Mair, C., Rezgui, D., & Titurus, B. (2018). Nonlinear stability analysisof whirl flutter in a rotor-nacelle system. Nonlinear Dynamics, 94(3),2013-2032. https://doi.org/10.1007/s11071-018-4472-y

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Nonlinear Dyn (2018) 94:2013–2032https://doi.org/10.1007/s11071-018-4472-y

ORIGINAL PAPER

Nonlinear stability analysis of whirl flutter in a rotor-nacellesystem

Christopher Mair · Djamel Rezgui ·Branislav Titurus

Received: 19 January 2018 / Accepted: 9 July 2018 / Published online: 2 August 2018© The Author(s) 2018

Abstract Whirl flutter is an aeroelastic instability thataffects propellers/rotors and the surrounding airframestructure on which they are mounted. Whirl flutteranalysis gets progressively more complicated with theaddition of nonlinear effects. This paper investigatesthe impact of nonlinear pylon stiffness on the whirlflutter stability of a basic rotor-nacelle model, com-pared to a baseline linear stiffness version. The useof suitable nonlinear analysis techniques to addresssuch a nonlinear model is also demonstrated. Threetypes of nonlinearity were investigated in this paper:cubic softening, cubic hardening and a combined cubicsoftening—quintic hardening case. The investigationwas conducted through a combination of eigenvalueand bifurcation analyses, supplemented by time simu-lations, in order to fully capture the effects of nonlinearstiffness on the dynamic behaviour of the rotor-nacellesystem. The results illustrate the coexistence of stableand unstable limit cycles and equilibria for a range ofparameter values in the nonlinear cases, which are notfound in the linear baseline model. These branches areconnected by a number of different bifurcation types:fold, pitchfork, Hopf, homoclinic and heteroclinic. The

C. Mair (B) · D. Rezgui · B. TiturusDepartment of Aerospace Engineering, University ofBristol, Queens Building, Bristol, UKe-mail: chris.mair@bristol.ac.uk

D. Rezguie-mail: djamel.rezgui@bristol.ac.uk

B. Tituruse-mail: brano.titurus@bristol.ac.uk

results also demonstrate the importance of nonlinearwhirl flutter models and analysis methods. Of particu-lar interest are cases where the dynamics of the nacelleare unstable despite linear analysis predicting stablebehaviour. A more complete stability envelope for thecombined model was generated to take account of thisphenomenon.

Keywords Bifurcations · Continuation · Stabilityboundary · Nonlinear stiffness · Whirl flutter

Abbreviations

FW Forward WhirlBW Backward WhirlADYN Advanced european tiltrotor DYnamics and

NoiseLCO Limit Cycle Oscillation

1 Introduction

The aeroelastic instability known as whirl flutter is animportant consideration in aircraft design. A propelleror rotor mounted in a wing nacelle may be suscepti-ble to whirl flutter. Typically associated with tiltrotorsand some fixed wing aircraft, the phenomenon is man-ifested in the hub whirling around its original position.Aerodynamic forces acting on the blades and gyro-scopic effects acting on the rotor as a whole couple

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with wing structural modes to produce this unstablemotion, which can damage or even destroy the aircraftstructure [1]. Two whirl flutter modes exist—forwardwhirl (FW) and backward whirl (BW)—identified bythe sense of the whirl relative to the rotor’s rotation;forward denotes that the whirl and the rotor are spin-ning in the same direction. With their large and flexibleblades, tiltrotors are particularly susceptible to whirlflutter. For given cruise speed requirements, whirl flut-ter stability considerations impact the design of theirwings, pylons and rotors [2].Designing to preventwhirlflutter becomes more complicated in the presence ofnonlinearity.Nonlinearities are often neglected for con-venience in modelling, contingent on the applicabilityof some assumption(s) given the scope of a particu-lar investigation, though aerospace structures regularlyexhibit nonlinear behaviours [3]. Furthermore, thereis little mention in the existing literature of nonlinearwhirl flutter studies being conducted.

The current literature has investigated methods ofimproving stability margins by alterations to existingrotor designs [4] and studied the impact of effects suchas control system stiffness [5]. However, in almost allcases, these studies restricted the structural modellingto linear approximations, which is contingent on theassumption of small deformations. Various kinds ofnonlinearity have been shown to have a non-negligibleeffect on system behaviour. A review of the impactof various types of structural nonlinearity on systemdynamics was provided by Breitbach [6], with furtherspecific investigations conducted by Dowell and Ilg-amov [7]. In both cases, analytical frameworks and theeffects of each nonlinearity on flutter predictions aresuggested. Masarati et al. [8] showed that nonlineareffects at the blade level can ultimately affect overallstability in a tiltrotor system. Moreover, Krueger [3]showed that nonlinearities introduced by the influenceof the drivetrain, free-play and backlash can create abehavioural discrepancy between rotors in windmilland thrust mode. While the main focus of Krueger’spaper is to present a multibody modelling approach ofa previous wind tunnel test performed as part of theEuropean ADYN project, the effects of nonlinearitywere investigated through the introduction of nonlin-ear springs in the computational model. Spring stopswere also added to provide hard limits onmodel deflec-tion, and a good agreement with the wind tunnel testdata was shown. Considering the repeated demonstra-tion that nonlinear effects have an impact on a system’s

behaviour, they are therefore an important modellingconsideration.

Park et al. [9] investigatedwhirl flutterwith a nonlin-ear structural model, though the focus of the paper wasan overall design optimisation framework as opposedto any impacts on the whirl flutter predictions made byusing nonlinear elements in the model. Additionally,the stability analysis was conducted through time sim-ulations alone rather than any dedicated direct methodsuch as bifurcation analysis. That is, the whirl flut-ter onset speed was determined through iterative timesimulations to find the maximum airspeed that didnot result in flutter. Furthermore, investigations byJanetzke et al. [10] used nonlinear aerodynamic mod-els adapted from aerofoil data, though the structuralaspects of the model did not appear to have benefittedfrom the same approach.

Moreover, Lee and Tron [11] showed that the inclu-sion in a model of certain known nonlinear effects incontrol surfaces can lead to the early onset of flutterbehaviours. A linear model incorrectly predicts a muchhigher onset speed.

