9. Example of use of an advanced aerodynamic coefficient model

The Department of Aeronautics, Naval architecture and Railway Vehicles at the Budapest University of Technology and Economics (operating two flight simulators, one air traffic management laboratory with several working environment for ATCOs, small gas turbines, water channel, etc.) is active in computational fluid dynamics [92, 93], vehicle design [24, 25, 94], vehicle motion simulation [95, 96], developing original and radically new technologies [97–99], and has worked on investigation of the thrust vectored aircraft motion at high angle of attack in poststall domain [100, 101], approximation of the motion after stall [102], and unconventional and critical flights [103, 104]. One of the excellent applications of the analytical models of the aerodynamic coefficients is their using in bifurcation analysis of the poststall motion of thrust vectored aircraft.

Only the longitudinal motion was investigated. The applied system of equations defined by the use of body axis was reduced to four dimensions given in the following form [101]:

$$\begin{aligned} \dot{u} &= -qw + \frac{X}{M} - g\sin\theta + \frac{T\_x}{M} \\ \dot{w} &= -q\mu + \frac{Z}{M} - g\cos\theta + \frac{T\_z}{M} \\ \dot{q} &= \frac{\mathbf{C}\_m \overline{q} \mathbf{S} \mathbf{c}\_A + Xl\_z + Zl\_x - T\_x L\_{xe}}{I\_y} \end{aligned} \tag{34}$$
 
$$\dot{\theta} = q$$

where <sup>X</sup> <sup>¼</sup> qS Cð Þ Lsin<sup>α</sup> � CDcos<sup>α</sup> , Z <sup>¼</sup> qS Cð Þ Lcos<sup>α</sup> <sup>þ</sup> CDsin<sup>α</sup> , CL <sup>¼</sup> CL<sup>0</sup> <sup>þ</sup> cA <sup>2</sup><sup>V</sup> CLα\_α\_ þ CLq q , CD <sup>¼</sup> CD<sup>0</sup> <sup>þ</sup> cA <sup>2</sup><sup>V</sup> CDα\_α\_ þ CDq q , Cm <sup>¼</sup> Cm<sup>0</sup> <sup>þ</sup> cA <sup>2</sup><sup>V</sup> Cmα\_α\_ þ Cmq q , Tx <sup>¼</sup> Tcosδvp, Tx <sup>¼</sup> Tsinδvp: Here X and Z are the force components to x and z axis, M is the pitching moment, q are q are the dynamic pressure and pitch rate,respectively, and T,Tx, and Tz are the thrust and its components.

Different types of simple and classic aerodynamic coefficient models were applied that could not result in stable and acceptable solutions. Therefore, the described system of equations and analytical models of aerodynamic coefficients were filled up by data of F/A-18 aircraft [68, 69, 78]. These models defined the hysteresis effects, as well, and they may be used in full region of the possible changes in angle of attack (see Figure 13).

The system of equation was solved by different numerical methods (Runge-Kutta and Adams-Moulton) with different step size. Software MATHLAB and ACSL were used in the simulations. The results received were stable and the same at time steps 10�<sup>2</sup> and 10�<sup>6</sup> s.

Figure 14 shows the equilibrium surface obtained in the thrust-thrust deflection parameter space (left side) and the bifurcation curves (right side) for flight regime V = 0.3 M and H = 15,000 ft. (T = 22.7 kN, δvp = 0�).

As it can be seen, the poststall domain of the thrust vectored aircraft motion can be divided into six different subspaces. The subspaces are divided by bifurcations. There were found two

9. Example of use of an advanced aerodynamic coefficient model

u\_ ¼ �qw þ

w\_ ¼ �qu þ

where <sup>X</sup> <sup>¼</sup> qS Cð Þ Lsin<sup>α</sup> � CDcos<sup>α</sup> , Z <sup>¼</sup> qS Cð Þ Lcos<sup>α</sup> <sup>þ</sup> CDsin<sup>α</sup> , CL <sup>¼</sup> CL<sup>0</sup> <sup>þ</sup> cA

, Cm <sup>¼</sup> Cm<sup>0</sup> <sup>þ</sup> cA

CD <sup>¼</sup> CD<sup>0</sup> <sup>þ</sup> cA

components.

<sup>2</sup><sup>V</sup> CDα\_α\_ þ CDq q 

180 Flight Physics - Models, Techniques and Technologies

H = 15,000 ft. (T = 22.7 kN, δvp = 0�).

the possible changes in angle of attack (see Figure 13).

