**3.1.1 Motion of supporting rotor blades**

To understand the flight dynamics of helicopters and determine dynamic moments and forces that act upon the helicopter, it is a necessity to preinvestigate the motion of supporting rotor blades. From a vast number of different types of helicopters, the single rotor helicopter was chosen that has its blades coupled with the main rotor by a hinge about which they can freely move. It should be noted that there are also rotors that have blade fixedly connected to the hub.

#### **3.1.1.1 Equations of blade fluttering**

Rotor blades are regarded to as rigid body. The horizontal hinge is placed at length eR from rotation axis. The shaft rotates at angle speed =const, and the blade flutters at angle speed of d/dt. The axis that passes through the blade is parallels to the axis of inertia of the blade and passes through the hinge (Figure 20).

Fig. 20. Blade fluttering scheme

Paremeter R represents the length of the blade, represents the flutter angle of the blade. Following a complex calculus, obtained are the equations for blade fluttering:

$$\begin{aligned} \ddot{\boldsymbol{\beta}} + \boldsymbol{\Omega}^2 \left( \mathbf{l} + \boldsymbol{\varepsilon} \right) \ \boldsymbol{\beta} &= \boldsymbol{M}\_{A\boldsymbol{y}} \ \boldsymbol{\beta} \ \mathbf{J}\_{\boldsymbol{y}}\\ \mathbf{J}\_{\times} \ \dot{\boldsymbol{\beta}} \ \boldsymbol{\Omega} \ \boldsymbol{\Omega} \ \cos \boldsymbol{\beta} &+ \mathbf{J}\_{\times} \left( - \dot{\boldsymbol{\beta}} \right) \ \boldsymbol{\Omega} \ \sin \boldsymbol{\beta} = \mathbf{0} \end{aligned} \tag{12}$$
 
$$\begin{aligned} -\boldsymbol{\Omega} \ \mathbf{J}\_{\times} \ \boldsymbol{\Omega} \ \dot{\boldsymbol{\beta}} \ \sin \boldsymbol{\beta} &= \boldsymbol{M}\_{z} \end{aligned} \tag{12}$$

turn in place, move forward and lateral, and can perform these movements in combinations.

In the present, problems in helicopter flight dynamics are mostly solved in aid of modern computers. Though inevitable in many complex problems, computers do not make it possible to understand the physical nature of the problem. Fortunately, many problems considering helicopters can be analyzed without overly complex calculus and usually it is possible to obtain simple formulas. Though not suitable for calculus, these formulas, when designing the helicopter, enable a satisfactory interpretation of required aerodynamic and

The mathematical model described in this paper is related to three-dimensional (space) geometry and kinematics, and rigid body dynamics and fluid dynamics through which it

To understand the flight dynamics of helicopters and determine dynamic moments and forces that act upon the helicopter, it is a necessity to preinvestigate the motion of supporting rotor blades. From a vast number of different types of helicopters, the single rotor helicopter was chosen that has its blades coupled with the main rotor by a hinge about which they can freely move. It should be noted that there are also rotors that have blade

Rotor blades are regarded to as rigid body. The horizontal hinge is placed at length eR from rotation axis. The shaft rotates at angle speed =const, and the blade flutters at angle speed of d/dt. The axis that passes through the blade is parallels to the axis of inertia of the blade

Paremeter R represents the length of the blade, represents the flutter angle of the blade.

*y z*

 

*J M*

J cos sin sin

*x*

*J*

/ J

*M*

*Ay*

0

(12)

y

 

Following a complex calculus, obtained are the equations for blade fluttering:

<sup>2</sup> 1

2

x

 

Because of this, helicopter dynamics modeling and testing is a very complex problem.

dynamic phenomena.

**3.1.1 Motion of supporting rotor blades** 

fixedly connected to the hub.

Fig. 20. Blade fluttering scheme

**3.1.1.1 Equations of blade fluttering** 

and passes through the hinge (Figure 20).

moves.

#### **3.1.1.2 Equations of blade lead-lag**

It is assumed =0 and that the blade is moving forward in relation to the vertical hinge by the lead-lag angle amount . The vertical hinge is placed at distance eR from the shaft axis. The coordinate system is placed as in the previous case.