Nonlinearity between load and displacement in astructure’s stiffness may be caused by non-uniformityin either geometry or material properties. Both sourcesof non-uniformity are likely to be present in anyaerospace structure, meaning that linear approxima-tions of stiffness are only acceptable when deflec-tions are very small. Realistic spring characteristicsmay include both softening and hardening phenom-ena at different points in the stiffness profile, visibleas decreases and increases, respectively, in the gradi-ent of the stress–strain curve for a given spring struc-ture [12]. Use of cubic terms for more representativestiffness modelling at large deflections can be found inThompson [13].

The previous studies either stopped short of a fullynonlinear analysis or avoided using nonlinear modelsaltogether. In order to understand the effect of nonlinearmodel aspects on a system’s behaviour, suitable anal-ysis methods must be used, namely continuation andbifurcation methods. Such methods have so far beenapplied in only a small number of rotorcraft dynamicalproblems, specifically flight mechanics [14], groundresonance [15] and rotor vortex ring state [16], thoughtheir inclusion in rotary-wing studies is steadily becom-ingmore prevalent as they are powerful in solving prob-lems such as the identification of instability scenariosof rotor blades [17]. Continuation methods were also

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used in the AW159/Wildcat Release to Service doc-ument [18,19], to investigate free-play effects on thebehaviour of the tail rotor. Salles et al. [20] used con-tinuation and bifurcation methods to investigate bifur-cations in the behaviour of whole engine rotordynamicmodels, due to the presence of nonlinearities, althoughtheir analysis was conducted in the frequency domain.

In this paper, a basicwhirl flutter system is presentedin Sect. 2. This model included linear and subsequentlynonlinear expressions for yaw stiffness, specificallypolynomial terms proportional to the cube and fifthpower of displacement. Section 3 describes the stabil-ity analysis methods used, and these are applied to thelinear and nonlinear models as appropriate. The anal-ysis was carried out for a number of cases to study theeffects of nonlinearity for a set of selected parameters.The stability results and bifurcation diagrams gener-ated are discussed in Sect. 4, along with revised stabil-ity boundaries that take account of the additional effectsfrom the nonlinearities introduced.

2 Whirl flutter model

A basic model given by Bielawa [21] and originallyformulated by Reed [22] was used. In this model, arotor of radius R, spinning at angular speed � withmoment of inertia about its rotational axis Ix , is ableto oscillate in pitch θ and yaw ψ about an effectivepivot point with moment of inertia In . The dynamicalcontributions of the wing structure are modelled withequivalent stiffness K and damping C properties in thepitching and yawing directions at the effective pivotpoint, to which the rotor is connected at a distance of amultiples of its radius. The rotor is subjected to the axialflow condition: a freestream velocity V is incident onthe rotor system along the x axis. The system schematicis shown in Fig. 1.

The original model in Reed [22] features linearstructural stiffness properties and was used as a base-line for comparison with the nonlinear stiffness ver-sions. The equations of motion governing the system,as given by Bielawa [21], are stated in Eq. (1).[In 00 In

] [θ

ψ

]+

[Cθ −Ix�Ix� Cψ

] [θ

ψ

]

+[Kθ 00 Kψ

] [θ

ψ

]=

[Mθ

Mψ

](1)

Fig. 1 Whirl flutter model schematic diagram

where Mθ and Mψ are aerodynamic moments in pitchand yaw, respectively, and are defined in Eqs. (2)and (3). They were derived in the manner employedin Ribner’s work [23] on forces and moments gener-ated by propellers experiencing yaw and yawing ratesat their hub. Ribner’s derivation is founded upon bladeelement theory and assumes quasi-steady aerodynam-ics, an aspect that some investigations, such as thatby Kim et al. [24], have built upon. A key aspect ofRibner’s work that separated it from existing theory atthe time was the inclusion of induction/inflow effects,“analogous to the downwash associated with a finitewing”. It can be seen from the equations that there iscoupling only at the angular displacement level.

Mθ = NB

2KaR

[−(A3 + a2A1)

θ

�

−A′2ψ + aA′

1θ

](2)

Mψ = NB

2KaR

[−(A3 + a2A1)

ψ

�

+A′2θ + aA′

1ψ

]

Ka = 1

2ρcl,αR

4�2 (3)

where Ka is a consolidation of terms for more con-cise presentation; ρ denotes air density and cl,α denotesthe blade section lift slope. The Ai terms are aerody-namic integrals that arise from integrating lift alongeach blade and summing the contributions from each,and are defined as:

A1 = c

R

∫ 1

0

μ2√μ2 + η2

dη (4)

A′1 = μA1 (5)

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Fig. 2 Nonlinear stiffnessprofiles used

A′2 = c

R

∫ 1

0

μ2η2√μ2 + η2

dη (6)

A3 = c

R

∫ 1

0

η4√μ2 + η2

dη (7)

μ = V

�R(8)

The A2 (without a hyphen) integral features only inthe derivation of these expressions [21]; however, theoriginal nomenclature has been retained here. For thenonlinear cases, the original linear expression for thestructural yaw stiffness (i.e. Kψψ) was replaced witha polynomial of the form given in Eq. (9), where “nl”denotes “nonlinear”. Here, the stiffness is a functionof angular deflection. The influence of each term iscontrolled via dedicated coefficients Ki . As the pitchand yaw degrees of freedom in the original formulationwere modelled in exactly the same way, either couldhave been selected for nonlinear adaption without anyqualitative impact on the results.

Kψ,nl (ψ)ψ = K1ψ + K2ψ3 + K3ψ

5

= (K1 + K2ψ2 + K3ψ

4)ψ (9)

The nonlinear stiffness expression can provide soften-ing behaviours by using negative values of K2 and/orK3, and hardening behaviours by using positive val-

ues. The cubic term is dominant at smaller deflections,while the quintic term is dominant at larger deflections,allowing both softening and hardening behaviours tobe observed in the same stiffness profile if K2 and K3

have opposite signs. In order to reflect the most preva-lent types of nonlinear spring stiffness, this researchselected the following three stiffness profiles for inves-tigation: cubic hardening (K2 = 10 Nm rad−3, K3 =0 Nm rad−5), cubic softening (K2 = −10 Nm rad−3,K3 = 0 Nm rad−5) and combined cubic softening—quintic hardening (K2 = −10 Nm rad−3, K3 =350 Nm rad−5). The linear coefficient K1 was var-ied between − 0.3 and 0.5 Nm rad−1 as the inde-pendent variable in each case. The overall shape ofthese profiles compared to the original linear modelis shown in Fig. 2. Hereafter, the model employingthe original linear yaw stiffness expression is referredto as the “linear model”, and the models employingthe nonlinear stiffness expressions as the “hardeningmodel”, “softening model” or “combined model” asappropriate.

