Document
Stability Analysis of the Slowed-Rotor
Compound Helicopter Configuration
Matthew W. Floros Wayne Johnson US Army Research Laboratory Army/NASA Rotorcraft Division Hampton, Virginia NASA Ames Research Center Moffett Field, California The stability and control of rotors at high advance ratio are considered. Teetering, articulated, gimbaled, and rigid hub types are considered for a compound helicopter (rotor and fixed wing). Stability predictions obtained using an analytical rigid flapping blade analysis, a rigid blade CAMRAD II model, and an elastic blade CAMRAD II model are compared. For the flapping blade analysis, the teetering rotor is the most stable, showing no instabilities up to an advance ratio of 3 and a Lock number of 18. A notional elastic blade model of a teetering rotor is unstable at an advance ratio of 1.5, independent of pitch frequency. Analysis of the trim controls and blade flapping shows that for small positive collective pitch, trim can be maintained without excessive control input or flapping angles.
Nomenclature configurations provide short takeoff or vertical takeoff capa- bility, but are capable of higher speeds than a conventional helicopter because the rotor does not provide the propulsive k blade pitch-flap coupling ratio p force. At high speed, rotors on compound helicopters and au- β rigid blade flap angle togyros with wings do not need to provide the vehicle lift.
γ Lock number The drawback is that redundant lift and/or propulsion systems δ blade pitch-flap coupling angle add weight and drag which must be compensated for in some ν fundamental flapping frequency β other way.
ω dominant blade flapping frequency μ rotor advance ratio One of the first compound helicopters was the McDon- ν blade fundamental torsion frequency θ nell XV-1 “Convertiplane,” built and tested in the early 1950s.
˙ ( ) derivative with respect to azimuth There are many novel design features in this remarkable air- craft (Refs. 1–4), which was tested in the NACA 40- by Introduction 80-Foot Wind Tunnel at the Ames Aeronautical Laboratory (Ref. 5) and flight tested near McDonnell’s St. Louis, Mis- souri facilities (Ref. 6). The aircraft successfully flew in its Recently there has been increased interest in expanding the three distinct operating modes, helicopter, autogyro, and air- flight envelope of rotorcraft, particularly in terms of speed, al- plane, and could transition smoothly between them.
titude, and range. Increased range allows attack, scout, and rescue aircraft to reach farther from their bases. Additional One of the features of the XV-1 was that in airplane mode, speed and altitude capability increases the survivability of the rotor would be slowed to a significantly lower speed to military vehicles and cost efficiency of civilian aircraft. Long reduce its drag in forward flight. The combination of high loiter times improve the effectiveness of scout aircraft, with forward speed and low rotor speed produced an advance ratio particular applications of interest being unmanned aerial ve- near unity, which is far above what is typical for conventional hicles (UAVs) and homeland security surveillance aircraft.
edgewise rotors.
Much work has been focused on tilt rotor aircraft; both Other prototype compound helicopters since the XV-1 in- military and civilian tilt rotors are currently in development.
clude the Fairey Rotodyne and the Lockheed Cheyenne. Pro- But other configurations may provide comparable benefits to totypes of both aircraft were built and flown, but never entered tilt rotors in terms of range and speed. Two such configura- production. Recently, CarterCopters and Groen Brothers have tions are the compound helicopter and the autogyro. These developed autogyro demonstrators and have proposed auto- gyros and compound helicopters for future heavy lift and un- Presented at the American Helicopter Society 60th Annual Forum, Baltimore, MD, June 7–10, 2004. Copyright c © 2004 manned roles.
by the American Helicopter Society International, Inc. All rights reserved. Previously, the performance of slowed-rotor compound aircraft was examined with isolated rotor and rotor plus fixed and for the coning equation, ν = 1 . 1/rev and δ = 65 . 6 deg β 3 wing analytical models (Ref. 7). The purpose of the current were used.
effort is to examine the stability of slowed-rotor compound A series of stability maps for an articulated rotor with flap aircraft, particularly at high advance ratios.
frequency ν = 1/rev is shown in Fig. 1. In each plot, the β damping contours are shown as solid lines, positive numbers In the present study, rigid blade flapping stability is exam- indicating positive damping, and negative numbers indicating ined with a simplified analysis and with the comprehensive an instability. Only the damping of the least stable root is analysis CAMRAD II. Elastic blade stability is also calculated shown. The dashed lines separate regions where the domi- with CAMRAD II. Finally, performance and trim are exam- nant frequency of the root is 1 ± n /rev, 0 . 5 ± n /rev, or non- ined for teetering and articulated rotors.
harmonic frequencies. Dominant system frequencies of 1/rev and 0.5/rev occur when the Floquet roots are on the real axis, Flap Stability whereas the frequency is non-harmonic when the roots are complex conjugates.
