiliy and conrol haper 3 (Lecures 1, 13 and 14) Longiudinal sick free saic sabiliy and conrol Keywords : inge momen and is variaion wih ail angle, elevaor deflecion and ab deflecion ; floaing angle of elevaor ; sick-free saic sabiliy and sickfree neural poin; sick force and sick force gradien and heir variaions wih fligh speed ; deerminaion of sick-free neural poin from fligh ess. Topics 3.1 Inroducion 3. inge momen 3..1 hanges in hinge momen due o α and e 3.3 Analysis of sick-free saic sabiliy 3.3.1 Floaing angle of elevaor ( efree ) 3.3. Saic sabiliy level in sick-free case (d' m / ) sick-free 3.3.3 Neural poin sick-free (x NP / c ) 3.3.4 Shif in neural poin by freeing he sick 3.4 Sick force and sick force gradien 3.4.1 Dependence of sick force on fligh velociy and airplane size 3.4.3 Tab deflecion for zero sick force 3.4.4 Requiremen for proper sick force variaion 3.4.5 Feel of he sabiliy level by he pilo 3.5 Deerminaion of sick-free neural poin from fligh ess Reference Exercises Dep. of Aerospace Engg., IIT Madras 1
Fligh dynamics II Sabiliy and conrol haper 3 Lecure 1 Longiudinal sick free saic sabiliy and conrol 1 Topics 3.1 Inroducion 3. inge momen 3..1 hanges in hinge momen due o α and e 3.3 Analysis of sick-free saic sabiliy 3.3.1 Floaing angle of elevaor ( efree ) 3.3. Saic sabiliy level in sick-free case (d' m / ) sick-free 3.3.3 Neural poin sick-free (x NP / c ) 3.1 Inroducion In he analysis of sick-fixed longiudinal saic sabiliy i is assumed ha he elevaor deflecion remains consan even afer he disurbance. The analysis of he longiudinal saic sabiliy when he elevaor is free o roae abou is hinge line is called sick-free sabiliy. The fligh condiion in which his may occur is explained below. To fly he airplane a differen speeds and aliudes, appropriae values of lif coefficien ( L ) are needed, e.g. in level fligh, L = W = (1/) ρ V S L or L = W / {(1/)ρV S} As seen in secion.1.3, differen values of rim are needed o bring he airplane in equilibrium a each L. To hold he elevaor a his rim, he pilo has o exer a force called sick force (F) a he conrol sick. F = G e where e is he hinge momen a he conrol surface hinge and G is he gearing raio which depends on he mechanism beween sick and he conrol surface. Figure 3.1 shows a schemaic arrangemen of he conrol surfaces and sick. Dep. of Aerospace Engg., IIT Madras
Fligh dynamics II Sabiliy and conrol Fig.3.1 Schemaic arrangemen of elevaor and sick inge momen and sick force are also shown To relieve he pilo of he srain of applying he sick force (F) all he ime, ab is used o bring he hinge momen o zero. To appreciae he acion of a ab noe is locaion as shown in Figs..16 a and b. I is observed ha by deflecing he ab in a direcion opposie o ha of he elevaor, a lif ΔL ab would be produced. This would slighly reduce he lif due o he elevaor, ΔL e, bu can make he hinge momen zero. This ype of ab is called a rim ab. Afer applying an appropriae ab deflecion, he pilo can leave he sick, i.e. sick is lef free as hinge momen is zero. The analysis of sabiliy when he sick is free or he elevaor is free o move afer he disurbance, is called sick-free sabiliy analysis. I will be explained laer ha his analysis also faciliaes sudy of aspecs like sick force and sick force gradien. The difference beween sabiliy when he sick is fixed and when he sick is free can be explained as follows. (1) When an airplane flying a an angle of aack α encouners a disurbance, is angle of aack changes o (α+δα). onsequenly, he angle of aack of he ail also changes along wih ha of he airplane. Now, he pressure disribuion on he elevaor depends on he angles of aack of ail (α ), elevaor deflecion ( e ) and ab deflecion ( ). ence, he momen abou he elevaor hinge line also depends on hese hree parameers viz. α, e and. Thus, when α changes as a resul of he disurbance, he hinge momen also changes. In sick-free case, Dep. of Aerospace Engg., IIT Madras 3
