**3.2.2 Non-linear mathematical models of flight dynamics**

Mathematical modelling of spin motion is a very complex task and, therefore, it is necessary to introduce series of assumptions and approximations. Mathematical model is based on the first and second non-linear equations systems, shown bellow. The first system (V=const and H=const) is:

$$\dot{\alpha} = a\_{11}\frac{1}{\cos\alpha} + \frac{\beta\,p}{\cos\alpha} + a\_{13}q + a\_{14}\frac{\cos\theta\cos\gamma}{\cos\alpha} + a\_{15}$$

$$\dot{\beta} = a\_{21}\beta + a\_{22}r\cos\alpha + a\_{23}p\sin\alpha + a\_{24}\cos\theta\sin\gamma + a\_{25}q$$

$$\dot{p} = a\_{31}rq + a\_{32}\beta + a\_{33}p + a\_{34}r + a\_{35} + a\_{36}$$

$$\dot{r} = a\_{41}pq + a\_{42}\beta + a\_{43}p + a\_{44}r + a\_{45}q + a\_{46} + a\_{47}$$

$$\dot{q} = a\_{51}pr + a\_{52}\Big{|}\quad\beta\ \ \left| + a\_{53}r + a\_{54}q + a\_{55} + a\_{56}\dot{\alpha} + a\_{57} \right| $$

movements of the airplane while spinning. It is form of "auto-rotation", which means that

In order to reach height, a sailplane pilot has to circle inside a thermal column in constant turns of a very small radius. If for any reason (severe turbulence, pilot's error, etc.) the speed of sailplane drops below the stalling speed, at such high angles of bank the sailplane will most probably fall into a spin. Spin is a very dangerous and unpleasant maneuver. Good

Vuk-T sailplane (Fig. 27) is a modern single seat, all composite sailplane for advanced pilot training and competitions, designed at the Belgrade Faculty of Mechanical Engineering. Beside the other complex analyses, spin characteristics have been analyzed thoroughly.

Major assumptions that were starting points in this subject are the adopted aerodynamic concepts of the sailplane and the full system of equations for the aircraft motion in case of spin. Evaluation of aerodynamic quotients and their derivatives, and thus, the equation system as a whole, is based on defined aircraft geometric characteristics and atmospheric conditions. The method developed for simulation of the real flight situation is based on the

Mathematical modelling of spin motion is a very complex task and, therefore, it is necessary to introduce series of assumptions and approximations. Mathematical model is based on the first and second non-linear equations systems, shown bellow. The first system (V=const and

> 11 13 14 15 1 cos cos cos cos cos *<sup>p</sup> <sup>a</sup> aq a <sup>a</sup>*

21 22 23 24 25

 

*a ar ap a* cos sin cos sin *a*

31 32 33 34 35 36 *p a rq a a p a r a a* 

41 42 43 44 45 46 47 *r a pq a a p a r a q a a*

51 52 53 54 55 56 57 *q a pr a a r a q a a a*

 

> 

> >

> 

 

there is a natural tendency for the airplane to rotate of its own accord.

spin recovery characteristics are the imperative for any modern sailplane.

Fig. 27. VUK-T sailplane

H=const) is:

first and second non-linear equation systems.

 

**3.2.2 Non-linear mathematical models of flight dynamics** 

$$\begin{aligned} \dot{\theta} &= a\_{61}r \sin \varphi + a\_{62}q \cos \varphi \\\\ \dot{\gamma} &= a\_{71}p + a\_{72}r \cos \varphi \tan \theta + a\_{73}q \sin \varphi \tan \theta \end{aligned}$$

The coefficients are given by:

