**7. Methods for recovery from inverted spin**

To recover modern aircrafts from an inverted spin there are three basic methods (letter "L" denotes the method for recovery from an inverted spin):


The N1L method is recommended for recovery from an unstable inverted spin, the N2L method from a stable wavering spin, and method N3L from an inverted stable uniform spin.

Deflection of controls and commanding surfaces during inverted spin recovery according to the three basic methods is shown in Fig.13, with the adopted notation shown in Fig.12.

For supersonic aircrafts, most often in use is the method N2L, because these aircrafts are most characterized by an unstable wavering inverted spin. In general, supersonic airplanes rarely fall into inverted spin; more often, it is a upright spin.

As a rule, airplanes of usual geometry more easily recover from an inverted spin than from a upright spin. This is explained by the fact that during this regime the autorotation is weaker, the rudder is more efficient (practically it is out of the wing and stabilizer flow), the efficient arrow angle of the vertical surfaces is decreased, and the average absolute values of attack angles are reduced.

However, despite of everything already said, for the pilot the inverted spin is always more difficult than the upright spin. This is conditioned by the unusual position of the pilot: hanging on restraint harnesses, head down, and a negative load (*nz <* 0) tends to detach him from the seat. In such conditions, the pilot could drop the yoke and release the pedals (he could lose control of the aircraft, especially if he is not firmly seated).

Sometimes, during aircraft wavering the pilot has great difficulty to visually determine type of spin - upright or inverted. This is expressed if the aircrafts longitudinal axis is close to the vertical axis - the plane will "swirl" at low, by absolute value, negative overcritical attack angles, which is typical for a stable inverted spin. In this case, in order to stop autorotation the rudders have to be put in neutral position.

In the absence of or inability to use visual landmarks (also, to have control over a classical spin), the pilot can easily determine the type of spin by sensation: if the seat is pressuring the pilot - it is a upright spin, if the pilot is detaching from his seat, i.e. hanging on restraint harnesses - it is an inverted spin). However, if the aircraft exits at high negative attack angles that randomly change during the inverted spin regime, determining spin type by sensation is inapplicable.

During an inverted spin, it is more difficult to determine the direction of rotation (whether the plane is rotating to the right or to the left). In a classical spin, when the position of the 14 Will-be-set-by-IN-TECH

The sequence of actions with the rudders for spin recovery depends on the nature of the interaction of inertial rolling, yawing, and pitching moments, with the inertial moments

To recover modern aircrafts from an inverted spin there are three basic methods (letter "L"

• Method N1L - spin recovery by simultaneous positioning elevator and rudder into neutral,

• Method N2L - spin recovery by deflecting the rudder fully opposite, followed by a delayed

• Method N3L - spin recovery by fully deflecting the both rudder and elevator (delayed for

The N1L method is recommended for recovery from an unstable inverted spin, the N2L method from a stable wavering spin, and method N3L from an inverted stable uniform spin. Deflection of controls and commanding surfaces during inverted spin recovery according to the three basic methods is shown in Fig.13, with the adopted notation shown in Fig.12.

For supersonic aircrafts, most often in use is the method N2L, because these aircrafts are most characterized by an unstable wavering inverted spin. In general, supersonic airplanes rarely

As a rule, airplanes of usual geometry more easily recover from an inverted spin than from a upright spin. This is explained by the fact that during this regime the autorotation is weaker, the rudder is more efficient (practically it is out of the wing and stabilizer flow), the efficient arrow angle of the vertical surfaces is decreased, and the average absolute values of attack

However, despite of everything already said, for the pilot the inverted spin is always more difficult than the upright spin. This is conditioned by the unusual position of the pilot: hanging on restraint harnesses, head down, and a negative load (*nz <* 0) tends to detach him from the seat. In such conditions, the pilot could drop the yoke and release the pedals (he

Sometimes, during aircraft wavering the pilot has great difficulty to visually determine type of spin - upright or inverted. This is expressed if the aircrafts longitudinal axis is close to the vertical axis - the plane will "swirl" at low, by absolute value, negative overcritical attack angles, which is typical for a stable inverted spin. In this case, in order to stop autorotation

In the absence of or inability to use visual landmarks (also, to have control over a classical spin), the pilot can easily determine the type of spin by sensation: if the seat is pressuring the pilot - it is a upright spin, if the pilot is detaching from his seat, i.e. hanging on restraint harnesses - it is an inverted spin). However, if the aircraft exits at high negative attack angles that randomly change during the inverted spin regime, determining spin type by sensation is

During an inverted spin, it is more difficult to determine the direction of rotation (whether the plane is rotating to the right or to the left). In a classical spin, when the position of the

(2 - 4 sec) elevator positioning into neutral, with ailerons in neutral position;

2 - 4 sec) opposite to spin direction, with ailerons in neutral position.

created by rudder deflection during autorotation.

**7. Methods for recovery from inverted spin**

with ailerons in neutral position;

denotes the method for recovery from an inverted spin):

fall into inverted spin; more often, it is a upright spin.

the rudders have to be put in neutral position.

could lose control of the aircraft, especially if he is not firmly seated).

angles are reduced.

inapplicable.

Fig. 13. Three basic methods of recovery from an inverted left spin for modern aircrafts (conditional annotation of yoke and pedal position is same as in Fig.12.)

nose of the aircraft to the horizon is practically unchanged, the direction of rotation can easily be determined according to the angular speed of rotation. However, when in inverted spin at high angular velocities of rolling, uneven motion of roll and pitch, it is impossible to determine the aircraft's direction of rotation with the mentioned method. The situation becomes more complex, because during inverted spin rolling motion is opposite to rotation. For a pilot seated in front cockpit look forward this means that, for example, in a right-hand spin the direction of rotation will into the left.

In a upright spin, the situation is opposite: directions of roll and yaw are the same. As so, for example, in a right upright spin the pilot can see the nose of the aircraft turning to the right and plane leaning to the same side. A more experienced pilot in this matter can distinguish a upright spin from an inverted.

