Flight Simulation and Control of a Helicopter Undergoing Rotor Span Morphing

Jayanth Krishnamurthi * Farhan Gandhi †

Rotorcraft, Adaptive, and Morphing Structures (RAMS) Lab Rensselaer Polytechnic Institute Troy, NY 12180, USA

Abstract

This study focuses on the flight simulation and control of a helicopter undergoing rotor span morphing. A model-following dynamic inversion controller with inner and outer loop Control Laws (CLAWS) is implemented, and radius change is introduced as a feedforward component to the inner loop CLAWS. Closed-loop poles associated with the low-frequency aircraft modes are observed to be robust to change in rotor span, eliminating the need for model updates due to span morphing during the dynamic inversion process. The error compensators in the CLAWS use PID control for roll and pitch attitude, PI control for yaw rate and lateral and longitudinal ground speed, but require PII control for vertical speed to avoid altitude loss observed with only PI control, during span morphing. Simulations are based on a span-morphing variant of a UH-60A Black Hawk helicopter at 18,300 lbs gross-weight and 40 kts cruise. From a baseline rotor radius of 26.8 ft, retraction to 22.8 ft, as well as extension to 31.5 ft is considered, nominally over a 60 sec duration. The controller is observed to regulate the longitudinal, lateral and vertical ground speeds well over the duration of the span morphing. Further, the controller is observed to maintain its effectivess in regulating the ground speeds when the span morphing duration is reduced to 30 sec.

1. INTRODUCTION power transfer to the rotating system, the potential pay- off is even greater. With the optimum rotor geometry known to vary de- Among the various “types” of rotor morphing consid- pending on the operating state, a fixed-geometry rotor ered in the literature, rotor span morphing is perhaps the can perform optimally in a specific set of conditions with most intriguing, with both maximum risk as well as re- significant penalties in other conditions, or alternatively, ward. Rotor span morphing was first considered as far represent a compromise design with adequate but sub- back as in the 1960’s1, 2, and received a second close optimal performance in most conditions. Recently, there look-in in the 1990’s for application to tiltrotor aircraft3, 4. has been significant interest in rotor morphing, or recon- While the benefit of rotor span variation is immediately figuration, to enhance performance in diverse operating evident on a tiltrotor aircraft (with a larger span pre- conditions, as well as expand the flight envelope and ferred for operation in helicopter mode, and reduced operational flexibility of rotary-wing aircraft. Although span for operation in propeller mode), span variation rotor morphing faces substantially greater challenges can also be highly advantageous for conventional edge- than morphing in fixed-wing aircraft due to a smaller wise rotors. It is well understood that the hover per- available area in which to fit the actuators and morph- formance of a conventional helicopter improves signifi- ing mechanisms, requirement for these to operate in the cantly with increase in rotor diameter5, 6, 7 but the larger presence of a large centrifugal field, and requirement for footprint may be limiting in space-constrained environ- *Graduate Research Assistant †Redfern Professor in Aerospace Engineering Presented at the 41st European Rotorcraft Forum, Munich, Germany, Sept. 1-4, 2015 ments (e.g., shipboard operation). Span variation al- ple 3-state generic turbine engine model given by Pad- lows operation in tight spaces, albeit at a reduced effi- field14 is used for the propulsion dynamics, with the gov- ciency, as well as at a significantly increased efficiency erning time constants approximated based on the Gen- when the aircraft moves to an unconstrained environ- Hel engine model11. ment. Span morphing can also offer significant bene- The nonlinear dynamics for the baseline aircraft are fit in very high-speed cruise. Slowed-rotor compound written as helicopters transfer the lifting function from the rotor to *˙ * * the fixed-wing in high-speed cruise and reducing rotor (1) x = f(x, u) * * * diameter reduces the overall rotor drag as well suscep- y = g(x, u) tibility to aeroelastic instability and gusts. The Boeing * * Company, funded by DARPA, recently developed the where y is a generic output vector. The state vector, x, DiscRotor technology where nested rotor blades were is given by retracted for high-speed operation. Other efforts have focused on design and demonstration of span-morphing * rotors5, 8,9, 10. (2) x = [u, v, w, p, q, r, φ, θ, ψ, X, Y, Z, ˙ ˙ ˙ ˙ The studies described on rotor morphing have thus β0, β1s, β1c, βd, β0, β1s, β1c, βd, λ0, λ1s, λ1c, far focused broadly on quantifying performance bene- T Ω, χf ,Qe] fits and demonstrating implementation methods. How- ever, transient behavior and control of the aircraft dur- The state vector comprises of 12 fuselage states (3 ing the morphing process has received little attention body velocities (u, v, w), 3 rotational rates (p, q, r), 3 at- so far. A helicopter undergoing rotor morphing will nat- titudes (φ, θ, ψ), and 3 inertial positions (X,Y,Z), 11 ro- urally tend to leave its trimmed flight condition. In order tor states (4 blade flapping states (β0, β1s, β1c, βd) and for the aircraft to maintain its current operating condi- their derivatives in multi-blade coordinates, and 3 ro- tion (speed, altitude, heading, etc.), the primary con- tor inflow states (λ0, λ1s, λ1c)) and 3 propulsion states trols would need to be simultaneously exercised as the (rotational speed (Ω), engine fuel flow (χf ) and engine rotor morphs. The current study addresses this gap torque (Qe). The control input vector is given by in knowledge and focuses on the design and applica- * T tion of a model-following dynamic inversion controller (3) u = [δlat, δlong, δcoll, δped, δtht] to maintain the aircraft’s current operating state during and is comprised of lateral, longitudinal, and collec- rotor span morphing. Simulation results are provided tive stick inputs to the main rotor, pedal input to the tail for a UH-60A Black Hawk helicopter at 40 knots cruise rotor, and throttle input to the engine. to demonstrate the effectiveness of the controller during morphing, and differences between the baseline and 2.1. Baseline Model Validation the morphed states are discussed as well. The baseline simulation model was validated against a trim sweep of flight test and GenHel data15. For valida- 2. SIMULATION MODEL tion purposes alone, elastic twist deformations on the blades, based on an empirical correction from the Gen- A simulation model for the UH-60A Black Hawk has Hel model11, were incorporated to improve the correla- been developed in-house, with components based on tion of the simulation model. Figure 1 shows represen- 11 Sikorsky's GenHel model .The model is a non-linear, tative results and the baseline simulation model corre- blade element representation of a single main rotor lates well with both flight test and GenHel. with articulated blades with airfoil table look-up. The For the design of control laws, the nonlinear equa- blades themselves are individually formulated as rigid tions of motion were linearized using numerical pertur- bodies undergoing rotations about an offset flapping bation at specific operating conditions. The linearized hinge. The lag degree of freedom is neglected. The version of Equation 1 can be written as 3-state Pitt-Peters dynamic inflow model12 is used to represent the induced velocity distribution on the ro- *˙ * * tor disk. The tail rotor forces and torque are based on (4) ∆x = A∆x + B∆u 13 * * * the closed-form Bailey rotor model . The rigid fuse- ∆y = C∆x + D∆u lage and empennage (horizontal and vertical tail) forces and moments are implemented as look-up tables based The linearized model was subsequently validated on wind tunnel data from the GenHel model11. A sim- against GenHel15 and flight data16 and Figure 2 shows representative results for hover and 80 knots forward 3. CONTROL SYSTEM DESIGN flight. The model correlates fairly well in the frequency range of 0.4-10 rad/sec for both cases, as shown. The design of the control system is based on model following linear dynamic inversion (DI)17. Model following 2.2. Variable Span Rotor Blade concepts are widely used in modern rotorcraft control systems for their ability to achieve task-tailored handling The baseline UH-60A simulation blade is now modified qualities via independently setting feed-forward and to create a span morphing variant, based on prior work feedback characteristics18. In addition, the dynamic done by Mistry and Gandhi7. The geometry is described inversion controller does not require gain scheduling in Ref.7 in detail but a brief overview is presented here. since it takes into account the nonlinearities and cross- Figure 3 shows the blade twist, chord, and mass dis- couplings of the aircraft (i.e. a model of the aircraft is tribution for the baseline and variable span blades. In built into the controller). It is thus suitable for a wide Figure 3(a), a significant portion of the baseline blade range of flight conditions17. has a constant twist rate. An implementation of variable A schematic of the overall control system is shown in span would work best for a constant twist rate and the Figure 4. The control system is effectively split into inner span morphing was thus limited to the linearly twisted and outer loop control laws (CLAWS). In designing the section of the blade. This is reflected in Figure 3(b) and CLAWS, the full 26-state linear model given by Equa- shows the twist distribution of the variable span blade in tion 7 was reduced to an 8-state quasi-steady model. its retracted, nominal, and extended configurations. In Firstly, the rotor RPM degree of freedom (Ω) is as- Figure 3(b), while the twist rate is identical between the sumed to be regulated via the throttle input determined inboard fixed and sliding sections, the nose-down twist by the RPM Governor (subsection 3.3), with the remain- at the tip of the blade relative to the root will vary with ing propulsion states (χf and Qe) coupling only with Ω. radius. Figure 3(c) shows the chord distribution of the Therefore, the propulsion states and throttle input are variable span blade. As seen in the figure, the sliding truncated from the linear model. Secondly, since the ro- section has a lower chord in order to enable full retrac- tor dynamics are considerably faster than the fuselage tion with some reasonable tolerance. Figure 3(d) shows dynamics, they can essentially be considered as quasi- the mass distribution of the variable span blade. The to- steady states and folded into the fuselage dynamics14, tal mass is increased by approximately 36% relative to which reduces computational cost. The resulting sys- the baseline blade to account for the actuation mecha- tem is an effective 8-state quasi-steady model whose nism/assembly. state and control vectors are given by With this modified blade, we now introduce a span morphing input into the dynamics given by Equation 1. * T The equations of motion now become (8) ∆x r = [∆u, ∆v, ∆w, ∆p ∆q, ∆r, ∆φ, ∆θ] * T ∆u 1r = [∆ (δlat) , ∆ (δlong) , ∆ (δcoll) , ∆ (δped)] * T *˙ * * * ∆u 2 = [∆Rspan] (5) x = f x, u 1, u 2 * * * * In this reduced-order model, the output vector is y = g x, u 1, u 2 set up such that it contains only the states themselves or contains quantities which are a function of where u1 is now the primary input vector given by Equa- only the states. Therefore, the matrices D1 and D2 tion 3 and u2 is the morphing input vector given by given in Equation 7 are eliminated from the model struc- * ture. In addition, note that while the controller uses (6) u = R 2 span a reduced-order linear model, its performance was ul- timately tested with the full nonlinear model given by Correspondingly, the linear model for the baseline Equation 5. aircraft given by Equation 4 now becomes