The simplified analysis predictions are based on rigid flap- Specific frequencies are identified by solving the flapping ping blade equations similar to those developed by Sissingh equation in hover, where the coefficients are constant rather (Ref. 8). These equations were used by Peters and Hohen- than periodic. The roots of the system are given by emser (Ref. 9) to examine flapping stability of an isolated blade and a four-bladed gimbaled rotor with tilt-moment feed- √ ( ) γ γ γ back. In the present study, they are used to compare different s = − ± i ν + k − (2) p β 16 8 16 hub configurations in order to assess suitability for high ad- vance ratio operation.
The frequency, ω , is the imaginary part, and can be solved for The analysis addresses only rigid blade flapping; lag and γ as torsion motion are not modeled. The aerodynamics are linear ( √ ) and aerodynamic coefficients are obtained by integrating ana- 2 2 2 γ = 16 k ± k + ν − ω (3) p p β lytically along the blade length. The flapping blade equations are integrated over a single rotor revolution and Floquet theory The hover Lock numbers for a blade frequency ν of 1.0 are β is used to determine the system stability. The homogeneous given in Table 1. Missing Lock numbers indicate that the roots flapping blade equation is given by are complex numbers.
The pitch-flap coupling varies from 0 to 65.6 deg in the ¨ ˙ β − γ M β + ( ν − γ M + γ k M ) β = 0 (1) ˙ p β θ β β four plots. The 65.6 deg angle was chosen because the con- ing hinges on the XV-1 have 65.6 deg of δ . Increasing δ 3 3 In this expression, M , M , and M are the aerodynamic coef- ˙ θ β (Figs. 1a-c) increases the flapping stability margin such that β ficients. The blade motion is thus defined by only the flap fre- at δ of 30 deg, there is no unstable region in this range of quency, Lock number, advance ratio (embedded in the aero- advance ratio and Lock number. Once δ exceeds about 45 dynamic coefficients) and pitch-flap coupling. The pitch-flap deg, the damping at high advance ratio declines again. Fig. 1d coupling ratio and the more commonly used δ angle are re- shows δ of 65.6 deg and includes several unstable regions lated by k = tan δ .
p 3 with the stability boundary occurring at a lower advance ratio than δ = 0 (Fig. 1a). The plots suggest that an articulated For the present study, multi-blade equations were derived blade can be used at advance ratios higher than 2 if appropri- for articulated and gimbaled (three bladed) rotors, as well as ate δ is included.
teetering and an XV-1-type gimbaled rotor. The latter two configurations were not addressed in Ref. 9. The teetering and Stability maps for a teetering rotor are shown in Fig. 2. The gimbaled rotors are straightforward. The teetering rotor has teetering rotor stability is quite different from that of the artic- only a single degree of freedom for the teeter motion; coning ulated blade. The stability is much less dependent on advance is not allowed. For the gimbaled rotor, there are two cyclic ratio throughout the entire δ and Lock number range. The degrees of freedom and a coning degree of freedom.
effect of δ on damping is also much less pronounced than in the single blade case. The damping magnitudes change with The XV-1 rotor is more complicated. It has a three-bladed changes in δ , but the characteristic shape remains the same.
gimbaled rotor with offset coning hinges. The gimbal motion The damping is level or slightly increasing up to an advance has a flap frequency of ν = 1/rev and pitch-flap coupling an- β ratio of unity, then gradually decreases at higher advance ra- gle δ = 15 deg. The coning motion has a flap frequency of tios. This simple analysis suggests that a teetering rotor is a ν = 1 . 1/rev and δ = 65 . 6 deg. To model the XV-1 rotor β 3 good candidate for a high advance ratio rotor.