Pilots lacking of sufficient flights for spin recovery training often determine spin direction according to rolling direction, but not yawing direction, because the rolling angular velocity

where:

coordinate system;

system;

system;

where:

• *Vx*, *Vy*, *Vz* correspond to projections of the velocity of aircraft center of gravity with respect

Spin and Spin Recovery 225

• *p*, *q*, and *r* represent projections of the aircraft angular velocities to the axes of the adopted

• *Ix*, *Iy*, *Iz* - aircraft axial inertia moments with respect to axes of the adopted coordinate

• *Rx*, *Ry*, *Rz* - projections of aerodynamic forces acting on the aircraft with respect to axes of

• *Gx*, *Gy*, *Gz* - projection of aircraft weight with respect to axes of the adopted coordinate

• *Mx*, *My*, *Mz* - projection of resulting moments from external forces acting on aircraft, with

In order to simplify the task during tests, following assumptions are made: the angle of sideslip is small so that *sinβ* ≈ *β* and *cosβ* ≈ 1, effects of Mach and Reynolds number are ignored, and motion is investigated with engines turn off. This study uses known kinematic

*<sup>H</sup>*˙ <sup>=</sup> *<sup>V</sup>* cos *<sup>α</sup>* sin *<sup>θ</sup>* <sup>−</sup> *<sup>V</sup>* sin *<sup>α</sup>* cos *<sup>γ</sup>* cos *<sup>θ</sup>* <sup>−</sup> *<sup>V</sup> <sup>β</sup>* sin *<sup>γ</sup>* cos *<sup>θ</sup>*

If it is assumed that velocity and altitude are unchanged, i.e. *V* = *const* and *H* = *const*, and if the right-hand sides of Eq.(3) are approximated by a Taylor polynomial, the following

*β*˙ = *a*<sup>21</sup> *β* + *a*<sup>22</sup> *r* cos *α* + *a*<sup>23</sup> *p* sin *α* + *a*<sup>24</sup> cos *θ* sin *γ* + *a*<sup>25</sup>

*q*˙ = *a*<sup>51</sup> *p r* + *a*<sup>52</sup> | *β* | +*a*<sup>53</sup> *r* + *a*<sup>54</sup> *q* + *a*<sup>55</sup> + *a*<sup>56</sup> *α*˙ + *a*<sup>57</sup>

+ *a*<sup>13</sup> *q* + *a*<sup>14</sup>

*r*˙ = *a*<sup>41</sup> *p q* + *a*<sup>42</sup> *β* + *a*<sup>43</sup> *p* + *a*<sup>44</sup> *r* + *a*<sup>45</sup> *q* + *a*<sup>46</sup> + *a*<sup>47</sup> (4)

*γ*˙ = *p* + (*q* sin *γ* − *r* cos *γ*) tan *θ* (3)

cos *θ* cos *γ* cos *α*

+ *a*<sup>15</sup>

to the axes of the adopted coordinate system;

respect to axes of the adopted coordinate system.

relations that are, with the adopted simplifications, equal to:

*θ* = *r* sin *γ* + *q* cos *γ*

• *θ* - pitch angle, angle between *X* axis and horizontal plane,

simplified system of equations is obtained, a so-called **System I**:

*θ* = *a*<sup>61</sup> *r* sin *γ* + *a*<sup>62</sup> *q* cos *γ*

+ *a*<sup>12</sup>

1 cos *α*

• *γ* - angle of transverse inclination, angle between *Y* axis and vertical plane.

*β p* cos *α*

*p*˙ = *a*<sup>31</sup> *r q* + *a*<sup>32</sup> *β* + *a*<sup>33</sup> *p* + *a*<sup>34</sup> *r* + *a*<sup>35</sup> + *a*<sup>36</sup>

*γ*˙ = *a*<sup>71</sup> *p* + *a*<sup>72</sup> *r* cos *γ* tan *θ* + *a*<sup>73</sup> *q* sin *γ* tan *θ*

• *Ixy* - aircraft centrifugal inertia moment;

the adopted coordinate system;

**8.1 Motion equations for modeling**

˙

*α*˙ = *a*<sup>11</sup>

˙

is usually higher than the yawing angular velocity (except for a flat spin). Determination of spin direction in this manner is applied more often (rolling angular velocity increasing). Use of this manner of determining spin during an inverted spin will only disorient an insufficiently trained pilot.

Therefore, when recovering from this regime, it is necessary to have safe means of control to facilitate easy conservation of spatial orientation and assure possibilities for effective and proper actions with wings. Such means could be the yaw indicator (it hand always turns in yaw direction regardless on spin type) and attack angle indicator, and if it is not present - a normal load indicator. Attack angle indicator allows the pilot to reliably determine the spin type (upright or inverted), and the yaw indicator - its direction (left or right).

#### **8. Spin modeling**

Modeling is the most accurate graph-analytical method for determining aircraft characteristics prior to flight tests. Modeling of flight conditions, with initial data correction, quite properly reflects timely development of aircraft motion and enables more complete conclusions about flight test results.

Fig. 14. Attached Coordinate System)

The differential equations of aircraft motion relative to its center of gravity is obtained from the Law of conversation of momentum. This is the moment equation. Projecting these equations to axes of the attached coordinate system (*X*1,*Y*1, *Z*1), shown in Fig.14, whose axes we denote as (*X*, *Y*, *Z*) for simplicity, the following system of differential equations is derived:

$$m\left(\frac{dV\_\mathbf{x}}{dt} + q\,V\_\mathbf{z} - r\,V\_\mathbf{y}\right) = R\_\mathbf{x} + G\_\mathbf{x}$$

$$m\left(\frac{dV\_\mathbf{y}}{dt} + r\,V\_\mathbf{x} - p\,V\_\mathbf{z}\right) = R\_\mathbf{y} + G\_\mathbf{y} \tag{1}$$

$$m\left(\frac{dV\_\mathbf{z}}{dt} + p\,V\_\mathbf{y} - q\,V\_\mathbf{x}\right) = R\_\mathbf{z} + G\_\mathbf{z}$$

$$I\_\mathbf{x}\,\frac{dp}{dt} + \left(I\_\mathbf{z} - I\_\mathbf{y}\right)q\,r + I\_\mathbf{x}y\left(p\,r - \frac{dp}{dt}\right) = \mathcal{M}\_\mathbf{x}$$

$$I\_\mathbf{y}\,\frac{dq}{dt} + \left(I\_\mathbf{x} - I\_\mathbf{z}\right)p\,r + I\_\mathbf{xy}\left(q\,r - \frac{dp}{dt}\right) = \mathcal{M}\_\mathbf{y} \tag{2}$$

$$I\_\mathbf{z}\,\frac{dr}{dt} + \left(I\_\mathbf{y} - I\_\mathbf{x}\right)p\,q + I\_\mathbf{xy}\left(p^2 - q^2\right) = \mathcal{M}\_\mathbf{z}$$

where:

16 Will-be-set-by-IN-TECH

is usually higher than the yawing angular velocity (except for a flat spin). Determination of spin direction in this manner is applied more often (rolling angular velocity increasing). Use of this manner of determining spin during an inverted spin will only disorient an insufficiently

Therefore, when recovering from this regime, it is necessary to have safe means of control to facilitate easy conservation of spatial orientation and assure possibilities for effective and proper actions with wings. Such means could be the yaw indicator (it hand always turns in yaw direction regardless on spin type) and attack angle indicator, and if it is not present - a normal load indicator. Attack angle indicator allows the pilot to reliably determine the spin

Modeling is the most accurate graph-analytical method for determining aircraft characteristics prior to flight tests. Modeling of flight conditions, with initial data correction, quite properly reflects timely development of aircraft motion and enables more complete conclusions about