*˙ * * * 3.1. Inner Loop CLAW (7) ∆x = A∆x + B1∆u 1 + B2∆u 2 * * * * ∆y = C∆x + D1∆u 1 + D2∆u 2 A diagram of the inner loop CLAW is shown in Fig- ure 5. In the inner loop, the response type to pilot where B1 and B2 are the control matrices that corre- input is designed for Attitude Command Attitude Hold spond to the primary and morphing input vectors, re- (ACAH) in the roll and pitch axis, where pilot input com- spectively. mands a change in roll and pitch attitudes (∆φcmd and ∆θcmd) and returns to the trim values when input is zero. The heave axis response type is designed for Vertical h i−1 h i Speed Command Height Hold (VCHH), where pilot in- * k−1 k * (11) ∆u 1r = CrAr B1r ν − CrAr ∆x r put commands a change in rate-of-climb and holds current height when the rate-of-climb is zero. The yaw axis h k−1 i * − CrA B2r ∆u 2 response type is designed for Rate Command Heading r Hold (RCHH), where pilot input commands a change in where k = 2 for the roll and pitch axes, and k = 1 for the yaw rate and holds current heading when yaw rate is * heave and yaw axes. The term C Ak−1B ∆u is an zero. These are based on ADS-33E specifications for r r 2r 2 additional feedforward component due to the morphing hover and low-speed forward flight (V ≤ 45 knots)19. input. The term ν is known as the "pseudo-command" The commanded values (shown in Figure 5) are vector or an auxiliary input vector, shown in Figure 5. given by The psuedo-command vector is a sum of feedforward   ∆φcmd and feedback components. It is defined as *  ∆θcmd    (9) ∆y inner,cmd =   νφ   ∆VZcmd  e ν *˙ ∆r (12) ν =  θ  = ∆y + [K K K ] e˙ cmd ν  ref P D I    VZ  R e dt They are subsequently passed through command fil- νr * ters, which generate the reference trajectories (∆y ref ) *˙ where the error vector, denoted as e (see Figure 5) is and their derivatives (∆y ref ) (see Figure 5). The pa- given by rameters of the command filter were selected to meet * * Level 1 handling qualities specifications (bandwidth and (13) e = ∆y ref − ∆y inner phase delay) given by ADS-33E for small amplitude response in hover and low-speed forward flight19. Table 1 shows the parameters used in the command filters in The variables KP ,KD, and KI indicate the proportional, the inner loop CLAW, where the roll and pitch axes use derivative, and integral gains in a PID compensator. second-order filters and the heave and yaw axes use Note that the application of dynamic inversion in first-order filters. Equation 10 is carried out in the body reference frame. In Equation 12, the pseudo-commands νφ, νθ, and νr Table 1: Inner Loop Command Filter Parameters are prescribed in the body frame, while νVZ is in the iner- Command Filter ωn (rad/sec) ζ τ (sec) tial frame. Therefore, a transformation was introduced Roll 2.5 0.8 - to change the heave axis pseudo-command to the body 21 Pitch 2.5 0.8 - frame prior to inversion, and is given by Heave - - 2 ˙ νV + uθ cos θ Yaw - - 0.4 (14) ν = Z w cos θ cos φ If the reduced-order model given by Equation 10 were In dynamic inversion, the technique of input-output a perfect representation of the flight dynamics, the re- feedback linearization is used, where the output equa- sulting system after inversion would behave like a set of * tion (∆y inner in Equation 10) is differentiated until the integrators and the pseudo-command vector would not input appears explicitly in the derivative17, 20. The in- require any feedback compensation. In practice, how- version model implemented in the controller uses the ever, errors between reference and measured values 8-state vector given by Equation 8. Writing the reduced- arise due to higher-order vehicle dynamics and/or ex- order linear model in state space form, we have ternal disturbances and therefore require feedback to ensure stability. *˙ * * * (10) ∆x r = Ar∆x r + B1r∆u 1r + B2r∆u 2 The PID compensator gains are selected to ensure * * that the tracking error dynamics due to disturbances or ∆y inner = Cr∆x r modeling error are well regulated. A typical choice for where the Ar matrix is 8x8, B1r is 8x4, B2r is 8x1, Cr the gains is that the error dynamics be on the same or- * matrix is 4x8, and the output vector ∆y inner is 4x1. der as that of the command filter model for each axis. Applying dynamic inversion on Equation 10 results Table 2 shows the compensator gain values used in in the following control law each axis. Table 2: Inner Loop Error Compensator Gains tion 10 and is given by K K K P D I *˙ * ∆φcmd 2 2 (16) ∆x r,outer = ATRC ∆x r,outer + BTRC Roll 10 (1/sec ) 5.75 (1/sec) 4.6875 (1/sec ) ∆θcmd Pitch 10 (1/sec2) 5.75 (1/sec) 4.6875 (1/sec2) 2 * ∆VX * Heave 1 (1/sec) 0 0.25 (1/sec ) ∆y outer = = CTRC ∆x r,outer ∆VY Yaw 1 (1/sec) 0 6.25 (1/sec2) * with ATRC , BTRC , and ∆x r,outer defined as