in the context of the simplified analysis, the appropriate con- stants were used in each of the multi-blade equations. For the Results for a rigid gimbaled rotor are shown in Fig. 3. For two cyclic equations, ν = 1/rev and δ = 15 deg were used, these results, a 3-bladed rotor with only the gimbal motion β Table 1. Hover Lock numbers for a rotor with flap frequency ν = 1 . 0 β k δ ω = 0 ω = 0 . 5 ω = 1 . 0 ω = 1 . 5 ω = 2 . 0 ω = 2 . 5 p 3 0 0 16 13.9 0 – – – 0.268 15 20.9 18.8 8.6 – – – 0.577 30 27.7 25.9 18.5 – – – 2.2 65.6 74.0 73.2 70.5 4.9, 65.5 13.5, 56.9 – 18 18 0/rev 0.5/rev 0.3 0.1 16 16 14 14 0.3 0.5/rev 12 12 0.6 0.6 10 10 1/rev 1/rev 0.1 8 8 Lock Number Lock Number 6 6 0.3 0.3 4 4 0.1 0.1 2 2 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (a) δ = 0 deg (b) δ = 15 deg 3 3 18 18 0.5/rev 1/rev 1.5/rev 0.6 0.6 0.6 16 16 0.6 1/rev 14 14 0.3 0.5/rev 12 12 2/rev 0.6 0.3 0.1 0.6 0.6 10 10 0.1 0.3 0.3 8 8 0.1 Lock Number Lock Number 6 6 0.3 0.3 0.3 4 4 0.1 1.5/rev 0.1 0.1 2 2 0.1 1/rev 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (c) δ = 30 deg (d) δ = 65 . 6 deg 3 3 Fig. 1. Stability maps of a rigid blade articulated rotor at 0, 15, 30, and 65.6 deg of δ , ν = 1.0.
β 0/rev 0.5/rev 1.5 1.5 0.75 1.25 0.5/rev 0/rev 0.5 0.75 0.75 0.5 1/rev 0.5 Lock Number Lock Number 1/rev 0.25 0.25 0.1 0.1 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (a) δ = 0 deg (b) δ = 15 deg 3 3 18 18 3/rev 16 16 1/rev 1.25 1.25 14 14 3.5/rev 1 1 0.75 1.25 0.75 1.5 12 12 10 10 3/rev 2.5/rev 0.5 0.5 8 8 Lock Number Lock Number 6 6 1.5/rev 0.25 0.25 2/rev 2/rev 4 4 0.1 0.1 2 2 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (c) δ = 30 deg (d) δ = 65 . 6 deg 3 3 Fig. 2. Stability maps of a rigid blade teetering rotor at 0, 15, 30, and 65.6 deg of δ .
(specifically two cyclic modes) is considered. Like the articu- only to capture the basic geometries of the rotor and wing lated and teetering rotors, the flap frequency is ν = 1 . 0. From of the aircraft as an alternative to inventing a geometry (see β these plots, an advance ratio limit near μ = 2 is evident. For Fig. 6). The maximum gross weight of the demonstrator is no pitch flap coupling, Fig. 3a, an instability occurs around approximately 4200 lb.
μ = 1 . 5. Increasing δ to 15-30 deg delays the onset of this Both rigid blade and elastic blade models were developed.
instability to about μ = 2 (Fig. 3b-c), but additional δ does The models were developed to investigate parameter varia- not delay the onset further (Fig. 3d). This suggests that an in- tions applicable to slowed-rotor vehicles in general rather than herent limit exists that can only be alleviated slightly with δ , to model the CCTD design specifically in detail. The rigid at least without coning motion.
blade analysis does not allow for elastic bending or torsion, so A production gimbaled rotor would not be rigid in coning.
many details of the mass and stiffness distributions and aero- It would either have coning hinges, like the XV-1, or it would dynamic center offsets are unnecessary. For the elastic blade have a coning mode due to elastic bending of the blades. In analysis, the rotor was made as simple as possible to avoid in- either case, the coning mode would have a frequency greater troduction of additional unknowns into the results. The prop- than 1. The coning mode of a 3-bladed gimbaled rotor is erties of the rotor and wing are shown in Table 2.
shown in Fig. 4. For this plot, the coning equation which was The CCTD prototype rotor has an extremely low Lock neglected for Fig. 3 was solved separately. To match the con- number caused by the presence of a 65 lb mass in each blade ing mode of the XV-1, the flap frequency for these plots has tip. These masses provide rotational inertia to store enough been increased to ν = 1.1.
β energy in the rotor for a jump take-off. For the present study, For this mode, no instability is seen for any of the plots.
variations in chordwise offset of masses were not considered.
The damping contours are relatively independent of advance The tip masses were placed on the quarter chord for both the ratio, and change very little with increasing δ . Although the rigid and elastic blade models.
frequency contours change dramatically with δ , the damping For the actual aircraft, the blade and wing use NACA 65- contours appear to change only in the vicinity of the frequency series airfoils. Airfoil tables were not available for the airfoils boundaries.
on the demonstrator, so the NACA 23012 was used as a substi- The stability map for the XV-1 rotor is shown in Fig. 5. If tute. The wing model is straightforward. The wing is swept, there were no coupling between the gimbal and coning modes, tapered, and untwisted, with an aspect ratio of 13.4. The lift- this plot would be the combination of Figs. 3b and 4d. There ing line aerodynamic model of the wing in CAMRAD II is are two large instability regions, the high Lock number re- identical to the aerodynamic model used for the rotor blades.
