Stability Analysis for an Ultra-Lightweight Glider Airplane with Electric Driven Two-Blade Propeller

Abstract

Safety is the most important requirement in flight operations. This also affects the application for an extreme lightweight glider in this paper. Essential properties are the target weight below 120 $\mathrm{k}\mathrm{g}$ and the electric propulsion. The unsymmetric inertia from the two-blade propeller at the rear in combination with the light and flexible aluminium tube support makes it necessary to investigate the risk of mechanical instability. Starting from the equations of motion, the time-variant system matrices are set up. The simulation of Floquet multiplier and Hill’s hyper-eigenvalue problem provide the necessary information about the system stability. The conclusion is that the potential instability due to structural damping in the observed system can be avoided in the range of operation. The damping, experimentally determined by approximately 2%, is sufficient.

1. Introduction

In structural mechanics, ordinary differential equations (ODEs) occur for the description of vibration problems. In most cases, the parameters in the describing matrices are constant. The base for the analysis is the solution of an eigenvalue problem.

If rotors are included, the ODEs have to be extended by rotational degrees of freedom for the tilt angle of the rotor. In the first step, gyroscopic effects will lead to first-order derivative terms that contain the rotor speed ${\omega }_e$. For constant or slowly varying speeds, the system matrix keep constant in time. Although the gyroscopic matrix $\mathbf{G}$ is skew-symmetric, the system can be analysed by left and right-hand-side eigenvalue problems.

The class of ODE becomes more demanding if the inertia of the rotor are different for the various tilt axes. Then, the mass and gyro matrix obtain components that are dependent of the angular position during rotation. This changes frequently over time, when the rotor rotates. For constant rotor speeds, that can be assumed in stationary conditions, the system matrices are periodic with a parameter period $T_p=2\pi /{\mathsf{\Omega }}_P=\pi /{\omega }_e$.

Special solution strategies have been developed in history. The references [1,2] are highlighted. Modern and technical applications can be found in

• Wind turbines,
• The ramair turbine in commercial aircraft as an emergency power supply,
• Helicopters,
• Sports aircraft.

In recent history, ref. [3] developed a basic understanding of systems with parametric excitation. Author [4] shows a very general approach to parametric excitation. The opposite research question is investigated. Instead of focusing on regions of instability, methods to use parametric excitation for stabilization are presented.

The search for stable regions in theoretical rotor systems with parametric excitation can be found in [5]. Author [6] includes magnetic bearings that are able to control the rotor vibrations.

The application of non-linear methods to parametrically excited systems is demonstrated in [7]. Here, the harmonic balance method finds out influences of the time-varying system properties.

The application of experimental modal analysis for a system with rotor disc and blades is investigated by [8]. On a rig test, the influence of periodic system behaviour was analysed. In the literature, wind turbines are especially under investigation. The focus of a theoretical and experimental investigation for turbine blades is presented in [9]. High-level applications like the damage detection in [10] due to un-symmetry serve the most important wind turbine industry.

The basic theoretical research towards two-blade rotor systems is given in [11]. Ramair turbines in commercial aircraft are one application. They have a generic construction that resembles the glider under investigation. However, for the ramair turbine, the design is much more massive. The object of this paper is an extremely lightweight glider with an electrically driven propeller and therefore a different class of application that has not yet been investigated.

2. The Glider Test Object

2.1. Glider Design

The Birdy, as shown in Figure 1, is a state-of-the-art ultra-lightweight glider with a rated maximum weight of 120 $\mathrm{k}\mathrm{g}$. This is including the electric propulsion system with battery, motor, driveshaft and propeller. The aerodynamic layout for a glider is decisive. Therefore, the propeller is positioned at the extreme rear and is foldable (see Figure 2).

The driveshaft is placed with four bearings into an aluminium tube. Then, the tube is fixed in three sleeves in the body (see Figure A1). The length of the tube is more than $3.5$ $\mathrm{m}$, so again, the weight is critical. The design is thin which offers flexibility and is susceptible for vibrations.

2.2. Basic Modal Analysis

In this paper, the driveshaft is under investigation. Critical vibration levels must be avoided. These are classical resonances and unstable system behaviour. The focus will be on the latter.

The classical resonances will be covered briefly. Measurements of structural frequency response functions (FRFs) are in Figure 3. They allow the proper estimation of natural frequencies and modal damping values. From methods of modal parameter estimation like least-squares complex exponential (LSCE) typical values for the natural frequency of $f_0=19.75$ and modal damping $\zeta =2\%$ can be derived [12]. Of course, they vary slightly for the horizontal and vertical motion and depend on the static position of the propeller. The frequency is used to validate the finite element model of the driveshaft tube. The damping affects significantly the stability and has to be included in the system matrices.

