Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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57► State equations:τdxi1,x+ xi= x0i−dt( α τ α )& 2, α i(2-2)► Output equations:CN= CN, l+ CN, r(2-3)Cl= CN lylb− CN ryrb= CN ly − ClN ryr,(2-4)whereCN i= CN+ CNα2( xi) αiC ( xi) αiC ( xi) ˆ αiC ( xi) ˆ 2+αiC ( xi) α ˆ2 + ˆ& + ˆ&ˆ&i i02Nα+N & αN & αNα& αα(2-5)with i = l (left), r (right) panel, and & ˆ α = & α tˆ .The spanwise positions y l,yrof the normal force point of application, the sigmoidlocation angle of attack∗αand the slope σ are made functions of the roll angle. Theremaining functions and parameters for this formulation are:
581x0 ( αi) =1+exp(2-6)N α( −σ( α −α∗)2( x ) = a + b x c xC +N αi112( x ) = a + b x c xC +22 ii i2( x ) = a + b x c xC +2ii12iN α& ˆ i 3 3 i 3 i(2-7)N α& ˆ2( x ) = a + b x c xC +N α ˆ & α42 ii i2( x ) = a + b x c xC +i545i45iThe parameters of this formulation that must be identified are:- the polynomial coefficients C N0 , a i , b i , c i of Eqs. (2-7);- transient time-constant τ1, xand time-delay constant τ 2 , α of Eq. (2-2), functions of thewing pitch angle θ0at which the dynamic experimental data are obtained;∗- the parameters α and σ of the logistic equation (2-6) that represent the static positionof the vortex breakdown over the panel. They are left free to vary with the roll angle atwhich the static experimental data was obtained and, therefore, there are one of each ofthese parameters for each static experimental value of the roll angle;- the normal force arms y , y , which are also varying with the roll angle at which thestatic experimental data are obtained.lrThe identification of these parameters is described in further detail in Section 3.3.As shown in Figures 3-4 to 3-9, the responses obtained for this, the first modelinvestigated did not match the experimental data very well. Because of that, a secondunsteady aerodynamic model was developed. Next, we present its description.
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An Investigation of Unsteady Aerody
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iAcknowledgmentsAll that I struggle
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Figure 3-19 Roll angle time history
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0( α )x static dependence function
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11 IntroductionThe atmospheric flig
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3aerodynamic forces in terms of the
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5another state-space representation
Page 20 and 21: 7a nonlinear indicial response theo
Page 22 and 23: 9Figure 1-2 Internal state-space va
Page 24 and 25: 11representation, we write it expli
Page 26 and 27: 13C& α cV() t = C ( α ) + C C ()
Page 28 and 29: 15(a)(b)*Figure 1-3 Influence of th
Page 30 and 31: 17( α )1x0 U eff=1+exp(1-13)[ −
Page 32 and 33: 19tˆ =c2V, (1-17)c being a charact
Page 34 and 35: 21representing forward and aft fuse
Page 36 and 37: 23In equation (1-19)∗αwdescribes
Page 38 and 39: 25Ntiα2( x ) = a + b x c xC +tit1
Page 40 and 41: 27The rolling moment of the aircraf
Page 42 and 43: 29are the A-4 Skyhawk, F-4 Phantom,
Page 44 and 45: 31Conventional wing rock is that oc
Page 46 and 47: 33Figure 1-8 Crossflow streamlines
Page 48 and 49: 35possible cases forC φ, where the
Page 50 and 51: 37Figure 1-10 Roll angle time-histo
Page 52 and 53: 39Figure 1-14 Rolling moment coeffi
Page 54 and 55: 41Figure 1-15 C l vs. roll angle hi
Page 56 and 57: 430.020-0.02C l-0.04-0.06-0.08φ =
Page 58 and 59: 451.2.3 Analytical and Computationa
Page 60 and 61: 47vertical fin inclusion, the oscil
Page 62 and 63: 49Table 1-1 Geometrical and physica
Page 64 and 65: 51Figure 1-22 Free to roll apparatu
Page 66 and 67: 53Both of these problems make it di
Page 68 and 69: 55experimental results the values o
Page 72 and 73: 592.2 The Second Unsteady Aerodynam
Page 74 and 75: 61With the help of the formulation
Page 76 and 77: 63tunnel tests results shown in Fig
Page 78 and 79: 65C2( ν ) = a + b ν c νN i 6 6 i
Page 80 and 81: 67Figure 2-2 Static model responses
Page 82 and 83: 69Figure 2-4 Variation of the norma
Page 84 and 85: 71When the quasi-static sequences o
Page 86 and 87: 73identification. The static data u
Page 88 and 89: 752( x ) - 7.293 x - 5.427 xCN i= 0
Page 90 and 91: 77Figure 3-1 Static normal force co
Page 92 and 93: Figure 3-3 Identified static values
Page 94 and 95: Figure 3-5 Rolling moment coefficie
Page 100 and 101: 87with the roll angle, and attains
Page 102 and 103: 890.060.04θ = 20 degModel response
Page 104 and 105: 9140θ = 27 deg20φ, deg0-20-4015.6
Page 106 and 107: 93C l0.150.1θ = 27 degModel respon
Page 108 and 109: 95C l0.150.1θ = 38 degModel respon
Page 110 and 111: 97100θ = 45 deg50φ, deg0-50-1000
Page 112 and 113: 990.060.04θ = 45 degModel response
Page 114 and 115: 101Table 3-1 Model parameters for t
Page 116 and 117: 103Table 3-3 Continuation of Table
Page 118 and 119: 105Table 3-6 Continuation of Table
Page 120 and 121:
107Table 3-9 Continuation of Table
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109q, r, the Euler angles time rate
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1114.3 Results of the SimulationsTh
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Figure 4-2 Phase-plane of the simul
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121Considering these characteristic
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123[10] Jones, R. T., “The Unstea
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125[28] Goman, M. G., Stolyarov, G.
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127[45] Nguyen, L. T., Yip L., Cham
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129AppendicesAppendix AA.1 The Conv
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131u = V cosθ 0v = Vsinθ0sinφw =
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Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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