2. Physical principle and analytical solution

The principle of the whirl flutter phenomenon is outlined on a simple mechanical system with two degrees of freedom [5]. The propeller and hub are considered to be rigid. A flexible engine mounting is substituted with a system of two rotational springs (of stiffnesses K<sup>Ψ</sup> and KΘ), as illustrated in Figure 1.

Such a system has two independent mode shapes of yaw and pitch, with respective angular frequencies of ω<sup>Ψ</sup> and ωΘ, as shown in Figure 2.

Figure 1. Two-degree-of-freedom gyroscopic system with a propeller.

Figure 2. Independent engine pitch (a) and yaw (b) mode shapes.

Considering the propeller rotation with angular velocity Ω, the primary system motion changes to the characteristic gyroscopic motion. The gyroscopic effect causes two independent mode shapes to merge into whirl motion as shown in Figure 3. The axis of a propeller makes an elliptical movement. The orientation of the propeller axis movement is backward relative to the propeller rotation for the lower-frequency mode (backward whirl mode) and is forward relative to the propeller rotation for the higher-frequency mode (forward whirl mode). The trajectory of this elliptical movement depends on both angular frequencies ω<sup>Ψ</sup> and ωΘ. Because the yaw and pitch motions have a 90 phase shift, the mode shapes in the presence of gyroscopic effects are complex.

The propeller whirl flutter phenomenon was analytically discovered by Taylor and Browne [1]. The next pioneering work was performed by Ribner, who set the basic formulae for the aerodynamic derivatives of propeller forces and moments due to the motion and velocities in pitch and yaw in 1945 [2, 3]. After the accidents of two Lockheed L-188 C Electra II airliners in 1959 and 1960 [4], the importance of the whirl flutter phenomenon in practical applications was recognised. This chapter is focused on turboprop aircraft whirl flutter; however, it may also occur in tilt-rotor aircraft. The whirl flutter phenomenon relates the mutual interactions of the rotating propeller with the aircraft deformations and the aerodynamic forces emerging during forward flight.

The principle of the whirl flutter phenomenon is outlined on a simple mechanical system with two degrees of freedom [5]. The propeller and hub are considered to be rigid. A flexible engine mounting is substituted with a system of two rotational springs (of stiffnesses K<sup>Ψ</sup> and KΘ), as

Such a system has two independent mode shapes of yaw and pitch, with respective angular

2. Physical principle and analytical solution

140 Flight Physics - Models, Techniques and Technologies

frequencies of ω<sup>Ψ</sup> and ωΘ, as shown in Figure 2.

Figure 2. Independent engine pitch (a) and yaw (b) mode shapes.

Figure 1. Two-degree-of-freedom gyroscopic system with a propeller.

illustrated in Figure 1.

The described gyroscopic motion causes the angles of attack of the propeller blades to change, which consequently leads to unsteady aerodynamic forces. These forces may, under specific conditions, induce whirl flutter instability. The most important terms regarding whirl flutter are yaw moment due to pitch MZ(Θ) and, similarly pitch moment due to yaw MY(Ψ). These moments are to be balanced by aerodynamic or structural damping terms. The state of neutral stability with no damping of the system represents the flutter state. The corresponding airflow (V<sup>∞</sup> = VFL) is called the critical flutter speed. In terms of flutter, the stable and unstable states of the gyroscopic system are applicable. Both states for the backward mode are explained in Figure 4. As long as the air velocity is lower than a critical value (V<sup>∞</sup> < VFL), the system is stable

Figure 4. Stable (a) and unstable (b) states of gyroscopic vibrations for the backward mode.

Figure 5. Kinematical scheme of the gyroscopic system.

and the gyroscopic motion is damped. When the airspeed exceeds the critical value (V<sup>∞</sup> > VFL), the system becomes unstable and the gyroscopic motion is divergent.

The main problem in obtaining the analytical solution is to determine the aerodynamic force caused by the gyroscopic motion on each of the propeller blades. The presented equations of motion were set up for the system described in Figure 1 by means of Lagrange's approach. The kinematical scheme, including gyroscopic effects, is shown in Figure 5.

