hapter 2 eture 5 ongitudinal stik fied stati stability and ontrol 2 Topis 2.2 mg and mα as sum of the ontributions of various omponent 2.3 ontributions of ing to mg and mα 2.3.1 orretion to mα for effets of horizontal omponents of lift and drag seondary effet of ing loation on stati stability Eample 2.1 Eample 2.2 Eample 2.3 2.2 mg and mα epressed as sum of the ontributions of various omponents of the airplane Using ind tunnel tests on a model of an airplane or by omputational Fluid Dynamis (FD), the mg vs α urve for the entire airplane an be obtained. Hoever, FD has not yet advaned enough to give aurate values of the moments and these omputations are not inepensive. Wind tunnel tests are very epensive and are resorted to only at the later stages of airplane design. Hene, the usual pratie to obtain the mg vs α urve is to add the ontributions of major omponents of the airplane and at the same time take into aount the interferene effets. The ontributions of individual omponents are based on the ind tunnel data or the analyses available in literature. Referenes 1.1,1.8,1.9, 1.12, 2.1 and 2.2 are some of the soures of data. The ontributions to mg and mα are due to the ing, the fuselage, the poer plant and the horizontal tail. Figure 2.8 shos the fores and moments produed by the ing and the horizontal tail. The ontributions of fuselage, naelle and the poer plant are shon as moments about.g. and denoted by M f,n,p. The fuselage referene line is denoted by FR. It may be realled that the Dept. of Aerospae Engg., IIT Madras 1
angle of attak (α) of the airplane is the angle beteen free stream veloity (V) and FR. The.g. of the airplane is also shon in the figure. The ing is represented by its mean aerodynami hord (m.a..). It is set at an angle of inidene i to the FR. Hene, the angle of attak of ing (α ) is α + i. Folloing the usual pratie, the lift of the ing ( W ) is plaed at the aerodynami entre of the ing (a..) along ith a pithing moment (M a ). The drag of the ing (D ) is also taken to at at the aerodynami entre of the ing. The ing a.. is loated at a distane a from the leading edge of the m.a.. The airplane.g. is at a distane g from the leading edge of the m.a.. Fig.2.8 ontributions of major omponents to mg The horizontal tail is also represented by its mean aerodynami hord. The aerodynami entre of the tail is loated at a distane l t behind the.g. The tail is mounted at an angle i t ith respet to the FR. The lift, drag and pithing moment due to the tail are t, D t and M at respetively. As the air flos past the ing, it eperienes a donash ε hih is shon shematially in Fig.2.8. Oing to this the angle of attak of the horizontal tail ould be (α + i t - ε ). Further, due to the interferene effets the tail ould eperiene a dynami pressure different from the free stream dynami pressure. These aspets ill be Dept. of Aerospae Engg., IIT Madras 2
elaborated in setion 2.4.2 and 2.4.3. With this bakground the pithing moment about the.g. an be epressed as: M = (M ) + (M ) + (M ) + (M ) + (M ) (2.11) g g g f g n g p g t M = = ( ) + ( ) + ( ) 1 ρvs 2 g mg mg mg f,n,p mg t 2 (2.12) mα = ( mα) + ( mα ) f,n,p+ ( mα ) t (2.13) Note: (i) For onveniene the derivative of mg ith α is denoted as mα. (ii) In Fig.2.8 the angle i t is shon positive for the sake of indiating the notation; generally i t is negative. The ontributions to and mα of the individual omponents are desribed in mg the net four setions. 2.3 ontributions of ing to mg and mα Figure 2.9 shematially shos the fores (lift, and drag, D ) and the moment (M a ) due to the ing and the relative loations of the.g. of the airplane and the aerodynami entre of the ing. The folloing may be realled / noted. i) The angle of attak of the airplane is the angle beteen the relative ind and the fuselage referene line (FR). This angle is denoted by α. ii) The ing is represented by its mean aerodynami hord (m.a..). iii) The ing is set at an angle i to the FR. This is done so that the fuselage is horizontal during ruising flight. Thus, α = α + i or α = α i. iv) a is the distane of the a.. from the leading edge of the m.a... v) g is the distane of the.g. from the leading edge of the m.a... vi) Z g is the distane of the a.. belo.g. Dept. of Aerospae Engg., IIT Madras 3
