Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

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5another state-space representation of aerodynamic characteristics that included nonlinearterms and that was capable of describing static hysteresis. An additional state-spaceapproach that could do this with fewer model parameters is the one presented by DeOliveira and Lutze [32].Next, some functional unsteady aerodynamic models of more interest, selected among theones mentioned before, are described in more detail.1.1.1 <strong>Stability</strong> Derivatives ApproachIn this method, the problem of airplane dynamics is formulated and the equations for sixdegree-of-freedommotion are derived through Newton’s laws. Because these equationsare coupled and nonlinear, it is difficult to obtain analytical solutions. In view of this, it isassumed that the motion following a disturbance has small amplitudes in all the disturbedvariables. With this assumption, it is possible to linearize the equations of motion aboutthe chosen flight condition. If the airplane configuration allows the definition of a verticalplane of symmetry, it is possible to decouple the equations of motion into two sets: onefor the longitudinal motion and another for lateral-directional motion. Nevertheless, thereare cases where the decoupling is not possible, like for example, those whose massdistribution is not symmetric. Even in such cases, the linear equations of motion can beused in the stability analysis [33]. After linearization, it is assumed that the aerodynamicforces and moments in the disturbed state depend only on the instantaneous values ofmotion variables. This allows them to be evaluated by using the method of Taylor seriesexpansion. With these approximations, the equations of motion become linear in all themotion variables. The aerodynamic coefficients appearing in the Taylor series expansionare called stability and control derivatives.For flight conditions where massively separated flow conditions prevail, the linearizedmethods become insufficiently accurate for the analysis and more accurate aerodynamicmodels are required.
61.1.2 Indicial ApproachThe concept of indicial aerodynamic functions may be defined briefly as “theaerodynamic response of the airfoil as a function of time to an instantaneous change inone of the conditions determining the aerodynamic properties of the airfoil in a steadyflow” [13]. Recent developments and applications of the indicial approach to therepresentation of unsteady aerodynamics can be found in references [18], [19], and [22].A short summary based on these references follows.In this approach, the combined vector of the total aerodynamic coefficientsC r T= [ C C C C C C ]DYLlmnof an aircraft undergoing an arbitrary motion is theindicial response obtained in conjunction with the superposition principle [15], [16] orDuhamel’s integral theorem (Appendix A.1). This indicial response is given byr t= A ( t −τ ) hd&r τ(1-1)C∫0hiWhere = { }A is a matrix of indicial response functions for stepwise variations ofA C jkinematic parameters and control surfaces deflections combined in the vectorh r T= [ α β p q r δ δ δ ] . That means that, in this approach, the variationsaerwith time of the aircraft kinematic variables such as angle of attack and angular velocityare replaced by a large number of small instantaneous step changes. The transientaerodynamic reactions to these step changes are named “indicial functions”, and the totalresponse is obtained by their superposition.Even though this approach is considered a good representation of unsteady aerodynamics,it is difficult to combine it with the differential equations of motion. The indicial conceptwas originally conceived for linear time-invariant systems, but it has been generalized to
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 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
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55experimental results the values o
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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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