The model equations were written in state-spaceform, as shown in (10) and (11):

Y = f (Y,p) , Y ∈ �4, p ∈ �n (10)

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Table 1 Datum parameter values

Description Symbol Value

Rotor radius R 0.152 m

Rotor angular velocity � 40 rad s−1

Freestream velocity V 6.7 ms−1

Pivot length to rotor radius ratio a 0.25

Rotor moment of inertia Ix 0.000103 kg m2

Nacelle moment of inertia In 0.000178 kg m2

Structural pitch damping Cθ 0.001 Nm s rad−1

Structural pitch stiffness Kθ 0.4 Nm rad−1

Structural yaw damping Cψ 0.001 Nm s rad−1

Structural yaw stiffness Kψ 0.4 Nm rad−1

Number of blades NB 4

Blade chord c 0.026 m

Blade lift slope cl,α 2π rad−1

and

Y =

⎡⎢⎢⎣

θ

ψ

θ

ψ

⎤⎥⎥⎦ (11)

whereY is the state vector andp is a vector of n parame-ters. The model was implemented in MATLAB 2016a[25], and time simulations were generated using theode45 solver, which implements an explicit Runge–Kutte (4, 5) formula, the Dormand–Prince pair [26].The parameter values used throughout the investigation(Table 1) were retained, where possible, from Reed.Where ranges of parameterswere used inReed [22], themidpoint value was taken for this parameter set. Thesevalues represent a scaled wind tunnel rotor-nacelle sys-tem; however, the qualitative results achieved from thefollowing analyses are applicable to full size aircraft.

3 Stability analysis methods

Initially, eigenvalue analysis was used to assess the sta-bility of the linear system. This standard method placesthe equations of motion of the system in state-spaceform in order to obtain the Jacobian matrix J about anequilibrium point, defined as

Y = JY (12)

where Y is the state vector, which for the whirl flut-ter model used in this paper is defined in Eq. (11). Ifthe various terms in the aerodynamic moment expres-sions [Eqs. (2) and (3)] are brought over to the left-handside of the equation and incorporated into the relevantmatrices, the equations of motion assume the form of

MX + CX + KX = 0

X =[

θ

ψ

](13)

and therefore the Jacobian matrix for this system is

J =[

0 I−M−1K −M−1C

](14)

where 0 and I are 2 × 2 zero and identity matrices,respectively. The eigenvalues of the Jacobian matrixcontain information about the frequency and damp-ing/decay rate (i.e. stability) of the system’s vibrationalmodes, and the corresponding right eigenvectors con-tain the mode shapes. The undamped natural frequencyω and damping ratio ζ for a given mode are calculatedfrom the real and imaginary parts of its eigenvalue λ

using Eqs. (15) and (16). The eigenvalue analysis wasalso completed in MATLAB.

ω =√Re (λ)2 + Im (λ)2 (15)

ζ = −Re (λ)

ω(16)

For nonlinear systems, numerical continuation andbifurcation theory are used. Given a known solutionas a starting point, continuation calculates the steady-state solutions of a dynamical system as one of itsparameters, called the continuation parameter, is var-ied [27]. The computed solutions construct a numberof branches that can be either stable or unstable. Todetermine their stability, either an eigenvalue or Flo-quet analysis is carried out at each computed solutionpoint, depending on the nature of the solution. For equi-libria (also known as fixed points), an eigenvalue anal-ysis can be used—requiring local linearisation in thecase of a nonlinear system—whereas periodic solutions(formally limit cycle oscillations, abbreviated to LCO)require Floquet theory to determine their stability [28].

A bifurcation is a qualitative point change in the sys-tem behaviour as a parameter is varied. In other words,when the stability of a system is changed or lost, thesystem bifurcates. The points at which these stabilitychanges happen are called bifurcation points. Another

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2018 C. Mair et al.

Fig. 3 Example frequency (left) and modal damping (right) plots for a sweep in freestream velocity V in the linear model. The shadedarea is unstable. “FW” denotes the forward whirl mode, while “BW” denotes the backward whirl mode

way to visualise this change is to consider the phaseportraits of the system either side of the bifurcation:they are topologically different and therefore one can-not bemapped to the other through a continuous one-to-one transformation [27].When the system is nonlinear,newsolutionbranchesmayemerge from thebifurcationpoints, leading to the presence of multiple solutions fora given set of system parameters. The identification ofthese different solution branches helps to uncover theglobal dynamics of the system. Of particular interestare instances where stability is dependent on the mag-nitude of a perturbation, a hallmark phenomenon ofnonlinear systems.

Each type of system (linear, cubic softening etc.)was analysed using the appropriate method. Bifurca-tion diagramswere produced using theDynamical Sys-tems Toolbox for MATLAB by Coetzee [29], whichuses an implementation of AUTO-07P [30]. Time sim-ulations were also used in both cases to corroborate thepredictions of the stability methods.

4 Results and discussion

4.1 Linear stability

The eigenvalue analysis described in Sect. 3 allows thestability of a linear system to be quantified in terms ofmargin, and the cause of any instability encountered tobe recognised through the location and movement ofthe eigenvalues on the complex plane. The eigenval-ues, damping ratio and frequency of the linear model’s

modes are shown in Fig. 3 as the freestream velocity Vis swept across a range of values. The two whirl flut-ter modes described in Sect. 1 were identified throughinspection of the eigenvector components. The middleplot shows that the system is predicted to encounterwhirl flutter at a V/Vtip value of approximately 1.25.The remaining parameter values used are those pre-sented in Table 1.

The concept of a stability boundarydiagrambetweentwo parameters can be useful for understanding a sys-tem’s sensitivity to changes in those parameters, par-ticularly parameters that are readily controllable in thedesign phase of a practical system, such as an aircraft.Such a diagram can be produced from a grid of thecombinations of different values for each parameter.The Jacobian matrix is calculated at each point, and asurface is overlaid where the level is determined by themaximum real component of the Jacobian’s eigenval-ues at each point. As the sign of the real componentof an eigenvalue determines the stability of the cor-responding mode—positive being unstable—and onlyone unstable eigenvalue is required for overall systeminstability, a horizontal plane cut of this surface at thelevel 0will produce a contour that denotes the boundarybetween the stable and unstable regions of the param-eter grid.

