Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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21representing forward and aft fuselage. When an airplane rotates in the air, the air velocityis different at each surface, making the local angles of attack also different. Since thestate-space model is based upon characterizing flow separation as a function of angle ofattack, the work in this model is done using the local angle of attack and its timederivative. The local forces and moments can be calculated on each panel and, whencombined with the forces and moments on the remaining panels, the total forces andmoments for the whole aircraft are obtained. The goal is not only to characterize therolling moment, but also the forces and moments along and about the other axes. Theidea is to have one model capable of the prediction of lateral/directional characteristics aswell as the longitudinal ones, even in the high angle of attack, unsteady aerodynamicregion.Next, the description of the basic model for each of the single lifting surfaces in thisformulation is reproduced with its original notation.► Normal Force on Left and Right WingsThe aerodynamic normal force acting on the left and right wing is represented in thismodel by the following state-space formulation [30],[34],[39],[40] :τdxdtwiw1 + xwi= xw0wi−C= C( α τ & α )w2wi( x ) + C ( x )0i+ CNwα wi wi Nw & wiα&Nwi Nwαiα iwi(1-18)where subscript “w” represents quantities of the wing or the forward group of liftingsurfaces, and i = l or r, respectively, standing for the left or right panel. In the stateequation of (1-18) τw1is the wing panel relaxation time constant defining the transientaerodynamic effects and τ w2quantifies the time delay due to the quasi-steadyaerodynamic effects about the wing panel. The variablevariable associated with the nonlinear flow effects, andxx wi( α )is the wing panel statew 0 wiis the static dependency
22of the state variable on the local angle of attack, given by the sigmoid-type functiondefined byx( α )=1∗( −σ( α −α) ww0 wi1+exp(1-19)w wiFigure 1-6 Experimental and predicted C N responses for large amplitude pitchoscillations of a rectangular wing with NACA 0018 airfoil. Experimental data weretaken from [31]
Page 1 and 2: An Investigation of Unsteady Aerody
Page 3 and 4: iAcknowledgmentsAll that I struggle
Page 10 and 11: Figure 3-19 Roll angle time history
Page 12 and 13: 0( α )x static dependence function
Page 14 and 15: 11 IntroductionThe atmospheric flig
Page 16 and 17: 3aerodynamic forces in terms of the
Page 18 and 19: 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 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 70 and 71: 57► State equations:τdxi1,x+ xi=
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
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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 96 and 97:
Page 98 and 99:
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:
Page 120 and 121:
Page 122 and 123:
109q, r, the Euler angles time rate
Page 124 and 125:
1114.3 Results of the SimulationsTh
Page 126 and 127:
Figure 4-2 Phase-plane of the simul
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
121Considering these characteristic
Page 136 and 137:
123[10] Jones, R. T., “The Unstea
Page 138 and 139:
125[28] Goman, M. G., Stolyarov, G.
Page 140 and 141:
127[45] Nguyen, L. T., Yip L., Cham
Page 142 and 143:
129AppendicesAppendix AA.1 The Conv
Page 144 and 145:
131u = V cosθ 0v = Vsinθ0sinφw =
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Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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