The tube is modelled with standard two-node Bernoulli beam elements. Each node has a bending displacement and an inclination angle. This multiplies by two as the spacial set up needs two bending planes. This satisfies the correct movement of the rotor. Together with the fixed boundary conditions for translation at the sleeves, it is a system with a low number of degrees of freedom (DOF) (four nodes with two DOF per node, minus three constraints, times two for horizontal and vertical plane, N=10 DOF), which allows later an efficient analysis of the system with parametric excitation (Figure A1) (number of physical DOF will be multiplied by two according state space and five according hyper-eigenvalue problem Equation (31) with $L=2$). It is mandatory to define the beam elements in both bending directions to cover the gyroscopic effects in matrix $\mathbf{G}$. From the design parameter, it follows the basic mass ($\mathbf{M}$) and stiffness ($\mathbf{K}$) matrix. The validation can be derived from the natural frequencies identified in the FRF measurements. Additionally, the identified modal damping values $\zeta$ serve to set up the damping matrix ($\mathbf{B}$):

$$\mathbf{diag}⌈\mathbf{2}\mathbf{\zeta }{\mathbf{\omega }}_{\mathbf{0}}⌋={\mathbf{M}}_0^{-1}·{\mathbf{B}}_0$$

(1)

The matrices with the index ${(\dots )}_0$ denote the diagonalized form from the pre- and post-multiplication with the matrix of eigenvectors.

2.3. Test in Operation

A test run for the propeller has been performed on the airfield in Renneritz, Saxony-Anhalt, Germany in 2024 [12]. The glider has been positioned in a chassis rig, so that it cannot move during the propulsion power. The end of the bearing tube was equipped with four accelerometers. The tube positions were at the end of the vertical tail and at the very end close to the propeller. The horizontal and vertical direction have been acquired.

For the test, the electric engine was manually controlled for slow runup and rundown measurements. The measurements took place during the operations, about 60 s for a rotational speed range from 300 $\mathrm{rpm}$ to 2500 $\mathrm{rpm}$. This covers the range of speeds in operation.

Figure 4 shows one example for the Campbell diagram. It reflects a rundown in horizontal direction at the very end of the tube. The acceleration has been integrated to a displacement. The expected first-order line is clearly visible and dominant. The second order line is also significant but low in amplitude. Some more higher orders have minor contribution at lower rpm speeds. In general, the influence of the parametric excitation cannot be seen in the test. Also, there is no indicator for instable behaviour. Some resonance effects can be observed in the range of 1250 $\mathrm{rpm}$.

A detailed investigation of stability is carried out in a simulation where there is no risk of destroying the system. Also, the simulation can cover the worst case of exact constant rpm speed.

The simulation would also allow to vary the critical parameters. Therefore, the theory of time-varying systems is explained in Section 3.

3. Introduction of Time-Varying Effects in Structure Dynamics

For the glider, structural dynamics and rotor dynamics have to be combined to set up the equations of motion. The solution strategy is explained further in Section 4.

3.1. Equation of Motion of the Rotor

The angular momentum theorem in mechanics delivers the equation of motion for the rigid rotor as in gyro mechanics (moment ${\overrightarrow{M}}^{\left(S\right)}$, angular momentum ${\dot{\overrightarrow{D}}}^{\left(S\right)}$, inertia tensor ${\mathcal{J}}^{\left(S\right)}$, all with respect to the center of gravity ${(\dots )}^{\left(S\right)}$; ${\overrightarrow{\mathrm{e}}}^T$ unit vector).

$${\overrightarrow{M}}^{\left(S\right)}={\dot{\overrightarrow{D}}}^{\left(S\right)}={\displaystyle \frac{d_F}{dt}}\left({\mathcal{J}}^{\left(S\right)}·\overrightarrow{\omega }\right)+{\overrightarrow{\mathsf{\Omega }}}_F\times \left({\mathcal{J}}^{\left(S\right)}·\overrightarrow{\omega }\right)$$

(2)

A special moving coordinate system is used for the attachment point of the rotor.

$${\overrightarrow{\mathsf{\Omega }}}_F=\left(\begin{array}{c}\overrightarrow{0}\\ \overrightarrow{{\omega }_{\eta }}\\ \overrightarrow{{\omega }_{\zeta }}\end{array}\right)={\overrightarrow{\mathrm{e}}}^T\left[\begin{array}{c}0\\ {\omega }_{\eta }\\ {\omega }_{\zeta }\end{array}\right]$$

(3)

The rotation around the constructive axis of rotation ${\omega }_e$ is part of $\overrightarrow{\omega }$. The angular tilt movements ${\overrightarrow{\mathsf{\Omega }}}_F$ are also the constraints to the elastic embedding structure used in Section 3.2.

3.2. The Equation of Motion of the Overall Structure

The general mechanical problem which is used in this paper is an elastic structure that includes a rigid, rotating rotor. The shape of the rotor is not restricted to a symmetric mass distribution. It fulfils the requirements of the lightweight glider with the two-blade propeller. The general assembly is shown in Figure 5.

