Chapter 2 Lecture 9 Longitudinal stick fixed static stability and control 6 Topics

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1 hapter Leture 9 Longitudinal stik fied stati stability and ontrol 6 Topis Eample.4 Eample.4 Referene.4 desribes the stability and ontrol data for ten airplanes. This inludes a general aviation airplane alled Navion. It seems an appropriate ase to illustrate the stati stability, dynami stability and response of an airplane without the ompliations of ompressibility effets. This airplane is dealt with in this hapter and also in hapters 8,9 and 10. The three-view drawing of the airplane is shown in Fig..9. The geometrial and aerodynami data and the flight ondition are given below. Some additional data given in Ref.1.1, hapter are also inluded therein. Remaining data are dedued by measuring dimensions from the three-view drawing. Though referenes 1.1 and.4 use FPS units, data are onverted to SI units for the sake of uniformity. Fig..9 Three views of a general aviation airplane (Adapted from Ref..4, setion 10) Dept. of Aerospae Engg., IIT Madras 1

2 Wing: Area (S) = m, Span (b) = m, Root hord ( r ) =.16 m, Tip hord ( t ) = 1.1 m, Taper ratio (λ) = 0.56, Aspet ratio (A w ) = 6.06, Mean aerodynami hord ( ) = m, i w = 1 0 (Ref.1.1, hapter ): harateristis of airfoil used on wing (dedued from Ref.1.1, hapter ): ma = , lαw = deg = 5.56 rad, α olw = - 6 0, a.. loation = 0.5. Fuselage: Length (l f ) = 8.3 m, Width of fuselage at maimum ross setion = 1.4 m, Height of fuselage at maimum ross setion = 1.6 m, The widths at different loations along the length of fuselage are shown in Tables E.4.1 & E.4.. Horizontal tail: Area (S t ) = 4.73 m, Span (b t ) = 4.01 m, Root hord ( rt ) = 1.54 m, Tip hord ( rt ) = 0.8 m. Aspet ratio of tail (A t )= 3.4, Distane between quarter hord of the mean aerodynami hords of wing and tail = 4.63 m, Distane l h as shown in Fig.. is 3.17 m. harateristis of the airfoil used on tail (dedued from Ref.1.1, hapter ): ma = 0, lαt = 0.1 deg = 5.73 rad, i t = 0. Flight ondition: Weight = 13.6 N. Altitude : sea level, ρ = 1.5 kg/m 3, speed of sound = m/s. flight veloity = m/s ; Mah no. (M) = W 13.6 Lift oeffiient = L = = = 0.406; 1 ρv S Ref..4 gives : L = 0.41,.g. loation: Dept. of Aerospae Engg., IIT Madras

3 ( Lα ) airplane = 4.44, ( mα ) airplane = Obtain : (i) ontributions of wing, horizontal tail, fuselage and power plant to the moment about.g. (ii) mα of the airplane (iii) loation of the neutral point and (iv) stati margin. Solution: i) Slopes of lift urve for wing, tail and airplane From Ref.1.8b, the slope of lift urve for unswept wing at low subsoni Mah number is given by : πa lα Lα = ; K = π A + +4 K where, lα is the lift urve slope of the airfoil. For wing: π 6.06 Lαw = = 4.17rad (5.56/6.8) For horizontal tail: π 3.4 Lαt = = 3.43rad (5.73/6.8) The slope of lift urve of the airplane ( Lα ) is obtained using Eq.(.60b): St Lα = Lαw + η Lαt (1- ) S η = 0.9 is assumed. / is estimated by the approimate method i.e πa Lαw = = = Hene, w 4.73 Lα = ( ) = 4.65rad This estimated value of Lα is only 4.7%higher than the atual values of 4.44 rad given in Ref..4. Thus, the values of Lαw, Lαt, / and η are onsidered to be reasonably aurate (see also Appendi ). Dept. of Aerospae Engg., IIT Madras 3

4 II) Wing ontribution: Following the simplified approah, the wing ontributions to m0 and mα are obtained from Eqs.(.19) and (.0): g a m0w = maw + L0w( - ) g a mαw = Lαw( - ) {1-(-6)} Low = Lαw(iw - α 0Lw ) = 4.17 = g a mow = maw + Low - = ( ) = mαw = Lαw - = 4.17 ( ) = rad Remark: g a For this partiular airplane and for the given onfiguration, the wing ontribution to mα is positive or destabilizing (Note:.g. is behind a..). III) Horizontal tail ontribution: The tail ontributions to m0 and mα are obtained from the following equations: = η V (i - ε ) m0t H Lαt t 0 ( mαt ) stik-fied = - η VH Lαt (1- ) The tail volume ratio is given by: St l t V H = = = S As estimated earlier : / = w 0lw ε = (i -α ) = 0.438{1-(-6)} = 3.07 mαt = - VH η Lαt(1- ) = ( ) =.8 rad m0t = - VH η Lt (it -ε 0) = ( ) = Dept. of Aerospae Engg., IIT Madras 4