Three angles (φ, Θ, Ψ) are selected as the independent generalised coordinates. The propeller is assumed to be cyclically symmetric with respect to both mass and aerodynamics (i.e., a propeller with a minimum of three blades). The angular velocity is considered to be constant (φ = Ωt). Nonuniform mass moments of inertia of the engine with respect to the yaw and pitch axes (J<sup>Y</sup> 6¼ JZ) are also considered. We will use a coordinate system X, Y, Z linked to the system. Then, kinetic energy is:

$$\mathbf{E\_K} = \frac{1}{2} \mathbf{J\_X} \boldsymbol{\alpha}\_X^2 + \frac{1}{2} \left( \mathbf{J\_Y} \boldsymbol{\alpha}\_Y^2 + \mathbf{J\_Z} \boldsymbol{\alpha}\_Z^2 \right) \tag{1}$$

The angular velocities will be

$$\begin{aligned} \omega\_{\bar{\lambda}} &= \Omega + \dot{\Theta} \sin \Psi \approx \Omega + \dot{\Theta} \Psi\\ \omega\_{\bar{\lambda}} &= \dot{\Theta} \cos \Psi \approx \dot{\Theta} \\ \omega\_{\bar{Z}} &= \dot{\Psi} \end{aligned} \tag{2}$$

Considering that <sup>Θ</sup>\_ <sup>2</sup> Ψ\_ 2 << Ω<sup>2</sup> , the equation for the kinetic energy becomes

$$\mathbf{E}\_{\mathbf{K}} = \frac{1}{2} \mathbf{J}\_{\mathbf{X}} \boldsymbol{\Omega}^2 + \mathbf{J}\_{\mathbf{X}} \boldsymbol{\Omega} \boldsymbol{\Psi} \dot{\boldsymbol{\Theta}} + \frac{1}{2} (\mathbf{J}\_{\mathbf{Y}} \dot{\boldsymbol{\Theta}}^2 + \mathbf{J}\_{\mathbf{Z}} \dot{\boldsymbol{\Psi}}^2) \tag{3}$$

The first part of Eq. (3) is independent of both Θ and Ψ; thus, it does not appear in Lagrange's equation. Then, the potential energy becomes

$$E\_P = \frac{1}{2} K\_{\Theta} \Theta^2 + \frac{1}{2} K\_{\Psi} \Psi^2 \tag{4}$$

To describe the damping, we assume the structural damping commonly used in the flutter analyses, with the damping force proportional to the amplitude of the displacement:

$$\mathbf{D} = \frac{1}{2} \frac{\mathbf{K}\_{\Theta} \boldsymbol{\gamma}\_{\Theta}}{\omega} \dot{\boldsymbol{\Theta}}^2 + \frac{1}{2} \frac{\mathbf{K}\_{\Psi} \boldsymbol{\gamma}\_{\Psi}}{\omega} \dot{\Psi}^2 \tag{5}$$

Then, we obtain from Lagrange's equations and Eqs. (3)–(5) a system of two mutually influencing differential equations:

$$\begin{aligned} \mathbf{J}\_{\mathbf{Y}}\ddot{\Theta} + \frac{\mathbf{K}\_{\Theta}\boldsymbol{\gamma}\_{\Theta}}{\omega}\dot{\Theta} + \mathbf{J}\_{\mathbf{X}}\boldsymbol{\Omega}\,\dot{\Psi} + \mathbf{K}\_{\Theta}\Theta &= \mathbf{Q}\_{\Theta} \\ \mathbf{J}\_{\mathbf{Z}}\ddot{\Psi} + \frac{\mathbf{K}\_{\Psi}\boldsymbol{\gamma}\_{\Psi}}{\omega}\dot{\Psi} - \mathbf{J}\_{\mathbf{X}}\boldsymbol{\Omega}\dot{\Theta} + \mathbf{K}\_{\Psi}\Psi &= \mathbf{Q}\_{\Psi} \end{aligned} \tag{6}$$

Generalized propeller forces and moments (see Figure 5) can be expressed as

and the gyroscopic motion is damped. When the airspeed exceeds the critical value (V<sup>∞</sup> > VFL),

The main problem in obtaining the analytical solution is to determine the aerodynamic force caused by the gyroscopic motion on each of the propeller blades. The presented equations of motion were set up for the system described in Figure 1 by means of Lagrange's approach. The

Three angles (φ, Θ, Ψ) are selected as the independent generalised coordinates. The propeller is assumed to be cyclically symmetric with respect to both mass and aerodynamics (i.e., a propeller with a minimum of three blades). The angular velocity is considered to be constant (φ = Ωt). Nonuniform mass moments of inertia of the engine with respect to the yaw and pitch axes (J<sup>Y</sup> 6¼ JZ) are also considered. We will use a coordinate system X, Y, Z linked to the system.

