*4.2. Dynamic Analysis*

Consider the equations of motion introduced in Section 3 and detailed in [25]. In nonlinear form, it is

$$
\dot{\mathfrak{x}} = f(\mathfrak{x}, \mathfrak{u}, t) \tag{13}
$$

provided *x* is rigid–body state vector, namely

$$\mathbf{x} = [\boldsymbol{u}, \,\,\boldsymbol{w}, \,\,\boldsymbol{q}, \,\,\boldsymbol{\theta}, \,\,\boldsymbol{v}, \,\,\boldsymbol{p}, \,\,\boldsymbol{\phi}, \,\,\boldsymbol{r}]^T \tag{14}$$

while time evolution of *xE*, *yE*, *zE*, and *ψ* is not accounted in the framework of system linearization. Control vector *u* has four components, expressed in terms of pilot commands as:

$$\mathfrak{u} = \begin{bmatrix} \delta\_{\mathfrak{c}\prime} \,\,\delta\_{\mathfrak{e}\prime} \,\,\delta\_{\mathfrak{a}\prime} \,\delta\_{\mathfrak{p}} \end{bmatrix}^{T} \tag{15}$$

Using small perturbation theory [29], helicopter motion is described in terms of perturbation from the equilibrium condition, *x<sup>e</sup>* = [*Ue*, *We*, *Qe*, Θ*e*, *Ve*, *Pe*, Φ*e*, *Re*] *<sup>T</sup>* and *u<sup>e</sup>* = [*U*1*e*, *U*2*e*, *U*3*e*, *U*4*e*, ] *<sup>T</sup>*, written in the form *x* = *x<sup>e</sup>* + *δx* and *u* = *u<sup>e</sup>* + *δu*. By following the approach and the nomenclature of [29], given the trim conditions in Section 4.1, the model in Equation (13) is linearized at all considered speeds to respectively obtain system and input matrices

$$A = \left(\frac{\partial f}{\partial \mathbf{x}}\right)\_{\mathbf{x}\_{\varepsilon}, \mathbf{u}\_{\varepsilon}} \quad \mathbf{B} = \left(\frac{\partial f}{\partial \mathbf{u}}\right)\_{\mathbf{x}\_{\varepsilon}, \mathbf{u}\_{\varepsilon}} \tag{16}$$

as a function of aerodynamic derivatives. The latter are estimated by numerical differencing in the Matlab/Simulink environment [29]. To this end, aerodynamic forces and moments are positively perturbed by each of the state and input vector components in turn, with amplitude equal to 0.02 (respectively intended in terms of m/s for *u*, *v*, *w*, rad/s for *p*, *q*, *r*, rad for *φ*, *θ*, and non-dimensional units for control inputs). State and control derivatives are written in the form:

$$X\_{\mathcal{U}} = \frac{1}{m} \frac{\partial X}{\partial u} \tag{17}$$

and

$$L\_p' = \frac{f\_{zz}}{f\_{xx}f\_{zz} - f\_{xz}^2} \frac{\partial L}{\partial p} + \frac{f\_{xz}}{f\_{xx}f\_{zz} - f\_{xz}^2} \frac{\partial N}{\partial p} \tag{18}$$

$$N\_r' = \frac{f\_{\rm xz}}{f\_{\rm xx} \, f\_{\rm zz} - f\_{\rm xz}^2} \frac{\partial L}{\partial r} + \frac{f\_{\rm xx}}{f\_{\rm xx} \, f\_{\rm zz} - f\_{\rm xz}^2} \frac{\partial N}{\partial r} \tag{19}$$

A total of 36 stability derivatives and 24 control derivatives are determined in the standard 6DOF representation for each flight condition. Due to the highly-classified nature of the data involved in the project, only one sample derivative is analyzed in the present paper at hover. A qualitative discussion about the behavior of the most significant derivatives is provided in what follows.

