3. Aerodynamic coefficients

The motion of aircraft can be described by a system of equations describing the motion of center of gravity of aircraft and its rotation around it [13–15]. The general form of the system contains stochastic, partial nonlinear differential equations with delays.

So, this motion can be defined by solving the inertial equations that should be coupled with the equations describing the aerodynamic (gas dynamic) and elastic phenomena (see central part of Figure 1) [9, 11, 12]. The latter equations are coupled through the aircraft shape and structure. The time-dependent aerodynamic equations describe the instantaneous aerodynamic effects on the aircraft assumed in the form of aerodynamic forces and moments depending on the state of the flow field surrounding the aircraft, the motion variables, the aircraft shape deformation, and the initial conditions.

In the first approximation, the instantaneous aerodynamic force depends on air density, r, and velocity, V, and mean geometrical parameter of the body, namely wing span, s. That by use of methods of dimensional analysis can be represented in the form:

$$F = \mathbb{C}\rho^{\alpha}V^{\beta}\mathbb{S}^{\gamma},\tag{1}$$

where C is the coefficient. The exponents α, β, and γ must be derived from the condition that the dimension of the different sides of equation should be equal. Using the results α = 1, β = 2, and γ = 2, the aerodynamic force can be calculated as:

$$F = \text{C} \rho^1 V^2 \text{S}^2 = 2 \text{C} \text{AR} \frac{\rho V^2}{2} \frac{\text{S}^2}{\text{AR}} = \text{C}\_F \frac{\rho V^2}{2} \text{S},\tag{2}$$

where cF is the so-called nondimensional force coefficient, AR is the aspect ratio (AR <sup>¼</sup> <sup>s</sup><sup>2</sup>=S), and S is the wing area. From here, the aerodynamic force and moment coefficients are:

$$\mathbf{C}\_{F} = \frac{F}{\frac{\rho V^{2}}{2}\mathbf{S}}, \qquad \mathbf{C}\_{M} = \frac{M}{\frac{\rho V^{2}}{2}\mathbf{S}\mathbf{c}\_{a}}.\tag{3}$$

Here Ca is the aerodynamic chord.

the CFD methods, (iv) optimizing shape for cruise flight mode, and (v) studying the most

Figure 6. The practical measurements: (a) An-225 Mria and space shuttle group model in the wind tunnel at the TsAGI [30], (b) lift coefficient distribution along the wing span (b/2) of deformed (left side) and nondeformed (right hand) wing

Figure 6b shows how the real lift distribution depends on the flight conditions, namely how the deformation of wing deformed under loads has influence on the actual lift distribution.

Generally, the differences in calculated and measured wind tunnel lift coefficient reach 7–8%, while, for example, the differences between the measured wind tunnel and flight test drag coefficient equal to 5–10% and up to 18% at the transition period from subsonic to supersonic flights [32]. During the periodic angle of attack oscillation of the wing, there is a large hysteresis in the lift coefficient—angle of attack function. So, there are considerable differences in

These thoughts on aerodynamic force and moment generation demonstrate that the theoretical calculation and the practical measurements cannot independently provide full and correct description for aerodynamic forces and moments. At first, the semiempirical methods were developed and applied for aircraft aerodynamic design and calculation of the aerodynamic characteristics [33–37]. Later, with gaining in prestige of CFD, the role of modeling of aerody-

The motion of aircraft can be described by a system of equations describing the motion of center of gravity of aircraft and its rotation around it [13–15]. The general form of the system

So, this motion can be defined by solving the inertial equations that should be coupled with the equations describing the aerodynamic (gas dynamic) and elastic phenomena (see central part of Figure 1) [9, 11, 12]. The latter equations are coupled through the aircraft shape and structure. The time-dependent aerodynamic equations describe the instantaneous aerodynamic effects on the aircraft assumed in the form of aerodynamic forces and moments

contains stochastic, partial nonlinear differential equations with delays.

dangerous flight mode, aircraft approach and landing.

a.) b.)

164 Flight Physics - Models, Techniques and Technologies

steady and unsteady regime.

[31].

namic coefficient increased.

3. Aerodynamic coefficients

The aerodynamic forces and moments, as well as their aerodynamic coefficients, can be represented by their components:

$$\mathbf{C}\_{\mathbf{F}} = \left[ \mathbf{C}\_{x} (= \mathbf{C}\_{D}), \mathbf{C}\_{y}, \mathbf{C}\_{z} (= \mathbf{C}\_{L}) \right]^{T}, \qquad \mathbf{C}\_{M} = \left[ \mathbf{C}\_{l}, \mathbf{C}\_{m}, \mathbf{C}\_{n} \right]^{T} \tag{4}$$

according to the axes of the applied reference system (here wind system) [1, 14].

The nondimensional aerodynamic coefficients fully describe the aircraft aerodynamics [1–8]. The basic aerodynamic characteristics of airfoils [38, 39] are shown in Figure 7. The left first figure shows the typical changes in lift coefficient with increase in angle of attack that begins with linear function, followed by nonlinear form at high angle of attack and dropping after separating the flow from the upper surface of aerofoil at the so-called critical angle of attack.

Figure 7. Aerodynamic characteristics of an airfoil: lift, drag, and moment coefficients as function of angle of attack, polar curve, and "goodness" factor.

This phenomenon is called stall. The drag coefficient increases with growing lift coefficient due to the induced drag. The moment coefficient follows the changes in lift and drag coefficients, especially after stall. The polar curve (CL ¼ f Cð Þ <sup>D</sup> ) and "goodness" factor (k ¼ ðCL=CD ¼ fð ÞÞ ∝ ) explain the relatively low angle of attack and must be realized during the most important flight regime, during the cruise flight for having minimum drag, minimum required thrust, and minimum fuel consumption.

As it is well known, the subsonic and supersonic aerodynamics is principally different. The "classic" airfoils with blunt leading edge cannot be applied, because their drag tends to the infinity nearing to Mach number (velocity related to the sound speed in the same condition) equals to one. At the supersonic speed only, the airfoils (wing and fuselage) with sharp leading edge can be applied (Figure 8a).

Figure 8b demonstrates how changes in some parameters may radically affect the aerodynamic characteristics. In case of high aspect ratio wing, the vortex generating the lift as a vortex tube along the wing span separates at the wing tips and causes the induced drag. The low aspect ratio delta wing has unique aerodynamic picture. The flow separating from the wing leading edge and the caused by this separated flow vortexes moving back on the top of the wing generate extra lift at high angle of attack, and the stall appears at 45–70�, only.