From this follows the equation for blade lead-lag:

$$
\ddot{\xi} + \Omega^2 \mathcal{E} \ \xi - 2 \ \Omega \ \beta \ \dot{\beta} = M\_z \ \ \ / \ \ f\_z
$$

Fig. 21. Blade lead-lag scheme

If the azimuth angle is described as =t, then follows:

$$\frac{d^2\xi}{d\nu^2} + \varepsilon \cdot \xi - 2\ \beta \frac{d\beta}{d\nu} = \frac{M\_z}{J\_z \Omega^2}$$

#### **3.1.1.3 Equation of blade climb**

It is assumed that flutter and lead-lag angles are equal zero. The blade step is the angle between the blade cross section chord and the plane of the hub, designated as k. Figure 22 shows the coordinate system attached to the blade.

Equations of blade motion about longitudinal axis are:

Fig. 22. Coordinate system at blade cross section

$$\ddot{\theta}\_{\mathbf{k}} + \Omega^2 \ \theta\_{\mathbf{k}} = M\_{\mathbf{x}} \ \text{J} \ \text{J}\_{\mathbf{x}} \qquad \qquad - \ \text{J}\_{z} \ \text{\small\Omega} \ \text{\small} \ \dot{\theta}\_{\mathbf{k}} \ \text{\small} \ \theta\_{\mathbf{k}} = M\_{z} \ \text{\small} \ \text{\small}\_{z} \ \text{\small\Omega} \ \text{\small}\_{k} \ \text{\small} \ \theta\_{\mathbf{k}} \cos \theta\_{\mathbf{k}} - \text{J}\_{y} \ \text{\small}\_{k} \ \text{\small} \cos \theta\_{\mathbf{k}} = 0$$

#### **3.1.2 Rotor forces**

To project forces the following axis may be use: control axis, rotor disc axis which is normal to the rotor plane, that is, to the plane on which reside blade tips, and shaft axis.

1 1

G f A - G hR M *R M b T hR S tt*

By replacing value A1 into corresponding equations, we obtain the value of angle , which

*t* 1 1 *S tt*

*G G hR M*

positioned vertically above the center of mass. All values of these determined angles are so

Mathematical modeling of helicopter motion is a very complex task and, therefore, it is necessary to introduce series of assumptions and approximations. Knowledge of motion of individual helicopter blades is not necessary for investigating dynamic characteristics of the helicopter, except in a special case, but rather for defining forces and moments in a disturbed flight it is sufficient enough to view the rotor as whole. Because of a great number of different helicopters, in this paper a single rotor helicopter was studied, that has its blades connected to the hub by hinges. As mentioned before, the helicopter can perform different motions and it would be very difficult to make a mathematical model that would combine all those motions. It is assumed the helicopter is airborne and in straightforward flight. It is required that the helicopter, at straight forward flight, has following velocity components: Wx, Wy, and Wz, at nominal values, and angle of turn , angle of roll , angle of climb , as long as the intensity of disturbance is in permitted limits. Figure 25 presents a schematic of the helicopter with a floating coordinate system tied to its center of mass, and Figure 26

*T G f R M b T hR*

*S*

*h* 

which means the rotor hub is

S

1

Fig. 24. Drawing for determining lateral equilibrium of forces

If MS=0 and ht=h, which can often be assumed, follows <sup>1</sup> , *<sup>f</sup>*

**3.1.3 Non-linear mathematical model of flight dynamics** 

determines the position of the fuselage.

presents a helicopter block diagram.

called *trimmed values*.

Once the axis is chosen, the remaining axis of the coordinate system will be normal to it and pointed lateral, that is, to the tail of the helicopter. Customarily, the force component along the chosen axis is said to be the tow force, the force component pointed towards the tail is said to be H force, and the force component pointed lateral is said to be Y force. If the force components are designated without subscripts, it is assumed they are determined relative to control axis, whereas subscripts "D" and "S" are used when relating to rotor axis, that is, shaft axis. Since flutter and mount angles are usually small (amounts greater than 10° are considered extreme), the relation between these components can be obtained:

$$T \approx T\_D \approx T\_S \qquad \quad \quad \quad H \approx H\_D + T\_D \; \; a\_1 \equiv H\_S + T\_S \; B\_1$$