Coupling the rotor to the base structure to form the overall structure leads to a linear differential equation of motion

$$(\mathbf{M}+\Delta \mathbf{M}\left(t\right))\ddot{\mathbf{y}}\left(t\right)+(\mathbf{B}+\mathbf{G}+\Delta \mathbf{G}\left(t\right))\dot{\mathbf{y}}\left(t\right)+\mathbf{K}\mathbf{y}\left(t\right)=\mathbf{f}\left(t\right).$$

(4)

The connection forces and moments between rotor and structure have become internal forces. The degree of freedom vector $\mathbf{y}$ of DOFs contains coordinates for translations and rotations with different physical units.

In addition to the components of the basic structure, the rotor generates contributions to the mass and gyroscopic matrix. The mass matrix $\mathbf{M}$ contains the constant inertia parameters of the rotor m and $J_m$ from the terms with second time derivative of $\mathbf{y}$. These are already present when the rotor is stationary (${\omega }_e=0$). Due to the rotation of the rotor, the matrices $\mathbf{G}, \Delta \mathbf{G}\left(t\right), \Delta \mathbf{M}\left(t\right)$ result, which depend on the angular velocity ${\omega }_e$. With the vector of degrees of freedom $\mathbf{y}=[\dots ,{\phi }_{\eta },{\phi }_{\zeta }]$, the skew-symmetric gyroscopic matrix

$$\mathbf{G}=\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ \dots & 0& 0& 0\\ \dots & 0& 0& J_e{\omega }_e\\ \dots & 0& -J_e{\omega }_e& 0\end{array}\right]$$

(5)

builds up for the symmetric rotor like a disc. It results from the cross product in the moment Equation (2). The equation of motion is therefore dependent on speed in every case. For the asymmetrical rotor, two time-varying matrices that are also dependent on speed

$$\Delta \mathbf{M}\left(t\right)=\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ \dots & 0& 0& 0\\ \dots & 0& -\Delta Jcos\left(2{\omega }_{e}t\right)& -\Delta Jsin\left(2{\omega }_{e}t\right)\\ \dots & 0& -\Delta Jsin\left(2{\omega }_{e}t\right)& \Delta Jcos\left(2{\omega }_{e}t\right)\end{array}\right] \mathrm{and}$$

(6)

$$\Delta \mathbf{G}\left(t\right)=2{\omega }_e\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ \dots & 0& 0& 0\\ \dots & 0& \Delta Jsin\left(2{\omega }_{e}t\right)& -\Delta Jcos\left(2{\omega }_{e}t\right)\\ \dots & 0& -\Delta Jcos\left(2{\omega }_{e}t\right)& -\Delta Jsin\left(2{\omega }_{e}t\right)\end{array}\right]$$

(7)

are added. Due to the time variance of the coefficients, we have a vibration system with parameter excitation. At a constant rotational frequency ${\omega }_e$, the matrices are periodically time-variant. The parameter circular frequency is ${\mathsf{\Omega }}_P=2{\omega }_e$.

If there are multiple rotor discs, the equation of motion with the degrees of freedom at the coupling point must be set up for each individual disc, so that the equation of motion of the entire system retains the form Equation (4).

In the case of unbalanced rotors, circulating forces and moments appear as excitation forces and moments in the vector $\mathbf{f}\left(t\right)$.

The mechanical description of the system is now complete and its solution is provided in Section 4.

4. Theoretical Modal Analysis

After the reduction of the ODE to first order, two methods for the stability analysis are presented. They are based on the Floquet and Hill approach. In this chapter, the general analysis is shown in detail. The application to the engineering problem of the glider is part of Section 5.

4.1. State Space Formulation

To solve the differential equation of motion (4), it is converted into a first-order differential equation by introducing the state vector

$$\mathbf{x}=\left[\begin{array}{c}\mathbf{y}\\ \dot{\mathbf{y}}\end{array}\right]$$

(8)

into a first-order differential equation,

$${\mathbf{A}}_{\mathbf{1}}\left(t\right)\dot{\mathbf{x}}\left(t\right)+{\mathbf{A}}_{\mathbf{0}}\left(t\right)\mathbf{x}\left(t\right)=\mathbf{p}\left(t\right),$$

(9)

with the time-varying system matrices

$$\begin{array}{cc}\hfill {\mathbf{A}}_{\mathbf{1}}\left(t\right)& =\left[\begin{array}{cc}\mathbf{B}+\mathbf{G}+\Delta \mathbf{G}\left(t\right)& \mathbf{M}+\Delta \mathbf{M}\left(t\right)\\ \mathbf{M}+\Delta \mathbf{M}\left(t\right)& \mathbf{0}\end{array}\right],\hfill \\ \hfill {\mathbf{A}}_{\mathbf{0}}\left(t\right)& =\left[\begin{array}{cc}\mathbf{K}& \mathbf{0}\\ \mathbf{0}& -\mathbf{M}-\Delta \mathbf{M}\left(t\right)\end{array}\right],\hfill \end{array}$$

and the excitation vector

$$\mathbf{p}\left(t\right)=\left[\begin{array}{c}\mathbf{f}\left(t\right)\\ \mathbf{0}\end{array}\right].$$