<sup>ω</sup><sup>X</sup> <sup>¼</sup> <sup>Ω</sup> <sup>þ</sup> <sup>Θ</sup>\_ sin<sup>Ψ</sup> <sup>≈</sup> <sup>Ω</sup> <sup>þ</sup> ΘΨ\_

<sup>ω</sup><sup>Y</sup> <sup>¼</sup> <sup>Θ</sup>\_ cos<sup>Ψ</sup> <sup>≈</sup> <sup>Θ</sup>\_

JXΩ<sup>2</sup> <sup>þ</sup> JXΩΨΘ\_ <sup>þ</sup>

<sup>ω</sup><sup>Z</sup> <sup>¼</sup> <sup>Ψ</sup>\_

<sup>Y</sup> <sup>þ</sup> JZω<sup>2</sup> Z

, the equation for the kinetic energy becomes

<sup>þ</sup> JZΨ\_ <sup>2</sup>

1 2 <sup>ð</sup>JYΘ\_ <sup>2</sup>

(1)

(2)

Þ (3)

the system becomes unstable and the gyroscopic motion is divergent.

Figure 5. Kinematical scheme of the gyroscopic system.

142 Flight Physics - Models, Techniques and Technologies

kinematical scheme, including gyroscopic effects, is shown in Figure 5.

EK <sup>¼</sup> <sup>1</sup> 2 JXω<sup>2</sup> <sup>X</sup> þ 1 <sup>2</sup> JYω<sup>2</sup>

Then, kinetic energy is:

Considering that <sup>Θ</sup>\_ <sup>2</sup>

The angular velocities will be

Ψ\_ 2

<< Ω<sup>2</sup>

EK <sup>¼</sup> <sup>1</sup> 2

$$\begin{aligned} \mathbf{Q}\_{\Theta} &= \mathbf{M}\_{\mathbf{Y},\mathbf{P}} - \mathbf{a} \mathbf{P}\_{\mathbf{Z}} \\ \mathbf{Q}\_{\Psi} &= \mathbf{M}\_{\mathbf{Z},\mathbf{P}} + \mathbf{a} \mathbf{P}\_{\mathbf{Y}} \end{aligned} \tag{7}$$

The index P means that the moment around the specific axis is at the plane of the propeller rotation. Employing the quasi-steady theory, the effective angles become

$$\begin{aligned} \Theta^\* &= \Theta - \frac{\dot{Z}}{V\_{\text{ov}}} = \Theta - \frac{a\dot{\Theta}}{V\_{\text{ov}}}\\ \Psi^\* &= \Psi - \frac{\dot{Y}}{V\_{\text{ov}}} = \Psi - \frac{a\dot{\Psi}}{V\_{\text{ov}}} \end{aligned} \tag{8}$$

Neglecting the aerodynamic inertia terms (Θ\_ <sup>∗</sup> <sup>≈</sup> <sup>Θ</sup>\_ , <sup>Ψ</sup>\_ <sup>∗</sup> ≈ Ψ\_ ), we obtain the equations for the propeller's dimensionless forces and moments as follows:

$$\begin{aligned} \mathbf{P}\_{\mathbf{Y}} &= \mathbf{q}\_{\mathrm{es}} \mathbf{F}\_{\mathrm{P}} \left( \mathbf{c}\_{\mathrm{Y}\Psi} \Psi^{\*} + \mathbf{c}\_{\mathrm{Y}\Theta} \Theta^{\*} + \mathbf{c}\_{\mathrm{Y}\Psi} \frac{\dot{\Theta}^{\*} \mathbf{R}}{\mathbf{V}\_{\mathrm{es}}} \right) \\ \mathbf{P}\_{\mathbf{Z}} &= \mathbf{q}\_{\mathrm{es}} \mathbf{F}\_{\mathrm{P}} \left( \mathbf{c}\_{\mathrm{x}\Theta} \Theta^{\*} + \mathbf{c}\_{\mathrm{x}\Psi} \Psi^{\*} + \mathbf{c}\_{\mathrm{x}\mathbf{r}} \frac{\dot{\Psi}^{\*} \mathbf{R}}{\mathbf{V}\_{\mathrm{es}}} \right) \\ \mathbf{M}\_{\mathrm{Y},\mathrm{P}} &= \mathbf{q}\_{\mathrm{es}} \mathbf{F}\_{\mathrm{P}} \mathbf{D}\_{\mathrm{P}} \left( \mathbf{c}\_{\mathrm{m}\Psi} \Psi^{\*} + \mathbf{c}\_{\mathrm{mq}} \frac{\dot{\Theta}^{\*} \mathbf{R}}{\mathbf{V}\_{\mathrm{es}}} \right) \\ \mathbf{M}\_{\mathrm{Z},\mathrm{P}} &= \mathbf{q}\_{\mathrm{es}} \mathbf{F}\_{\mathrm{P}} \mathbf{D}\_{\mathrm{P}} \left( \mathbf{c}\_{\mathrm{r}\Theta} \Theta^{\*} + \mathbf{c}\_{\mathrm{nr}} \frac{\dot{\Psi}^{\*} \mathbf{R}}{\mathbf{V}\_{\mathrm{es}}} \right) \end{aligned} \tag{9}$$