#### **3.1.2.1 Longitudinal equilibrium of forces**

Angle 1 is the longitudinal amplitude of a cyclic change in blade step; angle a1s is the angle between shaft and axis of rotor disc. After extensive calculus the expression for longitudinal amplitude of cyclic change in blade step is obtained:

$$B\_1 = \frac{M\_f - G \cdot f\mathcal{R} + H \cdot h\mathcal{R} + M\_S \cdot a\_1}{T \cdot h\mathcal{R} + M\_S}$$

Fig. 23. Drawing for determining longitudinal equilibrium of forces

For e=0, it can be adopted that Ms=0 and Mf=0, and since T=G, follows:

$$B\_1 = -\frac{f}{h} + \frac{H}{G} \quad \text{ } \quad \theta\_p = -\frac{D}{G} \text{ } \cos\pi - \frac{f}{h} + \frac{M\_f}{G \cdot hR}$$

Above equation has a simple physical interpretation: the amplitude of the longitudinal cyclic control must have such a value in order to position the direction of the resultant rotor force through the center of mass.

#### **3.1.2.2 Lateral equilibrium of forces**

Angle A1 presents the amplitude of lateral cyclic change in blade step of the supporting helicopter rotor:

Once the axis is chosen, the remaining axis of the coordinate system will be normal to it and pointed lateral, that is, to the tail of the helicopter. Customarily, the force component along the chosen axis is said to be the tow force, the force component pointed towards the tail is said to be H force, and the force component pointed lateral is said to be Y force. If the force components are designated without subscripts, it is assumed they are determined relative to control axis, whereas subscripts "D" and "S" are used when relating to rotor axis, that is, shaft axis. Since flutter and mount angles are usually small (amounts greater than 10° are

1 1 *T T T H H T a H TB D S* , *DD SS*

Angle 1 is the longitudinal amplitude of a cyclic change in blade step; angle a1s is the angle between shaft and axis of rotor disc. After extensive calculus the expression for longitudinal

*f S*

*<sup>M</sup> G fR H hR M a <sup>B</sup> T hR M* 

*S*

1

considered extreme), the relation between these components can be obtained:

**3.1.2.1 Longitudinal equilibrium of forces** 

amplitude of cyclic change in blade step is obtained:

1

Fig. 23. Drawing for determining longitudinal equilibrium of forces

*B*

force through the center of mass.

helicopter rotor:

**3.1.2.2 Lateral equilibrium of forces** 

For e=0, it can be adopted that Ms=0 and Mf=0, and since T=G, follows:

1 , cos *<sup>f</sup> p*

Above equation has a simple physical interpretation: the amplitude of the longitudinal cyclic control must have such a value in order to position the direction of the resultant rotor

Angle A1 presents the amplitude of lateral cyclic change in blade step of the supporting

 

*f f H D M*

*h G G h G hR*

 

Fig. 24. Drawing for determining lateral equilibrium of forces

By replacing value A1 into corresponding equations, we obtain the value of angle , which determines the position of the fuselage.

$$\varphi = -\frac{T\_t}{G} + \frac{G \cdot f\_1 R + M\_S \cdot b\_1 + T\_t \cdot h\_t R}{G \cdot h R + M\_S}$$

If MS=0 and ht=h, which can often be assumed, follows <sup>1</sup> , *<sup>f</sup> h* which means the rotor hub is positioned vertically above the center of mass. All values of these determined angles are so called *trimmed values*.

#### **3.1.3 Non-linear mathematical model of flight dynamics**

Mathematical modeling of helicopter motion is a very complex task and, therefore, it is necessary to introduce series of assumptions and approximations. Knowledge of motion of individual helicopter blades is not necessary for investigating dynamic characteristics of the helicopter, except in a special case, but rather for defining forces and moments in a disturbed flight it is sufficient enough to view the rotor as whole. Because of a great number of different helicopters, in this paper a single rotor helicopter was studied, that has its blades connected to the hub by hinges. As mentioned before, the helicopter can perform different motions and it would be very difficult to make a mathematical model that would combine all those motions. It is assumed the helicopter is airborne and in straightforward flight. It is required that the helicopter, at straight forward flight, has following velocity components: Wx, Wy, and Wz, at nominal values, and angle of turn , angle of roll , angle of climb , as long as the intensity of disturbance is in permitted limits. Figure 25 presents a schematic of the helicopter with a floating coordinate system tied to its center of mass, and Figure 26 presents a helicopter block diagram.