The aerodynamic coefficients depending on the flight modes and flight maneuvers are managed by use of control surfaces and motion devices as flaps, slots, and generally by all the devices deviating and changing the geometry like undercarriage system, braking parachutes, etc. [1–8]. For example, the flaps at wing trailing edge and slat at the leading edge (making slots between the slat and mean wing) are used for increasing the lift (and drag) allowing to reduce the take-off and landing speeds for making safer flight modes. The flaps increase the lift coefficients ("moving" the lift coefficient and angle of attack curve left and up in Figure 9a), while the slats/slots increase the critical angle of attack, (because they do not change the airfoil/ wing chamber). In Figure 9a, at low lift coefficient region, the effect of a special leading edge flap, called Krueger flap, is shown, too.

The flaps are deflected on the lower angle during take-off than during landing, because they increase the drag, too. Figure 9b demonstrates these changes in polar curve diagrams depending on the flap deflection.

Figure 8. Further specific aspects: (a) drag coefficient of the supersonic airfoil depending on the Mach number and (b) lift coefficients generated on the high and low aspect ratio (delta) wings.

Figure 9. An integrated aerodynamic characteristic of the wing (airfoil) and flap/slats (a), polar curve diagrams of a middle size passenger aircraft (b), and drag coefficient breakdown (c). (In figure b, the numbers at curves depict the flight conditions as 1. cruise flight (all devices are closed), 2. undercarriage system is open, only, 3. take-off regime (flaps deflected near 300 ), 4. landing condition (flaps deflected up to 45�), 5. all the wing mechanisms are opened (slats and interceptors).

Figure 9c calls attention to the final aerodynamic coefficients that always are composed from the coefficient generated on/by the aircraft elements.

These examples underline that the aerodynamic characteristics depend on the state of the flow field surrounding the aircraft, like air viscosity, motion variables, e.g., linear and angular velocities, real geometrical characteristics reflecting the effect of the deflection of the control elements and the deformation of aircraft, and they may have a sensitive dependence on the initial conditions (Figure 1). Therefore, the aerodynamic coefficients are given in the form of functions of different variables, like position angles and velocities of aircraft, flow characteristics, namely Reynolds number, Mach number (speed), deflection angles of aerodynamic control surfaces, control forces, etc. [1–8]. These functions are very nonlinear and very complicated. In case of dynamic changes in basic parameters (like angle of attack) and especially in case of oscillation motion, the aerodynamic coefficients contain the hysteresis-type nonlinearities depending on the frequencies and amplitudes of oscillation. So, different simplified, more complex, and special models and mathematical representations are needed.

#### 4. The first (simple) aerodynamic models

This phenomenon is called stall. The drag coefficient increases with growing lift coefficient due to the induced drag. The moment coefficient follows the changes in lift and drag coefficients, especially after stall. The polar curve (CL ¼ f Cð Þ <sup>D</sup> ) and "goodness" factor (k ¼ ðCL=CD ¼ fð ÞÞ ∝ ) explain the relatively low angle of attack and must be realized during the most important flight regime, during the cruise flight for having minimum drag, minimum required

As it is well known, the subsonic and supersonic aerodynamics is principally different. The "classic" airfoils with blunt leading edge cannot be applied, because their drag tends to the infinity nearing to Mach number (velocity related to the sound speed in the same condition) equals to one. At the supersonic speed only, the airfoils (wing and fuselage) with sharp leading

Figure 8b demonstrates how changes in some parameters may radically affect the aerodynamic characteristics. In case of high aspect ratio wing, the vortex generating the lift as a vortex tube along the wing span separates at the wing tips and causes the induced drag. The low aspect ratio delta wing has unique aerodynamic picture. The flow separating from the wing leading edge and the caused by this separated flow vortexes moving back on the top of the

The aerodynamic coefficients depending on the flight modes and flight maneuvers are managed by use of control surfaces and motion devices as flaps, slots, and generally by all the devices deviating and changing the geometry like undercarriage system, braking parachutes, etc. [1–8]. For example, the flaps at wing trailing edge and slat at the leading edge (making slots between the slat and mean wing) are used for increasing the lift (and drag) allowing to reduce the take-off and landing speeds for making safer flight modes. The flaps increase the lift coefficients ("moving" the lift coefficient and angle of attack curve left and up in Figure 9a), while the slats/slots increase the critical angle of attack, (because they do not change the airfoil/ wing chamber). In Figure 9a, at low lift coefficient region, the effect of a special leading edge

The flaps are deflected on the lower angle during take-off than during landing, because they increase the drag, too. Figure 9b demonstrates these changes in polar curve diagrams

Figure 8. Further specific aspects: (a) drag coefficient of the supersonic airfoil depending on the Mach number and (b) lift

wing generate extra lift at high angle of attack, and the stall appears at 45–70�, only.

thrust, and minimum fuel consumption.

166 Flight Physics - Models, Techniques and Technologies

edge can be applied (Figure 8a).

flap, called Krueger flap, is shown, too.

a.) b.)

coefficients generated on the high and low aspect ratio (delta) wings.

depending on the flap deflection.

The mathematical descriptions of the aerodynamic coefficients are called as aerodynamic models [1–8]. First models were based on the work of Bryan [40], who used two principal assumptions: the aerodynamic forces and moments depend only on the instantaneous values of the motion variables, and their dependence is of linear character. Therefore, the simple models of the aerodynamic coefficients can be expanded into a Taylor series about the reference states.

$$\mathbb{C}\_{A}(t) = \mathbb{C}\_{A\_{0}} + \sum\_{i=1}^{n} \mathbb{C}\_{a\_{\overline{i}\_{i}}} p\_{i}(t), \tag{5}$$

where CA is the aerodynamic coefficient, pi , i ¼ 1, 2, …, n are the parameters, CA0 is the aerodynamic coefficient at pi ¼ 0, ∀i and the Capi is the partial derivative coefficient.

$$\mathbf{C}\_{A\_{p\_i}} = \left(\partial \mathbf{C}\_A / \partial p\_i\right)\_{p\_i=0}.\tag{6}$$

For example, the pitching moment in simplified case can be represented by the following term:

$$\mathbf{C}\_{m}(t) = \mathbf{C}\_{m\_0} + \mathbf{C}\_{m\_V}V(t) + \mathbf{C}\_{m\_q}\boldsymbol{q}(t) \tag{7}$$

where the CmV and Cmq are the moment coefficient derivatives:

$$\mathsf{C}\_{m\_V} = (\mathsf{\partial C}\_m / \partial V)\_{V=0\prime} \qquad \qquad \mathsf{C}\_{m\_q} = (\mathsf{\partial C}\_m / \partial q)\_{q=0}.\tag{8}$$

