**2. Discrete vortex cylinders method**

The method of discrete vortex cylinders is based on the linear disk theory of Shaidakov described above. It allows you to calculate the induced velocity from the rotor at any point in the space around the rotor. Consider a main rotor with an infinite number of blades [2]. Imagine a vortex system that descends from the rotor in the form of a vortex column, starting at the plane of the disk and going to infinity. The vortex column is supported by a circle with a radius equal to the radius of the rotor. The angle of vortex column inclination to the disk plane depends on the helicopter forward flight speed and the thrust of the rotor. It is calculated using the Shaidakov formula for the angle of inclination of the vortex column.

Each partial volume of the vortex column can be considered as an elementary column of dipoles with a constant density of circulation. Alternatively consider it as an elementary vortex cylinder of arbitrary shape with a linear circulation of closed vortices *γ* along the generatrix.

In case of the beveled cylinder filling with dipoles, to calculate the inductive velocity from the entire vortex column, it is necessary to make integral sums from the *n* number final volumes at *n* ! ∞ for the limit case. In this case, the area of the base of the cylinder is divided into *n* number areas *dσ*1, *dσ*2, … , *dσn*. The area of the cylinder base is divided into finite regions, and the entire vortex column is divided into infinite volumes.

When filling a vortex column with vortex cylinders, the area of the base is filled with closed contours of a specific shape (**Figure 1**), and the column is entirely filled with vortex cylinders of linear circulation *γ* along the generatrix. The generatrices of the vortex cylinders are parallel to the axis of the vortex column and inclined at an angle *δ* to the plane of the disk (**Figure 3**).

We propose to consider a vortex column as a collection of a finite number of vortex cylinders resting on the plane of the rotor disk. The plane of the disk is filled with closed vortex contours on two sides by arcs of circles and on the other two sides by radial segments (**Figure 4**). When the disk is split in this way the size of the contours will depend on the number of calculated points along the blade radius and

*vy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

*DOI: http://dx.doi.org/10.5772/intechopen.93186*

1 sin *δ* ð

*<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

where *lz* ¼ ð Þ *ζ* � *z*<sup>1</sup> , *lx* ¼ ð Þ *ξ* � *x*<sup>1</sup> (**Figure 2**).

<sup>d</sup>*<sup>s</sup>* , <sup>d</sup>*<sup>ζ</sup>*

(**Figure 4**). This angle is denoted by

*vx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

*vy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

The derivatives <sup>d</sup>*<sup>ξ</sup>*

Finally, we will have

ð Þ *ab dv* <sup>þ</sup> <sup>Ð</sup>

**107**

ð Þ *bc dv* <sup>þ</sup> <sup>Ð</sup>

*s*

*lz dξ ds* � *lx*

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity*

*vz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>*

<sup>1</sup> � cos <sup>2</sup>*<sup>δ</sup>* sin *δ*

<sup>4</sup>*<sup>π</sup>* sin *<sup>δ</sup>*

ð

*s lx dζ ds* � *lz*

element *ds* to the axis *Ox*1. Their values do not depend on the position of the point

A, but only on the direction of the vector *ds* in the base coordinate system

<sup>¼</sup> cos *<sup>τ</sup>*, <sup>d</sup>*<sup>ζ</sup>*

d*s*

d*ξ* d*s*

*s*

*s*

*vz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>*

<sup>1</sup> � cos <sup>2</sup>*<sup>δ</sup>* sin *δ*

<sup>4</sup>*<sup>π</sup>* sin *<sup>δ</sup>*

ð

*s*

convenience, we will calculate the closed loop integral as the sum of integrals *<sup>v</sup>* <sup>¼</sup> <sup>Ð</sup>

angle *τ* is constant along the contour and is equal to the azimuthal angles *ψab* and *ψcd* þ *π*, respectively. On segments bc and da, the angle *τ* changes along the contour length, equal to the angle of inclination of the tangent in the middle of the arc *ds* to the axis *Ox*1. It depends only on the midpoint of the arc *ds* azimuthal position. With the proposed natural partition of the disk into discrete vortex cylinders, the problem of calculating inductive velocities in the disk plane is reduced to multiplying the matrix of influence on the column of linear circulations *γ* at the calculated points along the rotor disk. Create a matrix ½ � *A* of dimension *N* � *N*. The elements *aij* of matrix ½ � *A* are the induced velocities are caused at the *i* calculated point on the axis of the *i* vortex cylinder by the unit-strength vortex cylinder *j*