The differential equations of aircraft motion relative to its center of gravity is obtained from the Law of conversation of momentum. This is the moment equation. Projecting these equations to axes of the attached coordinate system (*X*1,*Y*1, *Z*1), shown in Fig.14, whose axes we denote

*dt* <sup>+</sup> *q Vz* <sup>−</sup> *r Vy*

*dt* <sup>+</sup> *r Vx* <sup>−</sup> *p Vz*

*dt* <sup>+</sup> *p Vy* <sup>−</sup> *q Vx*

 *<sup>p</sup>*<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> 

*p r* <sup>−</sup> *dp dt* 

*q r* <sup>−</sup> *dp dt* 

= *Rx* + *Gx*

= *Rz* + *Gz*

= M*<sup>x</sup>*

= M*<sup>z</sup>*

= *Ry* + *Gy* (1)

= M*<sup>y</sup>* (2)

as (*X*, *Y*, *Z*) for simplicity, the following system of differential equations is derived:

*m dVx*

*m dVy*

*m dVz*

*dt* + (*Iz* <sup>−</sup> *Iy*) *q r* <sup>+</sup> *Ixy*

*dt* + (*Ix* <sup>−</sup> *Iz*) *p r* <sup>+</sup> *Ixy*

*dt* + (*Iy* <sup>−</sup> *Ix*) *p q* <sup>+</sup> *Ixy*

type (upright or inverted), and the yaw indicator - its direction (left or right).

trained pilot.

**8. Spin modeling**

flight test results.

Fig. 14. Attached Coordinate System)

*Ix dp*

*Iy dq*

> *Iz dr*


#### **8.1 Motion equations for modeling**

In order to simplify the task during tests, following assumptions are made: the angle of sideslip is small so that *sinβ* ≈ *β* and *cosβ* ≈ 1, effects of Mach and Reynolds number are ignored, and motion is investigated with engines turn off. This study uses known kinematic relations that are, with the adopted simplifications, equal to:

$$\begin{aligned} \dot{\theta} &= r \sin \gamma + q \cos \gamma \\ \dot{\gamma} &= p + (q \sin \gamma - r \cos \gamma) \tan \theta \\ \dot{H} &= V \cos \kappa \sin \theta - V \sin \kappa \cos \gamma \cos \theta - V \beta \sin \gamma \cos \theta \end{aligned} \tag{3}$$

where:


If it is assumed that velocity and altitude are unchanged, i.e. *V* = *const* and *H* = *const*, and if the right-hand sides of Eq.(3) are approximated by a Taylor polynomial, the following simplified system of equations is obtained, a so-called **System I**:

$$\begin{aligned} \dot{m} &= a\_{11} \frac{1}{\cos \alpha} + a\_{12} \frac{\beta \, p}{\cos \alpha} + a\_{13} \, q + a\_{14} \frac{\cos \theta \, \cos \gamma}{\cos \alpha} + a\_{15} \\\\ \dot{\beta} &= a\_{21} \, \dot{\beta} + a\_{22} \, r \cos \alpha + a\_{23} \, p \sin \alpha + a\_{24} \cos \theta \, \sin \gamma + a\_{25} \end{aligned}$$

$$\begin{aligned} \dot{p} &= a\_{31} \, r \, q + a\_{32} \, \dot{\beta} + a\_{33} \, p + a\_{34} \, r + a\_{35} + a\_{36} \\\ \dot{r} &= a\_{41} \, p \, q + a\_{42} \, \beta + a\_{43} \, p + a\_{44} \, r + a\_{45} \, q + a\_{46} + a\_{47} \\\ \dot{q} &= a\_{51} \, p \, r + a\_{52} \, \left| \, \beta \, \big|\, + a\_{53} \, r + a\_{54} \, q + a\_{55} + a\_{56} \, \dot{\alpha} + a\_{57} \right. \\\ \dot{\theta} &= a\_{61} \, r \sin \gamma + a\_{62} \, q \cos \gamma \\\ \dot{\gamma} &= a\_{71} \, p + a\_{72} \, r \cos \gamma \, \tan \theta + a\_{73} \, q \sin \gamma \, \tan \theta \end{aligned}$$

*r*˙ = *b*<sup>41</sup> *p q* + *b*<sup>42</sup> *ρ V*<sup>2</sup> *β* + *b*<sup>43</sup> *ρ V*<sup>2</sup> *p* + *b*<sup>44</sup> *ρ V*<sup>2</sup> *r* + *b*<sup>45</sup> *q* + *b*<sup>46</sup> *ρ V*<sup>2</sup> + *b*<sup>47</sup> *ρ V*<sup>2</sup> *<sup>q</sup>*˙ <sup>=</sup> *<sup>b</sup>*<sup>51</sup> *p r* <sup>+</sup> *<sup>b</sup>*<sup>52</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> <sup>|</sup> *<sup>β</sup>* <sup>|</sup> <sup>+</sup>*b*<sup>53</sup> *<sup>r</sup>* <sup>+</sup> *<sup>b</sup>*<sup>54</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> *<sup>q</sup>* <sup>+</sup> *<sup>b</sup>*<sup>55</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>56</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> *<sup>α</sup>*˙ <sup>+</sup> *<sup>b</sup>*<sup>57</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

Spin and Spin Recovery 227

*H*˙ = *b*<sup>91</sup> *V* cos *α* sin *θ* + *b*<sup>92</sup> *V* sin *α* cos *θ* cos *γ* + *b*<sup>93</sup> *V β* cos *θ* sin *γ* (6)

*Cz*(*α*) *b*<sup>12</sup> = −1 *b*<sup>13</sup> = 1 *b*<sup>14</sup> = *g*

*Cz<sup>δ</sup>hk* (*α*) <sup>Δ</sup>*δhk*(*t*) *<sup>b</sup>*<sup>21</sup> <sup>=</sup> *<sup>S</sup>*

*Ix*

2 *Ix Clδk*

*Cn<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>43</sup> <sup>=</sup> *S l*

2 *Iy Cn<sup>δ</sup><sup>k</sup>*

*Cm<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>53</sup> <sup>=</sup> *Ip <sup>ω</sup><sup>p</sup>*

*b*<sup>92</sup> = −1 *b*<sup>93</sup> = −1 (7)

*Cm*(*α*) *<sup>b</sup>*<sup>56</sup> <sup>=</sup> *S b*

*Iy*

*Clp* (*α*) *<sup>b</sup>*<sup>34</sup> <sup>=</sup> *S l*

*V r β* cos *α*

+ *b*<sup>85</sup>

sin *θ* cos *α*

2 *m*

2 *Ix Clr* (*α*)

*Cnp* (*α*)

(*α*) Δ*δk*(*t*)

(*α*) Δ*δk*(*t*)

*Iz*

2 *m*

2 *Iz*

*Cx*(*α*)

*Cmα*˙ (*α*)

