Abstract
This work presents the modeling and preliminary Whirl–Flutter stability results achieved within the Advanced Testbed for TILtrotor Aeroelastics (ATTILA) CleanSky2 project. The project addresses the design, manufacturing, and testing of a semispan windtunnel model of the Next Generation Civil TiltRotor. The preliminary multibody models developed in support of the windtunnel testbed design are described, illustrating the modeling technique of each subcomponent of the model, namely the wing, the rotor, the blades, and the yoke. The methodologies used to analyze the stability of systems subjected to periodic aerodynamic excitation when the problem is modeled using fullfeatured multibody solvers are presented in support of Whirl–Flutter identification during windtunnel testing.
Introduction
Tilting rotor aircraft, or tiltrotor, with their outstanding capability of taking off and landing vertically like helicopters and, at the same time, of achieving high speeds—up to twice that of helicopters—in forward flight when operating like conventional turboprop aircraft, represent one of the few successful examples of advanced vertical lift configurations.
After a long development phase that encompassed few experimental aircraft that successfully made it to flight—noticeably the Transcendental Model 1G (a tilting rotor concept), which flew about 100 h without ever completing a full conversion [26] and the Bell XV3 (a tilting rotor configuration) in the 1950s [12], and the Bell XV15 (a tilting rotor/nacelle configuration) in the 1980s [26], unlike other less fortunate technology demonstrators—the concept finally proved its soundness with the BellBoeing V22 (a tilting rotor/nacelle configuration) [46], a military aircraft that after an almost 25 years long development phase, has been operated by the US Marine Corps with an increasing reward for the last 15 years, and is entering service for carrier onboard delivery (COD) in the US Navy. Remaining in the military arena, the Bell V280 Valor (a tilting rotor configuration) after a test campaign started in 2017 is currently under evaluation by the US Army for its Future LongRange Assault Aircraft (FLRAA) program [13]. However, with the then BellAgusta and now Leonardo AW609 (a tilting rotor/nacelle configuration) [36] about to become operational after long and thorough development [15], the tiltrotor design appears to be mature enough to enter also the civil air transport market [1].
Nevertheless, tiltrotor design remains a rather challenging engineering task, considering the various operating conditions and multipurpose missions that are expected to be accomplished by this complex type of aircraft. In particular, the problem of assessing Whirl–Flutter stability limits is at the same time fundamental and challenging. Whirl–Flutter is an aeroelastic stability phenomenon that affects both turboprop and tiltrotor aircraft [6]. When a rotor mounted on a flexible structure rotates, the normal vibration modes associated with the supporting structure may interact with the precession motion of the rotor.
When its motion is perturbed, each point on the rotor axis of rotation draws paths about its reference position. This motion changes the way each rotor blade is affected by the incoming airspeed, correspondingly altering the overall aerodynamic loads. At the verge of whirl flutter, when this phenomenon is triggered, perturbations result in a periodic orbit. The resulting forces can lead to the divergence of the system response, in what represents a truly aeroelastic instability [45].
Nowadays, although the key aspects of the phenomenon in tiltrotor aircraft are understood, the capability to predict it is still limited, despite the availability of sophisticated aeroelastic analysis tools. The difficulty lies in its dependence on several factors, including the geometrical design, the structural properties, the dynamics of the actuators, etc. which can all contribute to the Whirl–Flutter characteristics in ways that are not always intuitive. The problem may be exacerbated when unconventional configurations are considered, as is the case of propeller and rotordriven advanced air mobility concepts [22].
Aeroelastic testing, especially when it concerns stability and flutter, can be extremely dangerous. For this reason, windtunnel testing is needed to understand phenomena and increase the level of trust in predictions in support of their mandatory verification in flight. As a consequence, the possibility to validate numerical predictions using experimental data is of paramount importance from an industry standpoint.
Windtunnel testing of tiltrotor configurations, and its use for validation and correlation with numerical predictions, has been widely used for tiltrotor development. Due to the need to aeroelastically scale also the wing portion of the model, of extreme miniaturization of the rotor—consider for example the need to reproduce in detail rather complex blade root attachments and pitch control mechanisms—aeroelastic windtunnel models of tiltrotor aircraft can be even more complex than helicopter ones.
Several windtunnel models have been developed to explore the concept of Whirl–Flutter in prop and tiltrotor configurations. However, specific windtunnel testing of configurations that were intended to be tested in flight only dates back to the early development of the V22 when a 1/5scale semispan model was designed and manufactured by Bell and used for windtunnel tests and correlation with simulation capabilities available at that time [42]. Eventually, this model evolved in the Wing and Rotor Aeroelastic Test System (WRATS), which has been in use for several years to explore various aspects of tiltrotor aeroelasticity, including aeroelastic tailoring of the wing [10, 35], several parametric studies (e.g., in [40]), active control for flutter suppression [17, 25], an interesting fourblade, articulated softinplane configuration for hub load reduction [33, 34], and several numerical studies of stiff and softinplane configurations successfully correlated with experiments [29, 30, 48, 49].
Another effort towards the windtunnel testing of the aeroelastic stability of tiltrotors took place in Europe, within the EUROFAR program [21]. Subsequent efforts aimed at the development of an original European tiltrotor concept, ERICA [32]. This concept is characterized by the tilting of the outer portions of the wing, affected by the downwash of the rotor. These efforts resulted among the others in the project ADYN [5, 24], aimed at studying the aeroelasticity and the noise characteristics of the concept.
Currently, in the USA, a U.S. Armycoordinated program called TRAST is ongoing for developing a testbed for tiltrotor aeroelasticity investigations [2, 23, 55,56,57]. Another effort is taking place at the University of Maryland, where a reconfigurable windtunnel tiltrotor model is being developed [11, 50, 53] and recently underwent preliminary Whirl–Flutter stability tests [19, 51].
At about the same time, yet another effort is ongoing in Europe, within the CleanSky2 initiative, within the project ATTILA, to design, manufacture and test a tiltrotor windtunnel model in support of the NextGeneration Civil Tilt RotorTechnology Demonstrator (NGCTRTD), a fixedengine tilting rotor configuration. This paper illustrates the preliminary modeling efforts of the windtunnel testbed, as initially presented in [8, 9]. Preliminary studies were also conducted in [18]. Numerical models of the reference testbed have been developed using FLIGHTLAB^{Footnote 1} and MBDyn^{Footnote 2} [28], respectively, by collaborating research teams at the Royal Netherlands Aerospace Centre (NLR) and Politecnico di Milano. To the authors’ knowledge, a direct correlation of Whirl–Flutter results from the two solvers has not been published elsewhere.
These numerical models are developed in support of the design of the physical windtunnel testbed. Pending experimental characterization of the ATTILA testbed dynamics and Whirl–Flutter stability properties, codetocode verification results are here presented, compared with data obtained from a reference CAMRAD II model of the fullscale aircraft. In addition to componentlevel models verification, preliminary Whirl–Flutter predictions are presented. In this context, the Matrix Pencil Estimation (MPE) and Periodic Operational Modal Analysis (POMA) eigensolution identification methods are used and compared for the first time, to the authors’ knowledge.
Materials and methods
FLIGHTLAB
FLIGHTLAB is a stateoftheart multibody, componentbased, selective fidelity modeling and analysis software package. It supports modeling and simulation of rotorcraft, fixedwing aircraft, compound aircraft, helicopters, multicopters, drones, flying cars and experimental aircraft configurations. Rotorcraft and other aircraft models can be developed to fit their application with the desired level of fidelity. The numerical models can be used for engineering analysis, real time simulation, or both. The development system is also used to generate runtime models for real time applications. The key capabilities of the software include:

Multiple bodies, multibody dynamics

Nonlinear unsteady aerodynamics

Flight dynamics and real time simulation (including piloted, full flight simulators)

Flight performance, stability, controllability, and handling qualities

Aeroelastic stability, vibration, and loads

Aircraft systems analysis and hardwareintheloop (HIL) simulation

Coupling with external programs, including CFD and Matlab/Simulink
The Whirl–Flutter analysis is usually performed through direct linearization of the problem about a steady trim solution, using central difference perturbation of the first and secondorder degrees of freedom. This technique permits to identify the wingpylon, drive system and rotor modes, providing valuable information about the underlying dynamics.
A nonlinear transient analysis resulting from the application of appropriate excitations, replicating the expected windtunnel test procedure, has also been implemented to verify the results of the linearized stability analysis, investigate various experimental excitation strategies, and determine the required excitation magnitude and the associated structural loads.
MBDyn
MBDyn is a generalpurpose multibody solver, developed at the Aerospace Science and Technology department of Politecnico di Milano and distributed as free software.
MBDyn automatically writes and solves the equations of motion of a system of entities possessing degrees of freedom (nodes) connected through algebraic constraints and subjected to internal and external loads. Constraint equations are explicitly accounted for, following a redundant coordinate set approach. Thus, the resulting system of DifferentialAlgebraic Equations (DAE) takes the form
where \({\varvec{\mathbf {x}}}\) is the vector of the kinematic unknowns, \({\varvec{\mathbf {p}}}\) that of the momentum unknowns, \(\varvec{\lambda }\) that of the algebraic Lagrangian multipliers, \({\mathbf {M}}\) is a configuration and timedependent inertia matrix, \({\varvec{\mathbf {f}}}_i\), \({\varvec{\mathbf {f}}}_e\) are arbitrary internal and external forces, \(\varvec{\phi }({\varvec{\mathbf {x}}})\) is the vector of the (usually nonlinear) algebraic equations that express kinematic (holonomic) constraints, and \(\varvec{\phi }_{/\mathbf {x}}\) is the Jacobian matrix of the constraints with respect to the kinematic unknowns. Each node instantiates the corresponding balance equations (1b), while only nodes with associated inertia properties instantiate the related momenta definitions (1a). Additional states associated with scalar fields (namely, hydraulic pressure, temperature, electric potential) and thus the corresponding balance equations, can be taken into account through dedicated sets of nodes.
Elements are responsible for the contributions to the balance equations through (visco)elastic, internal forces \(\mathbf {f}_i\), possibly statedependent external force fields \(\mathbf {f}_e\) (e.g., aerodynamic forces), and reaction forces \({\varvec{\mathbf {f}}}_c = \varvec{\phi }_{/{\varvec{\mathbf {x}}}}^T \varvec{\lambda }\), introduced using the Lagrange multipliers \(\varvec{\lambda }\) and the Jacobian matrix of the algebraic constraint equations in Eq. (1c).
The DAE system can be integrated using several different A/L stable integration methods, among which is an original multistep method with tunable algorithmic dissipation, specifically designed for the class of problems usually solved with MBDyn.
The stability analysis of a nonlinear multibody model, formulated in an absolute reference frame, is critical since the problem may be timevariant. In principle, since in the configuration under analysis, the flow is essentially axial, this specific problem could be reduced to timeinvariant through the socalled multiblade coordinates, with the possibility to determine a steady solution, when asymptotically stable, for its linearization and analysis using methods for linear, timeinvariant (LTI) problems. However, owing to the flexibility of the wing and the opportunity to trim the aircraft at nonzero angles of attack, at the trim point the flow might not be perfectly axial; furthermore, aerodynamic interferences between the rotor and the wing may break the symmetry of the axial flow, introducing 1/rev periodicity in the motion of each blade, and \(N_b\)/rev periodicity in the overall response of the system.
It is thus convenient to use methods specifically designed to address problems of this kind. A typical approach, based on Floquet’s theory as introduced in the stability analysis of rotorcraft by Peters & Hohenemser in 1971 [38], evaluates the stability of linear, timeperiodic (LTP) problems. It requires the integration of the state transition matrix of the problem over a revolution, which is called the monodromy matrix; this is often impractical when using multibody solvers, which formulate the problem as a large number of nonlinear DAEs, as discussed in [4]. A generalization of Floquet’s approach to nonlinear, aperiodic problems, possibly subjected to random excitation, has been recently proposed in [52], based on Lyapunov’s theory of characteristic exponents (LCE); however, the applicability of this method to systems of DAE is still debated [27], and hardly practical from a computational viewpoint for large systems. An interesting approach, based on the proper orthogonal decomposition (POD) of snapshots of large sets of coordinates of the problem, taken during freeresponse transients at each rotor revolution has been proposed in [43] and successfully used with MBDyn. The equivalents of Floquet’s characteristic coefficients are obtained from the eigenanalysis of a matrix that corresponds to Floquet’s monodromy matrix, resulting in a leastsquares fitting of the snapshots.
All the previously mentioned methods can, to a different extent, evaluate the “contraction” characteristic of the free response of the system (e.g., the real part of the eigenvalues, when defined, or the characteristic coefficients of Floquet’s theory, or the characteristic exponents of Lyapunov’s theory), but some are unable to determine its periodicity, when the free response is oscillatory [4, 43, 52], while others suffer from limitations in the interpretation of the imaginary part of the eigenvalues of the monodromy matrix [38], although attempts have been made to overcome them (see for example [39]).
In this work, two somewhat complementary approaches are followed:

Matrix Pencil Estimation (MPE), used to retrieve the main wing modes;