gion with a 0.5/rev frequency, and the low Lock number re- gion, whose frequency is not locked to 0.5/rev or 1/rev. The Before discussing trim, some definitions should be noted.
low Lock number region extends down to an advance ratio of The CCTD is an autogyro, so while it is flying, there is no about 1.4. The Lock number at this minimum point is very torque applied to the rotor shaft. The XV-1 also operated in close to the 4.2 Lock number of the XV-1.
this mode at high speed. In the context of this paper, the word autorotation describes the trim state of the rotor, where rotor Ref. 3 identified a 0.5/rev instability in a test model at μ ≈ speed is maintained with no torque input to the shaft. For a 1 . 5. Such a stability boundary agrees well with the current helicopter, autorotation of the rotor implies that an emergency prediction, but the frequencies do not agree. The thin areas landing is in process, but for an autogyro, the rotor is in an enclosed by the dashed lines in the lower right of Fig. 5 are autorotation state for normal cruise flight. These should not frequency locked at 0.5/rev, but outside these small regions be confused. Rotor power , when used in reference to an au- the frequency is not locked.
torotating rotor, is defined here as the rotor drag multiplied by its velocity. This power is indirectly supplied by the aircraft’s CAMRAD II Teetering Rotor Model Description propulsion system (which overcomes the drag) and not shaft torque.
The flapping blade analysis provides a broad picture of the sta- bility of a number of rotor configurations, Lock numbers, and Several trim variables were used. The CCTD is controlled advance ratios, but is limited in usefulness by its many sim- only with collective pitch and tilt of the spindle to which the plifications. To go beyond the guidance provided by the flap- rotor is attached. For the calculations, spindle tilt was mod- ping blade analysis, a slowed-rotor vehicle model based on the eled by tilting the rotor shaft. If the rotor is trimmed in autoro- CarterCopter Technology Demonstrator, or CCTD (Ref. 10), tation, the shaft torque must be zero. The spindle tilt was used was developed for the comprehensive analysis CAMRAD II to control the shaft torque. The incidence angle of the wing (Ref. 11). The model was previously used to examine the was used to trim the vehicle lift. By using wing incidence and performance (Ref. 7) of the slowed-rotor concept and in the spindle tilt, the controls are largely independent of each other.
present study is used to examine stability and control. Since Shaft angle affects both rotor lift and shaft torque, but wing little detailed information is publicly available about the pro- incidence does not have any effect on the rotor lift or power.
totype, the analytical model is relatively simple. It is intended Cyclic pitch was not used for trim in any of the calculations.
18 18 0.75 16 16 0.5 0.25 14 14 0.25 0.75 0 0.5 0.75 12 12 0.5 10 10 0.5/rev 0.5/rev 8 8 1/rev Lock Number Lock Number 6 6 0.25 0.25 4 4 1/rev 2 2 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (a) δ = 0 deg (b) δ = 15 deg 3 3 18 18 1/rev 2/rev 16 16 0.5 0.5 14 14 0.75 0.25 0.75 0.75 12 12 0.5 10 10 0.5/rev 0.5/rev 0.5 8 8 Lock Number Lock Number 6 6 1/rev 0.25 0.25 4 4 1/rev 1.5/rev 2 2 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (c) δ = 30 deg (d) δ = 65 . 6 deg 3 3 Fig. 3. Stability maps of cyclic modes of a rigid blade gimbaled rotor at 0, 15, 30, and 65.6 deg of δ .
18 18 0.5 0.5/rev 1 1 16 16 1/rev 14 14 0.75 0.75 0.75 12 12 1.5/rev 1.5/rev 10 10 0.5 0.5/rev 0.5 8 8 Lock Number Lock Number 6 6 1/rev 0.25 0.25 4 4 2 2 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (a) δ = 0 deg (b) δ = 15 deg 3 3 18 18 1/rev 1 1 16 16 1.5/rev 14 14 0.75 0.75 1.5/rev 12 12 10 10 0.5 0.5 8 8 2/rev 0.5 1 Lock Number Lock Number 2/rev 6 6 1.5/rev 0.25 0.25 0.25 4 4 2 2 1.5/rev 0 0 0 1 2 3 0 1 2 3 Advance Ratio Advance Ratio (c) δ = 30 deg (d) δ = 65 . 6 deg 3 3 Fig. 4. Stability maps of only the coning mode of a rigid blade gimbaled rotor with coning hinge at 0.062R and 0, 15, 30, and 65.6 deg of δ , ν = 1.1.