(10)

The time-varying mass matrix $\mathbf{M}\left(t\right)$ is non-singular at all times, so that the differential equation with

$${\mathbf{A}}_{\mathbf{1}}^{-\mathbf{1}}\left(t\right)=\left[\begin{array}{cc}\mathbf{0}& {(\mathbf{M}+\Delta \mathbf{M})}^{-1}\\ {(\mathbf{M}+\Delta \mathbf{M})}^{-1}& -{(\mathbf{M}+\Delta \mathbf{M})}^{-1}(\mathbf{B}+\mathbf{G}+\Delta \mathbf{G}){(\mathbf{M}+\Delta \mathbf{M})}^{-1}\end{array}\right]$$

(11)

(from the multiplication of the term in Equation (11) by ${\mathbf{A}}_{\mathbf{1}}$ the validity can be shown. The condition of a regular $\mathbf{M}+\Delta \mathbf{M}$ is given in this class of systems) can alternatively be shown in the form

$$\dot{\mathbf{x}}\left(t\right)=\underset{{\displaystyle \mathbf{A}\left(t\right)}}{\underbrace{-{\mathbf{A}}_{\mathbf{1}}^{-1}\left(t\right){\mathbf{A}}_{\mathbf{0}}\left(t\right)}}\mathbf{x}\left(t\right)+{\mathbf{A}}_{\mathbf{1}}^{-1}\left(t\right)\mathbf{p}\left(t\right).$$

(12)

The standardized form of Equation (9) is the starting point for the stability analysis. All terms are defined or will be used as varying parameters.

4.2. Floquet Stability

The natural vibration behaviour of the structure is described by the solution of the homogeneous equation of motion ($\mathbf{p}\left(t\right)=0$). For a linear system with f mechanical degrees of freedom and $2f$ state variables, there are exactly $2f$ linearly independent solutions, which are arranged column by column in the fundamental matrix $\mathsf{\Phi }\left(t\right)$. The natural vibrations occur as a linear combination of the solutions:

$$\mathbf{x}\left(t\right)=\mathsf{\Phi }\left(t\right){\mathbf{c}}_0.$$

(13)

The matrix $\mathsf{\Phi }\left(t\right)$ thus satisfies the homogeneous form of the differential Equation (9) or (12). It is regular for all times t. For given initial conditions ${\mathbf{x}}_0=\mathbf{x}(t=t_0)$, the constant vector ${\mathbf{c}}_0$ can be eliminated, resulting in the solution adapted to the initial conditions with the state transition matrix ${\mathsf{\Phi }}_T(t,t_0)$:

$$\mathbf{x}\left(t\right)=\underset{{\displaystyle {\mathsf{\Phi }}_T(t,t_0)}}{\underbrace{\mathsf{\Phi }\left(t\right){\mathsf{\Phi }}^{-1}\left(t_0\right)}}{\mathbf{x}}_{\mathbf{0}}.$$

(14)

Further fundamental matrices

$$\mathsf{\Psi }\left(t\right)=\mathsf{\Phi }\left(t\right){\mathbf{C}}_{\mathbf{0}}$$

(15)

can be formed by linear combinations. The regularity of the constant matrix ${\mathbf{C}}_{\mathbf{0}}$ is required for the linear independence of the columns of $\mathsf{\Psi }\left(t\right)$. For periodic system matrices, ${\mathbf{A}}_{\mathbf{1}}(t+T_P)={\mathbf{A}}_{\mathbf{1}}\left(t\right),{\mathbf{A}}_{\mathbf{0}}(t+T_P)={\mathbf{A}}_{\mathbf{0}}\left(t\right)$, with $\mathbf{x}\left(t\right)=\mathbf{x}(t+m{T}_P)$ ($m\in \mathcal{N}$) is also a solution to the differential equation, so that Equation (15)

$$\mathsf{\Phi }(t+T_P)=\mathsf{\Phi }\left(t\right)\mathbf{C}$$

(16)

follows. The constant, non-singular matrix $\mathbf{C}$ is also called the monodromy matrix [14]. If one asks for solutions that result in

$$\mathbf{x}(T_P+t_0)=\mu {\mathbf{x}}_0$$

(17)

after a period $T_P$, this leads to Floquet’s eigenvalue problem ($\mathbf{I}$: identity matrix)

$$\left[\mathbf{C}-\mu \mathbf{I}\right]\mathbf{c}=\mathbf{0}.$$

(18)

The $2f$ solutions for the eigenvalues $\mu$ are called characteristic multipliers ${\mu }_k$. They provide information about the stability and periodicity of the natural vibrations. To calculate the eigenvalue problem (18), the state transition matrix ${\mathsf{\Phi }}_T(T_P,0)$ must be known. This generally requires numerical integration over a period $T_P$. In practice, this is only feasible for systems with few degrees of freedom and is further complicated by the implicit form of the differential Equation (9). With the help of the eigenvalues ${\mu }_k$ and the eigenvectors ${\mathbf{c}}_k$, which form the matrix of eigenvectors ${\mathbf{C}}_c$, a special representation of the monodromy matrix follows:

$$\mathbf{C}={\mathbf{C}}_c\mathrm{diag}⌈{\mu }_k⌋{\mathbf{C}}_c^{-1}.$$

(19)

The following applies to the stability of natural vibrations. The system is

• Unstable for $|{\mu }_k| >1$ for at least one eigenvalue,
• Marginally stable for $|{\mu }_k| =1$ for one and $|{\mu }_k| \le 1$ for all other eigenvalues,
• Asymptotically stable for $|{\mu }_k| <1$ for all eigenvalues.