Fig.2.9 Wing ontribution Taking moment about.g., gives the ontribution of ing (M g ) to the moment about.g as: M = os(α -i )[ - ]+Dsin(α - i ) [ - ] g g a g a Noting that, +sin(α- i )Zg - Dos(α -i )Z g +M a (2.14) Mg D Ma mg= ; = ; D= ; ma=, (2.15) 1 2 1 2 1 2 1 2 ρv S ρv S ρv S ρv S 2 2 2 2 yields: g a g a mg = os(α-i )[ - ]+Dsin(α-i )[ - ] Z Z + sin(α -i ) - os(α -i ) + g g D ma Remark: (α i ) is generally less than 10 0.Hene, os (α i ) 1; and sin(α i ) (α i ). Further >> D. Negleting the produts of small quantities, Eq.(2.16) redues to: (2.16) Dept. of Aerospae Engg., IIT Madras 4
g a mg = ma + [ - ] No, = (α - α ) α 0 α 0 = (α +i -α ) = (i -α )+ α α 0 α (2.17) = + α (2.18) 0 α here, α 0 is the zero lift angle of the ing and 0 = α(i - α 0) Hene, g a g a mg = ma + 0[ - ] + α α [ - ] Differentiating ith respet to α, gives the ontribution of ing to mα as : g a mα = α[ - ] Remark: (2.19) (2.20) The ontribution of ing ( mg ) as approimately alulated above and given by Eq.(2.19) is linear ith α. When the a.. is ahead of.g., the term positive and onsequently mα is positive (Eq.2.20). Sine, mα should be negative for stati stability, a positive ontribution to g a [ - ] mα is alled destabilizing ontribution. When the a.. is ahead of.g. the ing ontribution is destabilizing. Figure 2.10 shos mg vs α in this ase. is Dept. of Aerospae Engg., IIT Madras 5
Fig.2.10 Approimate ontribution of ing to mg 2.3.1 orretion to mα for effets of horizontal omponents of lift and drag seondary effet of ing loation on stati stability In the simplified analysis for the ontribution of ing to mg, the ontributions of the horizontal omponents of lift and drag to the moment about.g., have been ignored (ompare Eqs. 2.16 and 2.17).et, the negleted terms be denoted by M gh. Equation (2.14) gives the folloing epression for M mgh M gh =sin(α- i )Zg - Dos(α - i )Z g (2.21) Dividing by ½ ρv 2 S and noting that os α -i 1yields : Zg mgh = [sin(α -i ) - D ] ; (2.22) DifferentiatingEq.(2.22) ith α gives: d d Z dα dα D mαh = [ sin(α -i )+os(α -i )- ] d No, sin(α-i ) α(α-i ) dα g (2.23) Dept. of Aerospae Engg., IIT Madras 6
α(α-i ) = α(α- α 0) - α(i - α 0 )= - 0 Further, os(α -i ) d d d d and = = dα d dα d D D D α d Z D Thus, = [2 - - ] mαh 0 α d The drag polar of the ing an be assumed as : g (2.24) (2.25) 2 D = D0 + πae, d d D Then, = 2 πae Substituting this in Eq.(2.25) yields: 2 = [2 - - ] mαh 0 α πae Z α = [2 {1 - } - ] m h 0 πae Z g The term [1 - (2 α / π Ae)] is generally positive. This an be seen as follos. An approimate epression for α is: α Hene, g A = 2π ; A = Aspet ratio of ing. A+2 (2.26) α A 1 2 =2π = πae A+2 πae (A+2)e (2.27) 2/{(A+2)e} is less than 1 for typial values of A and e. Further, for lo ing airraft, here the a. of the ing is belo.g., the term Z g / is positive (Fig.2.9). Hene, mαh as given by Eq.(2.26) is positive or destabilizing (Fig.2.11). For high ing airraft, Z g / is negative onsequently mαh is negative and hene stabilizing (Fig.2.11). Dept. of Aerospae Engg., IIT Madras 7
Fig.2.11 Effet of ing loation on mg An important aspet of the above derivation may be pointed out here. The epression for mαh involves or the slope of mg vs α urve depends on or α (see eample 2.3). Hene, mg beome slightly non-linear. The usual pratie, is to ignore the ontributions of the horizontal omponents to mα. Hoever, the folloing aspets may be pointed out. (a) A high ing onfiguration is slightly more stable than a mid-ing onfiguration. A lo ing onfiguration is slightly less stable than the mid-ing onfiguration. (b) In the simpler analysis the mg vs α urve is treated as straight line but the mg vs α urves, obtained from flight tests on airplanes, are found to be slightly non-linear. One of the reasons for the non-linearity in atual urves is the term M egh. Dept. of Aerospae Engg., IIT Madras 8