Fligh dynamics II Sabiliy and conrol he elevaor is no resrained by he pilo, and i auomaically akes a posiion such ha he hinge momen is zero (see iem below). () I may be poined ou here, ha a surface which is free o move abou a hinge will always ake such a posiion ha he momen abou he hinge is zero. For example, consider he rod OB which is hinged a O and aached o anoher rod OA resing on a able (Fig.3.). When lef o iself, he rod OB will ake he verical posiion. In his posiion, he weigh of he rod passes hrough he hinge and he momen abou he hinge is zero. If he rod is o be held in anoher posiion OB (shown by doed lines in Fig.3.), hen a force F has o be applied o overcome he momen abou he hinge which is due o he weigh of he rod. owever, an imporan difference beween he moion of he rod and ha of he elevaor mus be noed. In he case of he rod he momen abou he hinge is due o he weigh of he rod. Whereas he momen abou he elevaor hinge is mainly aerodynamic in naure i.e. due o he pressure disribuion on he elevaor which depends on α, e and. The influence of he weigh of he elevaor on sabiliy is discussed in sick free dynamic sabiliy (secion 8.15). (3) Finally, he deflecion of an elevaor, which is free o move, depends on he disurbance viz. Δα. This would produce change in he ail lif and consequenly, momen abou c.g.. Thus, an addiional change in mcg is brough abou when he elevaor is free. This also resuls in change in mα and hence he longiudinal saic sabiliy. The expression for he hinge momen in erms of α, e and, is obained in secion 3.. The changes in longiudinal saic sabiliy are discussed in secion 3.3. Dep. of Aerospace Engg., IIT Madras 4
Fligh dynamics II Sabiliy and conrol Fig.3. Equilibrium posiions of a hinged rod 3. inge momen To analyze sick-free sabiliy, he dependence of he hinge momen, on α, e and needs o be arrived a. As menioned earlier, he hinge momen ( e ) is he momen abou he conrol surface hinge due o he pressure disribuion on i (conrol surface). The inge momen coefficien ( he ) is defined as: 1 = ρ V S c e e e he Or e he = 1ρ V S e c e where, S e is he area of elevaor af of he hinge line and ce is he m.a.c. of he (3.1) (3.) elevaor area af of he hinge line. By convenion, nose up hinge momen is aken as posiive (Fig.3.1). 3..1 hanges in he hinge momen due o α and e To examine he effecs of α and e, consider he changes in pressure disribuion on he ail due o hese wo angles. Figure 3.3a shows he disribuion of pressure coefficien ( p ) in poenial flow pas a symmeric airfoil a zero angle of aack. I may be recalled ha p is defined as: p p - p = 1 ρ V (3.3) Dep. of Aerospace Engg., IIT Madras 5
Fligh dynamics II Sabiliy and conrol where, p = local saic pressure, p = free sream saic pressure and (1/) ρ V = free sream dynamic pressure. Fig.3.3 hanges in disribuion of pressure coefficien wih α and e (Adaped from Dommasch, D.O., Sherby, S.S., onnolly, T.F. Airplane aerodynamics, haper 1 wih permission from Pearson Educaion, opyrigh 1967) Since, he airfoil used for he ail is symmeric, he p disribuion, a α = 0, is symmeric boh on he airfoil and he elevaor. ence, here is no force on he elevaor and no hinge momen abou he elevaor hinge. Figure 3.3b shows he same airfoil a a posiive angle of aack (α ). The disribuion of p shows ha i is negaive on he upper surface and posiive on he lower surface of he elevaor. This resuls in a posiive ΔL e on he elevaor and a nose down (i.e. negaive) hinge momen. The hinge momen becomes more negaive as α increases. Noe ha α < α sall. Figure 3.3c shows he changes in p disribuion due o posiive elevaor deflecion. This also causes p o be negaive on he upper surface and Dep. of Aerospace Engg., IIT Madras 6