The effect of linear velocity on aerodynamic forces is principally taken into account by *Xu*, *Yv*, and *Zw*. The force damping derivatives *Xu* < 0 and *Yv* < 0, which respectively reflect the drag and side force on rotor–fuselage combination, steadily increase in absolute value and are practically linear with speed beyond 50 km/h. At low speed, the effect of disc tilt following perturbations in *u* and *v* becomes predominant. Similar considerations hold for the heave damping derivative *Zw*, which is mostly influenced by the fuselage and horizontal empennage in high-speed flight. At low speed, the MR tends to dominate *Zw* through a reduction in *CT* determined by a vertical speed perturbation. In order to validate the numerical linearization routine, a comparison is performed with the analytical results obtained for the stability and control derivatives according to formulas available in the literature. As an example, the MR contribution only to *Zw* can be analytically estimated as [29]

$$Z\_w = -\frac{\rho(\Omega \, R) \pi R^2}{m} \frac{\partial \mathcal{C}\_T}{\partial \mu\_z} \tag{20}$$

where *μ<sup>z</sup>* = *w*/(Ω *R*) is MR climb ratio and

$$\frac{\partial \mathbb{C}\_T}{\partial \mu\_z} \approx \frac{2 \, a \, \sigma |\lambda|}{16|\lambda| + a \, \sigma} \tag{21}$$

Based on the data in Tables <sup>1</sup> and 3, it is *<sup>λ</sup>* <sup>=</sup> <sup>−</sup>0.0371 at hover, such that *<sup>∂</sup>CT*/*∂μ<sup>z</sup>* <sup>≈</sup> 0.018. It follows *Zw* ≈ −0.317 1/s, which is close to the value numerically obtained in the same condition for the full helicopter, namely −0.345 1/s. In such a case, the estimation error obtained according to literature results is −8.1%, provided that fuselage and appendages contributions are disregarded.

The speed stability effect is observed in *Mu* > 0 and *L v* < 0, the latter showing a practically linear behavior with speed. *Mw* > 0 is representative of the incidence static stability effect, which increases non-monotonically with speed and approximately tracks *Mu*, being influenced by MR inflow on helicopter components. Finally, *N v* > 0 accounts for the weathercock effect by means of TR and vertical fins (stabilizing with speed) and the fuselage (destabilizing).

The damping derivatives *L <sup>p</sup>* < 0, *Mq* < 0, and *N r* < 0 reflect short-term, small, and moderate-amplitude handling characteristics. If, on the one hand, *L <sup>p</sup>* and *Mq* principally account for MR flapping motion in the presence of roll and pitch rate perturbations, *N r* is dominated by loads on TR and vertical fins, with a stronger yaw-damping effect at high forward speeds.

Given the stability and control derivatives obtained above, the complete system and input matrices *A* and *B* are generated according to the structure provided on page 277 in [29]. Note that, with the idea of designing closed-loop control systems, the input matrix *B* is configured for application to the non-dimensional pilot commands. The formulation in terms of blade pitch control angles is however possible by means of the mapping functions *C*1, *C*2, *C*3, and *C*<sup>4</sup> introduced in Section 3.3.1.

For the aim of the present work, however, the results of a decoupled analysis are first discussed. Based on the approximate separation between the longitudinal and the lateral-directional dynamics, the decoupled representation is available in ref. [29], where input matrices are applied to blade pitch control angles. The longitudinal dynamics are described by the forced system:

$$\begin{aligned} \frac{d}{dt} \begin{bmatrix} u \\ w \\ q \\ \theta \end{bmatrix} &= \underbrace{\begin{bmatrix} X\_u & X\_w & X\_q - W\_\varepsilon & -g \cos \Theta\_\varepsilon \\ Z\_u & Z\_w & Z\_q + \mathcal{U}\_\varepsilon & -g \sin \Theta\_\varepsilon \\ M\_u & M\_w & M\_q & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}}\_{A\_{luv}} \begin{bmatrix} u \\ w \\ q \\ \theta \end{bmatrix} \\ &+ \underbrace{\begin{bmatrix} X\_{\theta\_0} & X\_{B\_{1s}} \\ Z\_{\theta\_0} & Z\_{B\_{1s}} \\ M\_{\theta\_0} & M\_{\theta\_{1s}} \end{bmatrix}}\_{B\_{luv}} \begin{bmatrix} \theta\_0 \\ B\_{1s} \end{bmatrix} \end{aligned} \tag{22}$$