Later, taking into account the more realistic characteristics of the nonsteady flow associated with the aircraft motion, the results received refused both assumptions of Bryan. The new models introduced by Glauert [41] contain additional elements taking into consideration the effect of the past history of the aircraft motion on the current aerodynamic forces and moments [42–44]. The flight dynamic, stability, and control had been applied to Glauert's idea in more general form. The aerodynamic coefficients were defined by the use of c linear air reaction theory outlined by Etkin [41, 45]. In this approach, the coefficients are linearized around the predefined operational points. The interactions between the angle of attack, the control surface deflection, and aerodynamic coefficient, as well as the time lag effect on the aerodynamics, were taken into account. The aerodynamic model was rewritten in form, like the following model of the lift coefficient:

$$\begin{aligned} \mathsf{C}\_{L}(t) &= \begin{array}{c} \mathsf{C}\_{L\_{0}} + \mathsf{C}\_{L\_{a}}a(t) + \mathsf{C}\_{L\_{a^{2}}}a^{2}(t) & + \mathsf{C}\_{L\dot{a}}\dot{a}(t) & + \mathsf{C}\_{L\_{b}}\delta(t) \\ \text{linear part} & + \text{non-linear part} + \text{time lag} & + \text{control effect.} \end{array} \end{aligned} \tag{9}$$

Here CLα\_ <sup>¼</sup> ð Þ <sup>∂</sup>CL=∂α\_ <sup>α</sup>\_¼α\_ <sup>0</sup> is the derivative of the lift coefficient, respectively, to the rate of change in angle of attack α\_ ¼ ∂α\_ =∂t and it represents the time lag addition determined by using the assumption that it is proportional to α\_ . The partial derivatives of the aerodynamic models are used as stability derivatives [13–15, 42, 45, 46] and they must be multiplied by changes in variables Δα; Δα<sup>2</sup>;… as deviations from the flight regime (operational point) at which the coefficients are determined. The derivatives should be independent. Principally, quantities α and α\_ are not independent. So the models like (9) approximate the aerodynamic coefficient in the form of a mathematically incorrect expansion [11].

#### 5. Classic aerodynamic models

The simplified aerodynamic coefficient representations adapted to the real situations and real problems today are the widely and most used aerodynamic models. The different types of simple classic aerodynamic models [1–8, 13–15, 42–46] are shown in Table 1.

The usual linearized formulations of the aerodynamic models and nonlinear models described above can only be used for detailed investigations where the aircraft motion is prescribed. This is the mean difficulty with such models.


Here δ<sup>e</sup> and δ<sup>r</sup> are the deflection angle of the control surface elevator and rudder, q is the dynamic pressure, β is the sideslip angle, i.e., angle between the x axes of the body and wind reference systems, and Δxcg and Δycg are the coordinates of deviated position of the center of gravity.)

Remarks. Often small c is used instead of capital C in aerodynamic coefficients. Sometimes mz is applied instead of Cm.

Table 1. Different aerodynamic models and their possible applications.

CApi ¼ ∂CA=∂pi 

where the CmV and Cmq are the moment coefficient derivatives:

168 Flight Physics - Models, Techniques and Technologies

model of the lift coefficient:

Here CLα\_ <sup>¼</sup> ð Þ <sup>∂</sup>CL=∂α\_ <sup>α</sup>\_¼α\_

5. Classic aerodynamic models

is the mean difficulty with such models.

For example, the pitching moment in simplified case can be represented by the following term:

Later, taking into account the more realistic characteristics of the nonsteady flow associated with the aircraft motion, the results received refused both assumptions of Bryan. The new models introduced by Glauert [41] contain additional elements taking into consideration the effect of the past history of the aircraft motion on the current aerodynamic forces and moments [42–44]. The flight dynamic, stability, and control had been applied to Glauert's idea in more general form. The aerodynamic coefficients were defined by the use of c linear air reaction theory outlined by Etkin [41, 45]. In this approach, the coefficients are linearized around the predefined operational points. The interactions between the angle of attack, the control surface deflection, and aerodynamic coefficient, as well as the time lag effect on the aerodynamics, were taken into account. The aerodynamic model was rewritten in form, like the following

CLðÞ¼ <sup>t</sup> CL<sup>0</sup> <sup>þ</sup> CL<sup>α</sup> <sup>α</sup>ð Þþ <sup>t</sup> CLα<sup>2</sup> <sup>α</sup><sup>2</sup>ðÞ þ <sup>t</sup> CLα\_α\_ðÞ þ <sup>t</sup> CL<sup>δ</sup> <sup>δ</sup>ð Þ<sup>t</sup>

coefficient in the form of a mathematically incorrect expansion [11].

simple classic aerodynamic models [1–8, 13–15, 42–46] are shown in Table 1.

change in angle of attack α\_ ¼ ∂α\_ =∂t and it represents the time lag addition determined by using the assumption that it is proportional to α\_ . The partial derivatives of the aerodynamic models are used as stability derivatives [13–15, 42, 45, 46] and they must be multiplied by changes in variables Δα; Δα<sup>2</sup>;… as deviations from the flight regime (operational point) at which the coefficients are determined. The derivatives should be independent. Principally, quantities α and α\_ are not independent. So the models like (9) approximate the aerodynamic

The simplified aerodynamic coefficient representations adapted to the real situations and real problems today are the widely and most used aerodynamic models. The different types of

The usual linearized formulations of the aerodynamic models and nonlinear models described above can only be used for detailed investigations where the aircraft motion is prescribed. This

pi

CmðÞ¼ t Cm<sup>0</sup> þ CmV V tð Þþ Cmq q tð Þ (7)

CmV <sup>¼</sup> ð Þ <sup>∂</sup>Cm=∂<sup>V</sup> <sup>V</sup>¼<sup>0</sup>, Cmq <sup>¼</sup> ð Þ <sup>∂</sup>Cm=∂<sup>q</sup> <sup>q</sup>¼<sup>0</sup>: (8)

linear part <sup>þ</sup> non � linear part <sup>þ</sup> time lag <sup>þ</sup> control effect: (9)

<sup>0</sup> is the derivative of the lift coefficient, respectively, to the rate of

¼<sup>0</sup>: (6)

The full aerodynamic description of the aircraft requires a lot of component models. These models are often defined as semiempirical models, such models are based on theoretical bases, adapted to measured data. Figure 10a shows an example for use of such methods developed for aircraft aerodynamic design. The derivative coefficient of the lift generated on the nose part of a fuselage depends on the flight Mach number (M∞Þ and ratio of lengths of central and nose parts of the fuselage.