Fligh dynamics II Sabiliy and conrol posiive on he lower surface of he elevaor, resuling in posiive ΔL e and negaive hinge momen. The effec of he deflecion of ab ( ) on he is similar in rend as ha due o he elevaor. Typical variaions of he wih α and e are shown in Fig.3.4. Noe ha when α > α sall he curves become non-linear. In he linear region of he curves, he can be expressed as: he = h0 + hα α + e+ h (3.4) h0 is zero for a symmeric airfoil and is omied in subsequen discussion; hα = he / α ; = he / e ; and h = he /. The quaniies hα, and h depend on he shape of he conrol surface, area behind he hinge line and he gap beween he main surface and he conrol surface. They are generally negaive. Discussion on hα and is aken up in chaper 6 afer laeral saic sabiliy is also covered.the discussion is common for elevaor, rudder and aileron. Fig.3.4 Variaion of hinge momen coefficien wih α and e 3.3 Analysis of sick-free saic sabiliy Dep. of Aerospace Engg., IIT Madras 7
Fligh dynamics II Sabiliy and conrol For he analysis of saic sabiliy wih sick free, i is assumed ha as soon as he disurbance is encounered, he elevaor akes such a posiion such ha he hinge momen is zero. As poined ou earlier, whenever a surface is free o roae abou a hinge line, i akes such a posiion such ha he hinge momen is zero. 3.3.1 Floaing angle of elevaor ( efree ) The elevaor deflecion when he hinge momen is zero is called he floaing angle and is denoed by efree. I can be obained by equaing l.h.s. of Eq.(3.4) o zero. i.e. 0 = hα α + efree + h (3.5) α + efree = - hα h 3.3. Saic sabiliy level in sick-free case (d mcg /) sick-free Assuming he elevaor o have aained efree, he lif on he ail becomes: = α + + L L L Lα efree e ab (3.6) = Lα {α + efree+ ab } (3.7) L ( ) = (3.8) Lα Subsiuing for efree, Eq.(3.7) becomes: hα α + h L = Lα(α - + ab ) hα h =Lα α (1- ) - Lα ( - ab ) Subsiuing α from Eq.(.45) in Eq.(3.9) gives: hα hα L = Lα(1 - )(α - ε 0 - α + i ) - Lα( - ab ) (3.9) (3.10) Denoing he momen abou c.g. by he ail, in he sick-free case, by mcg i can be expressed as : mcg = - V η L Dep. of Aerospace Engg., IIT Madras 8
Fligh dynamics II Sabiliy and conrol Subsiuing for L from Eq.(3.10) gives: ' mcg = V η Lα(ε 0 - i ) - V η Lα[α (1- )+ ab ] - [ hα { α - ε 0 - α + i } + h ] hα hα Or ' mcg = V η Lα (ε0-i - ab )(1- )- V η Lα α(1- )(1- ) Differeniaing Eq.(3.11) wih α and denoing he conribuion, of ail o sick-free sabiliy, by mα gives: hα ' mα = -V η Lα (1- )(1- ) Noing ha mα = -V η Lα (1- ), yields : ' mα = - V η Lα (1- )f = mα f hα f = (1- ) f is called free elevaor facor. (3.11) (3.1) (3.13) The conribuions of wing, fuselage, nacelle and power do no change by freeing he sick, hence, ( mα) sick-free = ' mα = ( mα) w + ( mα) f,n,p- V η Lα (1- ) f cg ac ' mα = Lαw( x - x ) + ( mα) f,n,p- V η Lα (1- ) f c c (3.14) d 1 x m cg xac 1 Lα ( ) sick-free = ' mα - ( mα) f,n,p V η (1- ) f d c c L Lαw Lαw Lαw d hα Lα ( m ) sick-fixed + V η (1- ) d L h e Lαw (3.14a) 3.3.3 Neural poin sick-free (X NP / c ) In he sick free case, he neural poin is denoed (Ref.1.1, chaper ) by x NP. I is obained by seing mα = 0 Dep. of Aerospace Engg., IIT Madras 9
Fligh dynamics II Sabiliy and conrol Or x ' NP c x ( ) ac mα f,n,p Lα = - - V η(1- ) f c Lαw Lαw (3.15) Dep. of Aerospace Engg., IIT Madras 10