One such stability boundary that uses parametersthat are controllable in the design phase of a rotary-wing aircraft’s rotor system is that between two struc-tural properties: yaw stiffness Kψ and pitch stiffnessKθ , shown in Fig. 4. To demonstrate the respective

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Nonlinear stability analysis of whirl flutter 2019

Fig. 4 Stability boundary for the linear model in the pitch stiff-ness and yaw stiffness plane

impacts of variations in some of the other physical sys-tem parameters, the same stability boundary is plot-ted for a number of such changes. Increasing thefreestream velocity V or the rotor’s moment of inertiaIx enlarges the unstable region symmetrically, whereasincreasing the damping in both degrees of freedom(Cθ , Cψ) diminishes the unstable region symmetri-cally. Altering the damping parameters asymmetricallyenlarges the stable unstable region in the direction ofthe reduced parameter and reduces it in the directionof the increased parameter. The datum case, using theparameter values given in Table 1, is similar to thatachieved by Reed [22].

4.2 Bifurcation analysis

Figure 4 can also be generated by continuation meth-ods, as the system has an equilibrium at Y =[0; 0; 0; 0] that can be used as a starting solution.Gen-erating the stability boundary this way in fact affordsdeeper insight than the contour cut method describedin Sect. 4.1. Key bifurcation types that are relevant tounderstanding the behaviour of this rotor-nacelle sys-tem, particularly when the nonlinear stiffness profilesare introduced, are Hopf bifurcations, branch pointsand fold bifurcations. At a Hopf bifurcation, the stabil-ity of a fixed point (i.e. an equilibrium) changes, anda periodic solution arises, caused by a pair of com-

plex conjugate eigenvalues crossing the complex planeimaginary axis. At a branch point, the solution changesstability, caused by a single eigenvalue crossing overthe complex plane imaginary axis. Because the branchpoints in this system are of the pitchfork type, two addi-tional equilibrium branches emanate from the bifurca-tion point. At a fold bifurcation (also known as a limitpoint), a solution branch changes direction within thesolution space and changes stability [27].

Two further bifurcation types that are also observedin themodel’s behaviour are the homoclinic bifurcationand the heteroclinic bifurcation. These are global bifur-cations that concern the collision of branches. A hete-roclinic trajectory is a path in the phase space that con-nects two equilibria,while a homoclinic trajectory joinsa single equilibrium to itself. Heteroclinic and homo-clinic bifurcations are points where a limit cycle makescontact with an equilibrium branch at some point alongitself, creating a heteroclinic/homoclinic trajectory andannihilating the periodic solutions branch of which itis part. These bifurcations are covered in greater detailin Sect. 4.6. For more information on the subject, thereader is referred to [27,31].

Choosing Kθ = 0.3 so that a continuation in Kψ willintersect the regions of interest in the contour-basedstability boundary shown in Fig. 4, the bifurcation dia-grams shown in Fig. 5 are obtained. In these diagrams,fixed point solutions in pitch θ and yaw ψ are plottedagainst the continuation parameter, termed pitch andyaw “projections”, respectively. A key to the symbolsand lines used in the bifurcation diagrams shown in thispaper is given in Table 2. Particular attention is drawnto periodic solution branches: it is common practice toindicate a branch by the maximum positive extent ofthe limit cycle at each point.

Note the Hopf bifurcations (square icon) at Kψ =0.28 and Kψ = 0.08, and the branch point (star icon)at Kψ = 0.03. The bifurcations are visible at the samepoints in the projections of θ and ψ . The same bifurca-tion diagram shown in Fig. 5 can be generated for dif-ferent values of Kθ , and the stability boundary (Fig. 4)built up iteratively. Alternatively, two-parameter con-tinuations in Kθ and Kψ can be performed on eitherof the Hopf bifurcations and the branch point to tracetheir loci in the Kθ − Kψ plane, and this method isemployed here. Plotting these continuations, shown inFig. 6, reconstructs the stability boundary obtained inFig. 4.Nowhowever, the significance of each part of the

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2020 C. Mair et al.

Fig. 5 Bifurcation diagram for Kθ = 0.3, for pitch θ (left) and yaw ψ (right) with Kψ as the continuation parameter

Table 2 Key to symbols and lines used in bifurcation diagrams

Graphic Description Meaning

Solid green line Stable equilibriumbranch

Dashed magenta line Unstable equilibriumbranch

Solid blue line Stable periodic solutionbranch (max. value ofLCO)

Dotted red line Unstable periodicsolution branch (max.value of LCO)

� Hollow square Hopf bifurcation

� Black star Branch point bifurcation

� Black circle Limit point (fold)bifurcation

� Black triangle Homoclinic bifurcation

� Hollow triangle Heteroclinic bifurcation

boundary is known, as well as the path of each segmentonce inside the unstable region.

Considering classical aeroelasticity, two types ofinstability often discussed are static divergence andflutter. Static divergence is a phenomenon concerningthe interaction of aerodynamic loads and internal elas-tic forces resulting in an exponentially growing non-oscillatory structural response. On the other hand, flut-ter is a dynamic phenomenon involving the interplaybetween aerodynamic, elastic and inertial forces result-ing in an exponentially growing oscillatory response.Both types of instability can be studied mathemati-

Fig. 6 Stability boundary generated by two-parameter continu-ation. All shaded areas are unstable

cally through eigenvalue analysis. Static divergence ischaracterised by a single real eigenvalue crossing theimaginary axis, from the negative to the positive half-plane. Flutter, however, involves a pair of complex con-jugate eigenvalues crossing the imaginary axis in thesame way. Both types of instability are observed in thisresearch.