A characteristic multiplier

• ${\mu }_k=1$ leads to periodic solutions of period $T_P$, and
• ${\mu }_k=-1$ leads to periodic solutions of period $2T_P$,

in each case with initial conditions in the form of an eigenvector ${\mathbf{c}}_k$.

The stability can be assessed on basis of the Floquet multipliers. Typically, parameters will be varied to obtain the dependence of the system with regard to the operating conditions.

4.3. Hill’s Hyper-Eigenvalue Problem

The regularity of the matrix $\mathbf{C}$ follows from the fact that it can also be written in the form:

$$\mathbf{C}={\mathrm{e}}^{\mathbf{R}{T}_P}\mathrm{with}\mathbf{R}=\mathrm{const}.$$

(20)

The eigenvalues of $\mathbf{R}$ are called characteristic exponents ${\rho }_k$ and are related to the characteristic multipliers:

$${\rho }_k={\displaystyle \frac{1}{T_P}}ln{\mu }_k+i{\displaystyle \frac{2\pi m}{T_P}},m\in \mathcal{N}.$$

(21)

The imaginary part is not unique. The real part is uniquely determined and provides information about stability.

The system is

• Unstable for $\Re \left({\rho }_k\right)>0$ for at least one characteristic exponent,
• Marginally stable for $\Re \left({\rho }_k\right)=0$ for at least one characteristic exponent and for all others $\Re \left({\rho }_k\right)<0$,
• Stable for $\Re \left({\rho }_k\right)<0$ for all characteristic exponents.

Floquet’s theorem is essential for further investigation of the solution structure, according to which the fundamental matrix can be represented in the form

$$\mathsf{\Phi }\left(t\right)=\mathbf{S}\left(t\right){\mathrm{e}}^{\mathbf{R}t}.$$

(22)

The matrix $\mathbf{S}\left(t\right)$ is a regular, continuously differentiable and periodic matrix with

$$\mathbf{S}(t+T_P)=\mathbf{S}\left(t\right).$$

(23)

It can be used for a coordinate transformation

$$\mathbf{x}\left(t\right)=\mathbf{S}\left(t\right)\mathbf{z}\left(t\right).$$

(24)

This is a so-called Lyapunov transformation, which has the important property that a linear differential equation with periodic coefficients can be transformed (reduced) into a differential equation with constant coefficients:

$$\dot{\mathbf{z}}\left(t\right)+\underset{{\displaystyle {-\mathbf{A}}_S=\mathrm{const}.}}{\underbrace{{\left({\mathbf{A}}_{\mathbf{1}}\left(t\right)\mathbf{S}\left(t\right)\right)}^{-1}[{\mathbf{A}}_{\mathbf{1}}\left(t\right)\dot{\mathbf{S}}\left(t\right)+{\mathbf{A}}_{\mathbf{0}}\left(t\right)\mathbf{S}\left(t\right)]}}\mathbf{z}\left(t\right)={\left({\mathbf{A}}_{\mathbf{1}}\left(t\right)\mathbf{S}\left(t\right)\right)}^{-1}\mathbf{p}\left(t\right).$$

(25)

Hill’s approach represents the solution based on Floquet’s theorem as a product of the scalar exponential function ${\mathrm{e}}^{\rho t}$ and the periodic vector $\mathbf{u}\left(t\right)$:

$$\mathbf{x}\left(t\right)={\mathrm{e}}^{\rho t}\mathbf{u}\left(t\right),$$

(26)

$$\mathbf{u}\left(t\right)=\mathbf{u}(t+T_P)=\sum _{l=-\infty }^{+\infty }{\mathbf{u}}_l{\mathrm{e}}^{il{\mathsf{\Omega }}_{P}t}.$$

(27)

If we substitute this approach into the differential Equation (9) and additionally specify the periodic system matrices in a Fourier series,

$${\mathbf{A}}_{\mathbf{1}}\left(t\right)=\sum _{a=-\infty }^{+\infty }{\mathbf{A}}_{1a}{\mathrm{e}}^{ia{\mathsf{\Omega }}_{P}t},{\mathbf{A}}_{\mathbf{0}}\left(t\right)=\sum _{a=-\infty }^{+\infty }{\mathbf{A}}_{0a}{\mathrm{e}}^{ia{\mathsf{\Omega }}_{P}t},$$

