State-Space Rotor Aeroelastic Modeling for Real-Time Helicopter Flight Simulation

Riccardo Gori1,a, Francesca Pausilli1, Marilena D. Pavel2 and Massimo Gennaretti1

1Università Roma Tre, Via della Vasca Navale 79, Roma, Italy

2TU Delft, Delft, Netherlands

ariccardo.gori@uniroma3.it

Keywords: Reduced-order model, Rotor aeroelasticity, State-space modeling, Flight simulation

Abstract. This paper introduces a new approach for the identification of linear state-space models

of dynamical systems of arbitrary complexity. The identification procedure is described and applied

for modeling aeroelastic response of helicopter main rotors. With the aim of developing a tool that

might be conveniently applied for real-time simulations of helicopter flight dynamics, the state-

space model considered is a reduced-order description of loads transmitted to the airframe due to

hub motion and blade pitch controls. In order to validate the proposed approach, loads from the

state-space, reduced-order model are compared with those predicted by the complete full-state,

nonlinear rotor model for prescribed helicopter maneuvers.

Introduction

This work deals with presentation and assessment of a novel procedure for the identification of

linear, state-space, reduced-order models (ROMs) representing aeroelastic response of helicopter

main rotors, with the aim of developing an accurate, as well as fast, helicopter model suitable for

real-time, flight dynamics simulations. It is of general validity and might be extended to

applications concerning nonlinear dynamical systems of arbitrary complexity.

Helicopter flight simulation is a challenging task that has been capturing the attention of industry

and academia since long time. Strong motivations come from the training community, with the

requirement of creating an illusion of reality for pilots to experience. The purpose is achieved when

suited fidelity of flight simulation is reached, which in turn requires real-time, accurate predictions.

Although traditional approach to flight simulation relies on empirical, test-based models, an

increasing interest has emerged on physics-based modeling, thanks to the significant development

of computational technologies [1, 2].

Main rotor aeroelastic behavior strongly affects the overall helicopter dynamics. It is the result

of coupling of complex phenomena; among these, it is worth mentioning the nonlinear dynamics of

the slender blade structure and the complex aerodynamic environment given by combination of

blade periodic motion and inflow induced by wake vorticity remaining in close proximity of the

rotor disk, or even coming in contact with it. Hence, accurate helicopter main rotor aeroelastic

response prediction is a nonlinear, time-periodic, coupled problem, for which real-time simulation is

a very hard task.

The rotor aeroelastic modeling proposed here aims at overcoming this limitation. The starting

point of the proposed model consists in the linear time invariant (LTI) state-space, reduced-order

modeling presented by some of the authors, through which it is possible to predict the complex

rotor aeroelastic behavior in the neighborhood of a given flight condition [3, 4]. This formulation

yields hub forces and moments generated by the rotor (and transmitted to the airframe) as functions

of both hub motion and rotor blade controls. However, it is not adequate for predicting rotor

aeroelastic responses within the whole flight envelope of interest for flight simulation purposes

(namely, when pilot controls and helicopter motion cannot be considered as small perturbations

with respect to a reference flight condition).

Advanced Materials Research Vol. 1016 (2014) pp 451-459

© (2014) Trans Tech Publications, Switzerland

doi:10.4028/www.scientific.net/AMR.1016.451

In order to overcome such limit of applicability, here it is proposed a methodology conceiving

the rotor state-space ROM as consisting of a database of models identified for a given finite set of

flight conditions within the flight envelope that might be of interest, to be interpolated during

helicopter flight dynamics simulations. Each element of the aeroelastic operator database is labeled

with a set of flight parameters that suitably characterize the rotor behavior, whose variation during a

simulated maneuver drives the change of the LTI, state-space rotor ROM to be applied.

Crucial issues affecting the possibility of simulating flight in real-time are the number of states

included to describe the rotor aeroelasticity and the numerical efficiency of the interpolation process.

Recent research work available in the literature has dealt with the problem of real-time operation

and robustness with respect to parameter changes of ROMs devoted to flight simulation of aircraft,

proposing novel numerically efficient ROM adaptation methods that rely on pre-computed ROM

databases [5-7].

Next section gives a description of the methodology applied for identification of the LTI, state-

space models collected in the aeroelastic operator database, as well as, an outline of the approach

suggested for model interpolation at arbitrary flight conditions. Then, results of a numerical

investigation aiming at validating the proposed model for real-time simulation purposes are

presented. They include the prediction of the aeroelastic response of the main rotor of the Bo-105

lightweight helicopter during a decelerated descent maneuver.