Periodic Operational Modal analysis (POMA), used to take into account the periodicity of the system to extract not only the wing’s normal modes, which represent fundamental harmonics of the system’s free response, but also the corresponding superharmonics, namely the fundamental harmonic plus or minus integer multiples of the rotor angular velocity, resulting from the interaction with the rotor itself.
The workflow of the two methods is sketched in Fig. 1, from the MBDyn simulation results to frequency and damping, identified by both methods, and participation factors and mode shapes, only reconstructed by POMA. The details of each process are discussed in the following sections.
Matrix pencil estimation
The Matrix Pencil Estimation method (MPE) was designed by Hua et al. [20] to estimate parameters of exponentially damped or undamped sinusoids in the presence of noise. A multipleinput algorithm was subsequently proposed by Favale et al. [41], who applied the methodology also to more complex problems, such as tiltrotor Whirl–Flutter analysis. The method can estimate the modal parameters of a system from its free responses to external input.
Starting from a multibody simulation of the complete semispan tiltrotor windtunnel model, excited using a sinusoidal input at the swashplate, all the available strain values in the beam elements and accelerations of the nodes are extracted. A selection process is then performed to ensure reliable identification and select the most meaningful data. After simple data preprocessing, consisting of filtering and resampling, the identification algorithm is applied to compute the system poles. A further postprocessing step involving the creation of stabilization diagrams, as suggested in [41], is finally executed to evaluate the quality of the identification process and retrieve the desired results, such as frequency and damping ratio.
The MPE method can estimate the modal parameters of a system starting from its free response to external finite input. A general response signal \({\varvec{\mathbf {y}}}(t)\) will be composed of two contributions: the system free decay \({\varvec{\mathbf {x}}}(t)\) and an unknown noise \({\varvec{\mathbf {n}}}(t)\). In discretetime, the signal can be written as:
where N is the total number of samples obtained using a sampling frequency \(f=1/dt\). According to the above hypothesis, the response signal of the system in discrete time can be approximated as a sum of complex exponentials:
where M is the order of the system and \(z_{i}\) is equal to:
The freeresponse of the system can be described by its modal parameters: the complex amplitude \(R_{i}\) and poles \(s_{i}\), with frequencies \(\omega _{i}\) and damping ratios \(\zeta _{i}\). The aim is then to find these parameters from the recorded response of the system. Starting from the noiseless signal x(k) a rectangular Hankel matrix with dimensions \((NL)\times (L+1)\) can be created:
where L is called the pencil parameter, which should be smaller than \((N1)\). Two matrices will be then created, excluding the first and the last column of \({\varvec{\mathbf {X}}}\):
The matrices \({\mathbf {X}}_{1}\) and \({\mathbf {X}}_{2}\) can be rewritten as:
where the matrices \({\varvec{\mathbf {R}}}\), \({\varvec{\mathbf {Z}}}_{0}\), \({\varvec{\mathbf {Z}}}_{1}\), and \({\varvec{\mathbf {Z}}}_{2}\) are functions of the complex amplitudes \(R_{i}\) and system’s poles \(z_{i}\):
Consequently, from Eq. (7):
It can be demonstrated that M is the rank of the matrix pencil if \(M\le L \le NM\) and the diagonal elements of \({\mathbf {Z}}_{0}\) are the eigenvalues of the matrix pair \({\mathbf {X}}_{2}\) and \({\mathbf {X}}_{1}\). Finding the system’s poles reduces to solving the following eigenvalue problem:
The same procedure described here holds for signals disturbed by noise. Hua et al. [20] suggested that to obtain the best results the parameter L should be chosen as high as possible but limited between N/3 and N/2. After creating the block Hankel matrix of the collected responses, \({\mathbf {Y}}\), a fundamental step is to perform its singular value decomposition. Then, a new set of filtered matrices can be obtained by truncating the matrices at the first dominant values. From their analysis, it is possible to determine, in first approximation, the order of the system M: when using very high signaltonoise ratio data, the first M singular values can be of orders of magnitude greater than the others. However, when very noisy signals are considered, the singular values relative to effective system contribution do not significantly stand out among all the others. In this case, after having normalized the singular values, a possible approach is to compute their first derivative and identify the order corresponding to its maximum value. For simplicity, the previous discussion was performed considering a single input response. However, the algorithm can be easily extended to multiple input channels considering a matrix of responses \({\mathbf {y}}\) instead of a column array.
Periodic operational modal analysis (POMA)