3 β 16 0.25 0.75 12 0.25 0.5 0.5 Lock Number 0.25 Fig. 6. Top view of CAMRAD II rotor and wing model, ψ = 0 deg, direction of flight to left.
0 1 2 3 Advance Ratio Fig. 5. Stability map for XV-1 rotor, δ = 15 deg δ = ing increases in the simplified analysis, but decreases in the 3 , g 3 , c 65.6 deg, ν = 1.1, ν = 1.0. CAMRAD II calculation.
β , c β , g The calculation was repeated, enforcing the autorotation condition. Here, the shaft angle was varied to maintain zero An additional, implicit trim condition for a teetering rotor is power on the rotor. This trimmed result is shown in Fig. 9.
that the hub moment must be zero. This condition is normally Note that the data only extends to an advance ratio of 2. It was accommodated by flapping.
difficult to find a stable autorotation condition at the higher Lock numbers. As the advance ratio approached 2, the anal- Ref. 7 presented correlation of CAMRAD II calculated ysis predicted a rapid change in trim shaft angle, suggesting trim and performance with wind tunnel measurements. While that the rotor stall was preventing autorotation.
in that work a vortex wake model was used, it was found that the induced drag of both the rotor and wing were small. Hence The damping contours for the trimmed case are also simi- a uniform inflow model (based on momentum theory) is used lar to the simplified analysis except in the high advance ratio, for the present results.
high Lock number region where the rotor begins to stall. This means that when the rotor is lifting, the damping is unaffected by nonlinear aerodynamics and dynamics, the introduction of Comparison of CAMRAD II Model to Simple Analysis a real airfoil, and trim. The simplified analysis is a good ap- proximation for a rigid flapping blade. Note that for a 230 The simplified analysis described above was compared with ft/sec tip speed, an advance ratio of 2 corresponds to nearly the rigid blade CAMRAD II model to determine what differ- 275 knots, which is very high speed for a rotary-wing vehicle.
ences would be introduced by more sophisticated aerodynam- ics and blade motion, airfoil tables, etc. To model the CCTD Control of Thrust and Autorotation using the simplified analysis, a δ of 10 deg was selected and the Lock number and advance ratio were varied as in the pre- vious results. The stability map for a teetering rotor with 10 The performance analysis in Ref. 7 suggested that there was a deg of δ is shown in Fig. 7.
narrow range of collective pitch where the rotor was autorotat- ing at the desired speed and producing positive lift. The most Stability calculations were performed for the CAMRAD desirable condition for low vehicle power is for the wing to lift II model with the rotor trimmed and untrimmed. For the the vehicle and for the rotor to produce no lift and as little drag untrimmed condition, the rotor collective was fixed at 1 deg as possible. Of course, the rotor must produce some thrust in and the rotor shaft was fixed at 0 deg. The rotor could flap order to maintain autorotation, so a more realistic condition freely and there was no zero torque constraint on the rotor.
is for the rotor to produce a small positive thrust. Conditions The tip speed was selected as 230 ft/sec to minimize com- where the rotor produces negative thrust or a significant por- pressibility effects at high advance ratio. The result is shown tion of the vehicle lift are undesirable.
in Fig. 8. For the majority of the plot, the damping levels are very similar to those in Fig. 7. At high Lock numbers and Producing too much lift rotor lift normally requires excess advance ratios above 2, the plots begin to differ, as the damp- power and reduces the vehicle efficiency, but does not pro- Table 2. Properties of the model rotor and wing 18 Rotor Number of Blades 2 0.25 0.5 0.1 Hub type teetering 0.75 Radius 22 ft Root chord 17 in 0.5 Tip chord 7 in Solidity 0.032 Lock number 2.3 Twist 0 deg Lock Number Airfoils NACA 23012 0.25 δ 10 deg Wing Span 32 ft 0.1 Root chord 45 in Tip chord 12.5 in 0 1 2 3 Aspect ratio 13.4 Advance Ratio Sweep angle 18 deg Incidence angle 5.2 deg Fig. 8. Stability map for CarterCopter rotor from CAM- Dihedral 6 deg RAD II rigid blade model, δ = 10 deg, ν = 1.0, no trim.
3 β Wash out none Airfoil NACA 23012 Position (8.9, 2.63) ft below, forward of rotor 0/rev 0.5 0.75 0.75 1.25 1.5 0.5 0.5 Lock Number 0.25 Lock Number 1/rev 0.25 0.1 0 1 2 3 Advance Ratio 0 0.5 1 1.5 2 2.5 3 Advance Ratio Fig. 9. Stability map for CarterCopter rotor from CAM- Fig. 7. Stability map for CarterCopter rotor from sim- RAD II rigid blade model, δ = 10 deg, ν = 1.0, trimmed 3 β plified analysis, δ = 10 deg, ν = 1.0 ( γ ≈ 2 . 5 for Carter- 3 β to autogyro condition.