Model description

The proposed aeroelastic rotor modeling for unsteady maneuvering simulation is inspired to the LTI,

state-space, ROM presented by some of the authors in Ref. [4]. It is a perturbation model relating

hub forces and moments, Hf , generated by the rotor to hub motion and rotor blade controls, x ,

through the following differential form

H H0 2 1 0= + δ + δ + δ +

= + δ

f f A x A x A x Cr

r Ar B x (1)

where x denotes hub motion and blade control perturbation with respect to a reference trim flight

condition state 0x , at which hub loads 0Hf hold. In addition, , , , ,2 1 0A A A B C are real, fully

populated matrices, whereas A is a block diagonal matrix of the poles related to the (internal)

dynamics of the additional states, r [3, 4]. In our problem, these additional states represent

dynamics of rotor blade degrees of freedom and of wake inflow on rotor disk. For a given trim

flight condition, Eq. 1 is derived starting from a suited sequence of rotor aeroelastic responses to

periodic perturbations within a prescribed range of frequencies, indicated by the characteristic

dynamics of the problem under examination [4]. The model identification process is independent on

the aeroelastic solver applied, but the accuracy of the identified model depends on it.

In order to apply the described aeroelastic rotor model for the simulation of an arbitrary

helicopter maneuver, as the operating conditions change along the trajectory, it is necessary to apply

stepwise updates of the matrix coefficients appearing in Eq. 1. The frequency of these updates

depends on the rate of variation of flight conditions. In principle, this can be obtained by an

interpolation procedure within a suited database that provides instantaneous values of matrices in

Eq. 1, for specific rotor flight parameters occurring during the unsteady maneuver. The database can

be generated through evaluation of matrices in Eq. 1 for a given set of operating trim conditions

(characterized by appropriate rotor flight parameters like advance ratio, thrust coefficient and tip-

path-plane angle, for instance) within the flight envelope.

Accurate and regular interpolation of differential model in Eq. 1 is not a straightforward task. As

explained in Appendix A, matrices , , , , ,2 1 0A A A A B C identified following the procedure

presented in Refs. [3, 4], might provide a discontinuous database, thus yielding inaccurate

interpolations. For this reason, a new model identification approach is developed with the aim of

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obtaining a smooth variation of model matrices within the range of operating condition to be

considered.

Akin to the modeling procedure presented in Refs. [3, 4], the proposed approach consists in: first,

the frequency-response functions, kH , involved in the aeroelastic model are sampled by

application of a time-marching aeroelastic solver for a given set of frequencies and operating

conditions within the flight envelope of interest, and then the following optimization problem is

solved

2

, ,

ˆ mink i j k i j

i j k

J H p , H p , (2)

In the above equation, ip denotes the set of rotor flight parameters considered, j is the set of

harmonic response sampling frequencies, whereas ˆkH denotes the approximated form of the k-th

transfer function. Coherently with the differential form of Eq. 1, all transfer functions involved in

the aeroelastic operator are approximated through the following matrix compact form

12ˆ ,i 2 1 0p = + j + j

H A A A C I A B (3)

with the matrices appearing in Eq. 3 that depend on flight parameters, ip . This dependence is

analytically approximated through linear combinations of b-spline functions [8]. The advantage in

using these interpolating functions lies in the possibility of adopting optimal sets of input data

locations in the space of parameters, that allow best fitting of matrix coefficients variation (for

instance, in our kind problems, input data are conveniently concentrated near hovering flight

condition, where the gradient of transfer functions is expected to be relevant).

The coefficients of b-splines combinations expressing matrix entries are the variables to be

identified by solution of the optimization problem in Eq. 2. The total number of them is usually of

the order of thousands. For this reason, the minimization problem is formulated as a nonlinear least-

square problem with separable variables [9], solved by means of a gradient based method (see

Appendix A for further details).

Numerical results

The numerical investigation carried out in this work has two main objectives: (i) validation of the

state-space rotor aeroelastic model proposed, and (ii) examination of its capability to predict loads

transmitted to the airframe by a helicopter main rotor during maneuvers. To these purposes, a

helicopter configuration representative of the lightweight Bo-105, with a four-bladed, hingeless

main rotor, is considered.

The state-space, aeroelastic ROM is identified by application of the interpolation procedure

described above within the flight envelope defined by flight parameters 0 ,80V kn kn and

21 ,3sh ; 25 transfer function samples (input data locations) are used.