Another example is the calculation of the fuselage friction drag coefficient appearing at zero angle of attack:

$$\mathbf{C}\_{D\_{0,f}} = \mathbf{C}\_f \boldsymbol{\eta}\_t \boldsymbol{\eta}\_M \frac{\mathbf{S}\_{fus\_{\text{net}}}}{\mathbf{S}\_{fus\_{\text{M}}}} \tag{10}$$

where Cf is the skin friction drag coefficient thin plate, η<sup>t</sup> and η<sup>M</sup> are coefficients taking into account the effect from body thickness and flow velocity, Sfuswet and Sf usM are the fuselage

Figure 10. Practical figures supporting the estimation and evaluation [5] of the aerodynamic coefficients: (a) lift slop coefficient for estimation of the lift generated on the fuselage nose, (b)–(d) figures defining the estimation of the friction drag generating on the fuselage.

so-called surface wetted area and area of the fuselage mean (maximum) cross-section area, and Df , lf , and lfn are the mean diameter, length of fuselage, and length of the nose section of the fuselage (in Figure 10). The Cf , η<sup>t</sup> , and η<sup>M</sup> coefficients can be estimated from Figure 10b–d, while the fuselage wetted area can be calculated with the use of the following formulas:

$$\mathbf{S}\_{\text{fps}\_{\text{net}}} = \pi \mathbf{D}\_{f} l\_{f} \left( 1 - \frac{2 \mathbf{D}\_{f}}{l\_{f}} \right)^{\frac{1}{2l\_{3}}} \left( 1 - \frac{2 \mathbf{D}\_{f}^{2}}{l\_{f}^{2}} \right) \text{if the } \mathbb{H}\_{f} /\_{\text{D}\_{f}} \geq \mathbf{4.5} \text{ or } \mathbf{S}\_{\text{fps}\_{\text{net}}} = \mathbf{2.53} \text{D}\_{f} l\_{f}. \tag{11}$$

As it had been outlined already, the first simplified models were adapted to the wide flight dynamics, stability, and control investigations and to the different form of aircraft [13–15, 42–46]. The modern control introduced the state space representation of the linearized system of equations describing the aircraft spatial motion:

$$
\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} \tag{12}
$$

x and u are the state and control (input) vectors, while A and B are the state and control matrices. In simplified case, when the aircraft is modeled as rigid body, the state vector contains the components of the linear and rotational (angular) velocities – x = [u, v, w, p, q, r] T . The control vector is composed of control inputs including the control surfaces' deflection, deflections of other moving elements as flaps, slat, as well as the changes in trust: u ¼ δe; δr; δa; δ<sup>f</sup> ; δs;…; nT � �<sup>T</sup> . Here the control elements are the deflection angle of elevator, rudder, aileron, flaps, slats, and engine revolution speed. Principally, because of the linearization, the state and control vectors contain the changes in velocity components and deflection angles related to the operational (initial) condition. Because of symmetry, the motion equations can be divided into two subsystems: longitudinal and lateral motion. The state vector of the longitudinal motion model (motion of aircraft in the vertical plane, only) contains the u, w, q and additionally the pitch (or climb) angle, θ. The A and B elements are special derivative coefficients.

The aircraft longitudinal motion can be modeled by

so-called surface wetted area and area of the fuselage mean (maximum) cross-section area, and Df , lf , and lfn are the mean diameter, length of fuselage, and length of the nose section of the

Figure 10. Practical figures supporting the estimation and evaluation [5] of the aerodynamic coefficients: (a) lift slop coefficient for estimation of the lift generated on the fuselage nose, (b)–(d) figures defining the estimation of the friction

As it had been outlined already, the first simplified models were adapted to the wide flight dynamics, stability, and control investigations and to the different form of aircraft [13–15, 42–46]. The modern control introduced the state space representation of the linearized system of equa-

x and u are the state and control (input) vectors, while A and B are the state and control matrices. In simplified case, when the aircraft is modeled as rigid body, the state vector contains the components of the linear and rotational (angular) velocities – x = [u, v, w, p, q, r]

The control vector is composed of control inputs including the control surfaces' deflection, deflections of other moving elements as flaps, slat, as well as the changes in trust:

rudder, aileron, flaps, slats, and engine revolution speed. Principally, because of the linearization, the state and control vectors contain the changes in velocity components and deflection angles related to the operational (initial) condition. Because of symmetry, the motion equations can be divided into two subsystems: longitudinal and lateral motion. The state vector of the

if the lf =Df

while the fuselage wetted area can be calculated with the use of the following formulas:

<sup>1</sup> � <sup>2</sup>D<sup>2</sup> f l 2 f

!

, and η<sup>M</sup> coefficients can be estimated from Figure 10b–d,

x\_ ¼ Ax þ Bu (12)

. Here the control elements are the deflection angle of elevator,

≥ 4:5 or Sfuswet ¼ 2:53Df lf : (11)

T .

fuselage (in Figure 10). The Cf , η<sup>t</sup>

170 Flight Physics - Models, Techniques and Technologies

drag generating on the fuselage.

Sfuswet <sup>¼</sup> <sup>π</sup>Df lf <sup>1</sup> � <sup>2</sup>Df

tions describing the aircraft spatial motion:

u ¼ δe; δr; δa; δ<sup>f</sup> ; δs;…; nT � �<sup>T</sup>

lf � �2=<sup>3</sup>

$$m\frac{du}{dt} = T\cos(\alpha + \varphi\_T) - D - W\sin\theta$$

$$m\frac{dw}{dt} = T\sin(\alpha + \varphi\_T) + L - W\cos\theta \,. \tag{13}$$

$$I\_y\frac{dq}{dt} = M$$

equations that are defined in body system of reference. Here m and W are the aircraft mass and weight, T is the trust and w<sup>T</sup> is the engine built angle, angle between the trust direction, xb is the axis of the body system of reference, and Iy is the inertia moment component. Supposing the cos α þ w<sup>T</sup> ð Þ ≈ 1, sin α þ w<sup>T</sup> ð Þ ≈ 0 and taking into account the X, Z, and M are the components of the total forces and moment component due to x, z, and y axes, respectively, Eq. (13) can be rewritten into the space state representation form:

$$
\begin{bmatrix}
\dot{u} \\
\dot{w} \\
\dot{q} \\
\dot{\theta} \\
\dot{\theta}
\end{bmatrix} = \begin{bmatrix}
\frac{X\_u}{m} & \frac{X\_w}{m} & -g\cos\theta\_0 & 0 \\
\frac{Z\_u}{m} & \frac{Z\_w}{m} & -g\sin\theta\_0 & 0 \\
\frac{M\_u}{I\_y} & \frac{M\_w}{I\_y} & \frac{M\_q}{I\_y} & 0 \\
0 & 0 & 1 & 0
\end{bmatrix} \begin{bmatrix} u \\ w \\ q \\ \theta \\ \theta \end{bmatrix} + \begin{bmatrix}
\frac{X\_{\delta\_\epsilon}}{m} & \frac{X\_{n\_r}}{m} \\
\frac{Z\_{\delta\_\epsilon}}{m} & \frac{Z\_{n\_r}}{m} \\
\frac{M\_{\delta\_\epsilon}}{I\_y} & \frac{M\_{\eta r}}{I\_y} \\
0 & 0
\end{bmatrix} \begin{bmatrix} \delta\_\epsilon \\ n\_T \end{bmatrix}.\tag{14}
$$

Here, the components Xu, Xw, …, Mu,…, Mq, are called stability derivatives and X<sup>δ</sup><sup>e</sup> , …, MnT are the control derivatives. Of course, the aerodynamic total forces and moments might be estimated by the sum of the derivatives of the force and moment components relevant to the given state and control vector elements.

