**2.3.2 Dynamics**

In this paper the rotor of helicopter SA 341 "Gazelle" is modeled, at which the blades are attached to the hub by flap, pitch and pseudo lead-lag hinges. Blade motion in lead-lag plane is limited by dynamic damper, which permits very small maximum blade deflection.

Fig. 13. Coordinate systems

Due to such small angular freedom of motion, we can assume that there is no led-lag motion at all. Pitch hinge is placed between flap hinge and pseudo lead-lag hinge.

According to that, the following frames have been selected for use:

This subchapter presents the numerical model developed for rotor blade aerodynamics loads calculation. The model is unsteady and fully three-dimensional. Helicopter blade is assumed to be rigid, and its motion during rotation is modeled in the manner that rotor

Helicopter rotor aerodynamic flow field is very complex, and it is characterized by remarkably unsteady behavior. The most significant unsteadiness appears during the forward flight. In that case, the progressive motion of helicopter coupled with rotary motion of rotor blades causes drastic variations of local velocity vectors over the blades, where the advancing or retreating blade position is of great significance. In first case, the local tip transonic flow generates, while in second, speed reversal appears. In addition, in forward flight, blades encounter wakes generated by forerunning blades and so encounter nonuniform inflow. The wake passing by the blade induces high velocities close to it causes changes in lifting force. Besides that, in horizontal flight blades constantly change pitch, i.e. angle of attack at different azimuths. Such angle of attack variations are very rapid, so that

In this paper the rotor of helicopter SA 341 "Gazelle" is modeled, at which the blades are attached to the hub by flap, pitch and pseudo lead-lag hinges. Blade motion in lead-lag plane is limited by dynamic damper, which permits very small maximum blade deflection.

Due to such small angular freedom of motion, we can assume that there is no led-lag motion

at all. Pitch hinge is placed between flap hinge and pseudo lead-lag hinge.

According to that, the following frames have been selected for use:

**2.3 Improved solution approach for aerodynamics loads of helicopter rotor blade in** 

presents a model of rotor of helicopter Aerospatiale SA 341 "Gazelle".

dynamic stall occurs, especially in case of retreating blades.

**forward flight** 

**2.3.1 Introduction** 

**2.3.2 Dynamics** 

Fig. 13. Coordinate systems


In derivation of the equations of motion, the following was assumed:


With assumptions above mentioned, equation of blade flapping motion is:

$$B\ddot{\beta} + \Omega^2 \left( B\cos\beta + m\_b e\_\beta \mathfrak{x}\_\chi R \right) \sin\beta = M\_A$$

where *B* is moment of inertia about *Px* , *mb* mass of the blade, *<sup>g</sup> x* position of blade center of gravity in *P* frame, and *MA* is aerodynamic moment.

#### **2.3.3 Aerodynamics**

The flow field is assumed to be potential (inviscid and irrotational) and incompressible. In that case, velocity potential satisfies the Laplace equation 0 .

The equation is the same, both for steady and unsteady flows. Owing to that, methods for steady cases can be applied for the solution of unsteady flow problems, as well. Unsteadiness is introduced by:


where: - is velocity potential,*V* - is absolute fluid velocity, *VT* - is lifting surface velocity, *n* - is the normal of the lifting surface at a certain point, and - is the bound circulation.

In case of inviscid problems, it is necessary to satisfy Kutta condition at the trailing edge. Based on unsteady Bernoulli equation, the difference between upper and lower surface pressure coefficients, in case of the thin lifting surface, is:

$$
\Delta \mathbf{C}\_P = \mathbf{C}\_{P\_{\rm II}} - \mathbf{C}\_{P\_{\rm L}} = -\frac{V\_{\rm II}^2 - V\_{\rm L}^2}{V\_{\rm \sigma}^2} - \frac{2}{V\_{\rm \sigma}^2} \frac{\partial}{\partial t} \left( \int\_{LE}^{\mathbf{M}} \mathbf{y} \, dl \right) \tag{10}
$$

The opposite side of the vortex is always placed at the trailing edge, while the other two sides are parallel to the flow. The wake is represented by quadrilateral vortex in the airflow behind the lifting surface. One side of it is connected to the trailing edge, while the opposite one is at the infinity. The other two sides (trailing vortices), which actually represent the wake, are placed parallel to the airflow. The vorticity of the quadrilateral vortex is equal to the sum of the vorticities of all bound vortices of the panels that correspond a certain lifting surface chord, but opposite in direction. Then the trailing edge vorticity is equal to zero.

Model established in such a manner corresponds to the steady flow case. On the other hand,

The variation of the lifting surface position in time induces variation of circulation around the lifting surface as well. According to the Kelvin theorem, this variation in circulation must also induce the variation around the wake. According to the unsteady Kutta condition,

Suppose that the lifting surface has been at rest until the moment t, when it started with the relative motion with respect to the undisturbed airflow. The vortex releasing, as a way of circulation balancing, is done continually, and in such a way a vortex surface of intensity

it can be very easily spread in order to include the unsteady effects.

this can be achieved by successive releasing of the vortices in the airflow.