Copter).
hibit operation. Excessive flapping or control input require- ments, however, might prevent the vehicle from operating safely. These represent flying qualities issues if they exceed the abilities of control actuators or of the pilot.
To determine the sensitivity of these variables to collective pitch and advance ratio, the rotor-wing combination described above was trimmed at tip speeds of 230, 345, and 460 ft/sec for teetering and articulated hubs. The articulated hub had no hinge offset, but results in (Ref. 12) showed that a 5% hinge offset produced nearly identical results to that with no hinge offset. Ref. 12 also presented results for a rigid rotor with no hinges or flap flexibility, but such a configuration could not be trimmed in rolland is not presented here. The rotors were identical in geometry to the model in the previous section; only the hub boundary condition was changed.
As in Ref. 7, only lift and rotor power were trimmed for !"#$ %$ these calculations. The lift of the rotor and wing combination was trimmed to 4200 lb and the rotor torque was trimmed to Fig. 10. Rotor thrust (open and dashed) and power (closed zero to model lifting the vehicle gross weight and an autorota- and solid) for an articulated rotor at 250 knots ( μ = 1.22) tion condition on the rotor. Trim controls were tilt of the wing vs. shaft angle, -2 to 2 deg collective, V = 345 ft/sec.
T and rotor shaft, but there was no cyclic pitch on the rotor.
Before proceeding, an interesting aspect of the autorota- cious selection of initial conditions was all that was necessary tion envelope must be discussed. The trim state in autorotation to reach the desired trim condition.
is not unique. Two conditions exist where the rotor can main- tain autorotation. To illustrate this phenomenon, isolated rotor power of an articulated rotor was considered while sweeping Teetering Rotor the shaft angle. Instead of trimming the rotor to zero power, the shaft angle was changed and the RPM held fixed. This The control issue raised in Ref. 7 was based on teetering ro- was intended to determine if the resulting power curve crosses tor performance calculations. The lift distributions for the ro- through zero in multiple places, indicating multiple autorota- tor and wing suggested that there was a narrow range of col- tion states.
lective pitch settings where the rotor produced an acceptable thrust level. Rotor lift as a function of airspeed and collective Fig. 10 shows thrust and power for an articulated rotor pitch for the teetering rotor model is shown in Fig. 11. The hinged at the root at 250 knots and a tip speed of 345 ft/sec.
contours indicate lines of constant lift and the dashed lines in- Collective pitch angles of -2, 0, and 2 deg are shown in the dicate negative lift. From these figures, there does seem to be figure. The rotor power (solid lines) peaks at different shaft a small range of acceptable collective pitch. At the lowest tip angles depending on the collective pitch. But for each shaft speed, Fig. 11a, there is a relatively large range of rotor lift angle, the power curve crosses zero power in two places about in the 4 deg collective pitch range shown. At 250 kts, the lift 4 deg apart. This means that autorotation can be maintained changes by approximately 1500 lb over that range. At very at either of these shaft angles. In addition, the overlaid rotor high speed, the lift becomes negative for collective pitch set- thrust (dashed lines) shows that for each collective pitch set- tings above 0.5 deg and the range of lift is on the order of the ting, one trim condition has positive thrust and the other has 4200 lb gross weight of the CCTD. Below 250 kts, the desired negative thrust. Note that the thrust difference between the small positive lift is realized over the entire range.
two points is on the order of 2000 lbs, a substantial amount for a 4200-lb vehicle.
The 345 ft/sec tip speed case, shown in Fig. 11b, shows similar behavior, albeit over a larger collective pitch range.
This raises questions about whether a maneuver could As with the lower tip speed case, the change in lift over the cause the rotor to switch abruptly between the two autoro- pitch range shown (6 deg for this tip speed) is also about 1500 tation points. Transient analysis of a full vehicle is beyond lb at 250 kts and increases thereafter. Also like the lower tip the scope of this paper, so this issue is not considered in de- speed, there does not appear to be any lift issue for airspeeds tail. For the purposes of this paper, the only consequence of below 250 kts.
multiple trim conditions is that care was taken to always trim to the higher thrust condition. The large difference in thrust For the highest tip speed, Fig. 11c, compressibility dom- between the two trim states makes it easy to identify when the inates the vehicle lift above 250 kts. Operating at high air- analysis has trimmed to the wrong thrust. Fortunately, judi- speeds for this tip speed is not practical due to the high power required (Ref. 7). In summary, while there is the potential for Advance Ratio some degradation in performance when operating at a non- 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 optimum collective, small variations will not radically alter -1500 1.5 the lift on the rotor.