Where:

longitudinal motion),

longitudinal motion),

from their nominal values are small.

present the equation at exit.

this mathematical model.

motion), and

Aeronautical Engineering 425

U1=B1 – amplitude of cyclic change in step in the longitudinal direction (regarding to

U2=0 – change of collective step of the helicopter rotor blade (regarding to

U3=A1 – amplitude of cyclic change in step in the lateral direction (regarding to lateral

In technical applications it has been shown that, with an acceptable accuracy, linearized mathematical models may be used under the condition that deviations of physical quantities

The outcome of adopted presumptions is that the output values, input values, and the

1 9 1 6 <sup>4</sup>

The vector equation of state for the linearized mathematical model with non-dimensional quantities, deviations, that is, quantities of state is shown in equation 13. Equation 14

000100000 0000

000000100 0000

11 12 13 14 11 12 21 22 23 24 21 22

*aaaa b b aaaa b b*

41 42 43 44 41 42

*aaaa b b X X aa aa b b*

0000 0 0 0

0000 0 0 0

000000001 0000 0 0

75 77 79

*aaa*

*aaa*

95 97 99

100000000 010000000 001000000 000010000 000001000 000000100

 

Xi X

Besides the way this is presented, in a form of common matrix, also should be noted that longitudinal and lateral motions are separated, because this was the condition for deriving

i 1 , X ,u u *T T XXX X X i i <sup>u</sup>* (13)

55 56 58 59 53 54

00000 0 0

00000 0 0 00000 0 0

T

73 74

*b b*

*b b*

*u*

(14)

0 0000

0 0

93 94

U4=t – change of collective step of the tail rotor (regarding to lateral motion).

**3.1.4 Linearized mathematical model of flight dynamics** 

vector of state for both longitudinal and lateral motion will be:

Fig. 25. Schematic of helicopter

Fig. 26. Helicopter dynamics block diagram

After introducing a series of assumptions, such as:


and we come to a non-linear mathematical model by deviations in the form:

$$\frac{d\left(\Delta\mathcal{W}\_{x}\right)}{dt} = \frac{1}{m} \left[ f\_{1} \left( \Delta\mathcal{W}\_{x'} \Delta\mathcal{W}\_{z'} \Delta\dot{\theta}\_{\prime} u\_{1}, u\_{2} \right) - \left( mg \cos \tau \right) \Delta\theta \right]$$

$$\frac{d\left(\Delta\mathcal{W}\_{z}\right)}{dt} = \frac{1}{m} \left[ f\_{2} \left( \Delta\mathcal{W}\_{x'} \Delta\mathcal{W}\_{z'} \Delta\dot{\theta}\_{\prime} u\_{1}, u\_{2} \right) + \mathcal{W}\_{zN} m \Delta\dot{\theta} - \left( mg \sin \tau \right) \Delta\theta \right]$$

$$\frac{d\left(\Delta\theta\right)}{dt} = \Delta\dot{\theta} \qquad , \quad \frac{d\left(\Delta\dot{\theta}\right)}{dt} = \frac{1}{I\_{y}} f\_{3} \left( \Delta\mathcal{W}\_{x'} \Delta\mathcal{W}\_{z'} \Delta\dot{\theta} \dot{\mathcal{W}}\_{z'} \Delta\dot{\theta}\_{\prime} u\_{1}, u\_{2} \right)$$

$$\frac{d\left(\Delta\boldsymbol{\mathcal{V}}\_{y}\right)}{dt} = \frac{1}{m} \left[ f\_{4} \left( \Delta\boldsymbol{\mathcal{V}}\_{y'} \Delta\dot{\boldsymbol{\phi}}, \Delta\dot{\boldsymbol{\nu}}, \boldsymbol{\mu}\_{3}, \boldsymbol{\mu}\_{4} \right) + \boldsymbol{\mathcal{V}}\_{z\boldsymbol{\mathcal{N}}} m \Delta\boldsymbol{\dot{\nu}} + mg \cos\pi \Delta\boldsymbol{\theta} + mg \sin\pi \Delta\boldsymbol{\nu} \right]$$