<sup>11</sup> <sup>2</sup> *<sup>z</sup> S V a C m* <sup>12</sup> *a* 1 <sup>13</sup> *a* 1 <sup>14</sup> *<sup>q</sup> <sup>a</sup> V* <sup>15</sup> <sup>2</sup> *nk z nk S V aC t m* <sup>21</sup> <sup>2</sup> *<sup>y</sup> S V a C m* <sup>22</sup> *<sup>a</sup>* <sup>1</sup> <sup>23</sup> *<sup>a</sup>* <sup>1</sup> <sup>24</sup> <sup>V</sup> *<sup>g</sup> <sup>a</sup>* <sup>25</sup> <sup>2</sup> *yk y k S V aC t m* 31 *y z x I I a I* 2 <sup>32</sup> 2 *<sup>l</sup> x Sl V a C I* 2 <sup>33</sup> 2 *lp x Sl V a C I* 2 <sup>34</sup> 2 *lr x Sl V a C I* 2 <sup>35</sup> 2 *vk l k x S V aC t I* 2 <sup>36</sup> 2 *<sup>k</sup> l k x Sl V aCt I* <sup>41</sup> *z x y I I a I* 2 <sup>42</sup> 2 *<sup>n</sup> y Sl V a C I* 2 <sup>43</sup> 2 *np y Sl V a C I* 2 <sup>44</sup> 2 *nr y Sl V a C I* 45 *p p y I a I* 2 <sup>46</sup> 2 *n k vk y Sl V aC t I* 2 <sup>47</sup> 2 *n k <sup>k</sup> y Sl V aC t I* 51 *x y z I I a I* 2 <sup>52</sup> 2 *<sup>m</sup> z Sb V a C I* 53 *p p z I a I* 2 <sup>54</sup> 2 *mq z Sb V a C I* 2 <sup>55</sup> 2 *<sup>m</sup> z Sb V a C I* 2 <sup>56</sup> 2 *<sup>m</sup> z Sb V a C I* 2 <sup>57</sup> 2 *m n nk <sup>k</sup> z Sb V aC t I* <sup>61</sup> *a* 1 <sup>62</sup> *a* 1 <sup>71</sup> *a* 1 <sup>72</sup> *a* 1 <sup>73</sup> *a* 1

The second system (V and H are not constant) is:

$$\dot{\alpha} = b\_{11} \frac{\rho V}{\cos \alpha} + b\_{12} \frac{p\beta}{\cos \alpha} + b\_{13} q + b\_{14} \frac{\cos \theta \cos \gamma}{V \cos \alpha} + b\_{15} \frac{V \tan \alpha}{V} + b\_{16} \frac{\rho \dot{V}}{\cos \alpha}$$

$$\dot{\rho} = b\_{21} \rho V \beta + b\_{22} r \cos \alpha + b\_{23} p \sin \alpha + b\_{24} \frac{\cos \theta \cos \gamma}{V} + b\_{25} \frac{\dot{V} \beta}{V} + b\_{26} \rho V$$

$$\dot{p} = b\_{31} r q + b\_{32} \rho V^2 \beta - b\_{33} \rho V^2 p + b\_{34} \rho V^2 r + b\_{35} \rho V^2 + b\_{36} \rho V^2$$

$$\dot{r} = b\_{41} pq + b\_{42} \rho V^2 \beta + b\_{43} \rho V^2 p + b\_{44} \rho V^2 r + b\_{48} q + b\_{46} \rho V^2 + b\_{47} \rho V^2$$

$$\dot{q} = b\_{51} pr + b\_{52} \rho V^2 \Big| \quad \beta \quad | + b\_{53} r + b\_{54} \rho V^2 q + b\_{55} q + b\_{56} \rho V^2 \alpha + b\_{57} \rho V^2$$

$$\dot{\theta} = b\_{61} r \sin \gamma + b\_{62} q \cos \gamma$$

by creating the SIMULINK model and implementing appropriate functions for the numeric

The program has been tested on the VUK-T sailplane project developed at the Institute of Aeronautics, at Faculty of Mechanical Engineering in Belgrade. The obtained numerical results fully agree, and in some cases are complementing with the flight test results obtained

solving differential equations.