Harmonic responses are evaluated through application of a rotor aeroelastic tool. Rotor blades

structural dynamics is simulated by a nonlinear, bending-torsion, beam-like formulation derived

from that presented in Ref. [10], valid for straight, slender, homogenous, isotropic, non-uniform,

twisted blades, undergoing moderate displacements [11, 12]. Distributed aerodynamic loads are

evaluated through the quasi-steady approximation of the Greenberg sectional theory [13], with

three-dimensional effects from wake vorticity taken into account through wake inflow given by the

boundary integral approach for potential flows presented in Ref. [14]. The set of aeroelastic integro-

partial differential equations derived by coupling blade structural dynamics equations with

aerodynamic loads modeling are spatially integrated through the Galerkin approach [11, 15].

Validation of state-space ROM identification process. First, the accuracy of the presented

methodology in identifying the LTI, finite-state, main rotor representation for operational conditions

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included in the considered flight parameters domain is examined. Figure 1 presents examples

concerning the effect of flight parameters, V and sh , on aeroelastic transfer functions amplitude.

In particular, in Fig. 1a it is shown the amplitude of the transfer function relating thrust force to

pitching angular velocity, for three different flight speeds, whereas Fig. 1b depicts the amplitude

of the transfer function relating rolling moment to blade collective pitch control, for three values of

the rotor shaft angle of attack. For all examined cases, the results obtained by ROM solutions are

compared with those directly identified through the aeroelastic numerical tool. These results reveal

that the considered transfer functions present a remarkable sensitivity to flight velocity, whereas

they are less affected by shaft angle variations. The influence of both velocity and shaft angle is

non-uniformly dependent on frequency. Further, these figures demonstrate the high level of

accuracy of the interpolation process proposed.

In order to appreciate the sensitivity of identified model coefficients on flight parameters

variation, Fig. 2 shows the location on the complex plane of the aeroelastic poles (eigenvalues of

matrix A in Eqs. 1 and 3), for forward flight ranging from 0 kn to 80 kn (Fig. 2a), and shaft angle

changing from 21 to 3 (Fig. 2b). It is worth noting that damping of poles is particularly

affected by flight parameters variation, while their frequency remains almost unchanged.

[a] thrust .vs pitching rate [b] roll moment .vs collective pitch

Fig. 1 Transfer functions variation with respect to forward velocity and shaft angle. Sampled vs

interpolated amplitude frequency distribution.

[a] flight velocity effect [b] shaft angle effect

Fig. 2 Influence of flight parameters on poles of identified aeroelastic ROM.

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Then, Fig. 3 shows the comparison between transfer function identified through the proposed

ROM and those obtained with the complete, full-state, nonlinear model, for a flight condition not

included in the database set. In particular, Fig. 3a depicts the transfer function between hub roll

moment and lateral cyclic pitch control, for forward flight velocity V = 30kn, whereas Fig. 3b

presents the transfer function between thrust and collective pitch, for advancing velocity V = 50kn

(in both cases, 7.5sh ). The very good correlation shown in Fig. 3 between interpolated

transfer functions and results directly given by the aeroelastic solver is a prove of the capability of

the presented ROM to accurately estimate system aeroelastic response for arbitrary flight conditions

inside the flight envelope of interest, which is an essential factor for the successful application of

the proposed method in flight simulations.

Application of ROM to maneuvering flight. Next, the performance of the proposed main rotor

model in predicting hub loads during maneuvering flights is assessed. First, it is applied to provide

hub loads response to a periodic cyclic lateral pitch (with 1 rad/s frequency and 1-degree amplitude)

and associated airframe motion (helicopter flight dynamics is perturbed from steady rectilinear,

uniform, level flight with V = 40kn). In Fig. 4 state-space, ROM predictions are compared with

those given by the nonlinear, time-marching solution (NLTM) provided by the rotor aeroelastic tool

also used for evaluation of harmonic responses needed to identify the model. More relevant hub

forces and moments from ROM simulation appear to be in good agreement with those determined

by NLTM analysis, whereas some limited discrepancies are observed between (small-amplitude)

lateral forces.

[a] thrust .vs collective pitch, V=50kn [b] roll moment .vs lat cyclic pitch, V=30kn

Fig. 3 Comparison between interpolated and sampled transfer functions for flight conditions not

included in the database.

Finally, the state-space, ROM is applied to evaluate hub forces and moments generated by main

rotor during a more complex unsteady Bo-105 maneuver. Starting from a steady level flight

condition, the helicopter model follows a descent trajectory, initially at constant velocity and then

operating a decelerated flight (Fig. 5 depicts time histories of forward velocity, V , and shaft

angle, sh ). Corresponding hub motion and blade pitch controls, previously determined through a

flight dynamics solver with lower-order rotor aeroelastic modeling, are used as input in this

validation analysis. The capability of the proposed state-space ROM to accurately predict hub loads

arising in unsteady helicopter maneuvering is assessed by comparison with loads given by NLTM

solution. The corresponding results are presented in Fig. 6, which shows correlations of time

histories of hub force and moment components (Figs. 6a and 6b, respectively) during the same time

interval of analysis considered in Fig. 1 to illustrate V and sh evolutions. These results

demonstrate that the proposed rotor LTI, state-space ROM is able to yield aeroelastic responses in

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good agreement with those provided by complex, full-state, nonlinear aeroelastic tools, and thus it

seems to be a good candidate for application in helicopter flight simulators, for which maneuver

high-performance computations are required.