* Xu Xv Finally, the vector ∆u 1r from Equation 11 is added * (17) ATRC = Yu Yv to the trim values of u 1r before being passed into the control mixing unit of the aircraft. 0 −g B = TRC g 0

* ∆u ∆x = 3.2. Outer Loop CLAW r,outer ∆v In order to maintain trimmed forward flight, an outer where u and v are body-axis velocities, Xu,Xv,Yu, and loop is designed to regulate lateral (VY ) and longitudi- Yv are stability derivatives and g is the gravitational ac- nal (VX ) ground speed while the aircraft is morphing. celeration. Applying dynamic inversion on this model A schematic of the outer loop CLAW is shown in Fig- results in the following control law ure 6. Note that the overall structure is similar to the inner loop. The response type for the outer loop is trans- ∆φcmd lational rate command, position hold (TRC/PH), where (18) = ∆θcmd pilot input commands a change in ground speed and −1 * holds current inertial position when ground speeds are (CTRC BTRC ) ν − CTRC ATRC ∆x r,outer zero. With the implementation of the outer loop, the pilot input does not directly command ∆φ and ∆θ ν cmd cmd The pseudo-command vector, ν = VX , is defined as in the inner loop CLAW. Rather, they are indirectly νVY commanded through the desired ground speeds (see similarly to Equation 12, with the error dynamics also Figure 4). defined in a manner similar to that of the inner loop. The The commanded values in the outer loop (shown in PID compensator gains for the outer loop are given in Figure 6) are given by Table 4.

Table 4: Outer Loop Error Compensator Gains * ∆VXcmd (15) ∆y outer,cmd = 2 ∆VYcmd KP (1/sec) KD KI (1/sec ) Lateral (VY ) 0.8 0 0.16 and passed through first-order command filters. Similar Longitudinal (VX ) 0.8 0 0.16 to the inner loop, the parameters of the command filter are selected based on ADS-33E specifications in hover 19 and low-speed forward flight . 3.3. RPM Governor Table 3: Outer Loop Command Filter Parameters A rotor that is undergoing span morphing will definitely impact the rotational degree of freedom (Ω). On the Command Filter τ (sec) UH-60A and in the GenHel model, the rotor RPM is reg- Lateral (V ) 2.5 Y ulated by a complex, nonlinear engine Electrical Control Longitudinal (VX ) 2.5 Unit (ECU)11. Since a simplified modeling of the propulsion dynamics is used in this study, a simple PI controller, with collective input feedforward from the inner In the outer loop, to achieve the desired ground loop CLAW, is implemented to regulate the rotor RPM speeds, the required pitch and roll attitude command in- via the throttle input, which is mapped to the fuel flow put to the inner loop (Equation 9) is determined through state (χf ). The controller is similar in structure to the model inversion17,22. A simplified linear model of the lat- one given by Kim23. A schematic of the RPM Governor eral and longitudinal dynamics is extracted from Equa- is shown in Figure 7. In Figure 7, the gains KP and KI were approximated based on the GenHel model's ECU. 4.2. Effect of Span Morphing on Dynamic The collective feedforward gain, KC , was approximated Modes and Robustness of Controller using a mapping between throttle and collective input, based on trim sweep results of the baseline aircraft in The introduction of span morphing is expected to affect subsection 2.1. Table 5 shows the gains used in the the dynamic modes of the aircraft. Figure 9 shows the governor. bare airframe (open-loop) eigenvalues of the full-order model (Equation 7) at 40 knots, for both retraction and Table 5: RPM Governor Gains extension. The change in rotor radius noticeably alters the rotor flap and inflow modes, as would be expected. KP (%/(rad/sec)) KI (%/rad) KC (nd) However, note that there is relatively very little move- Ω 3 5 1.2 ment of the fuselage modes. Since the characteristic frequencies of the fuselage and rotor dynamics are sep- arated by 1-2 orders of magnitude, there is relatively light coupling between the rotor and fuselage modes. 4. RESULTS & DISCUSSION Consequently, little movement of the fuselage modes is observed. 4.1. Heave Axis Error Compensator - PI vs PII The dynamic inversion controller uses a reduced- order version of the bare airframe model (see Equa- The aircraft was trimmed in hover with a gross weight tion 8) containing quasi-steady fuselage dynamics. As of 18,300 lbs. and the blades were retracted from a mentioned in subsection 3.1, the application of model nominal radius of R = 26.8 ft to R = 22.8 ft span span inversion in the inner loop CLAW effectively changes over 60 seconds. It was observed that the controller the plant dynamics into a decoupled set of integrators, was not able to compensate adequately in the heave which are then stabilized with feedback control. Using a axis. This is because of the parabolic nature24 of the quasi-steady model at the nominal radius (R = 26.8 ft), disturbance in vertical velocity due to span morphing. the closed-loop dynamics of the system with this con- The vertical speed and inertial position of the aircraft troller are shown in Figure 10 for 40 knots, where for with the PI controller is shown in Figure 8, indicated by clarity, only the eigenvalues close to the imaginary axis the black dashed lines. Though the overall deviation are shown. In both cases (retraction and extension), the in vertical speed is small in Figure 8(a) for the PI com- controller is robust to changes in span, particularly for pensator, the aircraft slowly descends approximately 27 the critical eigenvalues that are very close to the imag- feet over the duration of the morphing (see Figure 8(b)). inary axis. Based on these observations, the controller As a solution, the PI compensator was replaced with a was run with a single quasi-steady model at the nominal proportional-double integral compensator (PII), which is radius and found to be sufficient for both retraction and of the form extension, without requiring any model updates during K K the inversion process. (19) K(s) = K + I + II P s s2