$$\frac{d\left(\Delta\boldsymbol{\phi}\right)}{dt} = \Delta\boldsymbol{\dot{\phi}} \qquad , \qquad \frac{d\left(\Delta\dot{\boldsymbol{\phi}}\right)}{dt} = \frac{1}{f\_{x}} \left[ f\_{5} \left( \Delta\boldsymbol{\mathcal{V}}\_{y'} \Delta\dot{\boldsymbol{\phi}}, \Delta\dot{\boldsymbol{\nu}}, \boldsymbol{\mu}\_{3}, \boldsymbol{\mu}\_{4} \right) + f\_{xz} \Delta\boldsymbol{\dot{\nu}} \right]$$

$$\frac{d\left(\Delta\boldsymbol{\mu}\right)}{dt} = \Delta\boldsymbol{\dot{\mu}} \qquad , \qquad \frac{d\left(\Delta\boldsymbol{\dot{\nu}}\right)}{dt} = \frac{1}{f\_{z}} \left[ f\_{6} \left( \Delta\boldsymbol{\mathcal{V}}\_{y'} \Delta\dot{\boldsymbol{\phi}}, \Delta\boldsymbol{\dot{\nu}} \right), \boldsymbol{\mu}\_{3}, \boldsymbol{\mu}\_{4} \right] + f\_{xz} \Delta\boldsymbol{\dot{\phi}}$$

Where:

424 Mechanical Engineering

Fig. 25. Schematic of helicopter

Fig. 26. Helicopter dynamics block diagram

helicopter mass is a constant value,

 helicopter is a rigid body, 0xz is a plane of symmetry,

After introducing a series of assumptions, such as:

1

4 3 4

 

*z*

1

*y*

*d W*

*x*

angle increments , , are too small, and so on;

1

and we come to a non-linear mathematical model by deviations in the form:

2 1 2

*d W f W W u u W m mg dt m*

*y zN*

 

> 

1

*f W u u W m mg mg dt m*

d d , , , ,, dt

1

*z*

*x*

d d , , , ,, dt

1 1 2

*x z*

*d W f W W u u mg dt m*

1

*y*

*x z zN*

 

<sup>d</sup> d , , , , ,, dt

3 1 2

 

> 

   

> 

*xzz*

5 3 4

 

*f W uu J dt J*

*f W uu J dt J*

*y xz*

*y xz*

, , ,, cos sin

6 3 4

 

*f W W W uu dt J*

, , , , cos

, , ,, sin


#### **3.1.4 Linearized mathematical model of flight dynamics**

In technical applications it has been shown that, with an acceptable accuracy, linearized mathematical models may be used under the condition that deviations of physical quantities from their nominal values are small.

The outcome of adopted presumptions is that the output values, input values, and the vector of state for both longitudinal and lateral motion will be:

$$\underline{\mathbf{X}} = \begin{pmatrix} X\_1 \dots X\_9 \end{pmatrix}^T \quad , \quad \underline{\mathbf{X}\_i} = \begin{pmatrix} X\_{i1} \dots X\_{i6} \end{pmatrix}^T \quad , \quad \underline{\mathbf{u}} = \begin{pmatrix} \mathbf{u}\_1 \dots \mathbf{u}\_4 \end{pmatrix}^T \tag{13}$$

The vector equation of state for the linearized mathematical model with non-dimensional quantities, deviations, that is, quantities of state is shown in equation 13. Equation 14 present the equation at exit.