For instance, Xu should be determined from the first equation of (13):

$$\begin{split} m\frac{du}{dt} &= T(V, \Omega, \delta\_\epsilon, n\_T) - \mathbb{C}\_D \frac{\rho V^2}{2} \mathbb{S} - W\sin\theta\\ mX\_{\rm u} &= \frac{\partial T}{\partial u} - \frac{\rho V\_0^2}{2} \frac{\partial \mathbb{C}\_D}{\partial u} - \mathbb{C}\_D \frac{\rho \mathbb{S}}{2} \frac{\partial V^2}{\partial u}, \end{split} \tag{15}$$

and in simple case, when V = u, the dimension-less derivative equals to:

$$X\_{\mu} = \frac{T\_{\mu}}{\frac{\rho V\_{0}S}{2}} - V\_{0}\mathbf{C}\_{D\_{V}} - 2\mathbf{C}\_{D}.\tag{16}$$

The drag coefficient, CD, can be represented by the models described earlier.

The static and dynamic stability, flight dynamics (as maneuvers, maneuverability, departure to the critical regimes, and recovery from there) and control design, and control synthesis are required to know the aerodynamic characteristics of the aircraft elements and aircraft devices, too. For instant, the hinge moment coefficient (m) of the control surfaces (elevator, rudder, and ailerons) can be represented by the following simplified models:

$$\begin{aligned} m\_{\varepsilon} &= m\_{\varepsilon\_{a}}\alpha + m\_{\varepsilon\_{b\_{r}}}\delta\_{\varepsilon} + m\_{\varepsilon\_{b\_{r}T}}\delta\_{\varepsilon T} \\ m\_{\mathcal{I}} &= m\_{\mathcal{I}\_{\beta}}\beta + m\_{\mathcal{I}\_{\delta\_{r}}}\delta\_{\mathcal{I}} + m\_{\mathcal{I}\_{\delta\_{r}T}}\delta\_{\mathcal{I}}r \\ m\_{a} &= m\_{a\_{a}}\alpha + m\_{a\_{\mathcal{I}}}p + m\_{a\_{\delta\_{a}}}\delta\_{a} + m\_{a\_{a\mathcal{I}}}\delta\_{\mathcal{I}T} \end{aligned} \tag{17}$$

where index T depicts the trim tabs and the p is the pitch rate.

Finally, another excellent example demonstrates the interaction between the different theories. As Figure 8b shows, the lift coefficient on the delta wing depends on the vortex generated at the leading edge. Polhamus [47] created and explained a special formula for lift coefficient calculation:

$$\mathbf{C}\_{\text{L}\_{4w}} = \mathbf{K}\_p \sin \alpha \,\,\cos^2 \alpha + \mathbf{K}\_V \,\cos \,\,\alpha \,\,\sin^2 \alpha \,\,\tag{18}$$

Here the first part comes from the small angle of attack potential lifting surface theory. The Kp is the lift curve slop, sin α accounts for true boundary condition, and the cos <sup>2</sup>α arises from the Kutta-type condition at the leading edge. In second part of the formula (18), the KV sin <sup>2</sup>α gives the potential flow leading edge suction, i.e., vortex normal force, and the cos α defines its component in the lift direction.

The classic models are well applied in identifying them from flight data and developing the flight simulation methods, too [29, 48].

#### 6. Developed aerodynamic models

The classic aerodynamic models cannot be applied to accurate description of the aircraft motion at high angle of attack, aircraft maneuvers, dynamic, oscillation motion or aerodynamic characteristics in flutter, etc. Tobak [49] introduced a model structure. He made a special assumption: the changes in aerodynamic coefficients are linear functions of changes in variables that are independent of the past history of these variables, namely on all values that these variables have taken over the course of the motion prior to time τ. For example, the change in pitching moment can be defined by following functions:

$$
\Delta \mathcal{C}\_m = \frac{\Delta \mathcal{C}\_m(t-\tau)}{\Delta \delta} \Delta \delta + \frac{\Delta \mathcal{C}\_m(t-\tau)}{\Delta(ql/V)} \Delta(ql/V) \tag{19}
$$

Here δ is the motion of aircraft along the z axis in body axis system (δ = z), q is the angular velocity around the y axis, and the derivatives depend on elapsed time t -τ rather than on t and τ.

The derivatives in Eq. (19) come from solution of linear equation of gas dynamics. However, the linearity assumption does not rest on the assertion that change in pitching moment, (ΔCmÞ, is linear dependent on changes in variables, Δδð Þ ¼ Δδ<sup>e</sup> and Δð Þ ql=V . So, these two increments must not be linear additives in Eq. (19).

Principally, the aerodynamic pitching moment coefficient response to variations δ and q. These variations can be broken into a large number of small step changes (Figure 11). No matter how large the values of δ and q at the beginning of steps, the derivatives depend on the t -τ, only. The limits of these functions

$$\lim\_{\Delta\delta \to 0} \frac{\Delta\mathbb{C}\_m(t-\tau)}{\Delta\delta} = \mathbb{C}\_{m\_\delta}(t-\tau), \qquad \lim\_{\Delta(ql/V) \to 0} \frac{\Delta\mathbb{C}\_m(t-\tau)}{\Delta(ql/V)} = \mathbb{C}\_{m\_q}(t-\tau) \tag{20}$$

are called as the linear indicial pitching moment responses per unit step changes in δ and ql/V, respectively [11, 49].