where F<sup>P</sup> is the propeller disc area and D<sup>P</sup> is the propeller diameter.

The aerodynamic derivatives representing the derivatives of the two aerodynamic forces and two aerodynamic moments with respect to the pitch and yaw angles and to the pitch and yaw angular velocities are then defined as follows:

cy<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cy <sup>∂</sup>Θ� cy<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cy <sup>∂</sup>Ψ� cyq <sup>¼</sup> <sup>∂</sup>cy <sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cyr <sup>¼</sup> <sup>∂</sup>cy <sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cz<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cz <sup>∂</sup>Θ� cz<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cz <sup>∂</sup>Ψ� czq <sup>¼</sup> <sup>∂</sup>cz <sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! czr <sup>¼</sup> <sup>∂</sup>cz <sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cm<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cm <sup>∂</sup>Θ� cm<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cm <sup>∂</sup>Ψ� cmq <sup>¼</sup> <sup>∂</sup>cm <sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cmr <sup>¼</sup> <sup>∂</sup>cm <sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cn<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cn <sup>∂</sup>Θ� cn<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cn <sup>∂</sup>Ψ� cnq <sup>¼</sup> <sup>∂</sup>cn <sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! cnr <sup>¼</sup> <sup>∂</sup>cn <sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> ! (10)

These aerodynamic derivatives can be obtained analytically [2, 3, 6] or experimentally. Considering the symmetry, they can be expressed as follows:

$$\begin{aligned} \mathcal{L}\_{z}\boldsymbol{\omega} &= \mathcal{c}\_{y\ominus} \text{; } \mathcal{c}\_{m}\boldsymbol{\upmu} = -\mathcal{c}\_{n\ominus} \text{; } \mathcal{c}\_{m\boldsymbol{\uprho}} = \mathcal{c}\_{m\boldsymbol{\uprho}} \text{; } \mathcal{c}\_{zr} = \mathcal{c}\_{y\boldsymbol{\uprho}};\\ \mathcal{c}\_{z\ominus} &= -\mathcal{c}\_{y\boldsymbol{\uprho}} \text{; } \mathcal{c}\_{n\boldsymbol{\uprho}} = \mathcal{c}\_{m\boldsymbol{\uprho}} \text{; } \mathcal{c}\_{mr} = -\mathcal{c}\_{n\boldsymbol{\uprho}} \text{; } \mathcal{c}\_{yr} = -\mathcal{c}\_{z\boldsymbol{\uprho}} \end{aligned} \tag{11}$$

Neglecting the low value derivatives, we can consider:

$$
\omega\_{mr} = -\mathfrak{c}\_{n\eta} = 0; \mathfrak{c}\_{yr} = -\mathfrak{c}\_{z\eta} = 0 \tag{12}
$$

By substituting Eq. (10) into the equations of motion (Eq. (6)) and considering the harmonic motion

$$[\Theta, \Psi] = [\overline{\Theta}, \overline{\Psi}] \mathbf{e}^{\mathrm{j}\omega t} \tag{13}$$

we obtain the final whirl flutter matrix equation

$$\left(-\omega^2[\mathbf{M}] + \mathbf{j}\omega\left([\mathbf{D}] + [\mathbf{G}] + \mathbf{q}\_{\bullet}\mathbf{F}\_{\mathbf{P}}\frac{\mathbf{D}\_{\mathbf{P}}^2}{\mathbf{V}\_{\bullet}}[\mathbf{D}^{\mathbf{A}}]\right) + \left([\mathbf{K}] + \mathbf{q}\_{\bullet}\mathbf{F}\_{\mathbf{P}}\mathbf{D}\_{\mathbf{P}}[\mathbf{K}^{\mathbf{A}}]\right)\right)\left|\cfrac{\overline{\Theta}}{\overline{\mathbf{V}}}\right| = \{0\}\tag{14}$$

