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SIMULATION MODELING OF UNSTEADY MANEUVERS USING A TIME ACCURATE FREE WAKE Maria Ribera∗

Roberto Celi†

Department of Aerospace Engineering University of Maryland College Park, Maryland

Abstract The transient response of a helicopter to unsteady maneuvers using a time accurate free wake model coupled with a comprehensive flight dynamics simulation model was investigated. The simulation model is a coupled rotor-fuselage model with flexible blade modelling in flap, lag and torsion. The free wake model uses a time marching scheme to solve the governing wake equations, and it is capable of predicting unsteady maneuvers without any assumptions being made on the wake geometry. Two maneuvers were considered, an arrested descent and a roll reversal maneuver. For both, the results obtained with the time marching free wake model were compared to those obtained with dynamic inflow and a relaxation free wake model. The results of the simulation show that the coupled flight dynamics-time marching free wake model can effectively predict the response of the helicopter to unsteady maneuvers. The time marching wake model compares positively with the dynamic inflow and the relaxation wake models, with the most siginificant differences occurring around peaks of maximum amplitude of the applied controls. Since the time marching free wake model can calculate the inflow accurately to all the changes in the controls as soon as they happen, it can better predict the behavior of the helicopter at the times when drastic changes in the input occur, while the relaxation wake, which is only updated once per revolution, cannot. In addition, the time marching wake ge∗

Doctoral Candidate, Alfred Gessow Rotorcraft Center, e-mail: [email protected]. † Professor, Alfred Gessow Rotorcraft Center, email: [email protected]. Presented at the 60th Annual Forum of the American Helicopter Society, Baltimore, MD, June 7-10, 2004. c 2004 by M. Ribera and R. Celi. PubCopyright
lished by the American Helicopter Society, Inc., with permission.

ometry accounts for all the maneuver transients that are lost when using a relaxation type wake model, which is converged once per revolution as if the instantaneous situation was a steady-state flight condition. The time marching procedure is not only more rigorously correct than the relaxation free wake implementation, but also computationally more efficient.

Nomenclature Cl Cl M 2 CT L M M Nb p, q r r t u, v, w u V(r) y y˙ Z β(ψ) β1c , β1s Γ ∆ψ ∆ζ ζ θ θ0

Lift coefficient Elemental lift Thrust coefficient Rolling moment, lb-ft Pitching moment, lb-ft Mach number Number of blades Roll and pitch rate of the helicopter, deg/sec Blade radial station, ft Position vector of a point on the filament time, sec Velocity components on body axes, ft/sec Vector of controls Total velocity at a point r on the filament Vector of states Vector of state derivatives Vertical force, lbs Flap angle; flap distribution, deg cosine and sine of the first flapping mode, rad Circulation Wake azimuth resolution, deg Vortex filament discretization resolution, deg Age of the vortex filament, deg Pitch attitude, deg Collective control input, in

θ1c µ σ φ ψ Ω

Lateral control input, in Advance ratio Rotor solidity Bank angle, deg Blade azimuth angle, deg Rotor speed, rad/sec Introduction

Recent flight dynamics simulations models have achieved levels of sophistication that allow for increasingly more accurate predictions in a broader range of flight conditions. Such models have demonstrated their utility by helping design better and more maneuverable helicopters and reliable flight control systems. An important aspect of the flight dynamics simulation is the type of aerodynamics model used to predict the main rotor induced velocities. The existing flight dynamics simulation models, both in industry and in academia, utilize a wide spectrum of aerodynamic models to provide the main rotor inflow. The level of sophistication of the aerodynamics in flight dynamic simulation models has increased rapidly in the last decade, especially for nonreal-time, research type simulation models. One of the drivers in the research community has been the problem of predicting the off-axis response (e.g., the pitch response to lateral cyclic). The problem remained unresolved until Rosen and Isser [1] demonstrated the role played by the distortion of the wake geometry due to the maneuver using a prescribed wake approach. Since then, several approaches have been followed to account for maneuvering effects in the calculation of the main rotor inflow distribution, though most use a dynamic inflow type model. Basset and Tchen-Fo [2] developed a set of coefficients using a dynamics vortex model to couple the angular rates to the inflow distribution, and used them in a dynamic inflow model. The popular Pitt-Peters dynamic inflow model has also been extended to capture the dynamic distortion of the wake due to maneuver and transition flight. The work by Prasad, Peters et al. [3, 4], where the appropriate time constants are extracted from a vortex tube analysis, is one of the latest developments of the model. Simple but accurate state-space inflow models suitable for maneuver analyses have also been extracted using frequency domain system identification, either from experimental [5] or simulation [6,7] data. A free vortex wake model to account for the effects of maneuvering from first principles, has been developed by Bagai and Leishman [8]. Note that, in the context of the present paper, “from first principles” is limited to meaning that no a priori assump-