Rotorcraft modal identification is typically performed in nonrotating coordinates by applying a multiblade coordinate transformation (MBC). However, when dealing with realworld problems, several practical issues may arise due to, e.g., small anisotropy of the rotor blades, or slightly different axial position of the sensors on each blade. Furthermore, when performing a periodic stability analysis, richer insight into the system behavior can be retrieved: for each harmonic contribution captured with the MBC transformation approach, a (theoretically) infinite number of sub and superharmonics is obtained, each with its participation factor.
The algorithm called Periodic Operational Modal Analysis (POMA), originally proposed by Wereley and Hall [54], is used in this work. It is based on the harmonic transfer function concept, namely the periodic counterpart of the frequency response function for LTI systems. Wereley defined a new fundamental signal space for periodic systems, containing the socalled exponentially modulated periodic signals (EMP). These have been defined as the complex Fourier series of a periodic signal of frequency \(\omega _p\), modulated by a complex exponential signal:
where \(s_n = s + nj\omega _p\), and \(u_n\) are the Fourier coefficients of u(t). The harmonic transfer function can be defined as:
The output can be expressed as:
where \(\Omega ={2\pi }/{T_\text {rev}}\) is the characteristic frequency of the system (i.e. the frequency of rotation of the rotor in the case of a tiltrotor), and \({\varvec{\mathbf {Y}}}(\omega )\) is the Fourier transform of \({\varvec{\mathbf {y}}}(t)\).
In theory, to consider all the harmonics that characterize a linear timeperiodic (LTP) system, \({\mathbf {G}}(\omega )\) should be an infinite matrix. However, for most applications, a satisfactory approximation can be obtained with (often quite) a limited number of harmonics. Wereley also showed the expression of the harmonic transfer function in terms of modal parameters of the state transition matrix:
where \({\mathbf {D}}\) is the direct transmission term. The terms \({\bar{\mathbf {C}}}_{r,l}\) and \({{\bar{\mathbf {B}}}}_{r,l}\) are the Fourier coefficients obtained from the expansion of the following matrices:
where \(\varvec{\psi }_{r}(t)\) and \(\varvec{\varrho }_{r}(t)^{T}\) are defined as:
and matrix \(\varvec{\varXi }(t)\) can be computed as the product of \({\mathbf {P}}(t)\) and a matrix \({\mathbf {V}}\) containing the eigenvectors of the Floquet [14] factor \({\mathbf {R}}\).
The power spectrum of the output \({\mathbf {S}}_{yy}(\omega )\) can be expressed as a function of the harmonic transfer function:
Note that in this case, \({\mathbf {S}}_{yy}(\omega )\) is the power spectrum of the output signals of the system which have been previously exponentially modulated to be able to apply the relation of Eq. (15).
Substituting Eq. (18) into (21) and setting \({\mathbf {D}}\) to zero:
where
\({\mathbf {W}}(\omega )\) is a function of the input spectrum and the input characteristics of the system. Equation (23) shows that the autospectrum of the output can be approximated by a sum of modal contributions if \({\mathbf {W}}(\omega )\) is reasonably flat at least for the band of interest of a specific mode. A further simplification can be introduced with the assumption of suitably separated modes reducing Eq. (22) to:
The expression in Eq. (24) can be compared to the one for the power spectrum of an LTI system:
Equations (25) and (24) show that the peak related to any superharmonic of a given mode can be viewed as the peak of a mode of a generic LTI system. Standard identification techniques can then be used to compute frequencies, damping factors, and modal shapes from the measured spectra.
Under the assumption that the input spectrum is relatively flat in the range of frequencies of the modes of interest, the ATTILA wingpylon model in MBDyn was excited by superimposing a white noise disturbance to all the components of the wind speed. The resulting strains and displacements were preprocessed to retrieve zeromean signals, which were then exponentially modulated. Finally, a Stochastic Subspace Identification (SSI) [7, 44] algorithm was applied to identify the frequency and damping from the strains to remove the rotor periodicity, and the mode shapes from the displacements. Using displacement signals allowed to extract mode shapes of the system and better visualize the system’s deformation. This result is not easy to achieve for a complex multibody system. However, the procedure can become increasingly computationally expensive when a huge number of degrees of freedom are considered. Additionally, to retrieve accurate and reasonable mode shapes, a selection of the input data for the algorithm is not always feasible, increasing the odds of encountering clusters of modes, also considering the possible presence of noisy data in the identification process. When performing the analysis using strain measurements is easier to isolate modes’ contributions, minimizing the possibility of overlapping modes by carefully selecting data among all available measurement channels. For the frequency and damping identification of wing, blade and yoke strain measurements were employed. Note that for the extraction of mode shapes and other modal parameters, both rotating and nonrotating measurements were used in the same identification process. As a consequence, the computed rotorrelated frequencies are expressed in a rotating reference frame.
In short, the steps of the identification algorithm are:

record the response \({\varvec{\mathbf {y}}}(t)\) of the system to a broadband input;

exponentially modulate the response \({\varvec{\mathbf {y}}}(t)\) to generate the signals \(\hat{{\varvec{\mathbf {y}}}}_n (t) = {\varvec{\mathbf {y}}}(t)\mathrm {e}^{in\omega _pt}\). The value of n is in the range \([(n_h1)/2, (n_h1)/2]\), where \(n_h\) is the maximum number of Fourier coefficients, equal to the number of harmonics, used to approximate the periodicity of the system;

compute the autospectrum of \(\hat{{\varvec{\mathbf {y}}}}(t)\) using a standard approach (e.g., Welch’s method);

apply classical identification techniques to extract the system modal parameters. In this work, a covariancedriven stochastic subspace identification (covSSI) algorithm has been applied.
Model description
The entire model can be divided into two main subcomponents: the wingpylon assembly and the rotor. A render of the ATTILA multibody model is showed in Fig. 2.
Wingpylon model
The wingpylon model has been developed to reproduce the fundamental frequencies and mode shapes of an equivalent finite element stick model tuned to match the fullscale aircraft dynamics at the rotor hub.
FLIGHTLAB
The FLIGHTLAB model contains 16 elastic beam segments for the wing and 4 beam segments for the nacelle. The wing airloads are modeled using an enhanced lifting line model with a PetersHe [37] finitestate dynamic inflow model.
Each structural beam segment is connected to a quasisteady airloads component, which uses 2D (AoA, Mach/Reynolds) lookup table data to calculate the airloads.
The connection between the wing and the nacelle is modeled by three torsional springdamper components, collocated and connected in series.
The shaft tilting hinge is modeled by a gimbal hinge, with a pitchyaw stiffness that can be changed at runtime to switch between the downstop ON and OFF configurations (see Figs. 3 and 4).
MBDyn
The wing model consists of 3 finite volume threenode beam elements [3, 16] for the stiffness part, and one body element for each node to model the inertial component. The nacelle part is divided into a tilting and a nontilting part, both modeled as rigid bodies.
The parts are connected with deformable joints, which represent the flexibility of downstop and wingpylon attachment. The aerodynamic loads are introduced through MBDyn’s aerodynamic beam elements, based on simple strip theory, each linked to the corresponding structural element.
Rotor model
The ATTILA proprotor is a threebladed stiffinplane rotor with a gimballed hub. It consists of the control chain, three blades, and the yoke.
FLIGHTLAB
In FLIGHTLAB, the blade is modeled using elastic beam components. The beam axis of each finite element is defined by the locus of shear centers (the elastic axis) of the physical blade. Appropriate sweep and droop rotations are applied to approximate the position of the elastic axis relative to the feathering axis.
The aerodynamics of the blade are modeled using lookup tables with a correction for unsteady circulatory effects. The transition between the airfoils is nonsmooth. Provisionally, the induced velocity is modeled as a uniform inflow with the Glauert distribution.
The blade is connected to the yoke at two locations, at the inner and outer bearings. This creates a dual load path. The FLIGHTLAB solver is single load pathbased, where calculations are performed sequentially from the blade tip towards the rotor hub. To facilitate the modeling of the dual load path resulting from the combination of blade and yoke, the root end of the blade is modeled as a separate beam, inverted and connected at the physical root via twoparent springs representing the inner bearing. In this setup, the root of the torque tube acts as a second “blade tip” as far as the solver is concerned.
The main “blade” load path consists of the yoke segments and the blade segments outboard from the outer bearing. The torque tube load path consists of the blade segments between the inner and outer bearings. The outer bearing is modeled as a series of three hinges with zero spring stiffness, allowing rotation around all three axes but constraining translation. Due to the single load path nature of the FLIGHTLAB solver, the inner bearing cannot be modeled as a hingeslide combination. Instead, two perpendicular rigid flap/lag offsets are located at the inner bearing, perpendicular to the yoke. The inboard end of the torque tube is then connected to the rigid offsets through two twoparent translational linear springdampers that only constrain translation. The springs have been assigned high stiffness to approximate a rigid constraint. Both the springs and the rigid offsets are identical in length. The length of the spring is arbitrary but large enough to minimize the spring restraint in the axial direction of the yoke in the presence of relative displacements/rotations.