The gains were selected empirically, with KP and KI 4.3. 40 knots - Retraction being close to the original PI compensator values. Ta- This section presents simulation results for a UH-60A ble 6 shows the gains used in the compensator. Com- Black Hawk helicopter at 18,300 lbs gross weight un- paring the two compensators in Figure 8, it can be seen dergoing rotor span retraction while cruising at 40 kts that the drift in altitude is considerably reduced. The airspeed. The span is reduced from Rspan of 26.8 ft small drift with the PII (less than 8 ft) is attributed to to 22.8 ft over a 60 second interval (see Figure 11(a)). an imperfect inversion. In the forthcoming sections of Figures 11(b) and 11(c) show the variation in main rotor results for 40 knots retraction and extension, the simu- thrust and torque associated with the span retraction. It lations were run with the PII compensator in the heave is observed that the thrust is regulated well during the axis. duration of the morphing. The increase in rotor torque Table 6: Heave Axis PII Compensator Gains bears greater scrutiny and is discussed in detail toward the end of this section. 2 2 KP (1/sec) KI (1/sec ) KII (1/sec ) Figure 12 presents the variation in main rotor col- VZ 1.1 0.35 0.025 lective, longitudinal and lateral cyclic pitch, tail rotor collective, and aircraft roll and pitch attitudes, over the duration of the rotor span retraction. Also presented in Table 8 are the control and fuselage attitude values cor- seen over the entire rotor disk when the rotor is retracted responding to the baseline span and retracted configu- (compare Figures 16(c) and 16(d)) allows generation rations. Figure 13 presents the variation in rotor coning of the necessary lift for aircraft trim. From a momen- and longitudinal and lateral cyclic flapping during rotor tum theory standpoint, an increased disk loading corre- span retraction. An increase in root collective pitch from sponding to a smaller rotor area is expected to produce 15.6 deg to 18.3 deg (Figure 12(a)) is required for the a larger overall downwash. Increased lateral and longi- contracted rotor to be able to generate the necessary tudinal flapping for the retracted rotor (Figures 13(c) and lift. Since the centrifugal force on the blades decreases 13(b)) further increases downwash velocities over por- as the span is reduced, the coning angle is seen to in- tions of the rotor disk. The combination of these factors crease, as expected (Figure 13(a)). results in an increase in downwash velocities and con- The increase in tail rotor collective (Figure 12(d)) sequently inflow most prominently in the fourth quadrant with span retraction is consistent with the increase in of the rotor disk when the span is retracted (compare main rotor torque seen in Figure 11(c). The larger lat- Figures 16(a) and 16(b)). eral force due to increase in tail rotor thrust, as well as Figure 17 shows disk plots of the induced and profile the increased roll moment it generates due to its loca- drag over the rotor disk, for both the baseline rotor as tion above the aircraft CG affects the lateral force and well as the rotor in the retracted configuration. Compar- roll moment equilibrium of the aircraft. Both the rotor lat- ing Figures 17(a) and 17(b), the induced drag for the re- eral flapping (Figure 13(c)) and lateral cyclic pitch (Fig- tracted rotor is seen to be significantly higher, predom- ure 12(c)) are seen to increase in magnitude, and a inantly over the fourth quadrant of the rotor disk, but slight change in aircraft roll attitude (Figure 12(e)) is ob- spilling over into the first and third quadrants, as well. served, as well. Due to the tail rotor cant angle on the The areas of largest increase in induced drag are gen- UH-60A Black Hawk helicopter11, the increase in tail ro- erally consistent with the areas displaying the largest tor thrust also produces a nose-down pitching moment increases in inflow angle (Figures 16(a) and 16(b)). Ta- on the aircraft. The rotor longitudinal cyclic flapping (for- ble 7 presents the induced, profile and total rotor power ward tilt of the tip path plane) is also seen to increase for the baseline (26.8 ft radius) case, as well as the (Figure 13(b)), and a corresponding increase in longi- retracted (22.8 ft radius) configuration. The induced tudinal cyclic pitch is observed (Figure 12(b)). How- power requirements for the retracted rotor is seen to in- ever, the forward tilting of the tip path plane along with crease to 1140 HP (compared to 853.9 HP for the base- an increased overall downwash for the retracted rotor line). The profile power, on the other hand, reduces to changes the main rotor wake skew angle resulting in a 252 HP (from 340.7 HP for the baseline). In Figure 17(d) reduced upload on the horizontal stabilator. This results although the profile drag over the outboard regions of in a net increase in nose-up pitch attitude of the aircraft, the retracted rotor are higher than the baseline rotor as seen in Figure 12(f). experiences over the same radial range (attributed to higher angles of attack), the absence of the outer rim op- Figure 14 shows time histories of the aircraft veloci- erating at the highest Mach numbers reduces the over- ties and altitude over the duration of the rotor retraction. all profile drag. The increase in total power from 1194.6 Note that the lateral and longitudinal ground speeds HP for the baseline to 1392.9 HP for the retracted ro- are controlled by the outer loop CLAW (subsection 3.2) tor is consistent with the increase in main rotor torque while the vertical speed is controlled by the inner loop presented in Figure 11(c) (with the rotor RPM remaining CLAW (subsection 3.1). As seen in the figure, they are unchanged). well regulated by the controller. The maximum loss in altitude during the morphing process, with the use of PII control in the heave axis, is observed to be about 8 4.4. 40 knots - Extension ft. Figure 15 shows the time history of rotor RPM during Next, simulations were conducted for the UH-60A Black blade retraction. Although only a very small drift in RPM Hawk helicopter undergoing rotor span extension (with is observed over the duration of the morphing, the drift the aircraft gross weight at 18,300 lbs and cruise speed is not fully rejected as rotor span morphing manifests as at 40 kts, as in the previous section). The span is ex- a higher-order disturbance to the PI compensator (Fig- tended from the baseline 26.8 ft to 31.5 ft over a 60 sec- ure 7). ond interval (see Figure 18(a)). Figures 18(b) and 18(c) Figure 16 shows disk plots of the inflow angle and show the variation in main rotor thrust and torque asso- blade sectional angle of attack over the rotor disk, for ciated with the span retraction. As in the case of span both the baseline rotor as well as the rotor in the re- retraction (Figure 11(b)) the thrust is regulated well over tracted configuration. An increase in angle of attack the duration of the span extension. An increase in rotor torque is observed in Figure 18(c) and is examined in span begins increasing, but as the angles of attack at further detail below. the blade tips become negative and the tips generate negative lift, the collective pitch increases to counter- Figure 16 shows comparisons of the blade sectional act this effect. The changes in longitudinal and lateral angle of attack over the rotor disk, for the baseline and cyclic pitch, and aircraft pitch and roll attitude, are of op- the fully extended rotor. From Table 8, it is observed posite sense to those seen in the case of span retraction that the reduction in rotor collective pitch accompany- (compare Figures 21(b),21(c),21(e) and 21(f) to the cor- ing span extension is significantly smaller than the in- responding plots in Figure 12). The increase in tail rotor crease in collective pitch that accompanied span retrac- collective with span extension (Figure 21(d)) is consis- tion. Further, Figure 19(b) shows that the outboard sec- tent with the torque increase discussed in the preced- tions of the extended rotor are operating at negative an- ing paragraph. Figure 22 presents the variation in rotor gles of attack, due to the increased negative twist in coning and longitudinal and lateral cyclic flapping during the vicinity of the blade tip compared to the baseline rotor span extension. The increase in centrifugal force blade (see Figure 3(b)). Because of the loss in lift in the on the extended rotor reduces the rotor coning (Figure outer regions of the rotor disk, the angles of attack in 22(a)), and the changes in longitudinal and lateral cyclic the inboard sections are actually slightly higher than the flapping are of opposite send those seen in the case of baseline (compare Figures 19(a) and 19(b)). Figure 20 span retraction (compare Figures 13 and 22). shows a comparison of the elemental induced and pro- Figure 23 shows time histories of the aircraft veloci- file drag distributions over the rotor disk for baseline ro- ties and altitude over the duration of the rotor extension. tor and the extended configuration. For the extended As in the case of span retraction, the velocities are well rotor, the lower disk loading generally results in lower regulated by the controller. While a loss in altitude is values of induced drag over the rotor disk (compare Fig- intuitively expected with rotor span retraction, altitude ures 20(a) and 20(b)). With span extension, the profile might be expected to increase with rotor span exten- drag at the tips is the highest due to the higher tip Mach sion. However, Figure 23(d) shows a maximum 5 ft loss numbers than the baseline, but the profile drag is also in altitude during the span extension process with the observed to be higher in the inboard sections due to the use of PII control in the heave axis. It is hypothesized higher angles of attack (Figures 20(c) and 20(d)). Ta- that this is due to the controller over-compensating for ble 7 shows a very large increase in profile power for the the increase in lift due to span extension. Figure 24 extended rotor (931.5 HP compared to 340.7 HP for the shows the time history of rotor RPM during blade ex- baseline), due to the increase in profile drag observed tension. As in the case of span retraction in the previ- in Figure 20. Although the induced power reduces for ous section (Figure 15) only a very small drift in RPM is the extended rotor, the increase in profile power domi- observed over the duration of the morphing. nates, resulting in an overall increase in total power to 1572.2 HP (compared to 1392.9 HP for the baseline). The increase in power requirement for the extended ro- 4.5. Rate of Morphing tor (with its higher advancing tip Mach number driving The results in the previous section considered rotor profile power increases) is greater than that observed span retraction and extension over a 60 second time for the retracted rotor (with its higher disk loading driv- interval. This section examines the impact of reduction ing induced power increases). Correspondingly, the in- in morphing duration to 30 seconds as well as an increase in rotor torque with span extension (in Figure crease to 90 seconds. Furthermore, the preceding re- 18(c)) is observed to be greater than the increase in ro- sults used a PII controller for vertical velocity in the inner tor torque with span retraction (in Figure 11(c)). loop CLAW. With PI controllers used for longitudinal and Figure 21 presents the variation in main rotor collec- lateral ground speeds and yaw rate, the effect of using tive, longitudinal and lateral cyclic pitch, tail rotor collec- the original PI controller for vertical velocity (Table 2), as tive, and aircraft roll and pitch attitude, over the duration well, is examined in this section. of the rotor span extension. The control and fuselage at- Figure 25 shows the three rotor span retraction pro- titude values in the fully extended configuration are also files considered, over 30, 60 and 90 second durations. included in Table 8. The change in rotor root collective Figures 25(b)-25(d) show time histories of the aircraft pitch in Figure 21(a) is particularly interesting, first de- altitude corresponding to cases of span retraction over creasing slightly from its baseline value of 15.6 deg to 30, 60 and 90 second intervals, respectively. While PI about 15.24 deg as the rotor expands, then reversing control on the heave axis eliminates steady-state error course and increasing to a value of 15.38 deg at full ex- on vertical velocity, a steady state error on altitude is tension. The reduction in collective is anticipated as the observed. Regardless of the duration of morphing, a total altitude loss of over 27 ft is observed with the PI control for roll and pitch attitude, PI control for yaw controller. A PII controller on vertical velocity is equiva- rate and lateral and longitudinal ground speed. lent to a PID type control on altitude, thereby eliminating However, using only PI control for vertical speed steady-state error in altitude. Although a small initial resulted in altitude loss of up to 27 ft over the du- loss in altitude is observed, Figures 25(b)-25(d) show ration of rotor span morphing. Instead, using PII that this is quickly recovered. The maximum transient control for vertical speed reduced the altitude loss loss in altitude decreases with increase in morphing du- to 8 ft (for a 60 sec morphing duration). ration (from 11 ft for a 30 second morph, to 8 ft for a 60 second morph, and 6 ft for a 90 second morph). 3. The controller regulates the longitudinal, lateral Figure 26 presents similar results for the case of ro- and vertical ground speeds well over the nominal tor span extension. With the PI controller, a steady-state 60 sec duration of the span morphing, and main- 20 ft loss in altitude is observed, for all three morph- tains its effectiveness in regulating the ground ing durations considered. With the PII controller, steady speeds even when the span morphing duration state error in altitude is eliminated, and as with the case is reduced to 30 sec. However, the PII controller of span retraction, the transient loss in altitude reduces does even better at limiting loss in altitude when with increase in duration over which span extension is the duration of span morphing is increased. introduced (from 9 ft for 30 a 30 second morph, to 5 4. For both span retraction as well as extension, the ft for a 60 second morph, and 3.5 ft for a 90 second main rotor torque requirements were seen to in- morph). It was verified that the other aircraft responses crease. In the case of span retraction this was such as rotor thrust, torque, RPM and lateral and lon- attributed to the large increase in induced power gitudinal ground speeds continue to be well regulated (although the profile power decreased slightly). In (results not included in the paper) even when the mor- the case of span extension, there was a large in- phing duration was reduced. crease in profile power (due to higher tip Mach numbers) although the induced power reduced 5. CONCLUSIONS somewhat.