tions are made on the distortion of the wake geometry caused by the maneuver, that is, the wake geometry is free to evolve from the time history of the motion of the rotor blades, and that no empirical or semiempirical correction coefficients are used in the definition of the rotor inflow or wake geometry. The model of Ref. [8] is based on the solution of the vorticity transport equations, and uses a relaxation technique to solve numerically the governing equations. Therefore, it is rigorously appropriate only for a steady trimmed flight condition, and not for transient conditions. The limitations of the relaxation wake have been subsequently removed in the time-accurate free wake model of Bhagwat and Leishman [9], which is therefore suitable for analyzing unsteady maneuvers of arbitrary amplitude. Another free wake capable of modeling maneuvering flight has been developed by Wachspress et al. [10], as part of the simulation model CHARM, and successfully applied to the analysis of flight in vortex ring state and to the prediction of off-axis response. This wake has been coupled with the Sikorsky GenHel simulation code, and used for the modeling of free wake-empennage aerodynamic interaction, with improvements in the correlation with flight test data [11]. The wake has also been coupled with the NASA version of the GenHel code [12], giving results similar to those of the Pitt-Peters dynamic inflow model. The same simulation code has also been used to obtain aircraft and blade motion data for wake dynamics and acoustics studies [13,14]. In this case, aircraft and blade motion data were provided as input to the time-accurate free wake model of Ref. [9] in an “open loop” fashion, as the NASA GenHel model is based on linear inflow, and rigid flap and lag motion with an empirical correction for torsion [15]. The relaxation wake of Ref. [8] was coupled with a flight dynamic simulation that included a flexible blade model, and the resulting model was used to study the dynamics of the Eurocopter BO-105, with a special focus on the prediction of the off-axis response [16]. One of the conclusions of the study was that an accurate prediction of the off-axis response from first principles was indeed possible, but it required the simultaneous modeling of: (i) blade flexibility in flap, lag, and torsion, and (ii) distortion of the wake geometry caused by the maneuver. The off-axis predictions with the correct wake distortions but with elastic flap only, and those with fully coupled flap-lag-torsion but with a conventional free wake without maneuver distortion, exhibited the traditional “wrong direction” error [16]. On the other hand, accurate predictions of the on-axis response did not require such a sophisticated modeling. Although

the relaxation free wake was not rigorously applicable to transient maneuvers, it performed well, probably because the flight test maneuvers were of small amplitude. This brief overview of flight dynamic simulation models with refined wake aerodynamics does not include most of the comprehensive analysis codes used by industry and government laboratories. Many of these codes include, or are being upgraded to include, maneuvering free wakes, and therefore the capability to support research studies in flight dynamics. They are not included here only because no published references are currently available that describe in detail the incorporation of the wake in the simulation model, or the use of the refined wake in flight dynamic studies, or both. For example, Ref. [17] describes the application of the comprehensive analysis code RCAS to several examples, including a large amplitude roll maneuver. RCAS includes the same time-accurate maneuver free wake used in the present study, but a much simpler aerodynamic model is used for the specific roll maneuver example. This paper describes a new flight dynamic simulation model, obtained by coupling the time-accurate free wake model of Bhagwat and Leishman [9], with the same rotor-fuselage model of Ref. [16], provides a simulation model suitable for analyzing maneuvers of arbitrary amplitude. The initial results obtained with the new model focus on the fundamental understanding of the dynamic behavior of the helicopter in unsteady maneuvers. Such understanding is currently incomplete, primarily because of the cost and difficulty of obtaining accurate flight test data, and because of the complexity of deriving appropriate simulation models. These models should be able to capture all the key physics of the problem, and should be built as much as possible from first principles. Understanding the basic physics is important to increase maneuverability, reduce maneuver loads, and achieve favorable handling qualities. More specifically, the primary objectives of the present paper are: 1. To describe the steps required to couple the timeaccurate maneuvering free wake of Ref. [9] and the rotor-fuselage model of Ref. [16], and the implementation and numerical issues associated with the calculation of trim and of the free flight response to arbitrary pilot inputs; and 2. To present the results obtained using the new simulation model to study the dynamic behavior of a helicopter in two large amplitude maneuvers, namely, an arrested descent maneuver, and

a roll reversal maneuver. This will be accomplished by analyzing in detail the transients of aerodynamic, structural, and inertia loads during the maneuver, and comparing the predictions obtained using the time-accurate free wake model with those obtained with the relaxation free wake and with dynamic inflow models. Mathematical Model Flight Dynamics Model The flight dynamics model used in the present study [18, 16] is based on a system of coupled nonlinear rotor-fuselage equations in first-order, state-space form. It models the rigid body dynamics of the helicopter with non-linear Euler equations. The aerodynamic characteristics of the fuselage and empennage are included in the form of look-up tables as a function of angle of attack and sideslip. The dynamics of the rotor blades are modeled with coupled elastic flap, lag and torsion, a finite element discretization and a modal coordinate transformation to reduce the number of degrees of freedom. There is no limitation on the magnitudes of the hub motions. In particular, the effects of large rigid body motions on the structural, inertia, and aerodynamic loads acting on the flexible blades are rigorously taken into account. The first step of the solution of the calculation of the free flight response to pilot inputs is the calculation of a trim condition. The trim procedure is described in detail in Refs. [19] and [20], in which a dynamic inflow model is used to obtain the induced velocities. Ref. [18] includes a relaxation free wake model to determine the inflow. The governing ODEs are formulated implicitly in the form f (y, ˙ y, u; t) = 0 (1) (as seen in Ref. [21]), and are solved with DASSL, a variable step, variable order method in which the derivatives are approximated by backward differentiation formulae [22]. Maneuvering Free Wake Model Two free vortex or free wake models are used in the present paper, which differ in the solution technique. The first is the Bagai-Leishman free wake model, described in Ref. [23], which uses an iterative or relaxation solution method. The second, the Bhagwat-Leishman time accurate free wake model [9], solves the governing equations with a time marching scheme. The main characteristic of free vortex models for the flight-dynamicist is that they can model the distortions of the wake geometry due to maneuvers. Free vortex methods model the flow field using vortex filaments that are released along the span of the

blade, although for the present study a single tip vortex is considered. The behavior of the vortex filaments is described by a convection equation of the form dr (ψ, ζ) = V [r (ψ, ζ)] (2) dt which for a rotor can be written as ∂r (ψ, ζ) ∂r (ψ, ζ) 1 + = V [r (ψ, ζ)] ∂ψ ∂ζ Ω