> *a*<sup>11</sup> *a*<sup>12</sup> … *a*1*<sup>N</sup> a*<sup>21</sup> *a*<sup>22</sup> … *a*2*<sup>N</sup>* … ……… *aN*<sup>1</sup> *aN*<sup>2</sup> … *aNN*

ð

*s*

To calculate integrals along the contour, we will use integrating matrices [3]. For

1 sin *δ* ð

1 sin *δ* ð

*<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

ð Þ *cd dv* <sup>þ</sup> <sup>Ð</sup>

*v*1 *v*2 … *vN* 9 >>>=

>>>; ¼

8 >>><

>>>:

*dζ ds* � �*J*<sup>1</sup> � *<sup>J</sup>*<sup>2</sup>

ð

*s J*2 *dξ*

*dζ ds* cos *<sup>δ</sup>* � �*ds* (38)

*dξ*

<sup>d</sup>*<sup>s</sup>* are the cosine and sine of the angle of inclination of the arc

ð Þ *lz* cos *τ* � *lx* sin *τ J*<sup>1</sup> cos *τ* � *J*<sup>2</sup> ½ � sin *τ* d*s* (42)

*J*1ð*lz* cos *τ* � *lx* sin *τ*Þ � *J*<sup>2</sup> ½ � sin *τ* cos *δ* d*s* (43)

ð Þ *da dv* over four contours. On segments ab and cd the

*γ*1 *γ*2 … *γN* 9 >>>=

>>>;

(46)

8 >>><

>>>:

*ds ds* (39)

¼ sin *τ* (41)

*J*<sup>2</sup> cos *τ*d*s* (44)

*J*1ð Þ *lx* sin *τ* � *lz* cos *τ* d*s* (45)

*ds* � �*J*1*ds*, (40)

**Figure 3.** *Coordinate system of a discrete vortex cylinder.*

#### **Figure 4.**

*Scheme for calculating the influence function from a vortex cylinder (the contour integral).*

along the circumference of the rotor disk. In this case, the induced velocities will be calculated at points located at the vortex cylinder's axis, and outside the vortex column at any point other than the vortex cylinders surface. This avoids computational difficulties when calculating contour integrals on the surface of cylindrical columns. This point is indicated by a letter *A* [3]. We will also follow to this designation.

To calculate the components of the inductive velocity at point A by the method of discrete vortex cylinders we will use Shaidakov's formulas (Eqs. (31)–(33), (35)), derived for a discrete vortex cylinder (**Figure 4**).

We assume that the circulation along the contour and along the generatrix of each discrete vortex cylinder is constant. Then in formulas (Eq. (36)) it is possible to take the linear circulation *γ* as an integral and calculate the induced velocities from the vortex cylinder of the unit circulation, integrating along four segments of the contour (**Figure 4**).

After the obvious transformations, we will have:

$$v\_{x1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \int\_{\mathfrak{s}} \left[ \left( l\_x \frac{d\xi}{ds} - l\_x \frac{d\zeta}{ds} \right) l\_1 \cos \delta - f\_2 \frac{d\zeta}{ds} \right] ds \tag{37}$$

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity DOI: http://dx.doi.org/10.5772/intechopen.93186*

$$v\_{\mathcal{V}1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \int\_{\mathcal{s}} \left[ \left( l\_x \frac{d\xi}{ds} - l\_x \frac{d\zeta}{ds} \right) l\_1 - f\_2 \frac{d\zeta}{ds} \cos \delta \right] ds \tag{38}$$

$$w\_{x1} = \frac{\gamma}{4\pi} \sin \delta \left| f\_2 \frac{d\xi}{ds} ds \right. \tag{39}$$

$$w\_0 = \frac{\chi}{4\pi} \frac{1 - \cos^2 \delta}{\sin \delta} \int\_{\mathbb{R}} \left( l\_x \frac{d\tilde{\xi}}{ds} - l\_x \frac{d\tilde{\xi}}{ds} \right) l\_1 ds,\tag{40}$$

where *lz* ¼ ð Þ *ζ* � *z*<sup>1</sup> , *lx* ¼ ð Þ *ξ* � *x*<sup>1</sup> (**Figure 2**).