This study focuses on the flight simulation and con- 5. The rotor collective pitch increased, as expected, trol of a helicopter undergoing rotor span morphing. in the case of span retraction. In the case of A model-following dynamic inversion controller with in- span extension the rotor collective pitch first de- ner and outer loop Control Laws (CLAWS) is imple- creases but then reverses course as the increas- mented, and radius change is introduced as a feedfor- ing washout results in negative lift in the outboard ward component to the inner loop CLAWS. Simulation sections of the rotor. results are presented based on a span-morphing vari- 6. Span extension reduced rotor coning due to ant of a UH-60A Black Hawk helicopter at 18,300 lbs higher centrifugal force while the reverse was ob- gross-weight and 40 kts cruise. From a baseline rotor served for span retraction. Similarly, changes in radius of 26.8 ft, retraction to 22.8 ft, as well as exten- aircraft pitch and roll attitude, longitudinal and lat- sion to 31.5 ft is considered, nominally over a 60 sec eral cyclic pitch and rotor flapping were of opposite duration. From the results presented in the paper the sense for rotor span retraction and expansion. following observations were drawn: 1. Closed loop poles associated with the low- frequency aircraft modes were robust to change ACKNOWLEDGEMENTS in rotor span, eliminating the need for model updates due to span morphing during the dynamic The authors would like to thank Dr. Joseph Horn from inversion process. the Pennsylvania State University for his helpful insights and suggestions in the development of the simulation 2. The error compensators in the CLAWS use PID model and design of the controller.