X

Z

<sup>q</sup>\_ <sup>¼</sup> CmqScA <sup>þ</sup> Xlz <sup>þ</sup> Zlx � TxLxe Iy

<sup>θ</sup>\_ <sup>¼</sup> <sup>q</sup>

Here X and Z are the force components to x and z axis, M is the pitching moment, q are q are the dynamic pressure and pitch rate,respectively, and T,Tx, and Tz are the thrust and its

Different types of simple and classic aerodynamic coefficient models were applied that could not result in stable and acceptable solutions. Therefore, the described system of equations and analytical models of aerodynamic coefficients were filled up by data of F/A-18 aircraft [68, 69, 78]. These models defined the hysteresis effects, as well, and they may be used in full region of

The system of equation was solved by different numerical methods (Runge-Kutta and Adams-Moulton) with different step size. Software MATHLAB and ACSL were used in the simula-

Figure 14 shows the equilibrium surface obtained in the thrust-thrust deflection parameter space (left side) and the bifurcation curves (right side) for flight regime V = 0.3 M and

As it can be seen, the poststall domain of the thrust vectored aircraft motion can be divided into six different subspaces. The subspaces are divided by bifurcations. There were found two

tions. The results received were stable and the same at time steps 10�<sup>2</sup> and 10�<sup>6</sup> s.

<sup>M</sup> � gsin<sup>θ</sup> <sup>þ</sup>

<sup>M</sup> � gcos<sup>θ</sup> <sup>þ</sup>

<sup>2</sup><sup>V</sup> Cmα\_α\_ þ Cmq q 

Tx M

Tz M

(34)

,

<sup>2</sup><sup>V</sup> CLα\_α\_ þ CLq q 

, Tx ¼ Tcosδvp, Tx ¼ Tsinδvp:

The Department of Aeronautics, Naval architecture and Railway Vehicles at the Budapest University of Technology and Economics (operating two flight simulators, one air traffic management laboratory with several working environment for ATCOs, small gas turbines, water channel, etc.) is active in computational fluid dynamics [92, 93], vehicle design [24, 25, 94], vehicle motion simulation [95, 96], developing original and radically new technologies [97–99], and has worked on investigation of the thrust vectored aircraft motion at high angle of attack in poststall domain [100, 101], approximation of the motion after stall [102], and unconventional and critical flights [103, 104]. One of the excellent applications of the analytical models of the aerodynamic coefficients is their using in bifurcation analysis of the poststall motion of thrust vectored aircraft. Only the longitudinal motion was investigated. The applied system of equations defined by the use of body axis was reduced to four dimensions given in the following form [101]:

Figure 14. Equilibrium surface on the thrust-thrust deflection parameter space (left side) and the bifurcation curves (right side).

different types of bifurcation, e.g., Hopf (H) and saddle-nodes (SN) bifurcations. The first region at the small thrust and small angle of thrust-deflection is characterized the phugoid motion of aircraft before the stall. Oscillation of speed is greater than changes in angle of attack. The fighter slowly returns to the stable position.

As chosen by increasing the thrust and thrust-deflection, the system reads the first Hopf bifurcation (H1) (a small amplitude limit cycle appears at the bifurcation point). Further by increasing thrust and thrust deflection, there is no stable state of the aircraft. Over this second region, changes in the thrust and thrust deflection cause lack of stability before and poststall oscillation of the aircraft. This oscillation tends to the limit cycle and the angle of attack can reach the 90�.

By another Hopf-bifurcation, the system gains back its stability in the poststall regimes. This is the narrow streak area inside the second zone. At high thrust and thrust-deflection, the saddlenode bifurcation (SN) emerges creating jump phenomena. The motion of aircraft in zone appearing after first saddle-node bifurcation curve is an oscillation motion in the poststall domain.

Finally, in the last zone at very high thrust and thrust-deflection, an overpulling appears, when the angle of attack reaches over 90� during the first period of motion after changes in the thrust or thrust deflection.

The bifurcations were followed by continuation method. The input was generated in the thrust deflection (not in the thrust), as it would have been usual nonlinear approach. Components Tx, Tz were computed by the following formulas:

$$T\_x = T \cos \left(\delta\_{vp} + \varepsilon \cos \omega t\right), \quad T\_z = T \sin \left(\delta\_{vp} + \varepsilon \cos \omega t\right). \tag{35}$$

In some cases, several interesting changes were found in angle of attack response on oscillation in thrust deflection (Figure 15). Little bit nicer representation of this chaotic changes in angle of attack is given in Figure 16. This is a 3D phase plot by redrawing of simulation results shown in Figure 15. Such phase plot represents the chaos in system output received as results of periodic excitations and it is called as chaotic attractors.

Figure 15. Angle of attack response initiated by thrust oscillation with amplitude 2 and frequency 0.33 rad/s applied to initial condition of equilibrium at T = 35 kN and δvp=10<sup>0</sup> .

Figure 16. 3D phase plot of chaotic attractor is described by simulation results given in Figure 15.

