Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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9Figure 1-2 Internal state-space variable in predominantly vortical flows.Taking into account the above features, a mathematical model was proposed [28],[29]where the internal dynamical variables (vector x r ) approximately describes the state ofseparated and vortex flow about an aircraft. These variables are additional informationrequired at a given instant of time to calculate the outputs (aerodynamic forces and= C C C C C C ) from the systemmoments, given by the vector C r [ ] TDinputs (motion variables and surfaces deflections, given by the vectorh r T= [ α β p q r δ δ δ ] ). The unsteady aerodynamic state-space approachis then given by a dynamical system (1-2)aerYLlmnrdxdt=fr r( x,h)r r( x,h)rC = g(1-2)To obtain the simplest mathematical model of this kind, Goman and Khrabrov [29] firstconsidered the flow about a wing in pitching motion only. Then, the first equation of thedynamical system (1-2) was defined as:
τ x() t x() t = x ( α() t τ & α()t )&1 0−2+ (1-3)10This definition was taken with two groups of unsteady fluid mechanics processes inmind. The first group concerns the different quasi-steady aerodynamic effects that havetime-delays, such as circulation and boundary-layer convection lags. Since the resultingdelay is approximately proportional to the variation of the angle of attack α& , the quasisteadyvalue of the internal state variable can be expressed through the function ( α )means of argument shift ( α τ & α )x0−2, where 2x byτ defines the total time delay associatedwith the above-mentioned effects. The second group of fluid mechanics processes definestransient aerodynamic effects such as the dynamics of the flow adjustment to any changein the angle of attack. This dynamics can be described by the first-order differentialequation (1-3), where τ1is the transient time-constant.0The steady state position ( α )x of the separation point is generally a nonlinear function0of the angle of attack. It can be obtained from static wind tunnel measurements, but, inorder to be applicable for identification purposes, a mathematical relation would be moresuitable.In references [30],[34], Fan and Lutze started from the state-space model as proposed byGoman and Khrabrov [29], and further developed it to facilitate the identificationprocess. Their first improvement was to propose the logistic or sigmoid function (1-4)x .shown below as the representation of ( α )0x0( α () t )effΔ=1+exp1( σ ( α () t − α∗)eff(1-4)where α () t α() t −τ& α()teff∗= , α is the angle of attack at which the flow separation is at2the mid-chord point, and σ is the slope factor. Parameters∗αand σ are expected to beidentified from wind tunnel experimental data. Figure 1-3 shows the influence of theseparameters in the shape of the sigmoids for negative (a) and positive (b) values of σ .Because in the current work the sign of σ plays such an important role for the wing rock
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 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 70 and 71: 57► State equations:τdxi1,x+ xi=
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592.2 The Second Unsteady Aerodynam
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61With the help of the formulation
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63tunnel tests results shown in Fig
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65C2( ν ) = a + b ν c νN i 6 6 i
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67Figure 2-2 Static model responses
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69Figure 2-4 Variation of the norma
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71When the quasi-static sequences o
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73identification. The static data u
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752( x ) - 7.293 x - 5.427 xCN i= 0
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77Figure 3-1 Static normal force co
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Figure 3-3 Identified static values
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Figure 3-5 Rolling moment coefficie
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87with the roll angle, and attains
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890.060.04θ = 20 degModel response
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9140θ = 27 deg20φ, deg0-20-4015.6
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93C l0.150.1θ = 27 degModel respon
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95C l0.150.1θ = 38 degModel respon
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97100θ = 45 deg50φ, deg0-50-1000
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990.060.04θ = 45 degModel response
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101Table 3-1 Model parameters for t
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103Table 3-3 Continuation of Table
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Page 120 and 121:
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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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