References

[1] Linden, A., Bausch, W., Beck, D., D'Onofrio, M., Flemming, R., Hibyan, E., Johnston, R., Murrill, R.J., and Unsworth, D., ``Variable Diameter Rotor Study,'' Air Force Flight Dynamics Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, OH, Report No. AFFDL-TR-71-170, AFFDL, January 1972.

[2] Fenny, C., ``Mechanism for Varying the Diameter of Rotors Using Compound Differential Rotary Transmis- sions,'' American Helicopter Society, 61st Annual Forum Proceedings, Grapevine, TX, June 13, 2005.

[3] Fradenburgh, E.A., and Matsuka, D.G., ``Advancing Tiltrotor State-of-the-Art with Variable Diameter Rotors,'' American Helicopter Society 48th Forum Proceedings, Washington, D.C., June 3-5, 1992.

[4] Studebaker, K., and Matsuka, D.G., ``Variable Diameter Tiltrotor Wind Tunnel Test Results,'' American Heli- copter Society, 49th Forum Proceedings, St. Louis, MO, May 19-21, 1993.

[5] Prabhakar, T., Gandhi, F., and McLaughlin, D., ``A Centrifugal Force Actuated Variable Span Morphing He- licopter Rotor,'' 63rd Annual AHS International Forum and Technology Display, Virginia Beach, VA, May 1-3, 2007.

[6] Bowen-Davies, G., and Chopra, I., ``Aeromechanics of a Variable Radius Rotor,'' American Helicopter Soci- ety, 68th Annual Forum Proceedings, Fort Worth, TX, May 1-3, 2012.

[7] Mistry M., and Gandhi, F., ``Helicopter Performance Improvement with Variable Rotor Radius and RPM,'' Journal of the American Helicopter Society, vol. 59, 2014. doi: 10.4050/JAHS.59.042010.

[8] Turmanidze, R.S., Khutsishvili, S.N., and Dadone, L., ``Design and Experimental Investigation of Variable- Geometry Rotor Concepts,'' Adaptive Structures and Materials Symposium, International Mechanical Engi- neering Congress and Exposition, New York, November 11-16, 2001.

[9] Turmanidze, R.S., and Dadone, L., ``New Design of a Variable Geometry Prop-Rotor with Cable and Hydraulic-System Actuation,'' American Helicopter Society 66th Forum Proceedings, Phoenix, AZ, May 11-13, 2010.

[10] Misiorowski, M., Gandhi, F., and Pontecorvo, M., ``A Bi-Stable System for Rotor Span Extension in Rotary- Wing Micro Aerial Vehicles,'' Proceedings of the 23rd AIAA/ASME/AHS Adaptive Structures Conference, AIAA Science and Technology Forum, Kissimmee, FL, January 5-9, 2015.

[11] Howlett, J.J., ``UH-60A Black Hawk Engineering Simulation Program: Volume I - Mathematical Model,'' NASA CR-166309, 1981.

[12] Peters, D.A., and HaQuang, N., ``Dynamic Inflow for Practical Applications,'' Journal of the American Heli- copter Society, vol. 33, pp. 64--66, October 1988.

[13] Bailey, F.J., ``A Simplified Theoretical Method of Determining the Characteristics of a Lifting Rotor in Forward Flight,'' NACA Report 716, 1941.

[14] Padfield, G.D., Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling. Blackwell Publishing, 2nd ed., 2007.

[15] Ballin, M.G., ``Validation of a Real-Time Engineering Simulation of the UH-60A Helicopter,'' NASA TM-88360, 1987.

[16] Fletcher J.W., ``A Model Structure for Identification of Linear Models of the UH-60 Helicopter in Hover and Forward Flight,'' NASA TM-110362, NASA, 1995.

[17] Stevens, B.L. and Lewis, F.L., Aircraft Control and Simulation. John Wiley & Sons, 2nd ed., 2003.

[18] Tischler, M.B. and Remple, R.K., Aircraft and Rotorcraft System Identification: Engineering Methods with Flight Test Examples. AIAA, 2nd ed., 2012.

[19] Anonymous, ``Aeronautical Design Standard Performance Specification, Handling Qualities Requirements for Military Rotorcraft,'' ADS-33E-PRF, USAAMCOM, 2000.

[20] Slotine, J.E. and Li, W., Applied Nonlinear Control. Prentice-Hall Inc., 1991. [21] Horn, J.F., and Guo, W., ``Flight Control Design for Rotorcraft with Variable Rotor Speed,'' Proceedings of the American Helicopter Society, 64th Annual Forum, Montreal, Canada, April 29-May 1, 2008.

[22] Ozdemir, G.T., In-Flight Performance Optimization for Rotorcraft with Redundant Controls. PhD Thesis, The Pennsylvania State University, December 2013.

[23] Kim, F.D., ``Analysis of Propulsion System Dynamics in the Validation of a High-Order State Space Model of the UH-60,'' AIAA/AHS Flight Simulation Technologies Conference, Hilton Head, S.C., August 24-26, 1992.

[24] Franklin, G.F., Powell, J.D., and Emami-Naeini, A., Feedback Control of Dynamic Systems. Pearson Prentice Hall, 6th ed., 2010.

Table 7: Induced, Profile, and Total Rotor Power

Case Pi (HP) Po (HP) Ptotal (HP) Baseline (Rspan = 26.8 ft) 853.9 340.7 1194.6 Retracted (Rspan = 22.8 ft) 1140 252.9 1392.9 Extended (Rspan = 31.5 ft) 640.7 931.5 1572.2

Table 8: Rotor Controls and Fuselage Attitudes

Case θ0 (deg) θ1s (deg) θ1c (deg) θ0TR (deg) φ (deg) θ (deg) Baseline (Rspan = 26.8 ft) 15.6 -3.38 1.74 13.80 -1.12 2.35 Retracted (Rspan = 22.8 ft) 18.3 -5.81 2.40 15.35 -0.93 3.16 Extended (Rspan = 31.5 ft) 15.37 -1.64 1.40 15.53 -2.23 1.18 100 100 Current Simulation Current Simulation GenHel GenHel 80 Flight Test 80 Flight Test

60 60

40 40 Lateral Stick (%)

20 Longitudinal Stick (%) 20

0 0 0 50 100 150 200 0 50 100 150 200 Velocity (knots) Velocity (knots) (a) Lateral Stick (b) Longitudinal Stick

10 2500 Current Simulation GenHel Flight Test 2000 5 (deg) θ 1500 0 1000 Current Simulation -5

Pitch Attitude, GenHel Main Rotor Power (hp) 500 Flight Test

-10 0 0 50 100 150 200 0 50 100 150 200 Velocity (knots) Velocity (knots) (c) Pitch Attitude (d) Main Rotor Power