Fig. 28. Basic diagram (left), SE greater 40% (right)

**3.2.4 Program results** 

for this sailplane.

71 72 73 *bp br bq* cos tan sin tan 

$$\dot{V} = b\_{81} \frac{\rho V^2}{\cos \alpha} + b\_{82} V \dot{\alpha} \tan \alpha + b\_{83} V q \tan \alpha + b\_{84} \frac{V r \beta}{\cos \alpha} + b\_{85} \frac{\sin \theta}{\cos \alpha} + b\_{86} \frac{\rho V^2}{\cos \alpha}$$

$$\dot{H} = h\_{91}V\cos\alpha\sin\theta + h\_{92}V\sin\alpha\cos\theta\cos\gamma + h\_{93}V\beta\cos\theta\sin\gamma$$

and the applied coefficients are:

$$b\_{11} = -\frac{S}{2m}C\_{\omega}\left(a\right)b\_{12} = -1 \ b\_{13} = 1 \ b\_{14} = g \ b\_{15} = -1 \ b\_{16} = -\frac{S}{2m}C\_{\omega\omega\_{\omega}}\left(a\right)\Delta\delta\_{\text{f}\omega}\left(t\right)$$

$$b\_{21} = \frac{S}{2m}C\_{\gamma g}\left(a\right) \ b\_{22} = 1 \ b\_{23} = g \ b\_{25} = -1 \ b\_{26} = -\frac{S}{2m}C\_{\omega\omega\_{\omega}}\left(a\right)\Delta\delta\_{\text{f}\omega}\left(t\right)$$

$$b\_{31} = \frac{I\_y - I\_z}{I\_x} \quad b\_{32} = \frac{Sl}{2I\_z}C\_{\omega\_{\omega}}\left(a\right) \ b\_{33} = \frac{sSl}{2I\_x}C\_{\omega\omega}\left(a\right)\ b\_{34} = \frac{sSl}{2I\_x}C\_{\omega}\left(a\right)$$

$$b\_{35} = \frac{Sl}{2I\_x}C\_{\omega\omega\_{\omega}}\left(a\right)\Delta\delta\_{\text{f}\omega}\left(t\right)\ b\_{36} = \frac{sSl}{2I\_x}C\_{\omega\omega\_{\omega}}\left(a\right)\Delta\delta\_{\text{s}\omega}\left(t\right)$$

$$b\_{41} = \frac{I\_z - I\_x}{I\_y} \quad b\_{42} = \frac{sSl}{2I\_y}C\_{\omega\_{\omega}}\left(a\right)\ b\_{43} = \frac{sSl}{2I\_y}C\_{\omega\_{\omega}}\left(a\right)$$

$$b\_{44} = \frac{sSl}{2I\_y}C\_{\omega\omega}\left(a\right)\ b\_{45} = -\frac{I\_x\rho\_{\text{f}\omega}}{I\_z} \quad b\_{46} = \frac{sSl}{2I\_y}C\_{\omega\omega$$

#### **3.2.3 The general model**

51

*b*

Modeling was carried out using the software package MATLAB. Simulation was performed by using the SIMULINK module, having the special feature to simulate a dynamic system within a graphic mode, where the linear, non-linear, time-continuous or discrete multivariable systems having concentrated parameters can be analyzed. Simulation is achieved

 *bp br bq* cos tan sin tan 

81 82 83 84 85 86

<sup>91</sup> <sup>92</sup> <sup>93</sup> *H bV bV* cos sin sin cos cos cos sin

cos cos cos cos *<sup>V</sup> Vr <sup>V</sup> V b b V b Vq b b b*

 

2 2

<sup>12</sup> *<sup>b</sup>* <sup>1</sup> <sup>13</sup> *<sup>b</sup>* <sup>1</sup> <sup>14</sup> *b g* <sup>15</sup> *<sup>b</sup>* <sup>1</sup> <sup>16</sup> <sup>2</sup> *vk z hk*