(28)

several summands always occur with the same frequency, a multiple of ${\mathsf{\Omega }}_P$. (The time-dependent matrices Equations (6) and (7), that are part of the hyper-eigenvalue matrices Equation (28), can be formulated as complex exponential functions: $sin\left({\mathsf{\Omega }}_{P}t\right)={\displaystyle \frac{1}{2i}}({\mathrm{e}}^{\mathrm{i}{\Omega }_{\mathrm{P}}\mathrm{t}}-{\mathrm{e}}^{-\mathrm{i}{\Omega }_{\mathrm{P}}\mathrm{t}})$, $cos\left({\mathsf{\Omega }}_{P}t\right)={\displaystyle \frac{1}{2}}({\mathrm{e}}^{\mathrm{i}{\Omega }_{\mathrm{P}}\mathrm{t}}+{\mathrm{e}}^{-\mathrm{i}{\Omega }_{\mathrm{P}}\mathrm{t}})$.) A coefficient comparison leads to an algebraic eigenvalue problem of infinite dimension,

$$(\rho {\widehat{\mathbf{A}}}_1+{\widehat{\mathbf{A}}}_0)\widehat{\mathbf{u}}=\mathbf{0},$$

(29)

which is therefore called a hyper-eigenvalue problem. (For differential equations of the form (12), a similar hyper-eigenvalue problem follows.) The hyper-eigenvector $\widehat{\mathbf{u}}$ has the structure

$${\widehat{\mathbf{u}}}^T=[\dots ,{\mathbf{u}}_{-2}^T,{\mathbf{u}}_{-1}^T,{\mathbf{u}}_0^T,{\mathbf{u}}_{+1}^T,{\mathbf{u}}_{+2}^T,\dots ]$$

(30)

The subvectors ${\mathbf{u}}_l$ are those from the approach Equation (27). For practical problems, it is sufficient to restrict the approach vector $\mathbf{u}\left(t\right)$ to a few Fourier terms, so that the hyper-eigenvalue problem is reduced to a finite dimension and can thus be solved numerically. The terms in the vicinity of the constant part $l=0$ are of utmost importance here. For a symmetric approach $-L\le l\le +L$, the eigenvalue problem has the dimension $(2L+1)2f$. For $L=2$, Equation (29) is

$$\left(\left[\begin{array}{ccccc}{A}_{10}& A_{1-1}& A_{1-2}& A_{1-3}& A_{1-4}\\ A_{1+1}& A_{10}& A_{1-1}& A_{1-2}& A_{1-3}\\ A_{1+2}& A_{1+1}& A_{10}& A_{1-1}& A_{1-2}\\ A_{1+3}& A_{1+2}& A_{1+1}& A_{10}& A_{1-1}\\ A_{1+4}& A_{1+3}& A_{1+2}& A_{1+1}& A_{10}\end{array}\right]\rho \right.+$$

(31)

$$\left.\left[\begin{array}{ccccc}-2i{\mathsf{\Omega }}_P{A}_{10}+A_{00}& -i{\mathsf{\Omega }}_P{A}_{1-1}+A_{0-1}& A_{0-2}& i{\mathsf{\Omega }}_P{A}_{1-3}+A_{0-3}& 2i{\mathsf{\Omega }}_P{A}_{1-4}+A_{0-4}\\ -2i{\mathsf{\Omega }}_P{A}_{1+1}+A_{0+1}& -i{\mathsf{\Omega }}_P{A}_{10}+A_{00}& A_{0-1}& i{\mathsf{\Omega }}_P{A}_{1-2}+A_{0-2}& 2i{\mathsf{\Omega }}_P{A}_{1-3}+A_{0-3}\\ -2i{\mathsf{\Omega }}_P{A}_{1+2}+A_{0+2}& -i{\mathsf{\Omega }}_P{A}_{1+1}+A_{0+1}& A_{00}& i{\mathsf{\Omega }}_P{A}_{1-1}+A_{0-1}& 2i{\mathsf{\Omega }}_P{A}_{1-2}+A_{0-2}\\ -2i{\mathsf{\Omega }}_P{A}_{1+3}+A_{0+3}& -i{\mathsf{\Omega }}_P{A}_{1+2}+A_{0+2}& A_{0+1}& i{\mathsf{\Omega }}_P{A}_{10}+A_{00}& 2i{\mathsf{\Omega }}_P{A}_{1-1}+A_{0-1}\\ -2i{\mathsf{\Omega }}_P{A}_{1+4}+A_{0+4}& -i{\mathsf{\Omega }}_P{A}_{1+3}+A_{0+3}& A_{0+2}& i{\mathsf{\Omega }}_P{A}_{1+1}+A_{0+1}& 2i{\mathsf{\Omega }}_P{A}_{10}+A_{00}\end{array}\right]\right)\left[\begin{array}{c}{u}_{-2}\\ u_{-1}\\ u_0\\ u_{+1}\\ u_{+2}\end{array}\right]=\mathbf{0}$$

From the $(2L+1)2f$ solutions of the eigenvalue problem, an equal number of solutions of the differential equation can be formed, which apparently leads to a contradiction, since the linear differential equation only has $2f$ linearly independent solutions. However, it turns out that the solutions obtained always occur in $(2L+1)$-fold redundancy. The sum of the two imaginary parts in the exponential functions (Equation (26) with (27)), which represent the oscillation frequency of a solution, is constant for redundant solutions. Thus, redundant eigenvalues ${\rho }_k$ differ in the imaginary part by multiples of ${\mathsf{\Omega }}_P$. At the same time, the subvectors ${\mathbf{u}}_{kl}$ of the k-th hyper-vector are shifted. A selection must therefore be made for the $2f$ solutions.