Fig. 1: Baseline UH-60A Trim Sweep Validation 0 0

-10

-20 -50 Current Linear Model -30 Current Linear Model GenHel Linear Model GenHel Linear Model Magnitude (dB) Magnitude (dB) -40 Flight Test Flight Test -100 -50 180 180

90 90

0 0 Phase (deg) -90 Phase (deg) -90

-180 -180 10-1 100 101 10-1 100 101 Frequency (rad/s) Frequency (rad/s)

(a) Hover - p/δlat (b) Hover - q/δlong

0 0

-10 -10

-20 -20 Current Linear Model Current Linear Model -30 GenHel Linear Model -30 Magnitude (dB) Magnitude (dB) GenHel Linear Model Flight Test Flight Test -40 -40 45 180 0 90 -45 0 -90 Phase (deg) -135 Phase (deg) -90 -180 -180 10-1 100 101 10-1 100 101 Frequency (rad/s) Frequency (rad/s)

Fig. 2: Baseline UH-60A Frequency Response Validation Blade Chord (ft) Blade Chord (ft) Blade Chord (ft) Blade Mass (slugs/ft) Blade Chord (ft) Blade Twist (deg) -20 -10 0.2 0.4 0.6 1.6 1.7 1.8 1.9 1.6 1.7 1.8 1.9 1.6 1.7 1.8 1.9 1.6 1.7 1.8 1.9 0 0 01 02 035 30 25 20 15 10 5 0 01 02 035 30 25 20 15 10 5 0 Root c aibesa ld hr Distribution Chord - blade span Variable (c) Inboard FixedSection a aeieU-0 blade UH-60A Baseline (a) Radial Position(ft) Radial Position(ft) i.3 ld rpris-Bsln sVral Span Variable vs Baseline - Properties Blade 3: Fig. Sliding Section Tip

Blade Mass (slugs/ft) Blade Mass (slugs/ft) Blade Mass (slugs/ft) Blade Twist (deg) Blade Twist (deg) Blade Twist (deg) -20 -10 -20 -10 -20 -10 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0 0 0 0 0 0 01 02 035 30 25 20 15 10 5 0 01 02 035 30 25 20 15 10 5 0 Root Root d aibesa ld asDistribution Mass - blade span Variable (d) b aibesa ld ws Distribution Twist - blade span Variable (b) Inboard FixedSection Inboard FixedSection Radial Position(ft) Radial Position(ft) Sliding Section Sliding Section Tip Tip  Morphing u  Input 2  lat  1c        1s  cmd   long         0  VX  cmd  coll   cmd Outer Loop       Inner Loop  ped  Control  0TR  y V CLAW   Actuators Ycmd V CLAW Mixing Aircraft    Zcmd  Pilot  r   cmd  tht Input Pilot coll RPM Input Governor   2-state vector xr  u v w p q r    8-state vector xr,outer  u v CLAW – Control Law

Fig. 4: Overview of the Control System

 lat     u2    long  u1r  y  coll  ref    cmd  Inner   u   ped  To   Inner Loop y Inner  Loop 1r Control  cmd  ref   yinner,cmd  Command Loop Error Model Mixing VZ     cmd  Filters  e,e,  e dt Dynamics Inversion  rcmd   u   1r TRIM xr

 y inner From C r  Aircraft xr

Fig. 5: Inner Loop CLAW yref

cmd    VX  Outer Loop Outer Loop  cmd Outer Loop yref    cmd  To inner loop youter,cmd    V Command Error  Model CLAWS  Ycmd  Filters  e,e,  e dt Dynamics  Inversion

 xr,outer  youter From CTRC  Aircraft xr,outer

Fig. 6: Outer Loop CLAW

From Inner Loop CLAW coll KC   e K tht  ref K  I To Aircraft  P s 

From  Aircraft

Fig. 7: RPM Governor 2 205 Commanded PI 200 1 PII 195 PI PII 190 0 (knots)

z 185 V Altitude (ft)

-1 180 Duration of Duration of Morphing Morphing 175 -2 170 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Vertical Speed (b) Inertial Position

Fig. 8: Heave Velocity and Inertial Position - Hover: PI vs PII 60 2 0.36 0.27 0.19 0.12 0.06 0.85 0.74 0.62 0.48 0.32 0.16 0.5 1.5 Coupled Roll/Pitch 40 Progressive Flap 0.66 0.93 1 Dutch Roll 20 0.98 0.88 0.5 Coning/Differential Flap Heave R = 26.8 ft Phugoid span R = 25.8 ft 0 0 span R = 24.8 ft span RPM Regressive Flap -0.5 R = 23.8 ft 0.88 0.98 span Yaw/Spiral -20 R = 22.8 ft span Imaginary Axis (rad/s) Imaginary Axis (rad/s) -1 0.66 0.93 -40 -1.5 0.5 0.36 0.27 0.19 0.12 0.06 0.85 0.74 0.62 0.48 0.32 0.16 -60 -2 -30 -20 -10 0 10 -3 -2 -1 0 Real Axis (rad/s) Real Axis (rad/s) (a) Open-loop eigenvalues - 40 knots retraction

60 2 0.58 0.44 0.32 0.23 0.15 0.07 0.85 0.74 0.62 0.48 0.32 0.16 Progressive Flap 1.5 Coupled Roll/Pitch 40 Coning/Differential Flap 0.74 0.93 1 Dutch Roll Heave 20 0.92 0.98 0.5 R = 26.8 ft Phugoid span Regressive Flap R = 27.8 ft span 0 0 R = 28.8 ft span R = 29.8 ft span RPM

Imaginary Axis -0.5 R = 30.8 ft Yaw/Spiral

0.92 Imaginary Axis -20 0.98 span R = 31.5 ft span -1 0.74 0.93 -40 -1.5

0.58 0.44 0.32 0.23 0.15 0.07 0.85 0.74 0.62 0.48 0.32 0.16 -60 -2 -40 -30 -20 -10 0 10 -3 -2 -1 0 Real Axis Real Axis (b) Open-loop eigenvalues - 40 knots extension

Fig. 9: Bare Airframe (open-loop) eigenvalues at 40 knots R = 26.8 ft 5 span R = 25.8 ft span R = 24.8 ft span R = 23.8 ft 2.5 span R = 22.8 ft span

-2.5 Imaginary Axis (rad/s)

-5 -3 -2 -1 0 1 Real Axis (rad/s) (a) Closed-loop eigenvalues - 40 knots retraction

R = 26.8 ft 5 span R = 27.8 ft span R = 28.8 ft span R = 29.8 ft span 2.5 R = 30.8 ft span R = 31.5 ft span

0 Imaginary Axis -2.5

-5 -3 -2 -1 0 1 Real Axis (b) Closed-loop eigenvalues - 40 knots extension

Fig. 10: Closed-loop eigenvalues at 40 knots ×104 26.8 1.86

1.85 25.8 1.84

24.8 1.83 Radius (ft) Thrust (lbs) 1.82 23.8 1.81

22.8 1.8 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Blade Radius (b) Aircraft Thrust