(3)

where r(ψ, ζ) defines the position of a point on the vortex filament and V(r(ψ, ζ)) is the local velocity at that point. The wake geometry is discretized in two domains, ψ and ζ. The first represents the time component and is obtained by dividing the rotor azimuth domain into a number of angular steps of size ∆ψ. The second represents the age of the vortex filament, which is discretized into a number of straight line vortex segments of size ∆ζ. The wake discretization is shown schematically in Fig. 1. The right hand side (RHS) of Eq. (3) is determined by the addition of the freestream velocity, the velocities induced by all the other vortex filaments and the blades, plus other external velocities such as those due to maneuvering. The induced velocity is the most complicated and expensive term to compute. Biot-Savart law is used to calculate the induced contribution of each vortex segment at any point in the wake. The discretization of the left hand side (LHS) of Eq. (3), the integration scheme and how it is implemented depends upon the type of free wake model used. Relaxation Free Wake The Bagai-Leishman model discretizes the LHS of the governing equation in both the ψ and ζ domains. For both, it uses a five-point central difference approximation, which is second order accurate in both ψ and ζ. The integration method is a pseudo-implicit predictor-corrector (PIPC) scheme, with some form of numerical relaxation. Relaxation methods enforce periodicity, with the consequent limitation of being applicable only for steady-state flight conditions. However, they are usually free of the numerical problems that affect time marching methods and converge rapidly. Time Marching Wake As for the relaxation model, a five-point central difference scheme is used for the spatial derivative, i.e., in the ζ direction. For the time derivative in the ψ direction, however, a predictor-corrector with second order backward (PC2B) scheme is used, which is also second order accurate. Although this method is potentially subject to numerical instabilities, it is not restricted by the flight condition. Because time marching methods do not need to enforce any type

of boundary condition, they are suitable for transient conditions in which periodicity can not be enforced, and therefore relaxation methods can not be used rigorously. Such flight conditions include maneuvering flight or operations in descending flight in vortex-ring state. Coupling of the time marching free wake and the rotor-fuselage model For both free wake models, the flight dynamics simulation must provide the following [16]: 1. The bound circulation distribution, Γ(ψ, ζ). 2. The equivalent rigid blade flapping angles, β(ψ). 3. The hub linear u, v, w, p, q.

and

angular

velocities,

The free wake model then provides the inflow distribution for that particular condition. For the flight dynamics model and either free wake model to interact successfully, several other details need to be taken into account. For example, the two models use different coordinate systems, and proper transformation in both directions is necessary [18]. In addition, the different rotor distribution values may be computed at different values of the azimuth for the wake and the flight dynamic model, and adequate interpolation needs to be performed. A difficulty in coupling the time-accurate wake model to the rest of the flight dynamics model arises from the nature of the different parts of the simulation, i.e., the trim calculation and the time integration. Because trim is a steady-state condition, it is best analyzed using a relaxation wake model such as that of Ref. [23]. On the other hand, for a time marching integration a time accurate scheme such as that of Ref. [9] is more appropriate. This section discusses the transition between the two methods, necessary when starting a free flight simulation after a trim calculation. The time accurate wake model needs to start from a geometry free of transients, to accurately predict the inflow needed for the subsequent integration of the equations of motion of the helicopter. This is necessary because the transients are numerical, and not physical. A trimmed wake geometry obtained with a relaxation scheme is a good initial solution, but in practice some transients will still be present when the time integration starts. The reason is that the trim procedure enforces periodicity of all the states only approximately [16], therefore the result is a flight condition that is very close to, but not exactly in, a