-1000 -500 Although the rotor lift was well-behaved over a range of airspeed and collective pitch, large gradients in flapping or 0.5 controls indicate a handling qualities and perhaps vehicle sta- bility problem. The spindle tilt and blade flapping angles are shown in Figs. 12 and 13. Both the spindle tilt and blade flap- -0.5 ping are well-behaved.
1000 Collective Pitch (deg) -1 The spindle tilt (positive aft) is shown in Fig. 12. It changes with airspeed at low collective pitch, but as speed in- -1.5 creases, it is relatively independent of airspeed for all three tip speeds. The reason for this is the vehicle trim. At low speed, -2 100 150 200 250 300 350 400 Airspeed (kts) the wing (and therefore fuselage) must be at a high angle of attack to carry most of the vehicle weight. As speed and dy- (a) V = 230 ft/sec T namic pressure increase, this angle decreases. For the rotor to Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 maintain its orientation in space, the spindle must be tilted aft -1500 to account for the wing angle of attack.
-1000 3.5 -500 3 The flapping angle (positive forward), shown in Fig. 13, 2.5 is also well-behaved. For the 230 and 345 ft/sec tip speeds, the contours are parallel and the range of flapping is about the same as the range of collective pitch. If possible, flapping 1.5 should be minimized, so for the range of collective pitch set- tings shown, lower collective pitch is better. For the 460 ft/sec 0.5 case (Fig. 13c), although the contours are inclined at a steeper angle and the flapping range is slightly larger, there are no Collective Pitch (deg) -0.5 steep gradients and the maximum flapping angle is approxi- -1 mately 10 deg. This tip speed is undesirable from a power -1.5 2500 standpoint, but does not appear to have control or flapping -2 100 150 200 250 300 350 400 problems.
Airspeed (kts) The orientation of the tip path plane, shown in Fig. 14, is (b) V = 345 ft/sec T another indication of the state of the rotor. It is the sum of Advance Ratio the hub angle of attack and the longitudinal flapping. It only 0.4 0.6 0.8 1 1.2 1.4 varies over a few degrees for the three tip speeds, but the con- 3.5 tours bear some similarity to the contours of lift in Fig. 11.
Where the lift increases in Fig. 11, the tip path plane angle in- 2.5 creases. The absence of steep gradients indicates that the ro- tor orientation changes slowly with changes in collective pitch 1.5 and airspeed.
Finally, rotor power, calculated as rotor drag multiplied by 0.5 velocity, is shown in Fig. 15. The contributions to drag and 1500 0 2000 Collective Pitch (deg) power for this rotor are discussed in detail in Ref. 7. For the 2500 -0.5 present study, the only interest is sharp gradients, especially -1 with horizontal contours that indicate rapid changes with col- -1.5 lective pitch. In Fig. 15, there are none. The rotor power is -2 100 150 200 250 300 350 400 nearly independent of collective pitch, so from a power stand- Airspeed (kts) point, any collective pitch setting is appropriate.
(c) V = 460 ft/sec T This is consistent with findings for a single collective pitch setting in Ref. 7 that power was dominated by profile power Fig. 11. Lift for a teetering rotor vs. airspeed and collec- and interference and induced power were minor in compari- tive pitch, V = 230–460 ft/sec.
T son. Because the lift is strongly dependent on collective pitch in Fig. 11, but the power is not, the induced power must be Advance Ratio Advance Ratio 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 2 2 -4 -2 -6 1.5 1.5 1 1 0.5 0.5 -4 0 0 -0.5 -0.5 Collective Pitch (deg) Collective Pitch (deg) -1 -1 -2 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (a) V = 230 ft/sec (a) V = 230 ft/sec T T Advance Ratio Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.8 1 1.2 1.4 1.6 1.8 4 4 -4 3.5 3.5 -2 3 -7 3 2.5 2.5 2 -6 2 2 1.5 1.5 1 1 0.5 0.5 -4 0 0 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -1.5 -1.5 8 -2 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (b) V = 345 ft/sec (b) V = 345 ft/sec T T Advance Ratio Advance Ratio 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 1.4 1.6 4 4 3.5 3.5 -10 3 3 2.5 2.5 -8 2 2 1.5 -6 1.5 1 1 0.5 0.5 -4 0 0 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -2 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 450 100 150 200 250 300 350 400 450 Airspeed (kts) Airspeed (kts) (c) V = 460 ft/sec (c) V = 460 ft/sec T T Fig. 12. Spindle tilt for a teetering rotor vs. airspeed and Fig. 13. Flapping angle for a teetering rotor vs. airspeed collective pitch, V = 230–460 ft/sec. and collective pitch, V = 230–460 ft/sec.