A small change in system parameters or in excitations can cause a relatively big change in system output (Figure 17). For example, reduction of excitation frequency from 0.33 to 0.32 rad/s involved reduction of chaotic behavior in response and resulted in periodic orbits (see Figure 17). In some cases, the periodic orbits are reduced to one (it may be strange) limit cycle. The other figure shows that around 0.9 rad/s another type of noninear phenomenon appears, which is called period doubling bifurcation. At this point, the time period becomes twice as long (no sudden catastrophic change). Decreasing the frequency, a cascade of period doubling bifurcation happens leading to chaos around 0.65 rad/s. Figure 17 demonstrates several chaotic regions can appear (see chaotic window at the ω = 0.35 rad/s in Figure 17).

Further investigation of the aerodynamic coefficient models had been studied by use of sensitivity analysis and changes in structure of the models. The sensitivity analysis had shown that the changes in aerodynamic derivatives for 5 or 1% did not result in considerable changes in response on the applied oscillated thrust deflection.

Using the same mathematical model, initial condition, and excitation (at T = 35 kN, δvp = 10 deg., Δδvp =2d and ω = 0.33 rad/s), the simulations were realized with the use of different aerodynamic coefficient models, in which different parts, or derivatives, were omitted. The results show that elements cause the changes in angle of attack responses (Table 3).

Goal- and Object-Oriented Models of the Aerodynamic Coefficients http://dx.doi.org/10.5772/intechopen.71419 183

Figure 17. Stroboscopic map and different phase plots demonstrate the results of simulation thrust oscillation with amplitude 2 with varied frequency applied to initial condition of equilibrium at T = 35 kN and δvp=10<sup>0</sup> .


Table 3. Influence of aerodynamic model structure on the aircraft poststall motion initiated by cosine excitation in thrust deflection angle (sign shows the omitted elements).

#### 10. Conclusions

A small change in system parameters or in excitations can cause a relatively big change in system output (Figure 17). For example, reduction of excitation frequency from 0.33 to 0.32 rad/s involved reduction of chaotic behavior in response and resulted in periodic orbits (see Figure 17). In some cases, the periodic orbits are reduced to one (it may be strange) limit cycle. The other figure shows that around 0.9 rad/s another type of noninear phenomenon appears, which is called period doubling bifurcation. At this point, the time period becomes twice as long (no sudden catastrophic change). Decreasing the frequency, a cascade of period doubling bifurcation happens leading to chaos around 0.65 rad/s. Figure 17 demonstrates several chaotic regions can appear (see chaotic window at the

Figure 16. 3D phase plot of chaotic attractor is described by simulation results given in Figure 15.

Figure 15. Angle of attack response initiated by thrust oscillation with amplitude 2 and frequency 0.33 rad/s applied to

.

Further investigation of the aerodynamic coefficient models had been studied by use of sensitivity analysis and changes in structure of the models. The sensitivity analysis had shown that the changes in aerodynamic derivatives for 5 or 1% did not result in considerable changes in

Using the same mathematical model, initial condition, and excitation (at T = 35 kN, δvp = 10 deg., Δδvp =2d and ω = 0.33 rad/s), the simulations were realized with the use of different aerodynamic coefficient models, in which different parts, or derivatives, were omitted. The

results show that elements cause the changes in angle of attack responses (Table 3).

ω = 0.35 rad/s in Figure 17).

response on the applied oscillated thrust deflection.

initial condition of equilibrium at T = 35 kN and δvp=10<sup>0</sup>

182 Flight Physics - Models, Techniques and Technologies

Aerodynamics deals with interaction of air and bodies moving in it. The major task of aerodynamics is to define and describe the aerodynamic forces and moments generated on the bodies. Because of the very complex ways of causing the aerodynamic forces and moments, the nondimensional aerodynamic force and moment coefficient and series of their models had been developed for the last hundred years. This short chapter tries to show the different aspects having influences on "burning" the aerodynamic forces and moments and their contributing elements.

The aerodynamic coefficient models can be classified as simple, classic, developed, and advanced models. The models use the partial derivative coefficients, indicial step responses, analytical models, interpolation and approximation of the available wind tunnel, flight test, or numerical simulation data, and models are generated by utilization of the soft computing methods.

There is no unique and well-applicable method to selecting the required and best coefficient models. Always the object- and goal-oriented models must be selected. The identification, evaluation, and selection process may use the general methodology: (i) definition of the object, objectives, and goals, (ii) identification of the applicable models, (iii) evaluation of the identified models, (iv) selection of the best models, (v) development of the systems applying the selected aerodynamic coefficient models (including the verification and validation, too), and (vi) final decision.

There are some recommendations supporting the selection of the aerodynamic coefficient models and an example demonstrates using a special model to complex motion of thrust vectored aircraft in poststall domain.