$$
\underline{\dot{X}} = \begin{bmatrix} a\_{11} & a\_{12} & a\_{13} & a\_{14} & 0 & 0 & 0 & 0 & 0 \\ a\_{21} & a\_{22} & a\_{23} & a\_{24} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ a\_{41} & a\_{42} & a\_{43} & a\_{44} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a\_{55} & a\_{56} & 0 & a\_{58} & a\_{59} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & a\_{75} & 0 & a\_{77} & 0 & a\_{79} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & a\_{95} & 0 & a\_{97} & 0 & a\_{99} \\ \end{bmatrix} \underline{X} + \begin{bmatrix} b\_{11} & b\_{12} & 0 & 0 \\ b\_{21} & b\_{22} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ b\_{41} & b\_{42} & 0 & 0 \\ 0 & 0 & b\_{53} & b\_{54} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & b\_{93} & b\_{94} \end{bmatrix} \underline{M} \tag{14}$$

$$
\underline{\mathbf{X}}\_{i} = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0
\end{bmatrix} \underline{\mathbf{X}}\_{i}
$$

Besides the way this is presented, in a form of common matrix, also should be noted that longitudinal and lateral motions are separated, because this was the condition for deriving this mathematical model.

2 1 3 2

> 1 2 3

*i*

*X*

*i*

*X*

<sup>56789</sup> 3 4

u

*X X XX*

*i i i*

0 15125 0 0734 0 0 0 3 0 0 1 00 64 42 0 2 948 0 1 378 0 0 0 01 55 297 0 0 413 0 1 64

. ..

. ..

*X AX Bu X X X X X X u u u*

It can be seen that matrix A is singular. From the point of automation control this means the

A further analysis of the mathematical model can be made in order to investigate the dynamic and static properties, and to determine control that would guaranty the object

The major assumptions that were a starting point in this subject are the adopted aerodynamic conception of the aircraft and the full system of equations for the aircraft motion in the 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 I and II non-linear equation systems. The aerodynimc coefficients

A spin is an interesting manouvre, if only for the reason that at one time there stood to its discredit a large proportion of all airplane accidents that had ever occurred. It differs from other manouvres in the fact that wings are "stalled". Wings are beyond the critical angle-offattack, and this accounts for the lack of control which the pilot experiences over the

*<sup>T</sup>*

. . .

 

*X u*

*X u*

1

*X*

*X*

Equation at exit is:

**3.1.6 Conclusion** 

**3.2.1 Introduction** 

4

*X*

Matrix A of the linearized model of helicopter for lateral motion is:

*A*

system has unlimited number of equilibrium states.

**3.2 Modeling and simulation of spin on the vuk-T sailplane** 

used in systems are estimated from flight information.

execution of the required dynamic behavior.

In equations above, equations for longitudinal motion are presented within the first for rows of the matrices, while the remaining five rows present the equation of state and equation of lateral motion. Designation used in above equation is:

2 14 24 59 1 1 \* 11 12 13 21 22 23 55 56 58 41 a a / a cos a ˆ a a sin a a cos a sin ˆ a *u w xz x z c qu w c xN q vc c xN u wu C x x i ii m ax z z m aW z ym m a W m mz* 11 0 0 1 1 0 0 1 1 1 44 22 54 42 43 11 12 21 41 42 53 73 a ˆ a sin b b b b b b b / *<sup>t</sup> w ww w c q w xN q B B wB B w A A A xz m mz mm a m m W z x z x bz mz m mz m y b y l ni* 1 1 94 95 93 \* \* \* 74 \* \* 75 77 \* \* 97 \* \* 99 79 b / / a / a / / a / a / / a *t t t t x xz x xz z v xz x v p p xz x v v xz z p p xz z r r xz z A A xz z r r i C l ni i C b n li i C ni i l C l ni i C a n li i C n li i C n li i C b n l i i C l ni* \* / *xz xi C*

#### **3.1.5 Program results**

The program was tested on the example of a single rotor helicopter which main rotor blades are tied to the hub over hinges.

The helicopter is described by the following input data: helicopter weight G=45042N, rotor abundance degree s=0.058, rotor radius R=8.1m, hub height coefficient h=0.25, drag coefficient =0.013, number of blades of the main rotor b=4, blade mass m=79.6kg, rotor operate mode coefficient =0.3, gradient of lift a=5.65, velocity of blade top R=208m/s, distance of blade mass center coefficient xg=0.45, distance of hinge from shaft eR=0,04R, and air density at flight altitude (100m) =1.215 kg/m3.