A useful criterion is to select the eigenvalue and eigenvector from the redundant ones for which the subvector ${\mathbf{u}}_{k0}$, which represents the constant part, has the largest magnitude of all subvectors. The magnitudes of the subvectors at the edge ${\mathbf{u}}_{k\pm L}$ should be small compared to that of ${\mathbf{u}}_{k0}$. If this is not the case, one have to extend the set up for a larger L.

The fundamental matrix can be formed from the solutions of Hill’s method (Equation (22)):

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left(t\right)& =& \mathbf{U}\left(t\right){\mathrm{e}}^{{\displaystyle \mathrm{diag}⌈{\rho }_k⌋t}}\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \mathrm{with}\mathbf{U}\left(t\right)& =& [{\mathbf{u}}_1\left(t\right),\dots ,{\mathbf{u}}_k\left(t\right),\dots {\mathbf{u}}_{2f}\left(t\right)].\hfill \end{array}$$

(33)

The solution is in a very special form, such that the Lyapunov transformation

$$\mathbf{x}\left(t\right)=\mathbf{U}\left(t\right)\mathbf{z}\left(t\right)$$

(34)

as in Equation (25) leads to a differential equation with constant coefficients, which is also diagonalized during the transformation. Their system matrix is

$${\left({\mathbf{A}}_{\mathbf{1}}\left(t\right)\mathbf{U}\left(t\right)\right)}^{-1}\left[{\mathbf{A}}_{\mathbf{1}}\left(t\right)\dot{\mathbf{U}}\left(t\right)+{\mathbf{A}}_{\mathbf{0}}\left(t\right)\mathbf{U}\left(t\right)\right]=\mathrm{diag}⌈-{\rho }_k⌋.$$

(35)

These properties justify denoting ${\mathbf{u}}_k\left(t\right)$ as a periodic eigenvector and $\mathbf{U}\left(t\right)$ as a associated modal matrix. Substituting the solution (32) calculated using Hill’s eigenvalue problem into the left side of Equation (35) offers additional insight into the solution quality. In this context, the ratio between the residual off-diagonal terms and the main diagonal terms serves as an indicative measure.

Considering the special structure of the state vector $\mathbf{x}\left(t\right)$ (Equation (8)) and dividing the Fourier coefficients of the eigenvector into

$${\mathbf{u}}_{kl}=\left[\begin{array}{c}{\mathbf{v}}_{kl}\\ {\mathbf{w}}_{kl}\end{array}\right],$$

(36)

then it follows for the structure of the eigenvectors:

$${\mathbf{u}}_k\left(t\right)=\left[\begin{array}{c}{\displaystyle \sum _{l=-L}^{+L}}{\mathbf{v}}_{kl}{\mathrm{e}}^{il{\mathsf{\Omega }}_{P}t}\\ {\displaystyle \sum _{l=-L}^{+L}}({\rho }_k+il{\mathsf{\Omega }}_P){\mathbf{v}}_{kl}{\mathrm{e}}^{il{\mathsf{\Omega }}_{P}t}\end{array}\right].$$

(37)

For real state variables, the eigenvalues and eigenvectors have a special structure, such that each mode consists of a pair of solutions to conjugate complex eigenvalues. In the complex plane, the individual solution components of the overall solution can be represented as pointers.

Hill’s approach provides a method that allows the solution of the homogeneous differential equation to be calculated with sufficient accuracy. The calculation refers to a parameter circle frequency ${\mathsf{\Omega }}_P$ with fixed Fourier coefficients of the system matrices. Parameter variations within the system matrices, such as the amplitudes of the parameter excitation, can increase the computational effort.

5. Simulation Results for the Glider

The aim of the study is the assessment of stability for the rotor system. Therefore, not only the simulation with the true construction parameter is conducted, but also variations of it. First step is the Hill’s hyper-eigenvalue problem and its eigenvalue solutions. As stated before, the real parts of the eigenvalue are responsible and need to be selected from the redundant solutions.