5 IV) Fuselage ontribution: The ontributions of fuselage to m0 and mα are obtained using the method eplained in setion.5.3 and.5.4. To obtain mof we divide the fuselage into nine equal divisions as shown in Fig..30. Fig..30 Subdivisions of fuselage for alulating mof Table E.4.1 presents Δ and w f at various stations along the fuselage. The quantity α 0Lf is i w + α 0Lw whih equals 1-6 = As the fuselage has no amber i f is taken as zero. Hene, α 0Lf + i f equals -5 0.The quantity w f (α olf + i f )Δ is given in the last olumn of the table E.4.1. The sum w(α + i )Δ is f 0Lf f Station Δ (m) w f (m) α 0Lf + i f w f (α 0Lf + i f ) Δ i f = 0 at every station Sum= Table E.4.1 Estimation of m0f Dept. of Aerospae Engg., IIT Madras 5

6 To obtain the term (k -k 1 ) from Fig..19, requires the fineness ratio of the fuselage whih is obtained below. The area of the maimum fuselage ross setion (A fma ) is : A fma = =.4 m Hene, equivalent diameter (d e ) is: d = A /(π/4) =.4/(π/4) = 1.69m e fma onsequently, fineness ratio = l f /d e = 8.3/1.69 = From Fig..19, (k -k 1 ) orresponding to fineness ratio of 4.87 is 0.8. Substituting various values, m0f is given as: lf k-k1 0.8 ( ) m0f f 0Lf f 36.5S =0 = w (α +i )Δ = = To obtain mαf the fuselage is subdivided as shown in Fig..31. The portion of the fuselage ahead of the root hord is divided into four equidistant portions eah of length m. These subdivisions are denoted as 1,, 3 and 4. The portion of fuselage aft of the root hord is divided into five equidistant setions eah of length m and denoted as 5,6,7,8 and 9. The root hord (Fig..31) has length = 1.98 m. Thus, the total fuselage length of 8.3 m is thus divided as: ( ). The length l h as shown in Fig.. is the distane of the aerodynami entre of horizontal tail behind the root hord of the wing. It is 3.17m. The alulations of the quantities needed to obtain mαf are shown in Table E.4.. The seond olumn shows Δ whih is the length of eah subdivision of the fuselage. The third olumn gives the width of the fuselage in the middle of the subsetion (see Fig..). The fourth olumn gives the distane for the setion 4 as defined in Fig... For rows 3, and 1 of this olumn the distane is i is as defined in Fig... For rows 5 to 9 of this olumn the distane i is as shown in Fig... The fifth olumn shows / for the fourth row and i / for other rows. The sith olumn is / the upwash and downwash at the subdivision. For row four the upwash value is based on urve b of Fig..3. For rows 3, and 1 the upwash value is based on urve a of Fig..3. Dept. of Aerospae Engg., IIT Madras 6

7 Fig..31 Subdivisions of fuselage for estimating mαf Station Δ (m) w f (m) i or ( i or )/ /* w f (/)Δ =1.98m l h =3.17m Sum = *For LαW = /deg. See Remarks (ii) at the end of setion.5.4 Table E.4. Estimation of mαf Dept. of Aerospae Engg., IIT Madras 7

8 The rows 5 to 9 of this olumn show the downwash for the orresponding subdivisions. As given in Fig.. and by Eq.(.58), / at the subdivisions behind the root hord is given by: i = [1 - ( ) tail] lh We note that / at tail is for this airplane. Using values of i and l h the values of downwash are tabulated in olumn 6. The last olumn shows values of w f (/)Δ. The sum, Σw f (/)Δ is Sine, LαW = 4.17/rad = 0.078/degree, the atual value of the sum is (see Remark (ii) at the end of setion.5.4): (0.078/0.0785) = Finally, lf k-k f = mαf = w Δ = = 0.1 rad 36.5S V) ontribution of power plant: It is diffiult to estimate this ontribution aurately. As mentioned in Remark (ii) in setion.6., this ontribution is taken as 0.04 Lα = = VI) m0 and mα The ontributions to m0 and mα from the wing, the horizontal tail, the fuselage and the power plant are shown in Table E.4.3. The values of m0 and mα for the entire airplane are the sums of the values for the omponents. These are also shown in Table E.4.3. Item m0 mα Wing Fuselage Power H.tail Airplane Table E.4.3 m0 and mα due to omponents and for the entire airplane Dept. of Aerospae Engg., IIT Madras 8

9 Figure.7 shows the ontributions graphially. mg and mα for the airplane are: mg = α α α α = α Hene, ( mα ) stik-fied = It is very interesting to note that the value of mα given in Ref..4 is Thus the estimates of the ontributions of various omponents to stati stability an be onsidered to be reasonably aurate. VII) Neutral point loation: The neutral point is given by: 1 NP a = - {( m ) f,n,p - V H η L t(1- )} L w Substituting various values, NP is given as: NP 1 = { } 4.17 = = VIII) The stati margin: The stati margin when.g. is at 0.95 is : NP - g = = Hene, (d m / d L ) = -(stati margin) = mα = Lαw (d m / d L ) = 4.17 ( ) = as it should be. Dept. of Aerospae Engg., IIT Madras 9