trim condition. This is sufficient to trigger numerical transients that negatively affect the accuracy of the solution because of a sort of feedback effect with the rest of the coupled rotor-fuselage model. For this reason, an intermediate “convergence” phase is introduced between the end of trim and the start of the time integration. During this phase, the state vector and the controls are fixed at the trim values, and the time accurate wake model is exercised until the resulting wake geometry is free of numerical transients. After that, the integration process can be started with the frozen trim values and the converged wake geometry as initial conditions. A block diagram describing the overall coupling between the time accurate wake and the rest of the simulation model is shown in Fig. 2. The process starts with the calculation of the trim state with a relaxation free wake model. The trimmed wake geometry and inflow are used as the initial solution for the time marching wake convergence process, in which the time marching free wake is iterated for a prescribed number of revolutions to obtain a converged wake geometry. The geometry includes the distortions caused by the maneuver if the helicopter is in a steady turn, and convergence is obtained when the numerical transients have disappeared. With this geometry and the trim states and controls, the time marching simulation starts with the computation of the inflow for the first time step. The following time steps of the integration are carried out using a loose coupling between time accurate wake and the rest of the simulation. The ODE solver used for the solution can be used only once, thereby returning the solution at the end of the integration, or inside a loop, in which the solution is returned at user-specified intervals [22]. The second approach is used for this study, with the solution returned every 10 degrees of azimuth (however, the variable-step ODE solver is free to further subdivide this azimuth step to achieve the required accuracy). At every 10-degree time step, the inflow computed by the free wake model is held fixed, while the ODE solver advances the equations of motion by another 10-degree step. At the end of the step, the relevant rotor and fuselage states, plus the circulation, are passed to the free wake model, which uses them to compute the inflow for the next time step. This process is repeated until the end of the simulation. In other words, time marching free wake and simulation model communicate every 10 degrees. For the free wake this is the overall integration step; for the simulation model this is only the time at which the solution is made available for the coupling the wake (the true integration step is determined by the solver

step size calculation algorithm). As a reference, the results will be compared with those obtained using the relaxation free wake for the response calculation as described in Ref. [16]. Because the relaxation method requires a steady-state periodic condition, it is usually updated only once per revolution. For that reason, the procedure with the relaxation wake consists of three steps: (i) obtain a converged inflow-circulation condition; (ii) freeze the inflow and integrate the coupled rotor-fuselage equations over one full revolution with that inflow distribution, and (iii) with the new states at the end of the revolution recompute the converged inflowcirculation state for the next revolution. Results The results presented in this section have been calculated for a configuration similar to the UH-60, except that the blade is assumed to have a straight tip, and the airfoil is constant along the blade. The main rotor blades flexibility is modeled with the five lowest frequency modes, which include the rigid body flap and lag modes, and the first elastic flap, lag, and torsion modes. Each blade is divided into four finite elements of equal spanwise length. The baseline flight condition for all the results presented is a descending, free flight condition at 80 knots (µ = 0.185) with a flight path angle of -6 degrees. The results are calculated for a weight of 16000 lbs (CT /σ = 0.071) at an altitude of 5250 ft. Two different maneuvers are studied: an arrested descent due to a sudden increase in collective input, and a roll reversal response to a lateral cyclic pitch maneuver. These two maneuvers are similar to those studied by Ananthan and Leishman in Ref. [24], where the same time accurate free wake model is used, but for an isolated rotor coupled only with flapping blade motion. In other words, in Ref. [24] the hub motion is prescribed, whereas in the present study it is the stick input to be prescribed, and the hub motion is determined by the overall free flight response of the helicopter to that input. Ref. [24] investigates the transient rotor wake aerodynamics after several types of maneuvers, and shows that the transient wake dynamics are followed by restructuring of the wake geometry structure which lasts for several revolutions after returning to the nominal operating state. Furthermore, the paper shows how the unsteady flow during the maneuver affects the lift distribution. The maneuvers are modeled after those performed experimentally by Kufeld et al. [25].

Time Marching Wake Convergence The two free wake models used in this paper, the relaxation technique for trim and the time marching scheme for the response, should be very similar because they are based on the same governing equations, but there are some inherent differences primarily due to the different solution methodologies and the fact that the relaxation method imposes periodicity. For this reason, the relaxation solution should not be used directly as an initial solution for the free flight response integration with time marching wake. Instead, starting from the relaxation solution, the time marching wake model should be exercised for several revolutions until the numerical transients caused by this and some other theoretical and implementation differences have disappeared. In the present study, the relaxation solution is used as a starting point for the time marching convergence, which is then run for about 40 iterations (corresponding to 10 rotor revolutions), which is typically sufficient to make the transient disappear. The inflow distribution for both cases, before and after the time marching convergence, is shown in Fig. 3, along with the difference between both. In all the rotor maps presented in this paper, the nose of the helicopter is on the left and the tail is on the right. While the overall agreement between the results of the two wakes is good, there are some differences, the most significant of which is observed at the tip of the blades at about 90 degrees (advancing side) and 180 degrees (retreating side). The explanation is given by the respective wake geometries, shown in Fig. 4, which shows the side view (top), the top view (center) and the rear view (bottom). For clarity, the figure only shows the filament of the wake of the first blade. Due to the differences stated above, both models exhibit slightly different behavior on the advancing and retreating side, related to the roll-up of the wake, and therefore the inflow at the tips of the advancing and retreating side will be affected. Arrested Descent This section presents the results of a simulation of an arrested descent maneuver. The time history of the collective control input applied for the arrested descent is shown in Fig. 5 (top). After 2 seconds, the collective is increased by 3.7 inches during a 0.5 second period, and decreased at the original value after another 0.5 seconds. Only the collective input is applied, and the other controls are held at their respective trim values. Therefore, because of the intrinsic couplings in the helicopter dynamics, all the degrees of freedom are to some extent excited by the collective input, and the maneuver is not purely lon-