<sup>22</sup> *<sup>b</sup>* <sup>1</sup> <sup>23</sup> *<sup>b</sup>* <sup>1</sup> <sup>24</sup> *b g* <sup>25</sup> *<sup>b</sup>* <sup>1</sup> <sup>26</sup> <sup>2</sup> *vk y vk*

 <sup>36</sup> <sup>2</sup> *<sup>k</sup> l vk x*

*I*

 <sup>33</sup> <sup>2</sup> *pl x Sl b C I*

 

sin

*<sup>m</sup>*

*<sup>m</sup>*

*Sl bC t*

 

 <sup>43</sup> <sup>2</sup> *np y Sl b C I*

<sup>61</sup> *b* 1 <sup>62</sup> *b* 1 <sup>71</sup> *b* 1 <sup>72</sup> *b* 1 <sup>73</sup> *b* 1

<sup>82</sup> *<sup>b</sup>* <sup>1</sup> <sup>83</sup> *<sup>b</sup>* <sup>1</sup> <sup>84</sup> *<sup>b</sup>* <sup>1</sup> <sup>85</sup> *b g* <sup>86</sup> <sup>2</sup> *nk x hk*

*b V*

*<sup>S</sup> bC t*

*<sup>S</sup> bC t*

 

 <sup>34</sup> <sup>2</sup> *<sup>r</sup> l x Sl b C I*

 <sup>47</sup> <sup>2</sup> *n k <sup>k</sup> y Sl bC t*

*<sup>S</sup> bC t*

 

 

 <sup>56</sup> <sup>2</sup> *<sup>m</sup> z Sb b C I*

 

*I*

 <sup>55</sup> <sup>2</sup> *<sup>m</sup> z Sb b C I*

*<sup>m</sup>*

 

 

71 72 73

tan tan

<sup>32</sup> <sup>2</sup> *<sup>l</sup>*

*x*

<sup>41</sup> *z x y*

*I I <sup>b</sup> I*

*I*

*x Sl b C I* 

<sup>35</sup> <sup>2</sup> *vk l vk*

*Sl bC t*

 

*p p z*

*I* 

*I*

*b*

<sup>57</sup> <sup>2</sup> *m n nk <sup>k</sup>*

*Sb bC t*

  <sup>42</sup> <sup>2</sup> *<sup>n</sup>*

*p p z*

*I* 

*I*

*y Sl b C I*

 <sup>46</sup> <sup>2</sup> *n v vk <sup>k</sup> y*

> <sup>54</sup> <sup>2</sup> *mn z Sb b C I*

<sup>91</sup> *b* 1 <sup>92</sup> *b* 1 <sup>93</sup> *b* 1

Modeling was carried out using the software package MATLAB. Simulation was performed by using the SIMULINK module, having the special feature to simulate a dynamic system within a graphic mode, where the linear, non-linear, time-continuous or discrete multivariable systems having concentrated parameters can be analyzed. Simulation is achieved

*I*

*Sl bC t*

 

and the applied coefficients are:

<sup>11</sup> <sup>2</sup> *<sup>z</sup> <sup>S</sup> b C <sup>m</sup>*

<sup>21</sup> <sup>2</sup> *<sup>y</sup> <sup>S</sup> b C <sup>m</sup>*

31

*b*

<sup>44</sup> <sup>2</sup> *nr y Sl b C I*

<sup>52</sup> <sup>2</sup> *<sup>m</sup>*

51

*b*

*x y z I I*

*I*

45

*z Sb b C I*

*I*

<sup>81</sup> <sup>2</sup> *<sup>x</sup> <sup>S</sup> b C <sup>m</sup>*

**3.2.3 The general model** 

*z*

*b*

 53

 

*y z x I I*

*I*

by creating the SIMULINK model and implementing appropriate functions for the numeric solving differential equations.