Eample 2.1 Given a retangular ing of aspet ratio 6 and area 55.8 m 2. The ing setion employed is an NAA 4412 airfoil ith aerodynami entre at 0.24 and ma = -0.088.The.g. of the ing lies on the ing hord, but 15 m ahead of the a.. alulate the folloing. (a) The lift oeffiient for hih the ing ould be in equilibrium ( mg = 0). Is this lift oeffiient useful? Is the equilibrium statially stable? (b) alulate the position of.g. for equilibrium at = 0.4. Is this equilibrium statially stable? Solution: The given data for the ing are : A = 6, S = 55.8m 2, Airfoil: NAA 4412; a.. at 0.24, ma = -0.088 Before solving the problem e orkout the additional data needed for the solution. (d l /dα) or lα or a 0 of the given airfoil: From Ref.1.7 p.484 a 0 is 0.106/degree For a 0 = 0.106 and A = 6, from Fig.5.5 of Ref.1.7, α = 0.081/degree. Note: Using α = (A/A+2) lα, e ould get: α = {6/(6+2)}(0.106) = 0.0795 deg -1 For a retangular ing, =S/b Further A = b 2 / S. Hene, b = (AS) 1/2 = (655.8) 1/2 = 18.30 m onsequently, = 55.8/18.3 = 3.05 m. Hene, a = 0.243.05 = 0.732 m, g = 0.732-0.15 = 0.582 m The onfiguration is shon in Fig.E2.1 Dept. of Aerospae Engg., IIT Madras 9
(a) For equilibrium Fig.E2.1 onfiguration for eample 2.1 - W = 0 ; M g = - 0.15 + ½ ρ V 2 S ma = 0 Or 0.15 + (- 0.088) = 0 Hene, = - 0.088 3.05/0.15 = - 1.77 This lift oeffiient is not useful. The equilibrium is stable as.g. is ahead of a.. (b) alulation of.g. loation for moment equilibrium at = 0.4 = 0.4 ( - ) + (- 0.088) = 0 mg g a 0.088 g- a = 3.05 + = 0.671m 0.4 0.671 g a or = + = (0.24 + 0.22) = 0.46 This equilibrium is unstable as a.. is ahead of.g. Eample 2.2 If the ing given eample 2.1 is rebuilt maintaining the same planform, but using refle ambered airfoil setion suh that ma = 0.02, ith the a.. still at 0.24. alulate the.g. position for equilibrium at = 0.4. Is this equilibrium statially stable? Dept. of Aerospae Engg., IIT Madras 10
Solution: For equilibrium at = 0.4 ith ma = 0.02; = 0.4 ( - ) + (0.02) = 0 mg g a 0.02 g a - = - 3.05 = -0.1525 m 0.4 0.1525 g =0.24- =0.19 3.05 Equilibrium is stable as.g. is ahead of a.. Remark: From the above to eamples e dra interesting onlusions about an airplane hih has an all ing onfiguration. (a) For suh a onfiguration, the stati stability onsideration requires that.g. should be ahead of a... (b) ma should be positive. Eample 2.3 An airplane is equipped ith a ing of aspet ratio 6 ( lα = 0.095) and span effiieny fator e of 0.78, ith an airfoil setion giving ma = 0.02. alulate, for beteen 0 and 1.2, the pithing moment oeffiient of the ing about the.g. hih is loated 0.05 ahead of a.. and 0.06 under a... Repeat the alulations hen hord ise fore omponent is negleted. Assume D0 = 0.008, α o = 1 0, i = 5 0. Solution: The given data about the ing are: A = 6, lα = 0.095, e = 0.78, ma = 0.02, α o = 1 0, D0 = 0.008, i = 5 0, From Fig.5.5 of Ref.1.7, α = 0.074 deg -1 = 4.24 rad -1 0 = 0.074 (5-1) = 0.296. 2 2 D = 0.008 + = 0.008 + = 0.008 + 0.068 πa e 3.14 6 0.78 2 Dept. of Aerospae Engg., IIT Madras 11
Fig.E2.3 Shemati of onfiguration for eample 2.3 ombining Eqs.(2.20) and (2.26), (g- a ) Z α g mα = α+[2 {1- } - 0] ; πae 4.24 = - 0.05 4.24 + [2 {1- } - 0.296] (- 0.06) 3.14 6 0.78 = - 0.212-0.0854 + 0.0178 = - 0.1942-0.0854 Hene, mg = 0.02 + ( - 0.1942-0.0854 )α = 0.02 + (- 0.1942-0.0854 ) {( - o ) / 4.24} 2 = 0.0336-0.0399-0.0201 The values of mg for different values of are presented in table E2.3. The approimate ontribution of ing after negleting the horizontal omponent from Eq.(2.17) is : g a mg= ma+ [ - ] or ( mg ) approimate = 0.02-0.05. These values are also inluded in table E2.3. Dept. of Aerospae Engg., IIT Madras 12
( mg ) ithout horizontal omponent ( mg ) ith horizontal omponent 0 0.02 0.0336 0.4 0 0.0141 0.8-0.02-0.0112 1.2-0.04-0.04314 Table E2.3 ontribution of ing to mg Remark: The.g. is ahead of a., hene the ontribution of ing, even ithout onsidering horizontal omponent, is stabilizing. Further the.g. is belo a.. hene the ontribution, onsidering the horizontal omponent, beomes more stabilizing. Dept. of Aerospae Engg., IIT Madras 13