The derivatives <sup>d</sup>*<sup>ξ</sup>* <sup>d</sup>*<sup>s</sup>* , <sup>d</sup>*<sup>ζ</sup>* <sup>d</sup>*<sup>s</sup>* are the cosine and sine of the angle of inclination of the arc element *ds* to the axis *Ox*1. Their values do not depend on the position of the point A, but only on the direction of the vector *ds* in the base coordinate system (**Figure 4**). This angle is denoted by

$$\frac{d\xi}{d\mathfrak{s}} = \cos\mathfrak{r}, \quad \frac{d\zeta}{d\mathfrak{s}} = \sin\mathfrak{r} \tag{41}$$

Finally, we will have

$$w\_{\mathbf{x}1} = \frac{\gamma}{4\pi} \frac{\mathbf{1}}{\sin \delta} \int\_{\mathcal{S}} [(l\_{\mathbf{x}} \cos \tau - l\_{\mathbf{x}} \sin \tau)l\_{\mathbf{1}} \cos \tau - f\_2 \sin \tau] \, \text{ds} \tag{42}$$

$$w\_{\mathcal{Y}1} = \frac{\chi}{4\pi} \frac{1}{\sin \delta} \left[ \left[ I\_1(l\_x \cos \tau - l\_x \sin \tau) - I\_2 \sin \tau \cos \delta \right] \text{ds} \tag{43}$$

$$w\_{x1} = \frac{\chi}{4\pi} \sin \delta \oint\_{\delta} l\_2 \cos \tau d\mathbf{s} \tag{44}$$

$$v\_0 = \frac{\gamma}{4\pi} \frac{1 - \cos^2 \delta}{\sin \delta} \int\_{\mathbb{R}} l\_1(l\_x \sin \tau - l\_x \cos \tau) \,\mathrm{d}s \tag{45}$$

To calculate integrals along the contour, we will use integrating matrices [3]. For convenience, we will calculate the closed loop integral as the sum of integrals *<sup>v</sup>* <sup>¼</sup> <sup>Ð</sup> ð Þ *ab dv* <sup>þ</sup> <sup>Ð</sup> ð Þ *bc dv* <sup>þ</sup> <sup>Ð</sup> ð Þ *cd dv* <sup>þ</sup> <sup>Ð</sup> ð Þ *da dv* over four contours. On segments ab and cd the angle *τ* is constant along the contour and is equal to the azimuthal angles *ψab* and *ψcd* þ *π*, respectively. On segments bc and da, the angle *τ* changes along the contour length, equal to the angle of inclination of the tangent in the middle of the arc *ds* to the axis *Ox*1. It depends only on the midpoint of the arc *ds* azimuthal position.

With the proposed natural partition of the disk into discrete vortex cylinders, the problem of calculating inductive velocities in the disk plane is reduced to multiplying the matrix of influence on the column of linear circulations *γ* at the calculated points along the rotor disk. Create a matrix ½ � *A* of dimension *N* � *N*. The elements *aij* of matrix ½ � *A* are the induced velocities are caused at the *i* calculated point on the axis of the *i* vortex cylinder by the unit-strength vortex cylinder *j*

$$\begin{Bmatrix} v\_1\\ v\_2\\ \dots\\ v\_N \end{Bmatrix} = \begin{bmatrix} a\_{11} & a\_{12} & \dots & a\_{1N} \\ a\_{21} & a\_{22} & \dots & a\_{2N} \\ \dots & \dots & \dots & \dots \\ a\_{N1} & a\_{N2} & \dots & a\_{NN} \end{bmatrix} \times \begin{Bmatrix} \gamma\_1\\ \gamma\_2\\ \dots\\ \gamma\_N \end{Bmatrix} \tag{46}$$

along the circumference of the rotor disk. In this case, the induced velocities will be calculated at points located at the vortex cylinder's axis, and outside the vortex column at any point other than the vortex cylinders surface. This avoids computational difficulties when calculating contour integrals on the surface of cylindrical columns. This point is indicated by a letter *A* [3]. We will also follow to this

*Scheme for calculating the influence function from a vortex cylinder (the contour integral).*

To calculate the components of the inductive velocity at point A by the method of discrete vortex cylinders we will use Shaidakov's formulas (Eqs. (31)–(33), (35)),

We assume that the circulation along the contour and along the generatrix of each discrete vortex cylinder is constant. Then in formulas (Eq. (36)) it is possible to take the linear circulation *γ* as an integral and calculate the induced velocities from the vortex cylinder of the unit circulation, integrating along four segments of

� �

*dζ ds*

� �

*J*<sup>1</sup> cos *δ* � *J*<sup>2</sup>

*dζ ds*

*ds* (37)

derived for a discrete vortex cylinder (**Figure 4**).