Using this indicial function concept to calculate the aerodynamic coefficients, Tobak [11, 49] replaced Bryan's function with a linear functional in the form of the linear superposition integral like:

$$\mathbf{C}\_{m}(t) = \mathbf{C}\_{m}(0) + \int\_{0}^{t} \mathbf{C}\_{m\_{\delta}}(t-\tau) \frac{d}{d\tau} \delta(t) \mathbf{d}\tau + \frac{1}{V} \Big|\_{0}^{t} \mathbf{C}\_{m\_{q}}(t-\tau) \frac{d}{d\tau} q(t) \mathbf{d}\tau. \tag{21}$$

In reality, the functions of aerodynamic coefficient and derivatives depend on all the past values of the motion variables. In accordance to Volterra's description, the aerodynamic coefficient as function can be given in the form of a functional:

$$\mathbf{C}\_{\mathfrak{m}}(t) = \mathbf{G}[\delta(\xi), \eta(\xi)] \tag{22}$$

Generally, the whole time/past history of motion variables is unknown. Therefore, the functional (22) can be replaced by a functional describing the dependence on the past in the form of analytical functions in the neighborhood of ξ ¼ τ reconstructed from the Taylor series expansions of the coefficients about ξ ¼ τ. This obtains for example:

Figure 11. Simulation of incremental responses.

The static and dynamic stability, flight dynamics (as maneuvers, maneuverability, departure to the critical regimes, and recovery from there) and control design, and control synthesis are required to know the aerodynamic characteristics of the aircraft elements and aircraft devices, too. For instant, the hinge moment coefficient (m) of the control surfaces (elevator, rudder, and

> me ¼ me<sup>α</sup> α þ me<sup>δ</sup><sup>e</sup> δ<sup>e</sup> þ meδeT δeT mr ¼ mr<sup>β</sup> β þ mr<sup>δ</sup><sup>r</sup> δ<sup>r</sup> þ mrδrT δrT ma ¼ ma<sup>α</sup> α þ map p þ ma<sup>δ</sup><sup>a</sup> δ<sup>a</sup> þ maaT δaT

Finally, another excellent example demonstrates the interaction between the different theories. As Figure 8b shows, the lift coefficient on the delta wing depends on the vortex generated at the leading edge. Polhamus [47] created and explained a special formula for lift coefficient

Here the first part comes from the small angle of attack potential lifting surface theory. The Kp is the lift curve slop, sin α accounts for true boundary condition, and the cos <sup>2</sup>α arises from the Kutta-type condition at the leading edge. In second part of the formula (18), the KV sin <sup>2</sup>α gives the potential flow leading edge suction, i.e., vortex normal force, and the cos α defines its

The classic models are well applied in identifying them from flight data and developing the

The classic aerodynamic models cannot be applied to accurate description of the aircraft motion at high angle of attack, aircraft maneuvers, dynamic, oscillation motion or aerodynamic characteristics in flutter, etc. Tobak [49] introduced a model structure. He made a special assumption: the changes in aerodynamic coefficients are linear functions of changes in variables that are independent of the past history of these variables, namely on all values that these variables have taken over the course of the motion prior to time τ. For example, the change in

<sup>α</sup> <sup>þ</sup> KV cos <sup>α</sup> sin <sup>2</sup>

, (17)

α (18)

ailerons) can be represented by the following simplified models:

172 Flight Physics - Models, Techniques and Technologies

where index T depicts the trim tabs and the p is the pitch rate.

CL<sup>Δ</sup><sup>w</sup> <sup>¼</sup> Kp sin <sup>α</sup> cos <sup>2</sup>

calculation:

component in the lift direction.

flight simulation methods, too [29, 48].

6. Developed aerodynamic models

pitching moment can be defined by following functions:

<sup>Δ</sup>Cm <sup>¼</sup> <sup>Δ</sup>Cmð Þ <sup>t</sup> � <sup>τ</sup>

<sup>Δ</sup><sup>δ</sup> <sup>Δ</sup><sup>δ</sup> <sup>þ</sup>

Here δ is the motion of aircraft along the z axis in body axis system (δ = z), q is the angular velocity around the y axis, and the derivatives depend on elapsed time t -τ rather than on t and τ. The derivatives in Eq. (19) come from solution of linear equation of gas dynamics. However, the linearity assumption does not rest on the assertion that change in pitching moment, (ΔCmÞ,

ΔCmð Þ t � τ

<sup>Δ</sup>ð Þ ql=<sup>V</sup> <sup>Δ</sup>ð Þ ql=<sup>V</sup> (19)

$$\mathbb{C}\_{m\_0}[\delta(\xi), \eta(\xi); t, \tau] = \mathbb{C}\_{m\_0}\left(t, \tau; \delta(\tau), \dot{\delta}(\tau), \dots, \eta(\tau), \dot{q}(\tau), \dots\right). \tag{23}$$

Hence, at most, only the first few coefficients of expansions of δ ξð Þ and qð Þ ξ need be retained to characterize correctly the most recent past [49], which is all the indicial response remembers. Using the two coefficients of δ ξð Þ, for example, implies matching the true past history of δ in magnitude and slope at the origin of the step, thereby approximating δ ξð Þ by a linear function of time

$$
\delta\delta(\xi) \approx \delta(\pi) - \dot{\delta}(\pi)(\pi - \xi). \tag{24}
$$

With application of this approach. Eq. (21) can be rewritten into the following form:

$$\begin{split} \mathbf{C}\_{m}(t) &= \mathbf{C}\_{m}(0) + \int\_{0}^{t} \mathbf{C}\_{m}\left(t, \tau; \delta(\tau), \dot{\delta}(\tau), q(\tau), \dot{q}(\tau)\right) \frac{d}{d\tau} \delta(t) \mathbf{d}\tau \\ &+ \frac{1}{V} \int\_{0}^{t} \mathbf{C}\_{m\_{\eta}}(t, \tau; \delta(\tau), \dot{\delta}(\tau), q(\tau), \dot{q}(\tau)) \frac{d}{d\tau} q(t) \mathbf{d}\tau \end{split} \tag{25}$$

This method of model definition is more attractive then (21) and gives the possibility of taking into account the considerable nonlinearities, time lag, and hysteresis, too. All the developed models follow from this model formation. For example, in case of slowly varying motion, Eq. (25) may be formalized in a more general form,

$$\mathbb{C}\_{m}(t) = \mathbb{C}\_{m}(0) + \int\_{0}^{t} \mathbb{C}\_{m\_{0}}(t - \tau; \delta(\tau), q(\tau)) \frac{d}{d\tau} \delta(t) d\tau + \frac{1}{V} \Big|\_{0}^{t} \mathbb{C}\_{m\_{q}}(t - \tau; \delta(\tau), q(\tau)) \frac{d}{d\tau} q(t) d\tau \tag{26}$$

still capable of embracing a fairly broad range of nonlinear problems of aerodynamics.

The use of indicial aerodynamic functions is a rather complex task even for 2D [50].