gitudinal. The equations are integrated over approximately 2.7 seconds, corresponding to 12 rotor revolutions. Forces and moments, and linear and angular velocities are expressed in terms of the body axis system. Rotor flapping is expressed in terms of the hub plane. Figure 5 also shows the time history of the Z-force (center plot) and the pitching moment M (bottom plot) obtained obtained for this maneuver. The predictions are calculated with three different methods for determining the induced velocities: a dynamic inflow model, the relaxation free wake model and the time marching model. All three models predict similar behaviors of the Z-force component, with the time marching method producing the largest amplitude increase of the three methods. As to the prediction of the pitching moment M , both free wake models predict similar results, reaching their maximum right after the peak of the maneuver, and decreasing accordingly when the stick is returned to the original position. In the latter phase of the maneuver, the time marching wake predicts a slightly stronger noseup pitching moment. The dynamic inflow model, on the other hand, shows a slower growth of the pitching moment, reaching a maximum half a second after the free wake models. Figure 6 shows the time histories of the heave velocity w (top), the pitch rate q (center) and the pitch attitude θ (bottom), obtained with the three inflow models. Both free wake models predict a similar drop in the descent velocities, reached slightly after the peak of the maneuver. When the collective stick is returned to its original position the descent velocity increases by a similar amount with both methods. The dynamic inflow model shows a significantly smaller change in velocity, both when the collective step is applied and after it is returned to the baseline. For the pitch rate and attitude, all three models predict a very similar perturbation behavior, but the time-accurate wake predicts slightly higher peak values than the relaxation wake and the dynamic inflow, which again shows the lowest change in amplitude. Observe that the peaks of both the descent velocity and the pitch rate occur within the sixth revolution; during this revolution, the inflow produced by the time marching wake is updated and passed to the rest of the rotor-fuselage model every 10 degrees of blade azimuth, while the inflow produced with the relaxation wake is updated only once per revolution, and therefore it is frozen at the value obtained before the revolution starts. The consequence is that the predictions with the time marching wake reach a higher peak in both vertical velocity and pitch rate than those with the relaxation wake.

The time histories of the cosine and sine of the first flapping mode, corresponding to rigid longitudinal and lateral flapping, respectively, are shown in Fig. 7. It is interesting to note that, while for the cosine of the flapping mode both wakes again behave very similarly, the sine of the flapping mode is significantly larger in magnitude with the relaxation wake model, while the time marching free wake model presents the smallest peak in lateral flapping of the three methods. It should also be noted that the trim condition predicted with dynamic inflow is different from those predicted with the free wakes. Figures 5 through Fig. 7 show that with dynamic inflow the pitch attitude θ is higher, the velocity w is also higher, indicating a higher aerodynamic angle of attack of the fuselage, and a higher longitudinal flapping (front portion of the tip path plane tilted further forward). To better identify the physical mechanisms that cause the differences in predictions with both free wake models, it is useful to identify a significant point in the maneuver, corresponding to the peak of the collective input, and study the differences in inflow and lift at that moment. For simplicity, the next full revolution after the peak is considered, which is the 5th revolution that starts 1.16 seconds after the beginning of the simulation. Figure 8 shows the distribution of elemental lift, Cl M 2 , at the end of the 5th revolution, right after the peak of collective input. For both wake models, the lift generation is concentrated at the tips, but the relaxation wake (bottom), has an area in the front of the advancing side in which more lift is produced than in the time marching counterpart. Note that, while the time accurate model (top) predicts the lift with an inflow that is updated at every single step of the integration, the relaxation model uses an inflow value throughout the fifth revolution that was calculated at the beginning of this period, and is held fixed. Comparing the lift distribution on the rotor disk with the flapping time histories in Figure 7, it is clear that the higher lift produced on the front of the rotor disk with the relaxation wake results in a higher amplitude lateral flapping. The lift coefficient distribution with both models is shown in Fig. 9. For both models, the retreating side is working closer to Clmax . The relaxation model (bottom) has overall higher values of Cl , with values near 1 over a larger portion of the rotor. Figure 10 shows the change in inflow after the 5 revolutions from the baseline trim case for both the time marching (top) and relaxation (bottom) free wake models. On average, the inflow is higher for the relaxation wake, more positive at the rear of the rotor,

and less negative in the front. This is reflected on the elemental lift prediction, which is larger in the front of the rotor for the relaxation wake model, exactly in the region where the inflow is lower. Remember that the inflow produced with the relaxation wake is “old” and not updated, corresponding to a point before the peak of the collective, while the time marching inflow is accurate and has been affected by every single step of the maneuver. Finally, the geometries predicted with both wake models are shown in Fig. 11 for three instants in the maneuver, after 5, 8 and 10 revolutions. Note that the relaxation wake geometry is lagging behind the time marching wake geometry, which is accurate in time while the relaxation is not. These differences in the wake geometry after the peak of the maneuver are reflected in the larger differences observed in Figures 5 to 7 after the peak of collective input investigated above. Roll Reversal The roll reversal maneuver consists of four consecutive and opposite steps of lateral cyclic. The first step consists of a 2.59 inch step to the right over the trim setting, and it is applied after 1 revolution (approximately 0.23 sec.). This step is actually a ramp input that extends over approximately 0.2 sec. The second step is a roll to the left, with a total absolute value of 7.72 inches, and it is applied at 2.6 sec. after the start of the maneuver. Again, this is in reality a ramp input that takes place in about 0.7 sec. The maneuver continues with a symmetric roll to the right, exact in magnitude but of opposite sign, that takes place at 4.8 sec. Finally, the helicopter is returned to level flight. The time history of the control input is shown in Figure 12 (top). The equations are integrated over approximately 8.38 seconds (36 revolutions). The instant after 15 revolutions (3.49 seconds), corresponding to the end of the first roll to the left, is chosen for an in depth analsysis of the inflow and lift produced and their effect on the maneuver. Assuming a total excursion of 10 inches and a rate saturation limit of 100% of the total control excursion in one second, the input of Figure 12 (top) corresponds to rate saturation. This maneuver is similar the one presented in Ref. [24]. In that study, a simplified, 1-DOF linear roll model is assumed to determine the roll rates and bank angles necessary to perform this maneuver. The lateral cyclic input used in the present study is the same as that of Ref. [24], but because it is used in the context of a full flight dynamics model, with all its inherent couplings, it obviously generates different roll rate and roll angle histories than those presented in Ref. [24], plus off-axis effects not present in that

