*3.3. Aerodynamic Forces and Moments*

The characterization of aerodynamic force, *<sup>F</sup>*(*a*) = [*X*, *<sup>Y</sup>*, *<sup>Z</sup>*] *<sup>T</sup>*, and moment, *<sup>M</sup>*(*a*) = [*L*, *M*, *N*] *<sup>T</sup>*, is performed on the basis of the model detailed in [25], whose nomenclature is adopted in the present work. A conventional single MR helicopter with teetering configuration and counterclockwise rotation are considered. Contributions are provided by the main rotor (MR), tail rotor (TR), fuselage (F), horizontal stabilizer (HS), upper and lower vertical fins (VF1 and VF2), main rotor hub (MRH), and parachute canopy (PC). Air parameters are calculated from the International Standard Atmosphere (ISA) model as a function of rotorcraft altitude [27].

#### 3.3.1. MR and TR Modeling

The following assumptions and simplifications are made about the MR model: (a) rotor blades are rigid in bending and torsion; (b) flapping angles are small, and the analysis follows the simple strip theory [28]; (c) the effects of aircraft motion on blade flapping are limited to those related to the angular accelerations *p*˙ and *q*˙, the angular rates *p* and *q*, and the normal acceleration component *w*˙ ; (d) blade flow stall is disregarded; (e) rotor inflow is uniform, and no inflow dynamics is modeled; (f) main rotor blade flapping is approximated by the first harmonic terms with time-varying coefficients, that is

$$\beta(t) = a\_0 - a\_1 \cos \tilde{\xi} - b\_1 \sin \tilde{\xi} \tag{8}$$

where *a*<sup>0</sup> is treated as a preset constant (coning angle) and *ξ* is blade azimuth. Coefficients *a*1(*t*) and *b*1(*t*) respectively represent the longitudinal and lateral tilt of the rotor tip–path plane, obtained as solutions to the equations in Appendix C of [25] with null hinge offset ratio, = 0, flapping spring constant, *K<sup>β</sup>* = 0, and pitch-flap coupling ratio, tan *δ*<sup>3</sup> = 0. Finally, the MR shaft is aligned with *zB*.

The tail rotor is modeled according to a teetering configuration without cyclic pitch. Provided that the flapping frequency is typically much higher than that of the MR system, TR tip-path plane dynamics are neglected, no flapping spring constant is considered, and the pitch-flap coupling ratio, *δ*3*TR*, is characterized by a non-null value (see Appendix D in [25]).

Contrary to some of the assumptions provided in [25], the blades of both MR and TR are characterized by cambered airfoils with a lift-curve slope *a* < 2*π* 1/rad and a zero-lift angle of attack *α*<sup>0</sup> -= 0. The rotor blade profile drag coefficient, *Cd*, is calculated as

$$\mathcal{C}\_d = 0.008 + 0.3 \left( \frac{6 \, \mathcal{C}\_T}{\sigma \, a} \right)^2 + \Delta \mathcal{C}\_d \tag{9}$$

where *CT* is the rotor thrust coefficient, *σ* is rotor solidity, and Δ*Cd* is the extra drag coefficient determined by flow compressibility effects. Let *M*<sup>90</sup> be the Mach number evaluated at the tip of the advancing blade, where *ξ* = 90 deg. In order to estimate the extra drag, the approximate model proposed by Prouty and described in [28] is adopted, where

$$\Delta \mathcal{C}\_d(M\_{90}) = \begin{cases} 12.5(M\_{90} - M\_{dr})^3 & \text{for } M\_{90} \ge M\_{dr} \\ 0 & \text{otherwise} \end{cases} \tag{10}$$

and *Mdr* = 0.74 is the drag-rise Mach number. With respect to the characterization of rotor inflow, a number of non-ideal effects are considered, based on the approach in [28], for the characterization of forces and moments. A constant tip-loss factor *B* < 1 is adopted to account for blade tip losses. Other non-ideal effects, including nonuniform inflow, wake swirl and contraction, and blade interference, are accounted for by an induced power factor *ki*, assumed to be a constant. The MR in-ground effect is provided by the model in ref. [29], and the inflow iterative scheme is solved according to Halley's method with a damping coefficient equal to 0.01 [30].

Cockpit/RC control of MR is provided by pilot commands in terms of lateral cyclic *δa*, longitudinal cyclic *δe*, and collective *δc*. All commands are expressed in terms of nondimensional variables, such that *<sup>δ</sup><sup>a</sup>* <sup>∈</sup> [−1, <sup>+</sup>1] (positive direction: right to generate *<sup>L</sup>* <sup>&</sup>gt; 0), *<sup>δ</sup><sup>e</sup>* <sup>∈</sup> [−1, <sup>+</sup>1] (positive direction: aft to generate *<sup>M</sup>* <sup>&</sup>gt; 0), and *<sup>δ</sup><sup>c</sup>* <sup>∈</sup> [−1, <sup>+</sup>1] (positive direction: up to generate *Z* < 0). Onboard control of the tail rotor is performed by pedal commands, expressed as *δ<sup>p</sup>* ∈ [−1, +1] (positive direction: right pedal forward to generate *N* > 0). The transformation of pilot commands into blade pitch angles is provided by a set of low-order polynomial functions, *A*1*<sup>s</sup>* = *C*1(*δa*), *B*1*<sup>s</sup>* = *C*2(*δe*), *θ*<sup>0</sup> = *C*3(*δc*), and *θ*0*TR* = *C*4(*δp*), provided by the manufacturer. *A*1*<sup>s</sup>* and *B*1*s*, respectively, represent the lateral and longitudinal cyclic pitch angles measured from the MR hub plane in F*B*. Rotor blades are modeled with a linear twist, such that *θ*<sup>0</sup> is the blade collective pitch ideally extrapolated to the rotor center and *θtw* is the total blade twist angle (tip minus root pitch angle). No twist characterizes TR blades, where collective pitch is identified by *θ*0*TR*.