The red curved region in the bottom left corner of thediagram is defined by the location in Kψ of the Hopfbifurcation for a given value of Kθ . In the sameway, theblue strips that are adjacent to the axes are defined bythe branch point. Recalling the definition of each bifur-cation, periodic solution branches emanate from Hopfbifurcations while two additional equilibrium branches

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Nonlinear stability analysis of whirl flutter 2021

Fig. 7 Bifurcation diagram for case 2 (Kθ = 0.3) with K1 as the continuation parameter, with time simulations started with a selectionof K1 values (pitch [deg] vs. time [s]). Initial conditions are shown with coloured dots. (Color figure online)

emanate from pitchfork branch points. Therefore, allpoints that lie within the red region have periodic solu-tions in θ (and in fact all the state variables). Similarly,all points that lie within the blue lobes have additionalequilibrium solution branches. These bifurcations canbe linked to the onset of different behaviours of thenacelle at the physical level: the periodic solutions con-stitute whirl flutter, while the additional equilibriumbranches constitute static divergence, where the reduc-tion in stiffness causes the nacelle to be pushed to theside by the aerodynamic forces andmoments generatedby the rotor.

Establishing away of describing the theoretical find-ings of bifurcation analysis in terms of the practi-cally oriented language of aeroelasticity requires spe-cial care, despite the purportedly qualitative nature ofboth fields. The principal issue is the stability of solu-tions. When observed in practice, static divergence andwhirl flutter are almost always immediate, irreversible“runaway” unstable motions. However, in continuationanalysis, where precise “knife-edge” cases are found,both stable and unstable solution branches may befound for both equilibrium and periodic solutions, as isshown later in the paper. This leads to apparently con-tradictory terminology being used to describe the vari-

ous behaviours observed in the model; the phrase “sta-ble static divergence” is a contradiction in terms whenviewed from the standpoint of aeroelasticity, though inthe domain of bifurcation theory it refers quite clearlyto a stable equilibrium branch that has emanated froma branch point. In order to preserve both the physi-cal meaning of predicted behaviours and the insightsafforded by bifurcation analysis, the terms “static diver-gence” and “whirl flutter” are used in direct conjunctionwith terms qualifying stability throughout the discus-sion sections of this work. A number of cases of variousvalues of Kθ are selected for further bifurcation analy-sis and are indicated in Fig. 6.

As the Hopf and branch point are both on the equi-librium branch, which lies at zero displacement, thepositions of the bifurcations do not change with theaddition of any nonlinear stiffness terms. However, thedynamic behaviour outside the equilibrium branch cal-culated in Fig. 6 (hereafter referred to as the “mainbranch”) is affected by nonlinear terms.

4.3 Cubic hardening

A nonlinear yaw stiffness profile with a K2 value of 10and a K3 value of 0 was used in the cubic hardening

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2022 C. Mair et al.

Fig. 8 Bifurcation diagram for Kθ = 0.3, K2 = [4, 10, 16], respectively, K1 as continuation parameter

model. To facilitate understanding, case 2 (Kθ = 0.3)is initially considered in detail due to the variety ofdynamical behaviours present. A bifurcation diagramof the pitch projection is presented in Fig. 7. Thefigure shows complex behaviour manifested in stableand unstable limit cycles and secondary equilibriumbranches. Note that the continuation parameter is nowK1 rather than Kψ .While the limit cycle branches illus-trate the behaviour of the rotor-nacelle system encoun-tering whirl flutter, the secondary branches emanatingfrom the branch point bifurcation quantify the staticdivergence in pitch and yaw. For limit cycles, solid bluedenotes stability and dashed red denotes instability. It iscommon practice to illustrate only the maximum posi-tive extent of a limit cycle branch.Moreover, it is typicalin bifurcation analysis to extend the continuation out-side the physical range to search for any bifurcationswhich result in secondary branches extending back tothe physical parameter range.

While bifurcation analysis is able to reveal complexbehaviours of a system, the best approach is to sup-plement continuation with time simulations at pointsof interest for a fuller understanding. Time histories inpitch θ for a number of values of K1 are also shown inFig. 7, with different initial conditions to demonstratethe stability of limit cycles by showing convergenceor divergence as relevant. From left to right, the areasof interest that are selected for time simulation are thestable static divergence branch at K1 = − 0.2 (demon-strating convergence on approximately 1.8◦), the stableflutter on the static divergence branch at K1 = − 0.05(demonstrating convergence on a limit cycle centred atapproximately 1.2◦), both stable and unstable regionsof flutter on the main branch at K1 = 0.075 (demon-strating divergence from a limit cycle with amplitude

of approximately 3◦, convergence on a limit cycle withamplitude of approximately 5◦ and convergence on alimit cycle of approximately 2◦) and stable flutter on themain branch at K1 = 0.14 (demonstrating convergenceon a limit cycle with amplitude of approximately 6.3◦).All four state projections for hardening model case 2are shown in Fig. 11 (left column) to provide a fullcomparison between the behaviours of each system.

The value of cubic stiffness coefficient K2 usedabove was selected rather arbitrarily in order to effectnonlinear stiffness behaviour. It is therefore prudentalso to understand the effect of the value of K2. Bifur-cation diagrams with Kθ set to 0.3 (as per case 2) fordecreased and increased values of K2 are shown inFig. 8 compared to the original value of K2 = 10. Asis evident from the plots, increasing K2 decreases theamplitude of both the flutter and the static divergencefor a given value of K1, due to increased structuralstiffness.

It is noted that the periodic solution branch con-nected to the main branch always leans over the Hopfbifurcation adjacent to the branch point. As a result, aportion of the branch connecting to this bifurcation isunstable, which is present for all positive values of K2.Furthermore, the effects of changing K2 could also beexplored for the other cases shown in Fig. 6, but thiswas deemed outside the scope of this paper.

4.4 Cubic softening

Using a softening yaw stiffness profile (K2 = − 10,K3 = 0), a bifurcation diagram for case 2 (Kθ = 0.3)showing the pitch projection is presented in Fig. 9. Thestatic divergence branches emanating from the branch

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Fig. 9 Bifurcation diagramfor Kθ = 0.3, K2 = −10,K1 as continuationparameter, complementedwith time simulationsstarted with a selection ofK1. Initial conditions areshown with coloured dots.(Color figure online)

point at K1 = 0.04, thoughunstable, overhang themainbranch, to the right of the Hopf bifurcation near K1 =0.3. While an unstable flutter solution emanates fromthis Hopf bifurcation, this branch eventually becomesstable through a limit point at approximately K1 =0.42, and both portions overhang the stable portion ofthe main branch (from K1 = 0.28 upwards) as far asthis limit point. Time simulations for selected pointsare shown in subplots.