×104 2.9

2.8

2.7

2.6

2.5 Main Rotor Torque (ft-lbs)

2.4 0 30 60 90 120 150 180 Time (sec) (c) Main Rotor Torque

Fig. 11: 40 knots retraction - Radius, Thrust, and Torque 18.5 -3 (deg)

18 1s -3.5 θ (deg) 0

θ 17.5 -4

17 -4.5

16.5 -5

16 -5.5 Collective Pitch -

15.5 -6 0 30 60 90 120 150 180 Longitudinal Cyclic Pitch - 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Collective Pitch (b) Longitudinal Cyclic Pitch

2.6 15.5

(deg) 2.4 (deg) 1c 15 θ TR 0 2.2 θ 14.5 2

14 1.8 Tail Rotor Pitch - Lateral Cyclic Pitch - 1.6 13.5 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Lateral Cyclic Pitch (d) Tail Rotor Collective Pitch

-0.8 3.4 Commanded Commanded Actual 3.2 Actual -0.9 3

-1 2.8 (deg) (deg) θ φ 2.6 -1.1 2.4

-1.2 2.2 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (e) Roll Attitude (f) Pitch Attitude

Fig. 12: 40 knots retraction - Controls and Attitudes (θ is positive nose up, φ is positive roll right) 3.8 2.6

3.6 ) (deg) 2.4 1c β 3.4

) (deg) 2.2 0

β 3.2 2 3 Coning ( 1.8 2.8

2.6 Longitudinal Flapping ( 1.6 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Coning (b) Longitudinal Flapping

-0.15

-0.2 ) (deg)

1s -0.25 β

-0.3

-0.35

-0.4 Lateral Flapping ( -0.45 0 30 60 90 120 150 180 Time (sec) (c) Lateral Flapping

Fig. 13: 40 knots retraction - Flapping 42 2 Commanded Commanded Actual Actual 41 1

40 0 (knots) (knots) y x V V Duration of Duration of Morphing 39 -1 Morphing

38 -2 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Longitudinal Ground Speed (b) Lateral Ground Speed

2 205 Commanded Actual 1 200

0 (knots) z V Duration of Altitude (ft) 195 Morphing Duration of -1 Morphing

-2 190 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Vertical Speed (d) Altitude

Fig. 14: 40 knots retraction - Inertial Velocities and Altitude

260

259.5

259

258.5 RPM

258

257.5

257 0 30 60 90 120 150 180 Time (sec) Fig. 15: 40 knots retraction - RPM (a) Elemental Inflow Angle (deg), Rspan = 26.8 ft (b) Elemental Inflow Angle (deg), Rspan = 22.8 ft

Fig. 16: 40 knots retraction - Induced inflow angle (φi) and Angle of attack (α) distribution (a) Elemental Induced Drag (lbs/ft), Rspan = 26.8 ft (b) Elemental Induced Drag (lbs/ft), Rspan = 22.8 ft

Fig. 17: 40 knots retraction - Induced and Profile Drag Distribution ×104 31.5 1.86

30.8 1.85

29.8 1.84

1.83 28.8 Radius (ft) Thrust (lbs) 1.82 27.8 1.81

26.8 1.8 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Blade Radius (b) Aircraft Thrust

×104 3.4

3.2

2.8

2.6 Main Rotor Torque (ft-lbs)

2.4 0 30 60 90 120 150 180 Time (sec) (c) Main Rotor Torque

Fig. 18: 40 knots extension - Radius, Thrust, and Torque (a) Elemental Angle of attack (deg), Rspan = 26.8 ft (b) Elemental Angle of attack (deg), Rspan = 31.5 ft

Fig. 19: 40 knots extension - Angle of attack (α) distribution

(a) Elemental Induced Drag (lbs/ft), Rspan = 26.8 ft (b) Elemental Induced Drag (lbs/ft), Rspan = 31.5 ft

Fig. 20: 40 knots extension - Induced and Profile Drag Distribution 15.7 -1.5 (deg) 1s

15.6 θ -2 (deg) 0 θ 15.5 -2.5 15.4

-3 15.3 Collective Pitch -

15.2 -3.5 0 30 60 90 120 150 180 Longitudinal Cyclic Pitch - 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Collective Pitch (b) Longitudinal Cyclic Pitch

1.8 16

(deg) 1.7 15.5 (deg) 1c θ TR 0 1.6 θ 15

1.5 14.5

1.4 14 Tail Rotor Pitch - Lateral Cyclic Pitch - 1.3 13.5 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Lateral Cyclic Pitch (d) Tail Rotor Pitch

-1 2.5 Commanded Commanded Actual Actual

-1.5 2 (deg) (deg) θ φ -2 1.5

-2.5 1 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (e) Roll Attitude (f) Pitch Attitude

Fig. 21: 40 knots extension - Controls and Attitudes (θ is positive nose up, φ is positive roll right) 2.8 1.8

2.6 ) (deg) 1.6 1c β 2.4 1.4 ) (deg) 0

β 2.2 1.2

2 1 Coning ( 1.8 0.8

1.6 Longitudinal Flapping ( 0.6 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Coning (b) Longitudinal Flapping

0.2

) (deg) 0.1 1s β

-0.1 Lateral Flapping ( -0.2 0 30 60 90 120 150 180 Time (sec) (c) Lateral Flapping

Fig. 22: 40 knots extension - Flapping 42 2 Commanded Commanded Actual Actual 41 1

40 0 (knots) (knots) y x V V Duration of Duration of 39 Morphing -1 Morphing

38 -2 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Longitudinal Ground Speed (b) Lateral Ground Speed

2 205 Commanded Actual 1 200

0 (knots) z V Duration of Altitude (ft) 195 -1 Morphing Duration of Morphing

-2 190 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Vertical Speed (d) Altitude

Fig. 23: 40 knots extension - Inertial Velocities and Altitude

259

258.5

258

257.5 RPM

257

256.5

256 0 30 60 90 120 150 180 Time (sec) Fig. 24: 40 knots extension - RPM 26.8 205 30 seconds PI 1 minute 200 PII 25.8 1.5 minutes 195

190 24.8 185 Radius (ft) Altitude (ft) 180 23.8 175 Duration of Morphing 22.8 170 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Blade Radius (b) Altitude

205 205 PI PI 200 PII 200 PII 195 195

190 190

185 185 Altitude (ft) Altitude (ft) 180 180 Duration of Morphing 175 Duration of 175 Morphing 170 170 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Altitude (d) Altitude

Fig. 25: Rate of Retraction - 40 knots 31.5 205 30 seconds PI 30.8 1 minute 200 PII 1.5 minutes 195 29.8 190

28.8 185 Radius (ft) Altitude (ft) 180 27.8 175 Duration of Morphing 26.8 170 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (a) Blade Radius (b) Altitude

205 205

200 200

195 195 PI PI PII 190 PII 190

185 185 Altitude (ft) Altitude (ft) 180 Duration of 180 Morphing Duration of Morphing 175 175

170 170 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Time (sec) Time (sec) (c) Altitude (d) Altitude

Fig. 26: Rate of Extension - 40 knots