As mentioned, there are two offset/spring combinations: one in the flapwise and one in the edgewise direction. In this way, translation of the inboard end of the torque tube is constrained to the yoke at the inner bearing location, in both the flapwise and edgewise directions. Due to the length of the springs and rigid offsets, the angle between them will be small, resulting in very small offaxis forces. As such, translation in the spanwise direction is nearly unobstructed, whereas rotation is free in all axes (see Fig. 5).
The control chain is modeled as a conventional swashplate arrangement. The rotor hub is rigidly connected to the tip of the pylon shaft when analyzing the halfwing model and rigidly connected to the inertial system when analyzing the isolated rotor. A bearing component drives the rotor rotation over the gimbal, which is connected to the yoke via an underslung offset.
The swashplate is located above the hub, connected via a rigid offset. The nonrotating swashplate node is connected to the pylon through a translational linear springdamper, representing the collective spring stiffness of the control chain. A collocated gimbal hinge with springdamper effects models the cyclic pitch spring stiffness.
The nonrotating swashplate node translates along the shaft axis, following the collective input through a controlled slider. Azimuthal rotation of the rotating swashplate node is achieved by a controlled hinge, which is slaved to the rotational speed of the hub. The swashplate mass is fixed to the rotating swashplate node and is required to avoid a singular matrix during linearization. Each pitch link is connected to the rotating swashplate and offset from the shaft through azimuthal rotation and rigid translation. Cyclic control inputs are introduced at the root end of the pitch links through a controlled slider. A twoparent linear springdamper represents the pitch link stiffness and is connected to the pitch horn on the blade. The pitch link spring does not constrain rotation.
MBDyn
The blade and the yoke are modeled in MBDyn using the threenode beam element, similar to the modeling of the wing (Fig. 6).
To describe the orthotropy of blade and yoke, the stiffness matrix of the beam sections incorporates the offsets and relative rotations between the feathering axis and the neutral and elastic axes.
The aerodynamic model is constructed using MBDyn’s aerodynamic beam. Each aerodynamic panel can incorporate aerodynamic twist variation and airfoil transitions. Along the blade, six airfoils are placed with a nonsmooth transition. The rotor aerodynamics are completed by a uniform inflow model, based on momentum theory with Glauert’s distribution and empirical corrections for tip loss and other standard airflow features. The resulting model represents a lowfidelity description of the rotor aerodynamics; however, in the tiltrotor aeromechanics literature, it is considered adequate for Whirl–Flutter. Future investigation will also consider midfidelity aerodynamics, involving the VortexParticle Method (VPM) presented in [47].
The blade is connected to the yoke in two locations, at the inner and outer bearing. In MBDyn, these two bearings are modeled with ideal rigid constraints: for the inner one, both flapwise and chordwise displacements are constrained, whereas for the outer, all three components of translation are constrained.
The control chain has a traditional helicopterlike configuration: it is formed by seven MBDyn nodes joined following the scheme of Fig. 7:

Pylon: this node physically connects the pylon extremity and the rotor; when the isolated rotor is analyzed, this node is clamped.

Airframe: this node receives the commanded collective and cyclic pitch controls. To decouple the two cyclic inputs, the node is positioned on a reference system that is rotated by the angle \(\psi _{sp} = \tan ^{1} (x_{sp}/y_{sp})\), where \(x_{sp}\) and \(y_{sp}\) are the locations of the pitch link attachment to the swashplate.

Fixed Swashplate: this node’s inplane displacement components and axial rotation are rigidly constrained to the airframe. To take into account the flexibility of the control chain, a collective spring, and two cyclic springs are positioned in between the airframe node and the fixed swashplate.

Rotating Swashplate: this node is connected to the fixed swashplate by a revolute hinge, and its axial rotation is constrained to the mast.

Engine: this node is connected to the mast by a torsional spring to reproduce the drivetrain dynamics.

Mast: this node transmits the rotation to the hub and the rotating swashplate. It is connected to the pylon node by a revolute hinge.