A rotor-nacelle mounted on an aircraft is subject toperturbations, frommanoeuvring or gusts, for example.Aperturbationof the rotor-nacellemayultimately bringit sufficiently close to either of these solution branchesto experience behaviour of either type. The time simu-lations show that for K1 = 0.4, where the main branchis stable, a pitch perturbation of 2◦ provides a decayingmotion, but a stable flutter LCOdevelops almost imme-diately with a perturbation of 3◦. In general, a perturba-tion may consist of any combination of individual stateperturbations (i.e. angles and angular rates). Providedsufficient angular rates, attraction to the stable flutterbranch overhanging K1 = 0.4 could be possible fromeven lower angles than 3◦, and conversely larger per-

turbations than 3◦ may converge on the main branch ifthe angular rate components are not sufficiently large.

The linear stability analysis is unable to predict theabove result. The flutter boundary it predicts is simplythe location of the Hopf bifurcation at K1 = 0.28,though both flutter and static divergence behavioursare shown to exist and may be encountered for valuesof K1 that lie within the stable region.

4.5 Combined cubic softening: quintic hardening

Introducing a positive quintic coefficient (K3) intothe stiffness profile used in Sect. 4.4 allows softeningeffects to dominate at lower deflections and harden-ing effects at higher deflections. Compared with thesoftening model’s results, the hardening effects bendthe static divergence branches back round to the left,allowing a small branch of flutter LCO’s to exist oneach, as seen in the hardening model. This type of flut-ter is termed “secondary flutter” in the remainder ofthis paper to distinguish it from flutter about the mainbranch. The value of K3 chosen was 350, so that thestiffness curve is close to the linear profile within the

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Fig. 10 Bifurcationdiagram for combinedsoftening-hardening, case 2(Kθ = 0.3)

angle range of [−10◦, 10◦] (see Fig. 2). Initially, thebifurcation diagram for case 2 is presented in Fig. 10.

To provide a level comparison between thebehaviours of each stiffness case (hardening, soften-ing, combined), the projections for all states for case 2are shown in Fig. 11. As before, the secondary equi-librium branches in the pitch and yaw projections (firsttwo rows of Fig. 11) show the static divergence for agiven value of K1. As static divergence does not involveany movement by definition, these secondary branchesappear to overlap the main branch in the pitch rate andyaw rate projections (last two rows of Fig. 11) in allthree models, as in both branches pitch rate and yawrate are zero. In the hardening and combined models(left and right columns of Fig. 11), each static diver-gence branch has its own secondaryflutter LCObranch.The pitch projections for all five Kθ cases from all threemodels are summarised in Fig. 12.

Considering a given diagram from right to left, i.e.for descending K1: case 1 (Kθ = 0.4) shows diver-gence only (Fig. 12a), case 2 (Kθ = 0.3) shows aseparate region of flutter only followed by divergence(Figs. 7, 9, 10), case 3 (Kθ = 0.2) shows flutter which

eventually coexists with static divergence (Fig. 12c),case 4 (Kθ = 0.05) shows flutter only (Fig. 12d), andcase 5 (Kθ = 0.037) shows a separate region of diver-gence followed by flutter (Fig. 12e). Only the projec-tion in pitch θ is shown in Fig. 12, though projectionsin any of the other state variables would present thesame qualitative results. Solutions above 60◦ in cases 4and 5 have been ignored as they are considered extremenon-physical behaviour.

Each of the diagrams can be cross-referenced withFig. 6 to confirm that the bifurcations present on themain branch correspond to the extent of the unstableregions at the relevant value of Kθ . As the value ofpitch stiffness is gradually decreased, the amplitude ofthe limit cycles increases significantly.

Interesting to note is the complex interaction in case3 (Kθ = 0.2) in the hardening and combined mod-els (Fig. 12c, left and right) that occurs between thelimit cycles emanating from the main branch (Hopf 1)and those emanating from the two secondary branches(Hopfs 2 and 3). In case 3, these limit cycle brancheshave collided due to a homoclinic bifurcation, coveredinmore detail in Sect. 4.6. On the other hand, a collision

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Fig. 11 Bifurcation diagrams for case 2 (Kθ = 0.3) in all state projections for hardening (left), softening (centre) and combinedsoftening–hardening (right) models

between a flutter branch and a static divergence branchoccurs in case 5 (Kθ = 0.037), due to a heteroclinicbifurcation.

Regarding the results from the softening model, thebifurcations on the main branch still occur in the sameleft-to-right order as in Fig. 7, as nonlinear stiffness

terms do not affect their location.Moreover, the regionsof stability of the main branch are unaffected. How-ever, both the static divergence and flutter branches arereversed left-to-right in the direction of their departurefrom the main branch. With the exception of case 5,all the static divergence branches are unstable and no

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Fig. 12 Summary of pitch projections for all cases, for hardening, softening and combined models, with K1 as the continuationparameter

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secondary limit cycle branches were found to emanatefrom them, as seen in cases 2 and 3 of the hardeningmodel. The crossing of the stable and unstable por-tions of the flutter branch in case 2 at approximatelyK1 = 0.35 in the pitch projection without a bifurcationmay seem unusual at a first glance. However, a separa-tion between the branches can be seen in other projec-tions (such as yaw, Fig. 11). The crossing is due only tothe two-dimensional projection into the K1 − θ plane,and the solution branch does not cross itself within thefour-dimensional state space. The flutter branches incase 3 are no longer bounded or stable as they were inthe hardening model. Furthermore, the flutter branchin case 5 is now connected to the secondary flutterbranches through a homoclinic bifurcation.

The values of K1 at which the bifurcations on themain branch occur is still unchanged in the combinedsoftening–hardening model, as is to be expected. Thestatic divergence branches depart from themain branchin the samemanner as in the softeningmodel in terms ofdirection and stability, though at larger deflections (i.e.further away from the main branch) the quintic hard-ening overpowers the cubic softening and the branchesare bent back in the direction of the hardening model’sbranches.