2 *Iy*

+ *b*<sup>86</sup>

*Cy<sup>β</sup>* (*α*)

*ρ V*<sup>2</sup> cos *α*

+ *b*<sup>82</sup> *V α*˙ tan *α* + *b*<sup>83</sup> *V q* tan *α* + *b*<sup>84</sup>

2 *m*

*Cy<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>31</sup> <sup>=</sup> *Iy* <sup>−</sup> *Iz*

(*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>36</sup> <sup>=</sup> *S l*

2 *Ix*

*b*<sup>22</sup> = 1 *b*<sup>23</sup> = 1 *b*<sup>24</sup> = *g b*<sup>25</sup> = −1

*<sup>b</sup>*<sup>42</sup> <sup>=</sup> *S l* 2 *Iy*

*Cnr* (*α*) *<sup>b</sup>*<sup>45</sup> <sup>=</sup> <sup>−</sup> *Ip <sup>ω</sup><sup>p</sup>*

*<sup>b</sup>*<sup>52</sup> <sup>=</sup> *S b* 2 *Iz*

*<sup>b</sup>*<sup>71</sup> <sup>=</sup> <sup>1</sup> *<sup>b</sup>*<sup>72</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>b</sup>*<sup>73</sup> <sup>=</sup> <sup>1</sup> *<sup>b</sup>*<sup>81</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>*

*b*<sup>82</sup> = 1 *b*<sup>83</sup> = −1 *b*<sup>84</sup> = −1 *b*<sup>85</sup> = −*g*

*Cx<sup>δ</sup>hk* (*α*)Δ*δhk*(*t*) *b*<sup>91</sup> = 1

*Cmq* (*α*) *<sup>b</sup>*<sup>55</sup> <sup>=</sup> *S b*

*Cn<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>47</sup> <sup>=</sup> *S l*

2 *Iz*

*Cm<sup>δ</sup>hk* (*α*) Δ*δhk*(*t*) *b*<sup>61</sup> = 1 *b*<sup>62</sup> = 1

*Cl<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>33</sup> <sup>=</sup> *S l*

The coefficients involved in the system of equations (6) are defined by expressions:

˙

*V*˙ = *b*<sup>81</sup>

*θ* = *b*<sup>61</sup> *r* sin *γ* + *b*<sup>62</sup> *q* cos *γ*

*ρ V*<sup>2</sup> cos *α*

*<sup>b</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>*

*<sup>b</sup>*<sup>26</sup> <sup>=</sup> *<sup>S</sup>* 2 *m*

*<sup>b</sup>*<sup>32</sup> <sup>=</sup> *S l* 2 *Ix*

*<sup>b</sup>*<sup>35</sup> <sup>=</sup> *S l* 2 *Ix Cl<sup>δ</sup>vk*

*<sup>b</sup>*<sup>44</sup> <sup>=</sup> *S l* 2 *Iy*

*<sup>b</sup>*<sup>46</sup> <sup>=</sup> *S l* 2 *Iy*

*<sup>b</sup>*<sup>54</sup> <sup>=</sup> *S b* 2 *Iz*

*<sup>b</sup>*<sup>57</sup> <sup>=</sup> *S b* 2 *Iz*

*<sup>b</sup>*<sup>86</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>*

Notation in previous equations are:

• *Cz* - lift coefficient; • *Cx* - drag coefficient;

2 *m*

*<sup>b</sup>*<sup>51</sup> <sup>=</sup> *Ix* <sup>−</sup> *Iy Iz*

*<sup>b</sup>*<sup>41</sup> <sup>=</sup> *Iz* <sup>−</sup> *Ix Iy*

2 *m*

*<sup>b</sup>*<sup>15</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>b</sup>*<sup>16</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>*

*γ*˙ = *b*<sup>71</sup> *p* + *b*<sup>72</sup> *r* cos *γ* tan *θ* + *b*<sup>73</sup> *q* sin *γ* tan *θ*

The coefficients involved in the system of equations (4) are defined by expressions:

*<sup>a</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m Cz*(*α*) *<sup>a</sup>*<sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>a</sup>*<sup>13</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>14</sup> <sup>=</sup> *<sup>g</sup> V <sup>a</sup>*<sup>15</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m Cz<sup>δ</sup>hk* (*α*) <sup>Δ</sup>*δhk*(*t*) *<sup>a</sup>*<sup>21</sup> <sup>=</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m Cy<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>22</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>23</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>24</sup> <sup>=</sup> *<sup>g</sup> <sup>V</sup> <sup>a</sup>*<sup>25</sup> <sup>=</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m Cy<sup>δ</sup>vk* (*α*) Δ*δvk*(*t*) *<sup>a</sup>*<sup>31</sup> <sup>=</sup> *Iy* <sup>−</sup> *Iz Ix <sup>a</sup>*<sup>32</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>* 2 *Ix Cl<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>33</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix Clp* (*α*) *<sup>a</sup>*<sup>34</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix Clr* (*α*) *<sup>a</sup>*<sup>35</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix Cl<sup>δ</sup>vk* (*α*) Δ*δvk*(*t*) *<sup>a</sup>*<sup>36</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix Clδk* (*α*) <sup>Δ</sup>*δk*(*t*) *<sup>a</sup>*<sup>41</sup> <sup>=</sup> *Iz* <sup>−</sup> *Ix Iy <sup>a</sup>*<sup>42</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy Cn<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>43</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy Cnp* (*α*) *<sup>a</sup>*<sup>44</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy Cnr* (*α*) *<sup>a</sup>*<sup>45</sup> <sup>=</sup> <sup>−</sup> *Ip <sup>ω</sup><sup>p</sup> Iy <sup>a</sup>*<sup>46</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy Cn<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>a</sup>*<sup>47</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy Cn<sup>δ</sup><sup>k</sup>* (*α*) Δ*δk*(*t*) *<sup>a</sup>*<sup>51</sup> <sup>=</sup> *Ix* <sup>−</sup> *Iy Iz <sup>a</sup>*<sup>52</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz Cm<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>53</sup> <sup>=</sup> *Ip <sup>ω</sup><sup>p</sup> Iz <sup>a</sup>*<sup>54</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz Cmq* (*α*) *<sup>a</sup>*<sup>55</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz Cm*(*α*) *<sup>a</sup>*<sup>56</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz Cmα*˙ (*α*) *<sup>a</sup>*<sup>57</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz Cm<sup>δ</sup>hk* (*α*) Δ*δhk*(*t*) *a*<sup>61</sup> = 1 *a*<sup>62</sup> = 1 *a*<sup>71</sup> = 1 *a*<sup>72</sup> = −1 *a*<sup>73</sup> = 1 (5)

If assumed that velocity and altitude are changeable over time, i.e. *V* = *f*(*t*) and *H* = *f*(*t*), and if the right-hand side of Eq.(1) and (3) (projections of aerodynamic forces, weights and moments) is approximated by Taylor polynomial, the following simplified system of equations can be obtained, a so-called **System II**:

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

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

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

18 Will-be-set-by-IN-TECH

*Cz*(*α*) *<sup>a</sup>*<sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>a</sup>*<sup>13</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>14</sup> <sup>=</sup> *<sup>g</sup>*

2 *Ix*

2 *Iy*

*Iy*

2 *Iz*

If assumed that velocity and altitude are changeable over time, i.e. *V* = *f*(*t*) and *H* = *f*(*t*), and if the right-hand side of Eq.(1) and (3) (projections of aerodynamic forces, weights and moments) is approximated by Taylor polynomial, the following simplified system of

2 *Iz*

*a*<sup>61</sup> = 1 *a*<sup>62</sup> = 1 *a*<sup>71</sup> = 1 *a*<sup>72</sup> = −1 *a*<sup>73</sup> = 1 (5)

cos *θ* cos *γ V* cos *α*

2 *m*

*Cl<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>33</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

*Cl<sup>δ</sup>vk*

*Iy*

*Cnp* (*α*)

2 *Iy*

*Cm<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>53</sup> <sup>=</sup> *Ip <sup>ω</sup><sup>p</sup>*

*Cm*(*α*)

*Cn<sup>δ</sup><sup>k</sup>*

*Cm<sup>δ</sup>hk* (*α*) Δ*δhk*(*t*)

+ *b*<sup>15</sup>

*<sup>V</sup>* <sup>+</sup> *<sup>b</sup>*<sup>25</sup>

cos *θ* cos *γ*

*V*˙ tan *α*

*<sup>V</sup>* <sup>+</sup> *<sup>b</sup>*<sup>16</sup>

*V*˙ *β*

*ρ V* cos *α*

*<sup>V</sup>* <sup>+</sup> *<sup>b</sup>*<sup>26</sup> *<sup>ρ</sup> <sup>V</sup>*

(*α*) Δ*δk*(*t*)

*Iz*

*<sup>V</sup> <sup>a</sup>*<sup>25</sup> <sup>=</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>*

*Cy<sup>β</sup>* (*α*)

2 *Ix*

(*α*) Δ*δvk*(*t*)

2 *m*

*V*

*Cy<sup>δ</sup>vk* (*α*) Δ*δvk*(*t*)

*Clp* (*α*)

The coefficients involved in the system of equations (4) are defined by expressions:

*Cz<sup>δ</sup>hk* (*α*) <sup>Δ</sup>*δhk*(*t*) *<sup>a</sup>*<sup>21</sup> <sup>=</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>*

*<sup>a</sup>*<sup>32</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>* 2 *Ix*

(*α*) *<sup>a</sup>*<sup>35</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

*Cn<sup>β</sup>* (*α*) *<sup>a</sup>*<sup>43</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

*Cnr* (*α*) *<sup>a</sup>*<sup>45</sup> <sup>=</sup> <sup>−</sup> *Ip <sup>ω</sup><sup>p</sup>*

*<sup>a</sup>*<sup>52</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz*

*Cmq* (*α*) *<sup>a</sup>*<sup>55</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

*Cmα*˙ (*α*) *<sup>a</sup>*<sup>57</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

+ *b*<sup>13</sup> *q* + *b*<sup>14</sup>

*<sup>p</sup>*˙ <sup>=</sup> *<sup>b</sup>*<sup>31</sup> *r q* <sup>+</sup> *<sup>b</sup>*<sup>32</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> *<sup>β</sup>* <sup>−</sup> *<sup>b</sup>*<sup>33</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> *<sup>p</sup>* <sup>+</sup> *<sup>b</sup>*<sup>34</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> *<sup>r</sup>* <sup>+</sup> *<sup>b</sup>*<sup>35</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>36</sup> *<sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

(*α*) <sup>Δ</sup>*δk*(*t*) *<sup>a</sup>*<sup>41</sup> <sup>=</sup> *Iz* <sup>−</sup> *Ix*

*Cn<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>a</sup>*<sup>47</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup>

*<sup>a</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m*

*<sup>a</sup>*<sup>15</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup> <sup>ρ</sup> <sup>V</sup>* 2 *m*

*<sup>a</sup>*<sup>31</sup> <sup>=</sup> *Iy* <sup>−</sup> *Iz Ix*

*<sup>a</sup>*<sup>34</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix*

*<sup>a</sup>*<sup>36</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Ix*

*<sup>a</sup>*<sup>42</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy*

*<sup>a</sup>*<sup>44</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy*

*<sup>a</sup>*<sup>46</sup> <sup>=</sup> *S l <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iy*

*<sup>a</sup>*<sup>51</sup> <sup>=</sup> *Ix* <sup>−</sup> *Iy Iz*

*<sup>a</sup>*<sup>54</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz*

*<sup>a</sup>*<sup>56</sup> <sup>=</sup> *S b <sup>ρ</sup> <sup>V</sup>*<sup>2</sup> 2 *Iz*

> *ρ V* cos *α*

*α*˙ = *b*<sup>11</sup>

equations can be obtained, a so-called **System II**:

+ *b*<sup>12</sup>

*p β* cos *α*

*β*˙ = *b*<sup>21</sup> *ρ V β* + *b*<sup>22</sup> *r* cos *α* + *b*<sup>23</sup> *p* sin *α* + *b*<sup>24</sup>

*<sup>a</sup>*<sup>22</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>23</sup> <sup>=</sup> <sup>1</sup> *<sup>a</sup>*<sup>24</sup> <sup>=</sup> *<sup>g</sup>*

*Clr*

*Clδk*

$$\begin{aligned} \dot{r} &= b\_{41}p\,q + b\_{42}\rho\,V^2\rho + b\_{43}\rho\,V^2\,p + b\_{44}\rho\,V^2\,r + b\_{45}\,q + b\_{46}\rho\,V^2 + b\_{47}\rho\,V^2\\ \dot{q} &= b\_{51}\,p\,r + b\_{52}\,\rho\,V^2\,\mid\,\dot{\rho}\,\mid + b\_{53}\,r + b\_{54}\,\rho\,V^2\,q + b\_{55}\,\rho\,V^2 + b\_{56}\,\rho\,V^2\,\dot{\alpha} + b\_{57}\,\rho\,V^2\\ \dot{\theta} &= b\_{61}\,r\,\sin\gamma + b\_{62}\,q\,\cos\gamma\\ \dot{\gamma} &= b\_{71}\,p + b\_{72}\,r\,\cos\gamma\,\tan\theta + b\_{73}\,q\,\sin\gamma\,\tan\theta\\ \dot{V} &= b\_{81}\,\frac{\rho\,V^2}{\cos\alpha} + b\_{82}\,V\,\dot{\alpha}\,\tan\alpha + b\_{83}\,V\,q\,\tan\alpha + b\_{84}\,\frac{V\,r\,\beta}{\cos\alpha} + b\_{85}\,\frac{\sin\theta}{\cos\alpha} + b\_{86}\,\frac{\rho\,V^2}{\cos\alpha}\\ \dot{H} &= b\_{91}\,V\,\cos\alpha\,\sin\theta + b\_{92}\,V\,\sin\alpha\,\cos\theta\,\cos\gamma + b\_{93}\,V\,\beta\,\cos\theta\,\sin\gamma \end{aligned} \tag{6}$$