After the obvious transformations, we will have:

1 sin *δ* ð

*s*

*lz dξ ds* � *lx*

*vx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

designation.

**106**

**Figure 4.**

**Figure 3.**

*Coordinate system of a discrete vortex cylinder.*

*Vortex Dynamics Theories and Applications*

the contour (**Figure 4**).

In hovering mode, the main diagonal (*i* ¼ *j*) of the influence matrix ½ � *A* in the disk plane is 0.5, and for the influence matrix in a plane far from the disk plane is 1. Non-diagonal elements of the matrix ½ � *A* are close to zero.

The angle of attack

*DOI: http://dx.doi.org/10.5772/intechopen.93186*

The angle of sliding

**3.3 Trimming model**

*<sup>X</sup>* <sup>¼</sup> *<sup>γ</sup>*, *<sup>ϑ</sup>*, *<sup>θ</sup>*0, *<sup>θ</sup>c*1, *<sup>θ</sup>s*1, *<sup>θ</sup>tp <sup>T</sup>*

behavior of the vortex surface in modeling.

**4. Results**

from it [5].

**109**

corresponding loads.

have got an adequate correlation.

*X <sup>f</sup>* ¼ *CDf α <sup>f</sup>*

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity*

*Y <sup>f</sup>* ¼ *CLf α <sup>f</sup>*

*Z <sup>f</sup>* ¼ *CZf β <sup>f</sup>*

*Mxf* ¼ *CMxf β <sup>f</sup>*

*Myf* ¼ *CMyf β <sup>f</sup>*

CFD-method and corrected by the results of the tests in the wind tunnel.

tions. Equating to zero the linear and angular accelerations of the helicopter

As a result of the system of equations (Eq. (52)) solving we obtain the vector

To illustrate the comparison of result, we distribute the induced velocity in the plane of the rotor disk and at a far distance from it (**Figure 5**). This confirms the results of the Momentum theory: the inductive velocities in the plane of the rotor disk are two times less than the inductive velocities at an infinite distance

The tip vortices structure of the rotor shown in **Figure 6**. Comparison of the structure under the main rotor with the blade theory results shows an adequate

Introducing the air flow configuration, we can see which areas of the helicopter

**Figure 7** shows a comparison of the position of blade vortex theory (disorderly line) and disk theory (black line) for horizontal flight. The tip vortex is shown only

The results of calculating the normal component of induced velocity were compared with experimental data for forward flight from 75 to 180 km/h. Comparison with the experiment gave good results (**Figure 8**). Experimental and calculated data

are influenced by the induced flow and used in the analysis of information

from one blade, but the influence of all blades is taken into account.

The state vector *X* ¼ *Vx*,*Vy*,*Vz*, ω*x*, ω*y*, ω*z*, Ω, γ*h*, ϑ*<sup>h</sup>*

*Mzf* ¼ *CQf α <sup>f</sup>*

� *<sup>ρ</sup>V*<sup>2</sup>

� *<sup>ρ</sup>V*<sup>2</sup>

� *<sup>ρ</sup>V*<sup>2</sup>

� *<sup>ρ</sup>V*<sup>2</sup>

� *<sup>ρ</sup>V*<sup>2</sup>

� *<sup>ρ</sup>V*<sup>2</sup>

In this case, the coefficients in equations (Eqs. (50) and (51)) are calculated with

The helicopter trimming equations derived from the helicopter dynamics equa-

*<sup>ω</sup>* � *<sup>V</sup> Mh* <sup>¼</sup> *RMR* <sup>þ</sup> *RTR* <sup>þ</sup> *Rf* <sup>þ</sup> *Gh* ½ � *ω* � ð Þ *Jω Jh* ¼ *MMR* þ *MTR* þ *Mf*

*S <sup>f</sup> =*2

*S fL <sup>f</sup> =*2*:*

*S <sup>f</sup> =*2

*S fL <sup>f</sup> =*2*:*

is defined as a data source.

. This vector is calculated by Newton's method.