where the mass matrix becomes

$$\begin{bmatrix} \mathbf{M} \end{bmatrix} = \begin{bmatrix} \mathbf{J}\_{\mathbf{Y}} & \mathbf{0} \\ \mathbf{0} & \mathbf{J}\_{\mathbf{Z}} \end{bmatrix} \tag{15}$$

the structural damping matrix becomes

$$\mathbf{[D]} = \begin{bmatrix} \frac{\mathbf{K}\_{\Theta} \boldsymbol{\gamma}\_{\Theta}}{\boldsymbol{\omega}} & \mathbf{0} \\ \mathbf{0} & \frac{\mathbf{K}\_{\Psi} \boldsymbol{\gamma}\_{\Psi}}{\boldsymbol{\omega}} \end{bmatrix} \tag{16}$$

the gyroscopic matrix becomes

where F<sup>P</sup> is the propeller disc area and D<sup>P</sup> is the propeller diameter.

<sup>∂</sup>Θ� cy<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cy

<sup>∂</sup>Θ� cz<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cz

<sup>∂</sup>Θ� cm<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cm

<sup>∂</sup>Θ� cn<sup>Ψ</sup> <sup>¼</sup> <sup>∂</sup>cn

sidering the symmetry, they can be expressed as follows:

Neglecting the low value derivatives, we can consider:

we obtain the final whirl flutter matrix equation

½M� þ jω ½D�þ½G� þ q∞FP

where the mass matrix becomes

angular velocities are then defined as follows:

cy<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cy

144 Flight Physics - Models, Techniques and Technologies

cz<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cz

cm<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cm

cn<sup>Θ</sup> <sup>¼</sup> <sup>∂</sup>cn

motion

�ω<sup>2</sup>

The aerodynamic derivatives representing the derivatives of the two aerodynamic forces and two aerodynamic moments with respect to the pitch and yaw angles and to the pitch and yaw

<sup>∂</sup>Ψ� cyq <sup>¼</sup> <sup>∂</sup>cy

<sup>∂</sup>Ψ� czq <sup>¼</sup> <sup>∂</sup>cz

<sup>∂</sup>Ψ� cmq <sup>¼</sup> <sup>∂</sup>cm

<sup>∂</sup>Ψ� cnq <sup>¼</sup> <sup>∂</sup>cn

These aerodynamic derivatives can be obtained analytically [2, 3, 6] or experimentally. Con-

cz<sup>Ψ</sup> ¼ cyΘ; cm<sup>Ψ</sup> ¼ �cnΘ; cmq ¼ cnr; czr ¼ cyq; cz<sup>Θ</sup> ¼ �cyΨ; cn<sup>Ψ</sup> ¼ cmΘ; cmr ¼ �cnq; cyr ¼ �czq

By substituting Eq. (10) into the equations of motion (Eq. (6)) and considering the harmonic

D2 P V<sup>∞</sup>

þ ½K� þ <sup>q</sup>∞FPDP <sup>K</sup><sup>A</sup> � � � � � � <sup>Θ</sup>

<sup>½</sup>M� ¼ JY <sup>0</sup>

0 JZ � �

<sup>D</sup><sup>A</sup> � � � �

<sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup>

<sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup>

> <sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup>

<sup>∂</sup> <sup>Θ</sup>\_ <sup>R</sup> V<sup>∞</sup>

! cyr <sup>¼</sup> <sup>∂</sup>cy

! czr <sup>¼</sup> <sup>∂</sup>cz

! cmr <sup>¼</sup> <sup>∂</sup>cm

! cnr <sup>¼</sup> <sup>∂</sup>cn

cmr ¼ �cnq ¼ 0; cyr ¼ �czq ¼ 0 (12)

<sup>½</sup>Θ, <sup>Ψ</sup>� ¼ <sup>Θ</sup>, <sup>Ψ</sup> � �ejω<sup>t</sup> (13)

Ψ

� � � � �

¼ f0g (14)

(15)

� � � � �

<sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> !

<sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> !

> <sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> !