Concluding remarks

Development and accuracy assessment of a helicopter main rotor aeroelastic ROM aimed at real-

time maneuvering flight simulations have been presented. It is based on a LTI, state-space matrix

representation of hub loads generated by hub motion and pilot blade pitch controls, combined with a

suited optimization process that allows the identification of model matrices for any flight condition

within the flight envelope of interest, starting from a pre-defined matrix database. The numerical

investigation has demonstrated the capability of the modeling process presented to identify with an

excellent level of accuracy main rotor aeroelastic transfer functions, in any operational condition

within the flight envelope considered. Moreover, the application of the proposed state-space ROM

for evaluating main rotor loads transmitted to the airframe during arbitrary unsteady maneuvers has

provided results in good agreement with those predicted directly by the complex, full-state,

nonlinear, time-marching aeroelastic solver. Although limited to a couple of arbitrary maneuvers,

these outcomes indicate that the presented rotor model seems to be suited for application in

helicopter flight simulators. However, the achievable computational performance of the

methodology needs further investigation, in order to examine its actual capability of real-time

predictions.

[a] hub forces [b] hub moments

Fig. 4 Response to periodic lateral cyclic pitch. ROM vs NLTM.

[a] advance ratio [b] shaft angle

Fig. 5 Time histories of flight parameters during maneuver.

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[a] hub forces [b] hub moments

Fig. 6 Time histories of hub loads during maneuver flight. ROM vs NLTM.

Appendix A

The proposed helicopter main rotor aeroelastic state-space ROM is based on the evaluation of

harmonic responses within a prescribed range of frequencies and flight parameters (that characterize

the database). Many approaches have been proposed in the literature to identify the model matrix

coefficients in Eqs. 1 and 3 that, for a single flight condition, provide the best approximation of

sampled transfer functions (see, for instance, Ref. [16]). However, when a further interpolation

among the set of flight conditions considered in the database is required, it is found that the

sequential interpolation procedure (i.e., interpolation of ROMs at several flight conditions,

following frequency response approximations) yields inaccurate results. This phenomenon can be

explained observing that ROM coefficients should be considered as defined in a confidence interval,

because of possible inaccuracies arising in the process leading to their definition. Coefficient

uncertainty inside this interval has a negligible effect on the overall quality of ROM, for a single

flight condition [3, 4]. However if the estimate of ROM coefficients is sought for an arbitrary flight

condition starting from those evaluated for a given set of flight conditions, the interpolated solution

is affected by uncertainties present in the database. Moreover, matrices B and C are not uniquely

defined, and hence if they are identified by independent frequency-domain approximations, their

variation with respect to flight parameters may be not regular and consequently accurate

interpolation is not achievable.

To solve these critical issues, the presented approach proposes to formulate the ROM

approximation problem as a unique optimization problem, where the variables to be identified are

the b-spline coefficients through which the variation of model matrices with flight parameters is

defined [8]. In other words, a nonlinear regression is carried out in a space defined by the frequency

range of frequency responses available and by the range of flight parameters considered,

considering the transfer functions approximation given by Eq. 3, with model coefficients variation

with respect to flight parameters expressed through combination of splines.

A fundamental condition to be satisfied to obtain accurate model interpolations is that the

number of variables of the unique problem is strictly less than the sum of the number of variables of

single approximations that would be introduced in the sequential interpolation problem. If this

condition were not met, then the optimal solution of the unique problem would coincide with the

solution given by the sequential solver. Reducing the variables space is a mean to impose a smooth

variation of the coefficients, while solving the issues related to the not uniqueness of B and C

matrices.

Advanced Materials Research Vol. 1016 457

Of course, the solution of an optimization problem comprising all interpolation variables at the

same time poses a bigger numerical challenge, with respect to solving a set of separate smaller

interpolation problems. For this reason, a dedicated minimization procedure is developed, which

improves the convergence speed and the numerical stability of the method. This procedure consists

in separating the variables the objective function, J, depends linearly on, from those it depends

nonlinearly on [9]. Then, the linear problem is solved using a linear least square approach, whereas

the nonlinear least square problem is solved using a gradient based quasi-Newton method [17].

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