For longitudinal motion the mathematical model in vector form is:

$$
\underline{\dot{X}} = A \begin{array}{c} \underline{X} + b \ \underline{u} \end{array}
$$

$$
\underline{\dot{X}} = \begin{bmatrix} -0.0509 & 0.1323 & -0.0734 & 0.00263 \\ 0.1216 & -1.2525 & 0 & 0.3 \\ 0 & 0 & 0 & 1 \\ 6.512 & 12.1 & 0 & -0.844 \end{bmatrix} \underline{\underline{X}} + \begin{bmatrix} 0.1344 & 0.066 \\ 0.3578 & -0.9477 \\ 0 & 0 \\ -28.329 & 17.88 \end{bmatrix} \underline{u}
$$

In equations above, equations for longitudinal motion are presented within the first for rows of the matrices, while the remaining five rows present the equation of state and equation of

11 12

11 0 0

*x z x bz*

1 1 0 0

*ni i l C l ni i C*

94

*l ni i C b n li i C*

*xz x xz z*

*w c q w xN q*

*mm a m m W z*

1 1 1

b / /

*t t t t*

a / a /

*A A A xz*

*v xz x v p p xz x*

*v v xz z p p xz z*

1 1

*r r xz z A A xz z*

The program was tested on the example of a single rotor helicopter which main rotor blades

The helicopter is described by the following input data: helicopter weight G=45042N, rotor abundance degree s=0.058, rotor radius R=8.1m, hub height coefficient h=0.25, drag coefficient =0.013, number of blades of the main rotor b=4, blade mass m=79.6kg, rotor operate mode coefficient =0.3, gradient of lift a=5.65, velocity of blade top R=208m/s, distance of blade mass center coefficient xg=0.45, distance of hinge from shaft eR=0,04R, and

*X AX bu*

0 0509 0 1323 0 0734 0 00263 0 1344 0 066 0 1216 1 2525 0 0 3 0 3578 0 9477 0001 00 6 512 12 1 0 0 844 28 329 17 88

. . .. . . .. . ..

*X X u* 

. . . ..

a / /

*a n li i C n li i C*

93

*n li i C b n l i i C*

97

/ a /

b b / *<sup>t</sup>*

*mz m mz m y b y l ni*

*wB B w*

 

 

a a

24

42

 

 

\* \*

\* \*

 

\* \*

\* \*

a

22

*x*

*i C*

 

\*

*w ww*

*m mz*

*u w*

14

*vc c xN u wu*

44

*ym m*

ˆ a sin b b b

*c qu w c xN q*

*z m aW z*

*m ax z*

13 21

 a cos a ˆ a a sin a a cos a sin ˆ a

*C x x*

 

55 56 58 41

54

75 77

*r r*

air density at flight altitude (100m) =1.215 kg/m3.

*l ni*

\* / *xz xi C*

For longitudinal motion the mathematical model in vector form is:

11 12 21 41 42 53 73

*a W m mz*

*B B*

b b

lateral motion. Designation used in above equation is:

2

22 23

1 \*

1

/

*i ii*

*xz x z*

59

43

95

99 79

a

**3.1.5 Program results** 

are tied to the hub over hinges.

74

 

$$\underline{\mathbf{X}} = \begin{bmatrix} \mathbf{X}\_1 \\ \mathbf{X}\_2 \\ \mathbf{X}\_3 \\ \mathbf{X}\_4 \end{bmatrix} \qquad \qquad \qquad \underline{\mathbf{u}} = \begin{bmatrix} u\_1 \\ u\_2 \end{bmatrix}$$

Equation at exit is:

$$\underline{\mathbf{X}\_{i}} = \begin{bmatrix} 1 & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & 1 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & 1 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & 1 \end{bmatrix} \underline{\mathbf{X}} \qquad \qquad \underline{\mathbf{X}\_{i}} = \begin{bmatrix} \mathbf{X}\_{i1} \\ \mathbf{X}\_{i2} \\ \mathbf{X}\_{i3} \end{bmatrix}$$

Matrix A of the linearized model of helicopter for lateral motion is:

$$A = \begin{bmatrix} -0.15125 & 0.0734 & 0 & 0 & 0.3\\ 0 & 0 & 1 & 0 & 0\\ -64.42 & 0 & -2.948 & 0 & 1.378\\ 0 & 0 & 0 & 0 & 1\\ 55.297 & 0 & 0.413 & 0 & -1.64 \end{bmatrix}$$

$$\underline{\dot{X}} = A \begin{bmatrix} \underline{X} + B \ \underline{u} \end{bmatrix} \qquad \underline{X} = \begin{bmatrix} X\_5 & X\_6 & X\_7 & X\_8 & X\_9 \end{bmatrix}^T \qquad \underline{\underline{u}} = \begin{bmatrix} u\_3 & u\_4 \end{bmatrix}$$

It can be seen that matrix A is singular. From the point of automation control this means the system has unlimited number of equilibrium states.