Fig. 14. The steady panel scheme

**2.3.4.1 Vortex releasing model** 

Fig. 15. Vortex releasing

where integral should be calculated from leading edge to a certain point *M* at the surface, and is the local bound vortex distribution. If we assume that spanwise velocities are small, the difference of velocity squares can be calculated as:

$$
\lambda V\_{\rm II}^2 - V\_{\rm L}^2 \approx 2V\_{\rm \alpha} \gamma \tag{11}
$$

By substituting (11) in (10), the following equation can be obtained:

$$
\Delta C\_P = -\frac{2}{V\_\infty^2} \left( V\_\infty \mathcal{Y} + \frac{\partial}{\partial t} \int\_{LE}^M \mathcal{Y} dl \right),
$$

The Kutta condition can be expressed as the uniqueness of pressure coefficients at the trailing edge:

$$
\Delta \mathbf{C}\_p = \frac{2}{V\_\infty^2} \left( V\_\infty \mathcal{V}\_{\rm TE} + \frac{\partial}{\partial t} \int\_{LE}^{\rm TE} \mathcal{V} \, dl \right) = \frac{2}{V\_\infty^2} \left( V\_\infty \mathcal{V}\_{\rm TE} + \frac{\partial \widetilde{\Gamma}}{\partial t} \right) = 0
$$

where is the contour circulation which covers the lifting surface. Since it is impossible to be*V* , the relation within the parentheses must be equal to zero and the expression for unsteady Kutta condition comes out directly as:

$$\frac{\partial \bar{\Gamma}}{\partial t} = -V\_{\Rightarrow} \mathcal{Y}\_{TE}$$

If the right hand-side part is substituted with (11) written for the trailing edge, we obtain:

$$\frac{\partial \tilde{\Gamma}}{\partial t} = -\frac{V\_{\mathcal{U}\_{\rm TE}}^2 - V\_{\mathcal{L}\_{\rm TE}}^2}{2} = -\left(V\_{\mathcal{U}\_{\rm TE}} - V\_{\mathcal{L}\_{\rm TE}}\right)\frac{V\_{\mathcal{U}\_{\rm TE}} + V\_{\mathcal{L}\_{\rm TE}}}{2}$$

From this equation, it can be clearly seen that the variation of the lifting surface circulation in time can be compensated by releasing vortices of magnitude *U L TE TE V V* at the velocity 2 *U L TE TE V V* .

#### **2.3.4 Discretization and numerical solution procedure**

The method for the solution of this problem is based on the coupling of the dynamic equations of blade motion with the equations of aerodynamics.

Discretization in time is done by observing the flow around the blade in a series of positions that it takes at certain times *kt* , which are spaced by finite time intervals *t* at different azimuths. Also, discretization of the thin lifting surface is done by using the panel approach. By this method, the lifting surface is divided in a finite number of quadrilateral surfaces – panels. Vorticity distribution is discretized in a finite number of concentrated, closed quadrilateral linear vortices, in such a way that one side of the linear vortex is placed at the first quarter chord of the panel, and represents the bound vortex of the corresponding panel.

where integral should be calculated from leading edge to a certain point *M* at the surface,

2 2 *VV V U L* 2

2

 

2 2

*TE p TE TE LE C V dl V*

the difference of velocity squares can be calculated as:

unsteady Kutta condition comes out directly as:

*t*

**2.3.4 Discretization and numerical solution procedure** 

equations of blade motion with the equations of aerodynamics.

By substituting (11) in (10), the following equation can be obtained:

*P*

is the local bound vortex distribution. If we assume that spanwise velocities are small,

2 *<sup>M</sup>*

2 2 <sup>0</sup>

*C V dl V t* 

The Kutta condition can be expressed as the uniqueness of pressure coefficients at the

*V V t t*

where is the contour circulation which covers the lifting surface. Since it is impossible to be*V* , the relation within the parentheses must be equal to zero and the expression for

> *<sup>V</sup> TE <sup>t</sup>*

2 2 *TE TE TE TE TE TE U L U L U L V V V V V V*

 

If the right hand-side part is substituted with (11) written for the trailing edge, we obtain:

From this equation, it can be clearly seen that the variation of the lifting surface circulation in time can be compensated by releasing vortices of magnitude *U L TE TE V V* at the velocity

The method for the solution of this problem is based on the coupling of the dynamic

Discretization in time is done by observing the flow around the blade in a series of positions that it takes at certain times *kt* , which are spaced by finite time intervals *t* at different azimuths. Also, discretization of the thin lifting surface is done by using the panel approach. By this method, the lifting surface is divided in a finite number of quadrilateral surfaces – panels. Vorticity distribution is discretized in a finite number of concentrated, closed quadrilateral linear vortices, in such a way that one side of the linear vortex is placed at the first quarter chord of the panel, and represents the bound vortex of the corresponding

2 2

 

 

*LE*

 

> 

(11)

and 

trailing edge:

2 *U L TE TE V V* .

panel.

Fig. 14. The steady panel scheme

The opposite side of the vortex is always placed at the trailing edge, while the other two sides are parallel to the flow. The wake is represented by quadrilateral vortex in the airflow behind the lifting surface. One side of it is connected to the trailing edge, while the opposite one is at the infinity. The other two sides (trailing vortices), which actually represent the wake, are placed parallel to the airflow. The vorticity of the quadrilateral vortex is equal to the sum of the vorticities of all bound vortices of the panels that correspond a certain lifting surface chord, but opposite in direction. Then the trailing edge vorticity is equal to zero.

Model established in such a manner corresponds to the steady flow case. On the other hand, it can be very easily spread in order to include the unsteady effects.