The next step in developing the aerodynamic models was made by Goman and his colleague [51, 52]. They had formulated the aerodynamic coefficient models in the form of a state space representation:

$$\mathbf{C}\_{a} = \mathbf{C}\_{a}(\xi(t)\eta(t)),\tag{27}$$

where

$$\dot{\boldsymbol{\eta}}(t) = \mathbf{g}\left(\boldsymbol{\eta}(t)\boldsymbol{\xi}(t)\dot{\boldsymbol{\xi}}(t)\right) \tag{28}$$

and

$$\boldsymbol{\mathfrak{E}}(t) = \begin{bmatrix} \boldsymbol{\mathfrak{x}}(t)^T \boldsymbol{\mathfrak{u}}(t)^T \end{bmatrix}^T. \tag{29}$$

Here η is an internal additional state vector and x and u are the state and control vectors from the aircraft motion models (see Eq. (12)). For instance, Ref. [51] described the aircraft longitudinal dynamics by introducing the internal state variable representing the vortex burst point location along the chord of a triangular wing.

#### 7. Advanced aerodynamic models

Cm<sup>δ</sup> <sup>½</sup>δ ξð Þ; <sup>q</sup>ð Þ <sup>ξ</sup> ; <sup>t</sup>; <sup>τ</sup>� ¼ Cm<sup>δ</sup> <sup>t</sup>; <sup>τ</sup>; δ τð Þ; \_

of time

Hence, at most, only the first few coefficients of expansions of δ ξð Þ and qð Þ ξ need be retained to characterize correctly the most recent past [49], which is all the indicial response remembers. Using the two coefficients of δ ξð Þ, for example, implies matching the true past history of δ in magnitude and slope at the origin of the step, thereby approximating δ ξð Þ by a linear function

δ ξð Þ <sup>≈</sup> δ τð Þ� \_

With application of this approach. Eq. (21) can be rewritten into the following form:

Cm<sup>δ</sup> <sup>t</sup>; <sup>τ</sup>; δ τð Þ; \_

δ τð Þ; <sup>q</sup>ð Þ<sup>τ</sup> ; <sup>q</sup>\_ð Þ<sup>τ</sup> � � <sup>d</sup>

This method of model definition is more attractive then (21) and gives the possibility of taking into account the considerable nonlinearities, time lag, and hysteresis, too. All the developed models follow from this model formation. For example, in case of slowly varying motion,

δð Þt dτ þ

The next step in developing the aerodynamic models was made by Goman and his colleague [51, 52]. They had formulated the aerodynamic coefficient models in the form of a state space

<sup>η</sup>\_ðÞ¼ <sup>t</sup> <sup>g</sup> <sup>η</sup>ð Þ<sup>t</sup> <sup>ξ</sup>ð Þ<sup>t</sup> \_

<sup>ξ</sup>ðÞ¼ <sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> <sup>T</sup>

<sup>u</sup>ð Þ<sup>t</sup> <sup>T</sup> h i<sup>T</sup>

1 V ðt

0

ξð Þt � �

d dτ

still capable of embracing a fairly broad range of nonlinear problems of aerodynamics.

The use of indicial aerodynamic functions is a rather complex task even for 2D [50].

δ τð Þ; <sup>q</sup>ð Þ<sup>τ</sup> ; <sup>q</sup>\_ð Þ<sup>τ</sup> � � <sup>d</sup>

dτ q tð Þdτ

ðt

0

Cmq <sup>t</sup>; <sup>τ</sup>; δ τð Þ; \_

CmðÞ¼ t Cmð Þþ 0

þ 1 V ðt

Eq. (25) may be formalized in a more general form,

ðt

174 Flight Physics - Models, Techniques and Technologies

0

CmðÞ¼ t Cmð Þþ 0

representation:

where

and

0

Cm<sup>δ</sup> ð Þ t � τ; δ τð Þ; qð Þτ

δ τð Þ;…; <sup>q</sup>ð Þ<sup>τ</sup> ; <sup>q</sup>\_ð Þ<sup>τ</sup> ; … � �: (23)

δ τð Þð Þ τ � ξ : (24)

(25)

dτ δð Þt dτ

Cmq ð Þ t � τ; δ τð Þ; qð Þτ

Ca ¼ Cað Þ ξð Þt ηð Þt , (27)

d dτ

: (29)

q tð Þdτ (26)

(28)

The collection of large databases of practical wind tunnel and flight test measurements and wide use of rapidly developing methods of computational fluid dynamics and a series of new methods have developed for modeling the aerodynamic coefficients. Three different approaches can be applied: (i) approximation and interpolation, (ii) analytical models and special models, and (iii) models developed using soft computing models.

The polynomial, and trigonometric interpolation, spline or regression models can be used for determining the aerodynamic coefficients or aerodynamic forces directly. For example, reference [53] uses the Lagrange interpolation to determine the lift and drag coefficient when studying the takeoff taxiing. The piecewise cubic Hermite interpolating polynomial and Spline are applied [54] to calculating the derivative of the pressure distribution on airfoil for determining the laminar-to-turbulent transition. The transition is identified as the location of maximum curvature in the pressure distribution. The Chebyshev polynomials and their orthogonality properties were applied [55] for approximation of unsteady generalized aerodynamic forces from the frequency domain into the Laplace domain, acting on a Fly-By-Wire aircraft. The results were compared with Padé method and validated on the aircraft test model.

The oscillation in changes of the aerodynamic forces and their coefficients contributes to the most interesting areas of developing the aerodynamic coefficient models. This area has two major parts: (i) oscillation of the aircraft elements, like flutter, and (ii) oscillation flight of aircraft. The first and today valued as fundamental studies were published in 1920s and 1930s. Wagner [56] dealt with unsteady lift on airfoil due to abrupt changes in angle of attack and he calculated the circulation around the airfoil in response to a step in angle of attack. Theodorsen [57] extending the Wagner concept developed a model for quasi-steady thin airfoil theory including added-mass forces and the effect of wake vorticity.

$$\mathbf{C}\_{L} = \pi \left[ \ddot{h} + \dot{a} - a\ddot{a} \right] + 2\pi \left[ a + h + a \left( \frac{1}{2} - a \right) \right] \mathbf{C}(k) \tag{30}$$

Here the added-mass force taken into account by the first addend, while the second one defines the quasi-steady lift from thin airfoil theory by a transfer function C(k) as lift attenuation by the wake vorticity. The h is the vertical position of airfoil, a is the pitch axis with respect to 1/2 chord, and the Theodorsen's transfer function C(k) is expressed in terms of Hankel functions:

$$\mathbb{C}\_{L}(k) = \frac{H\_1^{(2)}(k)}{H\_1^{(2)}(k) + iH\_0^{(2)}(k)},\tag{31}$$

where Hð Þ<sup>2</sup> <sup>n</sup> ð Þ¼ k Jn � Yn, n ¼ 0, 1 are Bessel function, and k ¼ ωc=2V∞, where ω is the motion frequency, c is the airfoil chord, and V<sup>∞</sup> is the free stream velocity. This approach is well applicable nowadays, too (see [58, 59]).