#### **3.1.6 Conclusion**

A further analysis of the mathematical model can be made in order to investigate the dynamic and static properties, and to determine control that would guaranty the object execution of the required dynamic behavior.

#### **3.2 Modeling and simulation of spin on the vuk-T sailplane**

The major assumptions that were a starting point in this subject are the adopted aerodynamic conception of the aircraft and the full system of equations for the aircraft motion in the 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 I and II non-linear equation systems. The aerodynimc coefficients used in systems are estimated from flight information.

#### **3.2.1 Introduction**

A spin is an interesting manouvre, if only for the reason that at one time there stood to its discredit a large proportion of all airplane accidents that had ever occurred. It differs from other manouvres in the fact that wings are "stalled". Wings are beyond the critical angle-offattack, and this accounts for the lack of control which the pilot experiences over the

61 62

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> *<sup>q</sup> <sup>a</sup> V*

*I*

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

> *p p z*

*I* 

 

11 12 13 14 15 16

cos cos cos cos *V V p V b b bq b b b*

21 22 23 24 25 26 cos cos cos sin *<sup>V</sup> b V br b <sup>p</sup> b b bV*

 

31 32 <sup>33</sup> <sup>34</sup> 35 36 *p b rq bV bV*

41 42 <sup>43</sup> <sup>44</sup> 45 46 47 *r b pq bV bV*

51 52 53 54 55 56 <sup>57</sup> *q b pr b V b r b V q b q b V b V*

61 62

 *br bq* sin cos 

 

*I*

 

*a*

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

*Sb V aC t*

 

<sup>32</sup> 2 *<sup>l</sup> x Sl V a C I*

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

*S V aC t*

2

 

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

> <sup>54</sup> 2 *mq z Sb V a C I*

cos cos tan

*V V*

2 2 2 22

2 2 2 22

2 22 2

 

 

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2

<sup>44</sup> 2 *nr y Sl V a C 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

 

> >

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*V V*

 

> 

*p b Vr b V b V*

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71 72 73

 

> 

 <sup>15</sup> <sup>2</sup> *nk z nk S V aC t m*

*<sup>g</sup> <sup>a</sup>* <sup>25</sup> <sup>2</sup> *yk y k S V aC t*

*m*

 

 

 

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*I*

*p p y*

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45

*a*

 2

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

> >

 *ar aq* sin cos 

<sup>12</sup> *a* 1 <sup>13</sup> *a* 1 <sup>14</sup>

<sup>22</sup> *<sup>a</sup>* <sup>1</sup> <sup>23</sup> *<sup>a</sup>* <sup>1</sup> <sup>24</sup> <sup>V</sup>

*y z x I I*

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31

*a*

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*I*

> 

53

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

*I*

2

 2

<sup>34</sup> 2 *lr x Sl V a C I* 

2

The coefficients are given by:

<sup>11</sup> <sup>2</sup> *<sup>z</sup> S V a C m* 

2

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

*I*

*x y z I I*

2

*I*

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

<sup>33</sup> 2 *lp x Sl V a C I* 

*a*

51

*a*

<sup>21</sup> <sup>2</sup> *<sup>y</sup> S V a C m*

<sup>52</sup> 2 *<sup>m</sup> z Sb V a C I*

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

<sup>42</sup> 2 *<sup>n</sup> y Sl V a C I*

movements of the airplane while spinning. It is form of "auto-rotation", which means that there is a natural tendency for the airplane to rotate of its own accord.

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 spin recovery characteristics are the imperative for any modern sailplane.

## Fig. 27. VUK-T sailplane

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 first and second non-linear equation systems.