study. However, because the lateral roll inputs are symmetric with respect to the trim values, at the end of the maneuver some of the off-axis response is canceled. Figure 12 shows the time history of the roll (center) and pitch (bottom) moments, with dynamic inflow as well as both free wake models, the relaxation and the time marching wake model. After 4 seconds, when the stick is quickly deflected to the left, a large decrease in the rolling moment occurs, while at the same instant there is a very sigificant increase in pitching moment. All three inflow models present very similar predictions throughout the time range, except at the time of the second roll to the right. At this same point, both free wake models estimate much larger pitching moments than the dynamic inflow method, with the relaxation wake model providing the larger peak of the two methods. The time history of the lateral fuselage states is shown in Figure 13. The evolution of the sideways velocity (top) shows good correlation between the three methods for most of the time range, but again it is at the second roll to the right that the relaxation wake model predicts a larger peak value than the time marching wake or the dynamic inflow model. The time history of the roll rate (center) displays a very good correlation between the three mothods. Both wake models offer almost identical predictions for the majority of the time range, and both are slightly higher at the peaks than the estimate provided by the dynamic inflow model. The bank angle history (bottom) describes the same behavior, both relaxation and time marching wakes overshoot the dynamic inflow result both at the leftmost position in the turn and the following rightmost position. The time histories of the rotor states (Figure 14) show again the three methods in agreement except for the larger peak values of the free wake models. The longitudinal flapping (top) is predicted similary by all three models until the second roll to the right, at which point the value estimated by the dynamic inflow model is much smaller than that obtained with both wake models. For the lateral flaping, the three methods produce a very similar response prediction. The lift contribution to the roll moment, Cl M 2 r sin ψ, after 15 revolutions in the maneuver, is shown in Figure 15. At this time, the helicopter has turned all the way to the left. The largest contribution to the roll moment comes from the outer part of the the rear of the rotor, with a positive influence in the advancing side, and a negative influence at the retreating side. The time marching free wake (top) predicts a larger contribution to the roll moment than the relaxation wake (bottom). This slightly larger

contribution is reflected in the time history of the rolling moment, described in Figure 12 (center), in which it can be appreciated that, though both free wake methods produce very similar moments, the roll moment produced with the time marching wake reaches a higher peak in magnitude than the relaxation wake prediction. Note that, for both methods, the contribution to the roll moment is not symmetrical but biased towards the back, contributing to the cross couplings present in the maneuver. The consequence is the large pitching moment that occurs at that instant and that is observed in the time histories in Figure 12 (bottom). Of both methods, the time marching results appear more slightly more skewed than the relaxation results. This, combined with the larger values obtained with the time marching wake, results in a higher nose-down pitching moment produced with this method than with the relaxation free wake mothod. The elemental lift, Cl M 2 , is shown in Figure 16 for an instant 15 revolutions after the start of the maneuver. The higher values of Cl M 2 are obtained at the rear part of the advancing side, and this is reflected in a larger contribution to the roll moment shown in Figure 15. The time marching wake (top) predicts larger values of Cl M 2 in magnitude than the relaxation wake (bottom), higher in the rear of the rotor and lower in the front part. This difference in the lift produced by the two methods is reflected in the contribution to roll moment, which is lower for the relaxation method in the same areas in which it produces less lift than the time marching scheme. Figure 17 shows the lift coefficient at the 15th revolution for the two methods considered, time marching (top) and relaxation (bottom). The maximum values of Cl occur at the rear half of the rotor, particularly in the inboard section of the blade. It is in this area that the lift coefficient reaches the values closer to Clmax . Of the two methods, the relaxation wake has the largest portion operating close to stall. The inflow variation from the trim values after 15 revolutions in the roll reversal maneuver is shown in Figure 18 as obtained with a time marching free wake model (top) and with a relaxation wake model (bottom). Though they are very similar for most of the rotor, the relaxation wake predicts higher induced velocities towards the tips at the rear of the rotor. The higher inflow values obtained with the relaxation wake justify the lower lift produced in that same section of the rotor, and as a consequence the lower contribution to the roll moment. The geometry of the wake in the roll reversal maneuver is far more interesting than the one for the arrested descent case. Figure 19 shows the side view