The flutter branches in the combined model casesmainly resemble those of the hardening model casesin terms of shape; however, the regions of stability onthe branches have more in common with the softeningcases. This seems to be another effect of the differingdominant regions of the cubic and quintic terms. Thecubic softening’s dominance at low deflections influ-ences the direction of branch’s departure from themainbranch. By contrast, the quintic hardening’s dominanceat higher deflections plays a greater part in influencingthe path of the branch through the state space, specifi-cally which other bifurcations the branch is connectedto. This affects the overall shape of the branch andcauses resemblance of the hardeningmodel’s diagrams.As the stability of periodic solution branches changesthrough limit points, it is the combination of departuredirection and overall shape that influences the regionsof stability along a given branch. For example, a branchthat departs a bifurcation in one direction, but even-tually connects to another bifurcation in the oppositedirection, will have both stable and unstable portions.In contrast, if the branch spanned the two bifurcationswithout a change in direction and therefore a limit point,

there would not necessarily have been a change in sta-bility.

Taking a broader view of the bifurcations and branchshapes in each system allows some links to becomeclear. The branch points with their stable equilib-rium branches in the hardening model can be directlyattributed to the hardening term (K2) in the stiffnessfunction due to the close resemblance of the super-critical pitchfork bifurcation normal form. Similarly,the softening term present in the softening and com-bined models closely resembles the subcritical pitch-fork bifurcation normal form.

In the same manner that the effect of the value ofthe cubic stiffening coefficient K2 was investigated inSect. 4.3, the effect of the value of the quintic stiffeningcoefficient K3 on the combined softening–hardeningmodel’s behaviour is investigated here. Bifurcation dia-grams for increased and decreased values of K3 areshown in Fig. 13.

The effect of K3 is similar to the effect of K2 inthat a larger value makes for a stiffer structure than alower value, and the effect is to restrict the extent of thestatic divergence branches and the amplitude of peri-odic solutions. As in Sect. 4.3, a more detailed inves-tigation into the influence of K3 could be carried out,varying other parameters such as Kθ and K2, thoughthis is deemed beyond the scope of the study presentedin this article.

4.6 Homoclinic and heteroclinic bifurcations

As the continuation parameter is varied, a portion ofa limit cycle may approach a fixed point. Althoughthe fixed point may be unstable, the vector field (asdescribed by the differential equations of motion) inits near vicinity will be increasingly flat approachingthe fixed point. Therefore, the period of the limit cyclewill increase as it approaches the fixed point, reach-ing infinity when the collision occurs and the hetero-clinic/homoclinic trajectory is created. This increasetherefore can be used as an indication of the presenceof such a bifurcation.

In the hardening and combined models of case 3(Kθ = 0.2; Fig. 12c, left and right), the behaviour of thelimit cycles is explained by the heteroclinic and homo-clinic bifurcations that they undergo. Taking the hard-ening model first, a phase portrait is shown in Fig. 14to demonstrate how the limit cycles collide with a fixedpoint.

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Fig. 13 Bifurcation diagram for Kθ = 0.3, K2 = −10, K3 = [200, 350, 500], K1 as continuation parameter

Fig. 14 Phase portrait (left) for case 3 (Kθ = 0.2), hardening model, showing limit cycles near the homoclinic bifurcation at K1 =−0.08. The magenta crosses indicate the three fixed points present. Enlarged bifurcation diagram (right) for cross-reference

The solutions for three values of K1 are illustrated—two limit cycles, one on either side of the homoclinicbifurcation (blue and black), and the homoclinic orbititself (red). The various elements of the phase portraitcan be cross-referenced with the excerpt of the bifurca-tion diagram provided on the right side of the figure. Inboth of these plots, the maximum (positive) amplitudeof each limit cycle and the position of the fixed pointbranches are visible. To the left of the bifurcation point,two separate limit cycles exist (black), each about oneof the static divergence branches. As K1 increases, theinnermost corner of each limit cycle nears the equilib-rium at the origin—the main branch mentioned in pre-vious sections. The limit cycles simultaneously makecontact with the origin fixed point at K1 = − 0.0802,fusing to form a homoclinic trajectory (red). Beyondthis value of a K1, a limit cycle forms and the trajec-

tory loosens, taking on the appearance of a bow tie(blue).

The homoclinic bifurcation itself is therefore at[ψ, θ, K1] = [0, 0, − 0.0802], as this is the pointat which the two limit cycles collide and fuse. On thebifurcation diagram shown in Fig. 12c (left), the limitcycle branches are indicated by their maximum ampli-tude, and therefore, the secondary flutter branches seemto disappear on this hyper-plane. To indicate their anni-hilation via a homoclinic bifurcation, the ends of thebranches are also marked with the homoclinic bifurca-tion symbol. The period of the larger single limit cycleapproaching the homoclinic bifurcation (from the right)is shown in Fig. 15. The stability and limit points arealso included for cross-referencing with Figs. 12c (left)and 14 (right). The characteristic asymptotic increasein period near the bifurcation is clearly visible.

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Fig. 15 Graph of limit cycle period over a subset of the continu-ation parameter range shown in Fig. 12 (left), near the homoclinicbifurcation at K1 = − 0.0802 for case 3 (Kθ = 0.2), hardeningmodel

The combined model’s results feature homoclinicbifurcations and heteroclinic bifurcation. The hetero-clinic bifurcation at K1 = 0.0779 is explored first.Unlike the homoclinic bifurcations seen previously, theheteroclinic bifurcation involves a heteroclinic trajec-tory that joins two equilibria. A phase portrait show-ing trajectories at and near the bifurcation is shown inFig. 16.

The two equilibria that are joined by the heteroclinictrajectory are the inner pair of fixed points at [ψ, θ ] =± [− 4.5, 1.5]. As the complete motion strictly com-

Fig. 17 Graph of limit cycle period over a subset of the con-tinuation parameter range shown in Fig. 12c (right), near theheteroclinic bifurcation at K1 = 0.0779

prises two trajectories, one from left to right and theother vice versa, it is termed a heteroclinic cycle [31].As the bifurcation point is approached from beneath(increasing K1), certain corners of the limit cycle movetowards the fixed points mentioned, eventually collid-ing with them simultaneously at K1 = 0.0779. Theperiod of the limit cycle approaching the heteroclinicbifurcation (from the right) is shown in Fig. 17.

There are also two homoclinic bifurcations at K1 =0.0828 where the secondary flutter branches collide

Fig. 16 Phase portrait showing a limit cycle near the heteroclinic bifurcation at K1 = 0.0779 (left). The magenta crosses indicate thefive fixed points present. Enlarged bifurcation diagram (right) for cross-reference

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with the same inner pair of fixed points involved in theheteroclinic bifurcation.