The base data for the non-rotating system comes from the design parameter of the propulsion system according Figure 2 and Figure A1. The resulting FEM Model has been validated by the workshop FRF measurements with the example shown in Figure 3. Solutions with parameter variation have been simulated to make the borders of stability regions visible in Figure 6:

• Damping $D=0.5\dots 2.0\dots 5.0$ in different subplots; the damping is varied from the real structure to an upper and lower extreme ($D=0.5$: lower extreme for a airplane body with fitted metal materials; $D=2.0$: glider under investigation; $D=5.0$: glider with extra damping treatments).
• $J_{sim}=f_{dJ}·\Delta J_{gl}$, where $f_{dJ}$ is a scaling factor to the effective $\Delta J_{gl}$ of the glider in Equations (6) and (7); this varies the parameter amplitude in the simulation; for a better spread in the diagram the $lo{g}_{10}$ is taken, so that the effective $\Delta J_{gl}$ corresponds to $lo{g}_{10}\left(f_{dJ}\right)=0$; steps are $f_{dJ}=0.1\dots 0.5\dots 1\dots 2$ as the sensitivity to the change of parameter amplitude is low ($\Delta J_{gl}=0.066\mathrm{kg}{\mathrm{m}}^2$ in the glider).
• Rotor speed in the relevant range of the glider from 300 $\mathrm{rpm}$ to 2500 $\mathrm{rpm}$ in 400 simulation steps; this covers the complete rpm range during the operation of the glider.

One challenge is to find the relevant results of the redundant eigenvalues. Therefore, two strategies have been applied as follows:

• To avoid inaccuracy in the Fourier series of the eigenvectors, the time-constant block ${\mathbf{u}}_0$ Equation in (30) should be in the centre according $l=0$. Then, the Fourier components between $L=-2\cdots +2$ are balanced. The constant block is expected to have the largest vector length, so that this is a criterion for the selection.
• The imaginary part of the hyper-eigenvalue is expected to be close to the imaginary part of the system without parametric excitation $\Delta J=0$ but with consideration of the rotational speed $J_e·{\omega }_e$. So, the vicinity of the two assigned eigenvalues are the criterion for selection.

Figure 6 shows the areas where at least one of the real parts are positive (positive values are printed by dark color) and indicate a point of unstability. The second method for the redundancy has been applied here. The expected trend can be observed that higher damping and lower parametric excitation amplitude reduce the risk of unstablity. For the parameters of the real rotor with $f_{dJ}=1$ and $D=2.0\%$, there remains a risk at lower rpm speeds around 1100 $\mathrm{rpm}$. In general, the mechanical stability is ensured. The risk of changes in the damping and parametric amplitude can be concluded from the diagrams.

The characteristic multipliers have been calculated according Equations (16) and (18). In this set up, the estimated damping $\zeta =2$ and the effective inertia $\Delta J_{gl}$ have been used. The required solutions from time integration over one period $T_P$ is challenging for the numerical algorithm as the ODE is changing by every step in time (details are in Appendix B). Two different rotational speeds are evaluated as follows:

• 1125 $\mathrm{rpm}$ in Figure 7 for a speed expected close to a critical region,
• 2500 $\mathrm{rpm}$ in Figure 8 for a speed at nominal power.

The speeds have been chosen according to the risk assessment in Figure 6. For 1125 $\mathrm{rpm}$, there is a slight risk of instability as one multiplier is minimal above 1.0. The speed of 2500 $\mathrm{rpm}$ shows normal behaviour with all multipliers below 1.0. Natural vibrations decay due to the structural damping.

The simulation clearly states that the glider structure needs a proper assessment for mechanical stability. Based on a dynamic model that has been validated by experiments, the simulation was performed with three changing parameters.

• The structural Damping D.
• The unsymmetry of the rotor in terms of $\Delta J$.
• The rotor speed in the relevant range of operation.

Center point has been the real construction of the glider. The more direct Hill’s method provides a good overall result. For two certain conditions, the Floquet multipliers could confirm the assessment. The stability is, in general, given, but is borderline around 1125 rpm.

6. Conclusions

The stability of a modern lightweight glider rotor system has been subject of investigation. The basis was a validated FEM model with a low number of DOFs. The validation has been conducted in a standard way of a classical experimental modal identification. The focus was the change of vibration behaviour due to the gyroscopic effects under rotation. In addition to the frequency split of a gyro with rotational speed, the stability of the parametric excitation was the main target investigation. So, the paper answered the question of mechanical stability for a new class of applications.

Methods that have been adapted from the classical literature are able to answer the research question. Based on the assumption of periodic eigenvectors, the system dynamics can be described in a modal form. The solution of Hill’s hyper-eigenvalue problem generates an extended interpretation of modal analysis, where the stability results from the real part of the exponential factor. Here, the glider investigation shows the borders of mechanical stability. The alternative method from the Floquet theorem, that observes the development after a time $T_P$, concluded with the same result. A low rpm region around 1120 $\mathrm{rpm}$ is borderline stable, where in other rpm regions, the stability is given for the parameters of the real construction. Parameter variations into critical areas can be simulated.

Side effects like non-linearities in the bearings or aerodynamic influences of the propeller are not in the scope of the work. It is assumed that this would lower the risk of instability. This type of investigation is currently out of scope of structural dynamics. It is a research question for future work.