at different times in the maneuver, precisely after 15, 18 and 21 rotor revolutions. At 15 revolutions (3.49 seconds), the helicopter has banked all the way to the left, but has also experienced a nose down pitching moment. In subsequent revolutions, the helicopter rolls to the right and pitches up. The time marching free wake has the capability of adapting to these changes as they occur. The relaxation wake, however, is converged to the conditions present at the end of every full revolution, and is not capable of predicting the maneuver transients as the time accurate wake does. This difference is clearly observed in the sequence in Figure 19, in which the time marching wake geometry is bending as the helicopter pitches and rolls, while the relaxation wake is converged as if each condition was a stead-state condition, and all the maneuver transients are lost. Cost Considerations In a rigorously steady-state flight condition, the uncoupled relaxation wake model is Nb times faster than the time marching model (because of the difference in the way they solve the governing equations) [26]. In an unsteady maneuver, the relaxation wake would still reach a similar, though less accurate, solution in less time than the time marching. In the coupled flight dynamics-wake model system, the situation is very different. The inflow calculated with the relaxation wake model can only be obtained once per revolution, at which time it has to be converged with the circulation. If the flight condition is a steady-state one, the cost of using a time marching or a relaxation wake model might not be very different. In an unsteady maneuver, however, the inflowcirculation loop necessary with the relaxation wake often does not converge, reaching the maximum number of iterations set by the user. When this happens, the coupled flight dynamics-free wake model may take up to 3 times longer with the relaxation wake than with the time marching wake, depending on the maximum number of iterations it is allowed to run without converging. One must be careful in that case not to reduce the number of iterations below the necessary for the inflow obtained with the relaxation wake to be accurate. Summary and Conclusion A flight dynamics simulation was coupled with a time marching free wake model, and the resulting model was used to investigate the effect of inflow modeling on the prediction of the free flight response to pilot inputs. Two large amplitude unsteady maneuvers, namely, an arrested descent and a roll reversal maneuver, were considered. The main conclusions

obtained from the present study are summarized below: 1. The different implementation and solution techniques of the relaxation and the time accurate wake models require an intermediate phase between the convergence of the relaxation wake for trim, and the beginning of the calculation to the response to pilot inputs. Without this intermediate phase, the numerical transients created by this mismatch would trigger incorrect and unphysical responses of the helicopter. Ten rotor revolutions were sufficient to allow the transient to disapper. 2. Because the relaxation-based inflow is communicated to the rest of the simulation only once per revolution, when the changes caused by the maneuver begin in the middle of a rotor revolution revolution, the relaxation free wake model is not updated to reflect the change until the revolution is complete. Instead, the time marching free wake model generates updated predictions of the inflow at the very instant the maneuver begins (within the 10-degree azimuth step used in the study). 3. In general, the relaxation free wake model predicts higher values of the inflow than the time marching model at the times considered for the analysis, which results in lower lift and lower contribution to the roll moments in the case of the roll reversal maneuver. 4. The coupled time marching free wake-flight dynamics model is not only more rigorous than the relaxation free wake implementation for unsteady maneuvers, but is also computationally more efficient, as no nested circulation-inflow loop is necessary, taking about one third less time than the integration with a relaxation wake model. 5. The relaxation free wake geometry, which behaves at each revolution as if it the corresponding instantaneous condition were instead steadystate, misses the maneuver transients which are instead observed in the time accurate free wake geometry. 6. Further comparison with experimental results will determine the accuracy of the simulation with the coupled time marching wake, as compared to the other methods discussed in this paper.

Acknowledgments This research was supported by the National Rotorcraft Technology Center under the Rotorcraft Center of Excellence Program. The authors would like to thank Dr. M. Bhagwat, Dr. J. G. Leishman, and Mr. S. Ananthan for providing a copy of the maneuvering free wake code and for many useful discussions. Acknowledgments This research was supported by the National Rotorcraft Technology Center under the Rotorcraft Center of Excellence Program. The authors would like to thank Drs. M. Bhagwat and J. G. Leishman and S. Ananthan for providing a copy of the maneuvering free wake code and for many useful discussions. References 1

Rosen, A. and Isser, A., “A New Model of Rotor Dynamics During Pitch and Roll of a Hovering Helicopter,” Journal of the American Helicopter Society, Vol. 40, No. 3, Jul 1995, pp. 17–28.

8 Bagai, A., Leishman, J. G., and Park, J., “Aerodynamic Analysis of a Helicopter in Steady Maneuvering Flight Using a Free-Vortex Rotor Wake Model,” Journal of the American Helicopter Society, Vol. 44, No. 2, Apr 1999, pp. 109– 120. 9

Bhagwat, M. J., and Leishman, J. G., “Stability, Consistency and Convergence of Time Marching Free-Vortex Rotor Wake Algorithms,” Journal of the American Helicopter Society, Vol. 46, No. 1, Jan 2001, pp. 59–71. 10

Wachspress, D. A., Quackenbush, T. R., and Boschitsch, A. H., “First-Principles Free-Vortex Wake Analysis for Helicopters and Tiltrotors,” American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 2003. 11

Spoldi, S., and Ruckel, P., “High Fidelity Helicopter Simulation using Free Wake, Lifting Line Tail, and Blade Element Tail Rotor Models,” American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 2003. 12

2

Basset, P. M. and Tchen-Fo, F., “Study of the Rotor Wake Distortion Effects on the Helicopter PitchRoll Cross-Coupling,” Proceedings of the 24th European Rotorcraft Forum, Marseilles, France, Sep 1998. 3