*S <sup>f</sup> =*2 (50)

*S fL <sup>f</sup> =*2 (51)

(52)

## **3. Helicopter trimming**

#### **3.1 The aerodynamic and inertia loads on the blades**

The aerodynamic load on the blades is calculated from the known equations of the aerodynamics of a helicopter rotor is the same for main and tail rotors of the helicopter for the spatial movements relative to the longitudinal and transverse axis of the helicopter

$$\begin{aligned} \overline{U}\_x &= \left(\overline{r} - \overline{l}\_\mathcal{g}\right)\cos\beta\_b + \overline{l}\_\mathcal{g} + \mu\sin\psi\_b + \left(\overline{r} - \overline{l}\_\mathcal{g}\right)\sin\beta\_b \left(\overline{w}\_x\cos\psi\_b - \overline{w}\_x\sin\psi\_b\right) \\ &+ \overline{v}\_x(r, \psi\_b) \end{aligned}$$

$$\begin{aligned} \overline{U}\_\mathcal{r} &= \left(\overline{V}\sin\alpha - \overline{v}\_\mathcal{r}(r, \psi\_b)\right)\cos\beta\_b - \left(\mu\cos\psi\_b + \overline{v}\_x(r, \psi\_b)\right)\sin\beta\_b - \left(\overline{r} - \overline{l}\_\mathcal{g}\right)\frac{d\beta\_b}{d\psi\_b} \\ &+ \left(\overline{r} - \overline{l}\_\mathcal{g} + \overline{l}\_\mathcal{g}\cos\beta\_b\right)\left(\overline{w}\_x\sin\psi\_b + \overline{w}\_x\cos\psi\_b\right) \end{aligned} \tag{47}$$

$$\overline{U}\_x = (\mu \cos \psi\_b + \overline{v}\_x(r, \psi\_b)) \cos \beta\_b + \left[\overline{v}\_\gamma(r, \psi\_b) + \overline{l}\_\mathbf{g}(\overline{\alpha}\_\mathbf{x} \sin \psi\_b + \overline{\alpha}\_\mathbf{z} \cos \psi\_b) \right] \sin \beta\_b.$$

We add inertial loads [4] to the distributed aerodynamic forces in the blade cross section

$$dt\_r dt\_r = \left(c\_L \overline{U}\_x + c\_D \overline{U}\_\chi\right) \overline{U} \overline{b}\_r d\overline{r} + d \overline{l}\_{yb}, \\ dq\_r = \left(c\_D \overline{U}\_x - c\_L \overline{U}\_\chi\right) \overline{U} b\_r d\overline{r} + d \overline{l}\_{xb}. \tag{48}$$

As a result, we obtain the blade flap equations. The integration over the length of the blade gives us the equation

$$\begin{aligned} \left[M\_t - \frac{d^2 \rho\_b}{dt^2} J\_\mathbf{g} - \left(\cos\beta\_b l\_\mathbf{g} + l\_\mathbf{g} \mathbf{S}\_\mathbf{g}\right)\right] \Omega^2 \sin\beta\_b + 2\Omega \left(\omega\_x \cos\psi\_b - \omega\_y \sin\psi\_b\right) \cos\beta\_b\Big] + \\\ \left(-\frac{d\omega\_x}{dt} \sin\psi\_b + \frac{d\omega\_x}{dt} \cos\psi\_b\right) \left(l\_\mathbf{g} + \cos\beta\_b l\_\mathbf{g} \mathbf{S}\_\mathbf{g}\right) - \mathbf{g} \cos\beta\_b \mathbf{S}\_\mathbf{g} - K\_\beta(\cos\beta\_b - \cos\beta\_k) = 0 \end{aligned} \tag{49}$$

With a trimmed helicopter flight, the flapping of the blades can be represented as a Fourier series. This uniquely determined by the blade flaps angular velocity *dβb=dt* and angular acceleration *d*<sup>2</sup> *βb=dt*<sup>2</sup> . The parameters of the helicopter state *X* ¼ *Vx*,*Vy*,*Vz*, ω*x*, ω*y*, ω*z*, Ω, γ*h*, ϑ*<sup>h</sup>* are set by the main and tail rotor control (*θ*0, *θc*1, *θ<sup>s</sup>*<sup>1</sup> and *θtp* ) and the load computation is reduced to the computation of the Fourier series coefficients *β*0, *βc*1, *βs*1, … , *βcn*, *βsn* of the blade flaps *βb*. The Fourier coefficients are determined from the equation (Eq. (49)) by Newton's method. The number of coefficients in this case should be equal to the number of rotor azimuth steps.