In 1980s and 1990s during the development of the supermanoeuvrable and thrust vectored aircraft, the hysteresis in aerodynamic coefficient was intensively studied. These aircrafts fly at critical regimes, near or at the border of the flight envelopes. Thrust vectored aircraft uses the controlled poststall flights.

The hysteresis effects in aerodynamic coefficients can appear in different forms depending on the oscillation frequency [60–63]. Figure 12 shows typical hysteresis caused by stall in normal force coefficient at the high angle of attack flight [64] and in steady-state pitching moment response [65].

The considerable nonlinearities in the aerodynamic coefficients that generate the hysteresis in aerodynamic characteristics near the critical angle of attack in stall and poststall domain, of course, are well investigated by practical methods in wind tunnels [66–69].

The formation of flow separation at the critical angle of attack is a quite complete process [70], and the hysteresis [64] shown in Figure 12a fundamentally depends on the frequency of changes in the angle of attack. Therefore, the approximation of these characteristics is a difficult task. The models described earlier cannot ensure the required accuracy in the full region of parameter variations. The aerodynamic models used in the early works were based on fitting polynomials [71] or cubic [72, 73] or bi-cubic [74, 75] splines as interpolation schemes for measured data given in the form of table. In some cases [76], the methods that worked out for bifurcation analysis did not require further smoothing and the linear interpolation had been applied.

In many cases, the aerodynamic coefficients are given in table form [68, 69] or directly estimated from the flight tests [29, 77]. Data can be obtained by special analytical models [78]:

$$\mathcal{C}\_{\mathcal{F}} = b\_0 + \sum\_{i=1}^{n} b\_i \arctan((a - c\_i)d\_i), \tag{32}$$

where b0, bi, ci are the constants.

Figure 12. Typical hysteresis in aerodynamic coefficients. (a) At high angle of attack [64], (b) moment coefficient estimated from the wind tunnel and flight test of F-16XL-1 [65] (at reduced frequency k = 0.054).

This model was developed especially for the approximation [78] of experimental data received from wind tunnel investigations [68, 69]. The aerodynamic models obtained in form (32) can be used in full AoA region from �10 to 90�. Analytical models of type (32) have a great advantage; namely, there is no α value, where the derivative of this function does not exist.

where Hð Þ<sup>2</sup>

response [65].

applicable nowadays, too (see [58, 59]).

176 Flight Physics - Models, Techniques and Technologies

controlled poststall flights.

where b0, bi, ci are the constants.

<sup>n</sup> ð Þ¼ k Jn � Yn, n ¼ 0, 1 are Bessel function, and k ¼ ωc=2V∞, where ω is the motion

frequency, c is the airfoil chord, and V<sup>∞</sup> is the free stream velocity. This approach is well

In 1980s and 1990s during the development of the supermanoeuvrable and thrust vectored aircraft, the hysteresis in aerodynamic coefficient was intensively studied. These aircrafts fly at critical regimes, near or at the border of the flight envelopes. Thrust vectored aircraft uses the

The hysteresis effects in aerodynamic coefficients can appear in different forms depending on the oscillation frequency [60–63]. Figure 12 shows typical hysteresis caused by stall in normal force coefficient at the high angle of attack flight [64] and in steady-state pitching moment

The considerable nonlinearities in the aerodynamic coefficients that generate the hysteresis in aerodynamic characteristics near the critical angle of attack in stall and poststall domain, of

The formation of flow separation at the critical angle of attack is a quite complete process [70], and the hysteresis [64] shown in Figure 12a fundamentally depends on the frequency of changes in the angle of attack. Therefore, the approximation of these characteristics is a difficult task. The models described earlier cannot ensure the required accuracy in the full region of parameter variations. The aerodynamic models used in the early works were based on fitting polynomials [71] or cubic [72, 73] or bi-cubic [74, 75] splines as interpolation schemes for measured data given in the form of table. In some cases [76], the methods that worked out for bifurcation analysis did

In many cases, the aerodynamic coefficients are given in table form [68, 69] or directly estimated from the flight tests [29, 77]. Data can be obtained by special analytical models [78]:

Figure 12. Typical hysteresis in aerodynamic coefficients. (a) At high angle of attack [64], (b) moment coefficient esti-

biarctanð Þ ð Þ α � ci di , (32)

course, are well investigated by practical methods in wind tunnels [66–69].

not require further smoothing and the linear interpolation had been applied.

CF <sup>¼</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup>X<sup>n</sup>

mated from the wind tunnel and flight test of F-16XL-1 [65] (at reduced frequency k = 0.054).

i¼1

Figure 13 shows some examples of developed analytical models defined for different speeds and elevator deflections with linear approximation between them. One example of these NASA-backed representation of the actual derivative involves four to eight arcus tangent functions:

$$\begin{split} \mathbf{C}\_{mi} &= -\frac{0.02}{\pi} \operatorname{arcclg}\left(-5\pi \frac{\alpha - 1}{18}\right) + 0.5 \operatorname{arcclg}(5(\alpha - 6)) - 0.8 \operatorname{arcclg}\left(\frac{\alpha - 18}{2}\right) \\ &+ 0.9 \operatorname{arcclg}\left(\frac{\alpha - 45}{2}\right) - 0.9 \end{split} \tag{33}$$

Since 1990s, by developing numerical aerodynamics, and applying the methods of soft computing, new types of aerodynamic coefficient representations have been developed. It seems the most applied method is based on using the neural network [79, 80]. The other papers predicted the aerodynamic coefficient of transport aircraft with the use of artificial neural networks [81], simulated the dynamic effects of canard aircraft aerodynamics [82], used genetic algorithm optimized neural networks for predicting the practical measurements [83], determined the global aerodynamic modeling with multivariable spline [84], and applied the fuzzy logic modeling to the aircraft model identification [85] and nonlinear unsteady aerodynamics [86]. Principally all the numerical methods might be applied. For instance, the aircraft stability and control can be modeled with the use of wavelet transforms [87] or even the computed stability derivatives can be applied directly in aerodynamic shape optimization [88]. Nowadays, the computer capacity and sizes allow to use the real-time on-board identification of the nonlinear aerodynamic models [89].

Figure 13. Several analytical models (lift and pitching moment coefficients and their derivatives, respectively, to pitch rate and rate of angle of attack) defined by [78].

Two specific aspects must be underlined: (i) the computational fluid dynamics may easily determine the aerodynamic coefficients by integration of the calculated surface pressure distribution and (ii) all the aerodynamic coefficient models described earlier can be applied, while better using the models as simple as possible depending on the goal and object of their application.