Prasad, J.V.R., Zhao, J., and Peters, D. A., “Helicopter Rotor Wake Distortion Models for Maneuvering Flight,” Proceedings of the 28th European Rotorcraft Forum, Bristol, UK, Sep 2002. 4

Zhao, J., Prasad, J. V. R., and Peters, D. A., “Investigation of Wake Curvature Dynamics for Helicopter Maneuvering Flight Simulation,” American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 2003. 5

Fletcher, J. W., and Tischler, M. B., “Improving Helicopter Flight Mechanics Models with Laser Measurements of Blade Flapping,” American Helicopter Society 53rd Annual Forum, Virginia Beach, VA, Apr 1997, pp. 1467-1994. 6

Rosen, A., Yaffe, R., Mansur, M. H., and Tischler, M. B., “Methods for Improving the Modeling of Rotor Aerodynamics for Flight Mechanics Purposes,” American Helicopter Society 54th Annual Forum, Washington, DC, May 1998, pp. 1337-1355. 7

Mansur, M. H., and Tischler, M. B., “An Empirical Correction for Improving Off-Axes Responses in Flight Mechanics Helicopter Models,” Journal of the American Helicopter Society, Vol. 43, No. 2, Apr 1998, pp. 94-102.

Kothmann, B. D., Lu, Y., DeBrun, E., and Horn, J. F., “Perspectives on Rotorcraft Aerodynamic Modeling for Flight Dynamics Applications,” AHS Fourth Decennial Specialists’ Conference on Aeromechanics, San Francisco, CA, Jan 2004. 13

Brentner, K. S., Lopes, L., Chen H.-N., and Horn, J. F., “Near Real-Time Simulation of Rotorcraft Acoustics and Flight Dynamics,” American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 2003. 14

Hennes, C. C., Chen, H.-N., Brentner, K. S., Ananthan, S., and Leishman, J. G., “Influence of Transient Flight Maneuvers on Rotor Wake Dynamics and Noise Radiation,” AHS Fourth Decennial Specialists’ Conference on Aeromechanics, San Francisco, CA, Jan 2004. 15

Howlett, J. J., “UH-60A Black Hawk Engineering Simulation Program: Volume I—Mathematical Model,” NASA CR-166309, Dec 1981. 16

Theodore, C. R., and Celi, R., “Helicopter Flight Dynamic Simulation with Refined Aerodynamic and Flexible Blade Modeling,” Journal of Aircraft, Vol. 39, No. 4, Jul-Aug 2002, pp. 577–586. 17

Saberi, H., Khoshlahjeh, M., Ormiston, R. A., and Rutkowski, “Overview of RCAS and Application to Advanced Rotorcraft Problems,” AHS Fourth Decennial Specialists’ Conference on Aeromechanics, San Francisco, CA, Jan 2004.

18 Theodore, C. R., “Flight Dynamics Modeling of Articulated and Hingeless Rotor Helicopters Including a Refined Aerodynamics Model,” Ph.D. Dissertation, Department of Aerospace Engineering, University of Maryland, College Park, MD, May 2000. 19 Celi, R., “Helicopter Rotor Dynamics in Coordinated Turns,” Journal of the American Helicopter Society, Vol. 36, No. 4, Oct 1991, pp. 39–47.

y z Ω

Kim, F. D., Celi, R., and Tischler, M. B., “Forward Flight Trim Calculation and Frequency Response Validation of a High-Order Helicopter Simulation Model,” Journal of Aircraft, Vol. 30, No. 6, Nov-Dec 1993, pp. 854-863.

∆ψ x

20

∆ζ

21

Celi, R., “Implementation of Rotary-Wing Aeromechanical Problems Using DifferentialAlgebraic Equation Solvers,” Journal of the American Helicopter Society, Vol. 45, No. 4, Oct 2000, pp. 253–262.

Figure 1: Free Wake discretization in the azimuth (ψ) and filament (ζ) directions.

22

Brenan, K. E., Campbell, S. L., and Petzold, L. R., The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Elsevier Science Publishing Co., New York, 1989. 23

Leishman, J. G., Bhagwat, M. J., and Bagai, A., “Free-Vortex Filament Methods for the Analysis of Helicopter Rotor Wakes,” Journal of Aircraft, Vol. 39, No. 5, Sep-Oct 2002, pp. 759–775. 24

Ananthan, S., and Leishman, J. G., “Helicopter Wake Dynamics During Tactical Maneuvers,” Proceedings of the 60th Annual Forum of the American Helicopter Society International, Baltimore, MD, Jun 2004. 25

Kufeld, R. M., Cross, J. L., and Bousman, W. G., “A Survey of Rotor Loads Distribution in Maneuvering Flight,” American Helicopter Society Aeromechanics Specialists Conference, San Francisco, CA, Jan 1994. 26

Bhagwat, M. J., “Transient Dynamics of Helicopter Rotor Wakes Using a Time-Accurate FreeVortex Method,” Ph.D. Dissertation, Department of Aerospace Engineering, University of Maryland, College Park, MD, 2001.

Figure 2: Time integration procedure for the coupling of a time-accurate wake model to the flight dynamics simulation.

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Figure 8: Elemental lift distribution after 5 revolutions for an arrested descent maneuver with a time marching free wake model (top) and a relaxation free wake model (bottom).

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