A pair of complex-conjugate poles is determined from *Alon*, which is related to an unstable phugoid mode with natural frequency *ωph* and time constant *τph* (calculated as the reciprocal of the real part of the poles in its absolute value). Two real stable modes are also evaluated, namely heave and pitch subsidence effects. The first pole, identified by *phv* < 0, is practically determined by the vertical damping derivative *Zw*. The second pole, *pps* < 0, accounts for the fundamental contribution of both *Zw* and *Mq* and is characterized by a time constant approximately estimated as *τps* ≈ −1/ *Zw* + *Mq* [29]. The decoupled lateral-directional dynamics are defined by the system:

$$
\begin{split}
\frac{d}{dt} \begin{bmatrix} v \\ p \\ r \\ \Phi \end{bmatrix} &= \underbrace{\begin{bmatrix} Y\_{\mathcal{V}} & Y\_{\mathcal{V}} + W\_{\mathcal{C}} & Y\_{\mathcal{V}} - L\_{\mathcal{C}} & \mathcal{g} \cos \Phi\_{\mathcal{C}} \cos \Theta\_{\mathcal{C}} \\\ L\_{\mathcal{V}}^{\prime} & L\_{\mathcal{V}}^{\prime} & L\_{\mathcal{V}}^{\prime} & 0 \\\ 0 & 1 & \cos \Phi\_{\mathcal{C}} \tan \Theta\_{\mathcal{C}} & 0 \\\ \end{bmatrix}}\_{\mathbf{A}\_{\mathrm{d}t}} \begin{bmatrix} v \\ p \\ r \\ \Phi \end{bmatrix} \\ &+ \underbrace{\begin{bmatrix} Y\_{A\_{\mathcal{I}s}} & Y\_{\theta \theta\_{TR}} \\ L\_{A\_{\mathcal{I}s}}^{\prime} & L\_{\theta\_{TR}}^{\prime} \\ \mathbf{N}\_{A\_{\mathcal{I}s}}^{\prime} & N\_{\theta\_{TR}}^{\prime} \\ 0 & 0 \end{bmatrix}}\_{\mathbf{B}\_{\mathrm{int}}} \begin{bmatrix} A\_{\mathcal{I}s} \\ \theta\_{0TR} \end{bmatrix} \end{split} \tag{23}
$$

A pair of complex-conjugate poles is derived from *Alat* with the natural frequency *ωdr*. Such poles characterize the dutch-roll mode, which is unstable but slowly develops with a time constant *τdr*. The roll subsidence mode, mostly determined by the damping derivative *L <sup>p</sup>*, is related to the real pole *proll* < 0. The spiral subsidence mode at hover is stable, *pspiral* < 0, and dampens with a time constant *τspiral*.

The analysis of coupled representation behind state matrix *A* is also considered, and the obtained poles are marked by a superscript '*c*'. A comparison with the corresponding values derived through the decoupled analysis is provided where possible. Two real poles are first extracted. The roll subsidence effect is recognized in the first pole, *<sup>p</sup>*<sup>1</sup> <sup>=</sup> 0.85 · *proll*, provided *<sup>p</sup>*<sup>1</sup> <sup>≈</sup> *<sup>L</sup> p*. The same consideration holds for vertical damping mode, identified by *p* (*c*) <sup>8</sup> <sup>≈</sup> *phv* <sup>≈</sup> *Zw*. Three pairs of complex-conjugate poles complete the analysis. The first pair, *p* (*c*) 2,3, is stable with a real part proportional to *Mq* <sup>+</sup> *Zw*. It is representative of a damped oscillation with a natural frequency *<sup>ω</sup>*(*c*) *py* and time constant *<sup>τ</sup>*(*c*) *py* , determined by the coupling of pitch and yaw subsidence modes. The second pair, *p* (*c*) 4,5, characterizes the unstable phugoid mode, which develops with a time constant *<sup>τ</sup>*(*c*) *ph* = 0.75 · *τph* and shows natural frequency *<sup>ω</sup>*(*c*) *ph* <sup>=</sup> 1.38 · *<sup>ω</sup>ph*. The last pair of complex poles, *<sup>p</sup>*6,7 characterizes the

dutch roll motion, which is unstable and develops with natural frequency *<sup>ω</sup>*(*c*) *dr* = 0.92 · *ωdr* and time constant *<sup>τ</sup>*(*c*) *dr* = 1.07 · *τdr*.