#### **3.2 The fuselage aerodynamic loads**

The fuselage aerodynamic loads depend on the angle of attack or the angle of sliding is evaluated by wind tunnel or calculation by CFD-method.

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity DOI: http://dx.doi.org/10.5772/intechopen.93186*

The angle of attack

In hovering mode, the main diagonal (*i* ¼ *j*) of the influence matrix ½ � *A* in the disk plane is 0.5, and for the influence matrix in a plane far from the disk plane is 1.

The aerodynamic load on the blades is calculated from the known equations of the aerodynamics of a helicopter rotor is the same for main and tail rotors of the helicopter for the spatial movements relative to the longitudinal and transverse axis

*Uy* <sup>¼</sup> *<sup>V</sup>* sin *<sup>α</sup>* � *vy <sup>r</sup>*, *<sup>ψ</sup><sup>b</sup>* ð Þ cos *<sup>β</sup><sup>b</sup>* � *<sup>μ</sup>* cos *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup> *vz <sup>r</sup>*, *<sup>ψ</sup><sup>b</sup>* ð Þ ð Þ sin *<sup>β</sup><sup>b</sup>* � *<sup>r</sup>* � *lg*

*Uz* <sup>¼</sup> *<sup>μ</sup>* cos *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup> *vz <sup>r</sup>*, *<sup>ψ</sup><sup>b</sup>* ð Þ ð Þ cos *<sup>β</sup><sup>b</sup>* <sup>þ</sup> *vy <sup>r</sup>*, *<sup>ψ</sup><sup>b</sup>* ð Þþ *lg <sup>ω</sup><sup>x</sup>* sin *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup> *<sup>ω</sup><sup>z</sup>* cos *<sup>ψ</sup><sup>b</sup>* ð Þ sin *<sup>β</sup>b:*

We add inertial loads [4] to the distributed aerodynamic forces in the blade cross

As a result, we obtain the blade flap equations. The integration over the length of

With a trimmed helicopter flight, the flapping of the blades can be represented as a Fourier series. This uniquely determined by the blade flaps angular velocity

 are set by the main and tail rotor control (*θ*0, *θc*1, *θ<sup>s</sup>*<sup>1</sup> and *θtp* ) and the load computation is reduced to the computation of the Fourier series coefficients *β*0, *βc*1, *βs*1, … , *βcn*, *βsn* of the blade flaps *βb*. The Fourier coefficients are determined from the equation (Eq. (49)) by Newton's method. The number of coefficients in this case should be equal to the number of rotor azimuth

The fuselage aerodynamic loads depend on the angle of attack or the angle of

sliding is evaluated by wind tunnel or calculation by CFD-method.

<sup>Ω</sup><sup>2</sup> sin *<sup>β</sup><sup>b</sup>* <sup>þ</sup> <sup>2</sup><sup>Ω</sup> *<sup>ω</sup><sup>x</sup>* cos *<sup>ψ</sup><sup>b</sup>* � *<sup>ω</sup><sup>y</sup>* sin *<sup>ψ</sup><sup>b</sup>*

<sup>þ</sup>

sin *<sup>β</sup><sup>b</sup> <sup>ω</sup><sup>x</sup>* cos *<sup>ψ</sup><sup>b</sup>* � *<sup>ω</sup><sup>x</sup>* sin *<sup>ψ</sup><sup>b</sup>* ð Þ

*Ubrdr* <sup>þ</sup> *dJxb:* (48)

cos *β<sup>b</sup>*

*βb=dt*<sup>2</sup> . The parameters of the helicopter state

� *g* cos *βbSg* � *K<sup>β</sup>* cos *β<sup>b</sup>* � cos *β<sup>k</sup>* ð Þ ¼ 0

*<sup>d</sup>β<sup>b</sup>*

*dψ<sup>b</sup>*

(47)

(49)

Non-diagonal elements of the matrix ½ � *A* are close to zero.

**3.1 The aerodynamic and inertia loads on the blades**

cos *<sup>β</sup><sup>b</sup>* <sup>þ</sup> *lg* <sup>þ</sup> *<sup>μ</sup>* sin *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup> *<sup>r</sup>* � *lg*

*<sup>ω</sup><sup>x</sup>* sin *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup> *<sup>ω</sup><sup>z</sup>* cos *<sup>ψ</sup><sup>b</sup>* ð Þ

*Ubrdr* <sup>þ</sup> *dJyb*, *dqr* <sup>¼</sup> *<sup>с</sup>DUx* � *cLUy*

*Jg* þ cos *βblgSg* 

**3. Helicopter trimming**

*Vortex Dynamics Theories and Applications*

of the helicopter

section

*Mt* � *<sup>d</sup>*<sup>2</sup>

*dω<sup>x</sup>*

þ

steps.