The coefficients involved in the system of equations (6) are defined by expressions:

*<sup>b</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>* 2 *m Cz*(*α*) *b*<sup>12</sup> = −1 *b*<sup>13</sup> = 1 *b*<sup>14</sup> = *g <sup>b</sup>*<sup>15</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>b</sup>*<sup>16</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>* 2 *m Cz<sup>δ</sup>hk* (*α*) <sup>Δ</sup>*δhk*(*t*) *<sup>b</sup>*<sup>21</sup> <sup>=</sup> *<sup>S</sup>* 2 *m Cy<sup>β</sup>* (*α*) *b*<sup>22</sup> = 1 *b*<sup>23</sup> = 1 *b*<sup>24</sup> = *g b*<sup>25</sup> = −1 *<sup>b</sup>*<sup>26</sup> <sup>=</sup> *<sup>S</sup>* 2 *m Cy<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>31</sup> <sup>=</sup> *Iy* <sup>−</sup> *Iz Ix <sup>b</sup>*<sup>32</sup> <sup>=</sup> *S l* 2 *Ix Cl<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>33</sup> <sup>=</sup> *S l* 2 *Ix Clp* (*α*) *<sup>b</sup>*<sup>34</sup> <sup>=</sup> *S l* 2 *Ix Clr* (*α*) *<sup>b</sup>*<sup>35</sup> <sup>=</sup> *S l* 2 *Ix Cl<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>36</sup> <sup>=</sup> *S l* 2 *Ix Clδk* (*α*) Δ*δk*(*t*) *<sup>b</sup>*<sup>41</sup> <sup>=</sup> *Iz* <sup>−</sup> *Ix Iy <sup>b</sup>*<sup>42</sup> <sup>=</sup> *S l* 2 *Iy Cn<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>43</sup> <sup>=</sup> *S l* 2 *Iy Cnp* (*α*) *<sup>b</sup>*<sup>44</sup> <sup>=</sup> *S l* 2 *Iy Cnr* (*α*) *<sup>b</sup>*<sup>45</sup> <sup>=</sup> <sup>−</sup> *Ip <sup>ω</sup><sup>p</sup> Iy <sup>b</sup>*<sup>46</sup> <sup>=</sup> *S l* 2 *Iy Cn<sup>δ</sup>vk* (*α*) <sup>Δ</sup>*δvk*(*t*) *<sup>b</sup>*<sup>47</sup> <sup>=</sup> *S l* 2 *Iy Cn<sup>δ</sup><sup>k</sup>* (*α*) Δ*δk*(*t*) *<sup>b</sup>*<sup>51</sup> <sup>=</sup> *Ix* <sup>−</sup> *Iy Iz <sup>b</sup>*<sup>52</sup> <sup>=</sup> *S b* 2 *Iz Cm<sup>β</sup>* (*α*) *<sup>b</sup>*<sup>53</sup> <sup>=</sup> *Ip <sup>ω</sup><sup>p</sup> Iz <sup>b</sup>*<sup>54</sup> <sup>=</sup> *S b* 2 *Iz Cmq* (*α*) *<sup>b</sup>*<sup>55</sup> <sup>=</sup> *S b* 2 *Iz Cm*(*α*) *<sup>b</sup>*<sup>56</sup> <sup>=</sup> *S b* 2 *Iz Cmα*˙ (*α*) *<sup>b</sup>*<sup>57</sup> <sup>=</sup> *S b* 2 *Iz Cm<sup>δ</sup>hk* (*α*) Δ*δhk*(*t*) *b*<sup>61</sup> = 1 *b*<sup>62</sup> = 1 *<sup>b</sup>*<sup>71</sup> <sup>=</sup> <sup>1</sup> *<sup>b</sup>*<sup>72</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>b</sup>*<sup>73</sup> <sup>=</sup> <sup>1</sup> *<sup>b</sup>*<sup>81</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>* 2 *m Cx*(*α*) *b*<sup>82</sup> = 1 *b*<sup>83</sup> = −1 *b*<sup>84</sup> = −1 *b*<sup>85</sup> = −*g <sup>b</sup>*<sup>86</sup> <sup>=</sup> <sup>−</sup> *<sup>S</sup>* 2 *m Cx<sup>δ</sup>hk* (*α*)Δ*δhk*(*t*) *b*<sup>91</sup> = 1 *b*<sup>92</sup> = −1 *b*<sup>93</sup> = −1 (7)

Notation in previous equations are:


**8.2 Methods of modeling**

angle of attack.

It is obvious that most of the coefficients in Exp.(6) are not constant, but vary with time, i.e.

Spin and Spin Recovery 229

If assumptions that speed and altitude are constant are rejected, a complete system of equations is obtained. In order to perform modeling a computer is used with its memory loaded with the simplified or complete system of equations, which depends on the regime that has to be studied or the accuracy of results. The computer loaded with data necessary to calculate the coefficients and other values in Eq.(6), for a given time *t*1, or the given angle of attack *α*1. Data is obtained by testing models in wind tunnel or in flight. Then, the data is entered for moment *t*<sup>2</sup> or attack angle *α*2, and so on. Data is entered until the end of the observed time interval, and it is entered point-by-point. The accuracy of obtained results depends on the magnitude of the time change (time difference between two points), i.e. the angle of attack. The more points are entered, the more accurate the results will be. The computer will integrate and associate values for angle of attack, angle of sideslip, pitching angle and angular velocities for the observed time interval, for which data is entered, and results are obtained as shown in Fig.15, i.e. following functions are defined: *α* = *f*(*t*), *θ* = *f*(*t*), *β* = *f*(*t*), *p* = *f*(*t*), *q* = *f*(*t*) i *r* = *f*(*t*). For System II (Eq.7), following the same analogy, the

Fig. 15. Diagram for a timely development of a left-hand spin (System I)

was entered, and results are obtained as shown in Fig.16.

computer will integrate and associate values from the observed time interval, for which data


These coefficients are entirely determined with the aid of DATCOM<sup>1</sup> reference, and in some additional literature<sup>2</sup> their values are defined for the category of light aircrafts.

<sup>1</sup> D. E. Hoak: *USAF Stability and Control DATCOM*, (N76-73204), Flight Control Devision, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1975.