Hub: This node is constrained to the mast node by a spherical hinge and a gimbal rotation constraint: these two combined elements create an ideal constant velocity joint.
Results
Model validation
Wingpylon
To validate the wingpylon model, a modal analysis has been performed comparing the frequencies and mode shapes of both the MBDyn and FLIGHTLAB models with a target NASTRAN stick model.
In the NASTRAN model, the wing is modeled with 12 CBAR elements (twonode beam elements), whereas the nacelle is modeled with 4 CBAR elements. Lumped stiffness elements are used to join the two subcomponents. Compared with the NASTRAN model, the MBDyn model of the wing is discretized with a smaller number of beam elements (3, with 6 nodes vs. 12); instead, the FLIGHTLAB model is a onetoone translation of the NASTRAN model.
At this stage, the comparison of the mode shapes is based on the Modal Assurance Criterion (MAC) defined by Eq. (26). In this equation, \(\psi ^{\mathrm {FEM}}_i\) represents the ith mode shape calculated using the original finite element stick model, whereas the term \(\psi ^{\mathrm {MBD}}_j\) represents the jth mode shape obtained from the two multibody codes.
The MAC matrix is presented in Fig. 8. The closer the matrix is to identity, the closer the results are. The differences between the results obtained from MBDyn and FLIGHTLAB and those from NASTRAN appear to be negligible. Close inspection of the 6DOF mode shape at the location of the rotor hub confirms the rather satisfactory correlation.
Rotor
To validate the dynamic behavior of the rotor, its frequencies in vacuo (i.e., neglecting aerodynamic loads) at different collective pitch settings have been evaluated and compared to the fan plots obtained with an equivalent fullscale CAMRAD II rotor model.
Moreover, the rotating blade mode shapes have been compared for different collective angles and with and without the presence of the drive train system. Remarkably good agreement has been obtained between the three models, despite their relative complexity.
Trim procedure
The trim targets depend on the configuration being investigated but are identical for both FLIGHTLAB and MBDyn.
Poweron trim is based on prescribing the desired rpm and finding the collective and cyclic pitch controls that achieve a target thrust and zero cyclic gimbal flap angle, increasing the airstream speed until the maximum torque is reached. From that point on, while the airstream speed is increased further, a constant torque trim is maintained to represent a steady powered descent.
During poweroff trim, the rotor is declutched from the engine (i.e., no torque is transmitted), and the rotor speed is held constant by acting on the collective pitch while the airstream speed is increased.
In FLIGHTLAB, trim is achieved by a gradientbased iteration process. In poweroff trim, the requested rotor speed is frozen at the target value while the swashplate controls are adjusted to achieve an average rotor azimuthal acceleration equal to zero. Posttrim, the rotor speed degree of freedom is released and the collective is finetuned to achieve the desired steady poweroff rotor speed.
In MBDyn, poweron trim is achieved by starting from an initial guess of the collective angle and then applying the desired torque at the engine side. The desired rotor speed is maintained using a PI controller that sets the required swashplate collective as a function of the errors on the rotor angular velocity and its integral. The poweroff condition is achieved by simply setting the applied torque to zero.
Figure 9 presents a comparison of the trim results for the poweron configuration. The results are presented in normalized form due to company confidentiality reasons.
Whirl–Flutter prediction
Whirl–Flutter results mainly concern the lowfrequency modes that characterize the aeroelasticity of the model in the configuration that will be tested in the wind tunnel, that is the wing flatwise and chordwise bending modes (respectively, called “wing bending” and ”wing chord’ in the following), the ”wing torsion” mode, and the “pylonyaw” mode. The latter mode is characterized by substantial participation of a rotation of the pylonnacelle body relative to the wing about an axis loosely normal to the wing surface. This latter mode is particularly sensitive to the locking of the downstop mechanism. The focus is on evaluating the damping associated with those modes as the wind tunnel speed is increased and its sensitivity to several parameters of the problem, with particular attention on the prediction capabilities and accuracy of the algorithms considered in this work.
Figure 10 shows the Whirl–Flutter stability predictions for the poweron configuration with the downstop engaged. The figure compares the modes obtained in FLIGHTLAB through a direct linearization and those obtained by applying the Periodic Operational Modal Analysis to MBDyn’s results.
Figure 11 compares the damping factors computed by MPE and POMA, the two identification algorithms applied to MBDyn’s results, in the poweron configuration.
Figures 12 and 13 show analogous results, but now in the poweroff configuration.
POMA mode shapes
The stability analysis of a complex nonlinear system modeled using the multibody approach cannot be easily addressed using analytical methods based on a linearized expression of the governing equations. This problem has been previously addressed, in analogy to what is done here, extracting the characteristics of the system from numerical experiments. In [31], the stability properties of a complete tiltrotor multibody model were computed using the previously mentioned PODbased approach [43].
The method used in this work, however, has the benefit of extracting a prescribed number of periodic mode shapes starting from the main harmonic contribution in the collected data. The results will be presented independently for each identified wing mode. For each of them, the visualized mode shape and the participation factors computed performing the analysis using strain measurements are compared. This additional check allowed to verify the agreement of the results of the two analyses. Furthermore, after having performed this comparison, it was easier to identify the modes looking at the patterns of the extracted participation factors \(\phi _{j_n}\) that are calculated as:
where \(\psi _{j_n}\) is the \(jth\) eigenvector associated to the \(nth\) harmonics.