**108**

*βb*

*dt* sin *<sup>ψ</sup><sup>b</sup>* <sup>þ</sup>

*Ux* ¼ *r* � *lg*

þ *vx r*, *ψ<sup>b</sup>* ð Þ

*dtr* ¼ *сLUx* þ *cDUy*

the blade gives us the equation

*dt*<sup>2</sup> *Jg* � cos *<sup>β</sup>bJg* <sup>þ</sup> *lgSg*

*dω<sup>z</sup> dt* cos *<sup>ψ</sup><sup>b</sup>*

*dβb=dt* and angular acceleration *d*<sup>2</sup>

*X* ¼ *Vx*,*Vy*,*Vz*, ω*x*, ω*y*, ω*z*, Ω, γ*h*, ϑ*<sup>h</sup>*

**3.2 The fuselage aerodynamic loads**

þ *r* � *lg* þ *lg* cos *β<sup>b</sup>*

$$\begin{aligned} X\_f &= \mathbf{C}\_{Df}(a\_f) \cdot \rho \mathbf{V}^2 \mathbf{S}\_f / 2 \\ Y\_f &= \mathbf{C}\_{Lf}(a\_f) \cdot \rho \mathbf{V}^2 \mathbf{S}\_f / 2 \\ M\_{\sharp f} &= \mathbf{C}\_{Qf}(a\_f) \cdot \rho \mathbf{V}^2 \mathbf{S}\_f L\_f / 2. \end{aligned} \tag{50}$$

The angle of sliding

$$\mathbf{Z}\_f = \mathbf{C}\_{\widetilde{\mathbf{z}}'} \left(\boldsymbol{\beta}\_f\right) \cdot \boldsymbol{\rho} \mathbf{V}^2 \mathbf{S}\_f / 2$$

$$\mathbf{M}\_{\mathbf{x}f} = \mathbf{C}\_{\mathbf{M} \mathbf{x}f} \left(\boldsymbol{\beta}\_f\right) \cdot \boldsymbol{\rho} \mathbf{V}^2 \mathbf{S}\_f \mathbf{L}\_f / 2 \tag{51}$$

$$\mathbf{M}\_{\mathcal{H}} = \mathbf{C}\_{\mathbf{M} \mathbf{y}f} \left(\boldsymbol{\beta}\_f\right) \cdot \boldsymbol{\rho} \mathbf{V}^2 \mathbf{S}\_f \mathbf{L}\_f / 2.$$

In this case, the coefficients in equations (Eqs. (50) and (51)) are calculated with CFD-method and corrected by the results of the tests in the wind tunnel.

#### **3.3 Trimming model**

The helicopter trimming equations derived from the helicopter dynamics equations. Equating to zero the linear and angular accelerations of the helicopter

$$\begin{aligned} (\overline{\boldsymbol{\alpha}} \times \overline{\boldsymbol{V}}) \mathbf{M}\_h &= \overline{\boldsymbol{R}}\_{\text{MR}} + \overline{\boldsymbol{R}}\_{\text{TR}} + \overline{\boldsymbol{R}}\_f + \overline{\boldsymbol{G}}\_h \\ [\overline{\boldsymbol{\alpha}} \times (\boldsymbol{f} \overline{\boldsymbol{\alpha}})] \boldsymbol{I}\_h &= \overline{\boldsymbol{M}}\_{\text{MR}} + \overline{\boldsymbol{M}}\_{\text{TR}} + \overline{\boldsymbol{M}}\_f \end{aligned} \tag{52}$$

The state vector *X* ¼ *Vx*,*Vy*,*Vz*, ω*x*, ω*y*, ω*z*, Ω, γ*h*, ϑ*<sup>h</sup>* is defined as a data source. As a result of the system of equations (Eq. (52)) solving we obtain the vector *<sup>X</sup>* <sup>¼</sup> *<sup>γ</sup>*, *<sup>ϑ</sup>*, *<sup>θ</sup>*0, *<sup>θ</sup>c*1, *<sup>θ</sup>s*1, *<sup>θ</sup>tp <sup>T</sup>* . This vector is calculated by Newton's method.