An additional degree of freedom is related to the power plant made of free turbines, MR, and TR transmissions. In particular, MR and TR rotational speeds vary according to the current torque requirements and the engine power available. Changes in speed cause the free turbine governor to vary fuel flow to change the available power and maintain the desired angular rate. The engine dynamic model is found in [25]. For the sake of brevity, details are not provided in the present paper. Modeling parameters in terms of maximum available power, engine dynamics, specific fuel consumption, and mechanical transmission efficiency are provided by the manufacturer.

#### 3.3.2. Fuselage, Empennages, and Miscellaneous Components

With respect to fuselage aerodynamics, it is assumed that longitudinal forces and moments are dependent on fuselage angle of attack and lateral forces and moments are dependent on angle of sideslip. The exception is the drag force, which is assumed to have a contribution from both angles of attack and sideslip. The modeling is based on a low- and a high-angle representation of forces and moments, according to Appendix F in [25], with data obtained through a detailed computational fluid dynamics characterization. Phasing between the two approximations is performed by means of cubic spline interpolation, with improved performance with respect to the proposed linear transition.

The modeling of the two vertical empennages and of the horizontal stabilizer also follows the approach in [25]. The aerodynamics of the MR hub and parachute pod are assessed by the equivalent flat plate area model. As an example, the force vector generated by MRH is expressed as:

$$F\_{MRH} = -\frac{1}{2}\rho \left( \left[ A\_{\text{xMRH}\prime} A\_{\text{yMRH}\prime} A\_{\text{zMRH}} \right] \mathbf{V}\_{MRH} \right) \mathbf{V}\_{MRH} \tag{11}$$

where *V MRH* = [*uMRH*, *vMRH*, *wMRH*] *<sup>T</sup>* is the velocity, relative to the air mass, of the main rotor hub and includes the contribution of MR downwash, according to [31]. *AxMRH*, *AyMRH*, and *AzMRH* are the equivalent flat plate drag areas, respectively orthogonal to *<sup>x</sup>B*, *<sup>y</sup>B*, and *<sup>z</sup>B*. The moment generated by *<sup>F</sup>MRH* about *CG* is given by *<sup>M</sup>MRH* <sup>=</sup> *<sup>d</sup>MRH* <sup>×</sup> *FMRH*, where

$$\mathbf{d}\_{MRH} = \begin{bmatrix} \mathbf{S}TA\_{\rm CG} - \mathbf{S}TA\_{MRH} \\ \mathbf{B}L\_{MRH} - \mathbf{B}L\_{\rm CG} \\ \mathbf{W}L\_{\rm CG} - \mathbf{W}L\_{MRH} \end{bmatrix} \tag{12}$$

is the vector directed from *CG* to MRH position, assumed to be coincident with its center of pressure, with constant components expressed in F*B*.

#### **4. Trim and Stability Analysis**

The nonlinear model described in Section 3 is implemented in the Matlab/Simulink environment, where differential equations are solved by the Dormand-Prince ode8 method with a frequency of 1000 Hz [32]. In what follows, (1) the trim conditions are determined for different cruise speeds, (2) a linearization procedure is applied to the complete model about such equilibria, and (3) an open-loop dynamic analysis is performed to investigate the helicopter control and stability properties.

## *4.1. Trim Analysis*

The helicopter model is numerically trimmed for straight-and-level flight at *h* = 50 m in standard atmospheric conditions. Different values of forward speed are considered, ranging from 0 km/h (hover) to 180 km/h (approximately the never-exceed speed), with steps of 5 km/h. For the sake of brevity, the results of both the static and the following dynamic analysis are summarized only for the hovering condition, for which dedicated flight tests were performed for validation purposes.

The main results of trim analysis for the hovering condition are given in Table 3 and compared with the data available from flight tests performed with the same vehicle configuration (deviations with respect to measured data are reported in terms of absolute values of percentage errors). To this end, the helicopter was equipped with a set of sensors, including: (a) potentiometers for cockpit command acquisition and blade pitch measurement, (b) torque–meters for MR and TR torque analysis, and (c) a AHRS providing rigid body attitude, angular rate, acceleration, speed, and position information.

According to Table 3, good agreement is found between predicted and measured values, showing the validity of the modeling approach. A major difference characterizes the longitudinal cyclic pitch, with a 42% error. It must be noted that a degree of uncertainty characterizes the knowledge of *CG* position (especially the *STACG* parameter) in the actual flight configuration, which is estimated by means of CAD analysis and suspension techniques. Uncertainty also characterizes the aerodynamics of the fuselage, especially in the case of hovering and low-speed forward flight, where MR wake envelops a large portion of the fuselage. For the aim of the present analysis, the model adopted for both MR inflow and fuselage aerodynamics necessarily represents a compromise solution, which allows for satisfactory accuracy in terms of the dynamic characterization of rotorcraft without the cost of excessively-complex aerodynamic models.

The match expected at hover between MR cyclic pitch angles and flapping coefficients, namely *<sup>A</sup>*1*<sup>s</sup>* <sup>=</sup> *<sup>b</sup>*<sup>1</sup> and *<sup>B</sup>*1*<sup>s</sup>* <sup>=</sup> <sup>−</sup>*a*1, holds almost exactly in Table 3. Slight differences occur for the simulated hover condition, which is actually obtained by flying the helicopter at a residual forward speed of 0.1 m/s. With respect to the experimental campaign, effective environmental conditions were also monitored, provided the helicopter was maintained in upwind hover while estimating a maximum wind speed of 20 km/h.


**Table 3.** Trim analysis for the hovering flight.