As an example, the wing bending mode is considered. The channels used in the identification process are the wing root out of plane bending moment and additional ones related to the rotor, the gimbal angle, and the blades out of plane bending moments, to also capture the rotor contribution in the superharmonics. Figure 14 shows the complex mode indicator function for the above listed channels for the powered condition at minimum speed.
The peaks associated with the wing bending mode and its relative superharmonics are clearly visible and distinguishable, assuring a highquality identification. The wing bending frequency, \(\omega _b\), is smaller than the rotor angular velocity, so the \(\omega _b\Omega\) superharmonic is reflected in the positive halfplane of the complex mode indicator function, resulting in the lower superharmonic in the Figure. Of particular interest is the response of the system at the identified superharmonics frequencies. Clearly, in the nonrotating frame, the wing subsystem is almost timeinvariant, while the measurements in the rotating frame, related to the rotor, exhibit a more periodic behavior dominating the two superharmonics. This is observed in the mode shapes identified using displacement measurements and by computing the participation factors of the identified eigenvectors using strain data.
As described in Fig. 15, it is evident that the wing contribution in the response is limited to the main harmonic; the participation of the wing bending moment to the eigenvectors is negligible in both superharmonics. Nevertheless, the rotor contribution is predominant and is mainly associated with a variation of the rotor gimbal angle. The bending of the wing and the flapping of the rotor dominate the overall response of this mode. The latter is induced by the transverse wind component the rotor experiences during wing outofplane motion.
Observing the mode shapes, depicted in Fig. 16, the behavior noticed from the computed participation factor is more easily visualized. In addition, the \(+\Omega\) superharmonic shows, not only a major contribution of gimbal mode, but also of inplane blade bending.
Discussion
In Fig. 10, the results from the two codes show similar trends for the wing bending mode (red) and the wing chord mode (blue). The FLIGHTLAB model predicts a lower damping value when considering the torsional mode (black) and pylonyaw mode (cyan).
The firsts three modes (wing bending, wing chord, and wing torsion) show a good match in Fig. 11. A nonnegligible difference is found for the pylon yaw mode. The damping curve predicted by the MPE method changes slope at a lower speed compared to the one obtained with the POMA method. This discrepancy may be due to the different types of excitation used to trigger the time responses analyzed by the two methods. Since the MBDyn model is nonlinear, what is interpreted as a modal response may differ when excited by turbulence rather than a deterministic excitation applied to the blade pitch through the swashplate.
The comparison of the Whirl–Flutter stability predictions for the poweroff configuration presented in Fig. 12 and obtained with MBDyn, identified using the POMA method, and FLIGHTLAB exhibits an overall good agreement. In this case, the wing bending mode identified in FLIGHTLAB shows a slope slightly steeper than that identified from MBDyn’s results, whereas the damping predicted for the pylon yaw mode is higher in the MBDyn analysis. The wing chord and torsion modes, instead, match almost perfectly.
The damping factors shown in Fig. 13, identified from MBDyn’s results with the two proposed algorithms, show an almost perfect agreement for the wing beam bending and the wing chord modes. When identified through the MPE method, the wing torsion mode does not show a change in slope. As in the poweron configuration, the pylonyaw mode shows different trends between the two identification methods.
Conclusions
Two multibody models of the NextGeneration Civil TiltRotorTechnology Demonstrator (NGCTRTD) windtunnel testbed, independently developed using FLIGHTLAB and MBDyn, are presented. Both models are in good agreement with a reference CAMRAD II rotor model and NASTRAN wingpylon stick model. After thorough validation of the aeromechanics predictions, aeroelastic characteristics of the experimental setup have been assessed utilizing three methods to identify its aeroelastic modes. The first method is based on state linearization as adopted in FLIGHTLAB. The Matrix Pencil Method and Periodic Operational Modal Analysis have been applied to time histories simulated using MBDyn. The results show good agreement for the wing beam bending, chord and torsion modes, with residual discrepancies for pylonyaw. They are attributed to residual differences in the underlying dynamics of the models rather than to the identification methodologies. The last method can easily extract the shapes of both wingpylon and rotor modes. Their visualization grants the possibility of a deeper understanding of the system behavior and instability mechanisms. The analyses confirm the absence of Whirl–Flutter instability in the airstream speed range of interest. The satisfactory agreement between the different approaches considered in this work gives us sufficient confidence in view of the continuation of the research towards the experimental verification of the aeroelastic stability of the system. Future work will address more accurate predictions using midfidelity aerodynamic models, model updating after preparatory and ground vibration testing of the windtunnel model, and correlation with test results when available.
Availability of data and material
Data used in this work may be requested to the ATTILA Consortium.
Notes
http://www.flightlab.com/, last accessed September 2021.
http://www.mbdyn.org/, last accessed September 2021.
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Funding
Open access funding provided by Politecnico di Milano within the CRUICARE Agreement. This research received funding from the Clean Sky 2 – H2020 Framework Program, under grant agreement N. 863418 (the ATTILA project).
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FLIGHTLAB is available from Advanced Rotorcraft Technology Inc., https://flightlab.com/. MBDyn is free software, available from https://mbdyn.org/.
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Cocco, A., Mazzetti, S., Masarati, P. et al. Numerical Whirl–Flutter analysis of a tiltrotor semispan wind tunnel model. CEAS Aeronaut J 13, 923–938 (2022). https://doi.org/10.1007/s13272022006052
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DOI: https://doi.org/10.1007/s13272022006052
Keywords
 Tiltrotor
 Windtunnel testing
 Whirl–Flutter
 Matrix pencil estimation
 Periodic operational modal analysis
 Aeroelasticity
 Multibody dynamics