<sup>2</sup> D. Cvetkovi´c: *The adaptive approach to modeling and simulation of spin and spin recovery*, PhD Thesis, Faculty of Mechanical Engineering, Belgrade University, 1997.

20 Will-be-set-by-IN-TECH

• *Cz<sup>δ</sup>hk* - derivative of lift coefficient with respect to angle of deflection of the elevator;

• *Cy<sup>δ</sup>vk* - derivative of sideslip force with respect to angle of deflection of the rudder; • *Cl<sup>β</sup>* - derivative of rolling moment coefficient with respect to angle of sideslip;

• *Clp* - derivative of rolling moment coefficient with respect to rolling angular velocity; • *Clr* - derivative of rolling moment coefficient with respect to yawing angular velocity;

• *Cn<sup>β</sup>* - derivative of yawing moment coefficient with respect to angle of sideslip;

• *Cm<sup>β</sup>* - derivative of pitching moment coefficient with respect to angle of sideslip;

• *Cx<sup>δ</sup>hk* - derivative of drag coefficient with respect to elevator angle of deflection;

additional literature<sup>2</sup> their values are defined for the category of light aircrafts.

Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1975.

Faculty of Mechanical Engineering, Belgrade University, 1997.

• *Cmq* - derivative of pitching moment coefficient with respect to pitching angular velocity; • *Cmα*˙ - derivative of pitching moment coefficient with respect to derivative of angle of attack

• *Cm<sup>δ</sup>hk* - derivative of pitching moment coefficient with respect to elevator angle of

These coefficients are entirely determined with the aid of DATCOM<sup>1</sup> reference, and in some

<sup>1</sup> D. E. Hoak: *USAF Stability and Control DATCOM*, (N76-73204), Flight Control Devision, Air Force Flight

<sup>2</sup> D. Cvetkovi´c: *The adaptive approach to modeling and simulation of spin and spin recovery*, PhD Thesis,

• *Cnp* - derivative of yawing moment coefficient with respect to rolling angular velocity; • *Cnr* - derivative of yawing moment coefficient with respect to yawing angular velocity; • *Cn<sup>δ</sup>vk* - derivative of yawing moment coefficient with respect to rudder angle of deflection; • *Cn<sup>δ</sup><sup>k</sup>* - derivative of yawing moment coefficient with respect to aileron angle of deflection;



• *Cy<sup>β</sup>* - derivative of sideslip force with respect to sideslip angle;

• *Cm* - pitching moment coefficient;

• *Cl<sup>δ</sup>vk*

• *Cl<sup>δ</sup><sup>k</sup>*

over time;

deflection;

• *Ip* - polar moment of inertia of engine rotor;

• Δ*δhk* - change in elevator angle of deflection; • Δ*δvk* - change in rudder angle of deflection; • Δ*δ<sup>k</sup>* - change in aileron angle of deflection.

• *ωp* - angular velocity of engine rotor;

#### **8.2 Methods of modeling**

It is obvious that most of the coefficients in Exp.(6) are not constant, but vary with time, i.e. angle of attack.

If assumptions that speed and altitude are constant are rejected, a complete system of equations is obtained. In order to perform modeling a computer is used with its memory loaded with the simplified or complete system of equations, which depends on the regime that has to be studied or the accuracy of results. The computer loaded with data necessary to calculate the coefficients and other values in Eq.(6), for a given time *t*1, or the given angle of attack *α*1. Data is obtained by testing models in wind tunnel or in flight. Then, the data is entered for moment *t*<sup>2</sup> or attack angle *α*2, and so on. Data is entered until the end of the observed time interval, and it is entered point-by-point. The accuracy of obtained results depends on the magnitude of the time change (time difference between two points), i.e. the angle of attack. The more points are entered, the more accurate the results will be. The computer will integrate and associate values for angle of attack, angle of sideslip, pitching angle and angular velocities for the observed time interval, for which data is entered, and results are obtained as shown in Fig.15, i.e. following functions are defined: *α* = *f*(*t*), *θ* = *f*(*t*), *β* = *f*(*t*), *p* = *f*(*t*), *q* = *f*(*t*) i *r* = *f*(*t*). For System II (Eq.7), following the same analogy, the

Fig. 15. Diagram for a timely development of a left-hand spin (System I)

computer will integrate and associate values from the observed time interval, for which data was entered, and results are obtained as shown in Fig.16.

In case of modeling spin with the complete system of equations, terms *Aij* and *A*¯

Spin and Spin Recovery 231

Fig. 17. Diagram of development for a flat spin (spin ongoing as a falling leaf)

*<sup>i</sup>*2, . . ., ends, value *A*¯

Figure 17 shows results obtained for a numerically modeled spin. From diagrams (b), (c), (d), (e), and (f), it can be seen which terms have the most effect on results given in diagram (a). Diagrams (b), (c), (d), (e), and (f), were constituted by showing time on the abscissa, while values of *A*¯*ij* (*i* = *const*) are "stacked" on the ordinate, one over the other. For example, value

ends at 100%. By doing so, these diagrams show which term *A*¯*ij* has the most impact on values of *α*, *β*, *θ*, *p*, *q* and *r*. For example, in diagram (b) it is shown that at moment *t* = 23 *s*,

on values of function *α* = *f*(*t*) at the given time. When observed what *A*<sup>13</sup> is equal to, it

*<sup>i</sup>*<sup>2</sup> is imposed from point where *A*¯*i*<sup>1</sup> ends, value *A*¯

<sup>14</sup> and *A*¯ 15. This means that *A*¯

*im* is imposed from point where *A*¯

introduced in the same manner.

*A*¯

*<sup>i</sup>*<sup>1</sup> is imposed from 0, value *A*¯

value of *A*¯ <sup>13</sup> is higher than of *A*¯ 11, *A*¯ 12, *A*¯

from point where *A*¯

*ij* are

*<sup>i</sup>*<sup>3</sup> is imposed

*<sup>i</sup>*,*m*−<sup>1</sup> ends, and

<sup>13</sup> has the most effect

Fig. 16. Diagram for a timely development of a left-hand spin (System II)

#### **8.3 Analysis of results from modeled flat spin (spin ongoing as a falling leaf On a spiral path)**

An analysis of relations is made if effects from individual terms in equations of motion need to be determined (Eq.6). To do this, each member will be designated by the letter "A" with appropriate numerical indexes *i*, *j* (*Ai*,*j*), with "i" being the ordinal number of the equation *i* = (1, *n*), and "j" being the ordinal number of the term in the equation *j* = (1, *m*). For example, the first equation of Eq.6 will look:

$$\dot{\mathfrak{a}} = A\_{11} + A\_{12} + A\_{13} + A\_{14} + A\_{15\prime}$$

where:

$$\begin{aligned} A\_{11} &= a\_{11} \frac{1}{\cos \alpha'} \quad A\_{12} = a\_{12} \frac{\not p \not p}{\cos \alpha'} \quad A\_{13} = a\_{13} \, q \dots \\\ A\_{14} &= a\_{14} \frac{\cos \theta \cos \gamma}{\cos \alpha} \quad A\_{15} = a\_{15} \end{aligned}$$