Homoclinic and heteroclinic bifurcations are alsovisible in case 5 (Kθ = 0.037). In the hardeningmodel (Fig. 12e, left), the periodic branch’s maximumand minimum extents simultaneously make contactwith the static divergence branches, annihilating afterforming a heteroclinic cycle between the two equi-libria. In the softening model (Fig. 12e, centre), theunstable flutter branch folds back to become stable atapproximately K1 = 0.38, and very shortly afterwardssplits into two limit cycles via a homoclinic bifurca-tion at [ψ, θ ] = [0, 0]. These two new limit cycles aresecondary flutter motions about the static divergencebranches. This is the same process that occurred in thehardening model for case 3 (Kθ = 0.2), interpreted inreverse. In the combined model (Fig. 12e, right), themain flutter branch collides with both static divergencebranches simultaneously, in the same manner as in thehardening model.

4.7 Implications for stability boundaries

As shown in Figs. 11 and 12, stable regions of theflutter branch emanating from the Hopf bifurcationon the main branch can overhang the main branchitself in the softening and combined models, mean-ing that flutter can be encountered despite the predic-tion of stability using linear analysis. In the softeningmodel, this overhang occurs in the approximate region0.28 < Kθ < 0.32. In this region, bifurcation dia-grams take the form of case 2 (Kθ = 0.3; Figure 12b,centre). That is, a stable flutter branch exists but is con-nected only to the main branch, and the static diver-gence branches each have a secondary flutter branchabout a small portion of them. The region is boundedby the existence of all the necessary bifurcations; atapproximately Kθ = 0.28 the left-most Hopf and thepitchfork collide and the left-most Hopf annihilates asdetailed in Sect. 4.5. For lower values of Kθ , the peri-odic branch no longer has a second main branch Hopfbifurcation to fold back to, and therefore while it con-tinues to overhang the main branch, it does not con-tain any stable regions (Fig. 12c–e, centre). The mainbranch is, however, overhung by two unstable equi-librium branches, which leads to a divergence if thesystem strays from the main branch.

Fig. 18 Additional unstable region area for the combinedmodel(red) compared to the original linear prediction (grey). (Colorfigure online)

In the combined model, stable overhang of the mainflutter branch exists for a much greater range of Kθ .Overhang exists from Kθ = 0.32 downwards as in thesoftening model (Fig. 12b, right). However, after theleft-most main branch Hopf bifurcation has collidedwith the branch point at approximately Kθ = 0.28,stable portions of flutter branch still overhang a sta-ble portion of the main branch. Continuing to descendin Kθ , this overhang exists until the static divergenceregion near the K1 axis is met. Here, at Kθ = 0.037,the main branch rightward of the branch point doesexperience stable flutter branch overhang (albeit con-nected to the static divergence branches), though themain branch itself is unstable (Fig. 12e, right).

A revised stability boundary accounting for therightward reach of any overhanging flutter branch witha stable portion can be generated. This can be achievedeither through iterated one-parameter bifurcation over avariety of Kθ values, or through two-parameter contin-uation of the right-most limit point found on the flutterbranch to trace its path in Kθ and K1 simultaneously.Such a revised stability boundary for the combinedmodel is shown in Fig. 18. The original linear modelboundary and the enclosed unstable region are shown ingrey. The additional unstable area due to the aforemen-tioned overhang phenomenon in the combined modelis shown in red. The boundaries are mostly coincident

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though the overhang extends the region of instabilityto the right of the Hopf loci.

5 Conclusions

This article has demonstrated the use of continua-tion and bifurcation methods to provide nonlineardynamic analysis of a basic rotor-nacelle systemmodel.Both linear and nonlinear stiffness profiles were usedfor the yaw degree of freedom through addition ofcubic and quintic stiffening terms. The cubic stiffen-ing behaviours investigated were softening and hard-ening, and quintic hardening was used in conjunctionwith cubic softening to create a combined softening–hardening model. Stability analysis methods weredescribed and employed for both linear and nonlin-ear models. Bifurcation diagrams were generated fora number of pitch stiffness cases, in each of the hard-ening, softening and combined models. It was shownthat in the softening and combined models, whirl flut-ter was possible in regions where linear analysis wouldhave predicted stability, due to stable portions of flutterbranches overhanging stable main branches. A revisedstability boundary based on this phenomenon was gen-erated for the combinedmodel,where this phenomenonexists over the greatest range of pitch stiffness. Wherewhirl flutter does not cause the loss of an aircraft, oscil-lations induced by whirl flutter mechanisms present afatigue hazard to aircraft nacelle mounts.

The study showed that the introduction of basicand smooth polynomial stiffness profiles into the rotor-nacelle system produced very complicated dynamics.These dynamics were manifested in the coexistence ofmultiple equilibrium and periodic branches, as well asthe various types of bifurcation, namely fold, Hopf,branch point, homoclinic and heteroclinic. These com-plex behaviours could not have been predicatedwithoutproper nonlinear analysismethods such as continuationand bifurcation methods.

These observations show that nonlinear aspects of asystem may have a significant impact on its dynamicbehaviour, particularly where stability is a focus. It istherefore advisable to model nonlinear aspects wherepossible and to employ proper nonlinear analysis tech-niques, allowing informed system design. Stabilityboundaries generated should also take into accountthe coexistence of dynamic behaviours over param-eter regions where equilibrium branches are stable.

Given the likely proliferation of continuation methodsand bifurcation analysis in aircraft certification, propercharacterisation of aircraft materials and sub-systemsis crucial, and nonlinear stiffness profiles should beused in full nonlinear models for any dynamic analysisconducted. Where analytical functions cannot be fit tomaterial or sub-system stiffness profiles, a table-basedapproach could be used.

Going forward, the model used in this paper is to bedeveloped further as part of an incremental approach.Refinements to existing aspects, such as unsteady ornonlinear aerodynamics, are to be made. Additionally,new featureswill also be incorporated, such as the influ-ence of the drivetrain and blade flap and lag dynamics.

Acknowledgements The authors would like to kindly thankthe UK Engineering and Physical Sciences Research Council(EPSRC) for funding this research project, and Dr. David Bartonfor sharing his expertise of bifurcation theory.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrest-ricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

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