(10)

(11)

<sup>∂</sup> <sup>Ψ</sup>\_ <sup>R</sup> V<sup>∞</sup> !

$$[\mathbf{G}] = \begin{bmatrix} 0 & \mathbf{J}\_{\mathbf{X}} \mathbf{\Omega} \\ -\mathbf{J}\_{\mathbf{X}} \mathbf{\Omega} & 0 \end{bmatrix} \tag{17}$$

the structural stiffness matrix becomes

$$[\mathbf{K}] = \begin{bmatrix} \mathbf{K}\_{\Theta} & \mathbf{0} \\ \mathbf{0} & \mathbf{K}\_{\Psi} \end{bmatrix} \tag{18}$$

the aerodynamic damping matrix becomes

$$\begin{bmatrix} \mathbf{D}^{\rm A} \end{bmatrix} = \begin{bmatrix} -\frac{1}{2}\mathbf{c}\_{\rm mq} - \frac{\mathbf{a}^{2}}{\mathbf{D}\_{\rm P}^{2}}\mathbf{c}\_{\rm r\Theta} & \frac{1}{2}\frac{\mathbf{a}}{\mathbf{D}\_{\rm P}}\mathbf{c}\_{\rm rq} - \frac{\mathbf{a}}{\mathbf{D}\_{\rm P}}\mathbf{c}\_{\rm r\Theta} - \frac{\mathbf{a}^{2}}{\mathbf{D}\_{\rm P}^{2}}\mathbf{c}\_{\rm r\Theta} \\\\ -\frac{1}{2}\frac{\mathbf{a}}{\mathbf{D}\_{\rm P}}\mathbf{c}\_{\rm yq} + \frac{\mathbf{a}}{\mathbf{D}\_{\rm P}}\mathbf{c}\_{\rm r\Theta} + \frac{\mathbf{a}^{2}}{\mathbf{D}\_{\rm P}^{2}}\mathbf{c}\_{\rm y\Theta} & -\frac{1}{2}\mathbf{c}\_{\rm mq} - \frac{\mathbf{a}^{2}}{\mathbf{D}\_{\rm P}^{2}}\mathbf{c}\_{\rm x\Theta} \end{bmatrix} \tag{19}$$

and the aerodynamic stiffness matrix becomes

$$\mathbf{I}\left[\mathbf{K}^{\mathcal{A}}\right] = \begin{bmatrix} \frac{\mathbf{a}}{\mathcal{D}\_{\mathcal{P}}} \mathbf{c}\_{\mathcal{x}\Theta} & \mathbf{c}\_{\mathbb{n}\Theta} + \frac{\mathbf{a}}{\mathcal{D}\_{\mathcal{P}}} \mathbf{c}\_{\mathcal{Y}\Theta} \\\\ -\mathbf{c}\_{\mathbb{n}\Theta} - \frac{\mathbf{a}}{\mathcal{D}\_{\mathcal{P}}} \mathbf{c}\_{\mathcal{Y}\Theta} & \frac{\mathbf{a}}{\mathcal{D}\_{\mathcal{P}}} \mathbf{c}\_{\mathcal{x}\Theta} \end{bmatrix} \tag{20}$$

Equation (14) can be solved as an eigenvalue problem. The critical state emerges for a specific combination of the parameters V<sup>∞</sup> and Ω, for which the angular velocity ω becomes real.

The influences of the main structural parameters are shown in the next figures. Figure 6 shows the influence of the propeller advance ratio (V∞/(ΩR)) on the stability of an undamped gyroscopic system. Increasing the propeller advance ratio has a destabilising effect. Another important parameter is the propeller hub distance ratio (a/R), the influence of which is documented in Figure 7. Figure 7 also shows the influence of the structural damping (γ), which is a significant stabilisation factor. In contrast, the influence of the propeller thrust is negligible. The most critical state is ω<sup>Θ</sup> = ωΨ, when the interaction of both independent yaw and pitch motions is maximal and the trajectory of the gyroscopic motion is circular. Considering rigid propeller blades, the whirl flutter inherently appears in the backward gyroscopic mode. The flutter frequency is the same as the frequency of the backward gyroscopic mode. The critical state can be reached by increasing either V<sup>∞</sup> or Ω. A special case of eq. (14) for ω = 0 is gyroscopic static divergence, which is characterised by unidirectional divergent motion.

Figure 6. Influence of the propeller advance ratio on the stability of an undamped gyroscopic system.

Figure 7. Influence of the propeller hub distance ratio.

The described model, which is based on the assumption of a rigid propeller, is obviously applicable for standard turboprop aircraft (commuters, utility aircraft, and military trainers), for which the natural frequencies of the propeller blades are much higher than the frequencies of the engine system suspension vibrations. In large turboprops, in particular military transport aircraft with heavy multiblade propellers, the solution requires taking into account the deformations of the propeller blades as well [7–10]. Obviously, whirl flutter investigation of tilt-rotor aircraft must include even more complex analytical models [11, 12].