Ratios of absolute values of terms in Eq.6 are assessed, as of these depend the magnitudes of effects of terms *α*, *β*, *θ*, *p*, *q* and *r*" and therefore are shown as fractions:

$$\bar{A}\_{\bar{i}\bar{j}} = \frac{|\!\!\/ \/ A\_{\bar{i}\bar{j}}\,|}{\sum\_{j=1}^{m} \left|\!\/ \/ A\_{\bar{i}\bar{j}}\,|} \, 100 \qquad [\%]$$

22 Will-be-set-by-IN-TECH

Fig. 16. Diagram for a timely development of a left-hand spin (System II)

1

effects of terms *α*, *β*, *θ*, *p*, *q* and *r*" and therefore are shown as fractions:

*A*¯

cos *θ* cos *γ*

cos *<sup>α</sup>* , *<sup>A</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>*<sup>12</sup>

*ij* <sup>=</sup> <sup>|</sup> *Aij* <sup>|</sup> ∑*<sup>m</sup>*

example, the first equation of Eq.6 will look:

*A*<sup>11</sup> = *a*<sup>11</sup>

*A*<sup>14</sup> = *a*<sup>14</sup>

**path)**

where:

**8.3 Analysis of results from modeled flat spin (spin ongoing as a falling leaf On a spiral**

An analysis of relations is made if effects from individual terms in equations of motion need to be determined (Eq.6). To do this, each member will be designated by the letter "A" with appropriate numerical indexes *i*, *j* (*Ai*,*j*), with "i" being the ordinal number of the equation *i* = (1, *n*), and "j" being the ordinal number of the term in the equation *j* = (1, *m*). For

*α*˙ = *A*<sup>11</sup> + *A*<sup>12</sup> + *A*<sup>13</sup> + *A*<sup>14</sup> + *A*15,

cos *<sup>α</sup>* , *<sup>A</sup>*<sup>15</sup> <sup>=</sup> *<sup>a</sup>*<sup>15</sup>

Ratios of absolute values of terms in Eq.6 are assessed, as of these depend the magnitudes of

*<sup>j</sup>*=<sup>1</sup> | *Aij* |

*β p* cos *α*

100 [%]

, *A*<sup>13</sup> = *a*<sup>13</sup> *q*,

In case of modeling spin with the complete system of equations, terms *Aij* and *A*¯ *ij* are introduced in the same manner.

Fig. 17. Diagram of development for a flat spin (spin ongoing as a falling leaf)

Figure 17 shows results obtained for a numerically modeled spin. From diagrams (b), (c), (d), (e), and (f), it can be seen which terms have the most effect on results given in diagram (a). Diagrams (b), (c), (d), (e), and (f), were constituted by showing time on the abscissa, while values of *A*¯*ij* (*i* = *const*) are "stacked" on the ordinate, one over the other. For example, value *A*¯ *<sup>i</sup>*<sup>1</sup> is imposed from 0, value *A*¯ *<sup>i</sup>*<sup>2</sup> is imposed from point where *A*¯*i*<sup>1</sup> ends, value *A*¯ *<sup>i</sup>*<sup>3</sup> is imposed from point where *A*¯ *<sup>i</sup>*2, . . ., ends, value *A*¯ *im* is imposed from point where *A*¯ *<sup>i</sup>*,*m*−<sup>1</sup> ends, and ends at 100%. By doing so, these diagrams show which term *A*¯*ij* has the most impact on values of *α*, *β*, *θ*, *p*, *q* and *r*. For example, in diagram (b) it is shown that at moment *t* = 23 *s*, value of *A*¯ <sup>13</sup> is higher than of *A*¯ 11, *A*¯ 12, *A*¯ <sup>14</sup> and *A*¯ 15. This means that *A*¯ <sup>13</sup> has the most effect on values of function *α* = *f*(*t*) at the given time. When observed what *A*<sup>13</sup> is equal to, it

**1. Introduction** 

have imposed research in this field.

**10** 

 *Serbia*

**Surface Welding as a** 

**Way of Railway Maintenance** 

Olivera Popovic and Radica Prokic-Cvetkovic *Faculty of Mechanical Engineering, University of Belgrade* 

Since its early days the development of railway systems has been an important driving force for technological progress. From the 1840s onward a dense railroad network was spread all over the world. Within a few decades railway became the predominant traffic system carrying a steadily increasing volume of goods and number of passengers. This rapid development was accompanied by substantial developments in many areas such as steel production, engine construction, civil engineering, communication, etc (Zerbst et al., 2005). The railway industry worldwide is introducing heavier axle loads, higher vehicle speeds, and larger traffic volumes for economic transportation of goods and passengers. Increasing demands for high-speed services and higher axle loads at the turn of the 21st century account for quite new challenges with respect of material and technology as well as safety issues. The main factors controlling rail degradation are wear and fatigue, which cause rails to become unfit for service due to unacceptable rail profiles, cracking, spalling and rail breaks. Degradation of rail is microstructure and macrostructure sensitive and there is a complicated interaction between wear mechanisms, wear rates, fatigue crack initiation and growth rates, which affect rail life (Eden et al.,2005; Kapoor et al.,2002). Defects such as squats and wheelburns occur even in the most modern and well maintained railway networks and, as a broad general rule, every network develops one such defect each year, every two kilometers. At least one European railway network suffers almost 4000 rail fractures every year. Although such fractures are rarely dangerous when actively managed, they entail a high replacement cost and can be disruptive to the network (Bhadeshia,2002). The replacement of such defects with a short rail section is expensive and not always desirable as it introduces two new discontinuities in the track in the form of two

aluminothermic weld that destroy the advantages obtained with long hot-rolled rail.

Given that an average cost per repair or short replacement rail can run into several thousands of euros and that the occurrence of wheel rail interface defects is likely to increase with the evident increase in levels of traffic on most railways, the importance of the surface welding is easy to understand. Growing need for reparation due to large financial demands,

Based on up-to date theoretical grounds and referencial facts, the aim of this paper is to show the possibilities of surface welding of the pearlitic high-carbon steel and the properties of the obtained joint. Discussion of the aquired results and conclusions indicate superior

can be noticed that the value of angle of attack, at this moment, mostly depends on *q*, since *a*<sup>13</sup> = 1. In this way, every term *Aij* on the right-hand side of Eq.(6) can be analyzed on its impact on the left-hand side of same equations. Analysis of every term *Aij*, brings about a deeper understanding of the physical image of the studied regime. In addition, it enables the determination of terms that have the most effect on such a state, in cases when the aircraft does not fulfill necessary requirements for spin, and by appropriate modifications obtain an aircraft with necessary technical characteristics for spin.

#### **9. References**