T T Advance Ratio Advance Ratio 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 50 100 300 Collective Pitch (deg) Collective Pitch (deg) -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (a) V = 230 ft/sec (a) V = 230 ft/sec T T Advance Ratio Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.8 1 1.2 1.4 1.6 1.8 4 500 3.5 3.5 50 100 3 3 2.5 2.5 2 2 1.5 1.5 0.5 0.5 0 0 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (b) V = 345 ft/sec (b) V = 345 ft/sec T T Advance Ratio Advance Ratio 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4 1.6 4 4 3.5 3.5 3 2.5 2.5 2 2 1.5 4 1.5 1 1 0.5 500 0.5 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 450 Airspeed (kts) Airspeed (kts) (c) V = 460 ft/sec (c) V = 460 ft/sec T T Fig. 14. Tip path plane angle of attack for a teetering rotor Fig. 15. Power required for a teetering rotor vs. airspeed vs. airspeed and collective pitch, V = 230–460 ft/sec. and collective pitch, V = 230–460 ft/sec.
T T small relative to the profile power on the rotor. Given this, it Advance Ratio 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 is not a detriment for the rotor to carry lift. 2.5 -1000 These results provide guidance for an optimum collective -500 pitch. The first clear conclusion is not to use the 460 ft/sec tip 1.5 speed. The increased power required is clearly undesirable.
For the lower tip speeds, the lift gradients do not translate into gradients in rotor power, so the optimum collective can be 0.5 chosen based on control and flapping angles. These results, Figs. 12–13, oppose each other. Spindle tilt is minimized as -0.5 collective pitch increases, but flapping is minimized for lower Collective Pitch (deg) collective pitch. Therefore a moderate value in the 0–1 deg -1 range is appropriate.
-1.5 -2 100 150 200 250 300 350 Articulated Rotor Airspeed (kts) (a) V = 230 ft/sec T The previous section described control calculations for a tee- Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 tering rotor. The same results for an articulated rotor hinged at the center of rotation are shown in Figs. 16–20. The model -1500 3.5 -1000 used to calculate these results is the same as the teetering rotor -500 except that the blades can now flap independently. The results 2.5 for the 230 and 345 ft/sec cases are indeed very similar to those for the teetering rotor. The rotor lift, Fig. 16, increases 1.5 at low collective pitch angles and high speed, and decreases to the point of being negative at high collective pitch angles 0.5 and high speed. The 460 ft/sec articulated case is also quite similar to the 460 ft/sec teetering case.
Collective Pitch (deg) -0.5 The flapping, spindle tilt, and tip path plane angle are also -1 similar to the teetering rotor. The flapping angle (Fig. 17) -1.5 decreases with positive collective, and the spindle tilt (Fig. 18) -2 100 150 200 250 300 350 400 decreases with negative collective. The change in slope of the Airspeed (kts) contour lines between the 345 and 460 ft/sec tip speed cases (b) V = 345 ft/sec T is also present. The tip path plane angle tracks the rotor lift as Advance Ratio well, and no steep gradients are present.
0.4 0.6 0.8 1 1.2 1.4 1.6 The power plots (Fig. 20) also look similar to those for 3.5 the teetering rotor, except the power differences between the tip speeds are more pronounced. In Fig. 15, the differences 2.5 between the 230 and 345 ft/sec tip speed cases were hardly noticeable. In Fig. 20, the differences are still not large but 1.5 it is clear that the power is higher for the 345 ft/sec tip speed case. The power required for the 460 ft/sec tip speed case is 0.5 significantly higher than that for the 345 ft/sec tip speed, again 1500 0 indicating that the rotor should not be operated at this speed.
Collective Pitch (deg) 2500 -0.5 The conclusion is that the optimum collective pitch should 4000 -1 be in the middle of the collective range, although the power -1.5 curves suggest that a bias toward lower collective pitch would -2 100 150 200 250 300 350 400 450 reduce the power required by the rotor. Depending on the Airspeed (kts) maximum speed for the vehicle, this would require a spindle (c) V = 460 ft/sec T tilt of 7–8 deg, which should be a tolerable control angle.
Fig. 16. Lift for an articulated rotor hinged at the center In summary, there do not appear to be any significant fly- of rotation vs. airspeed and collective pitch, V = 230–460 ing qualities or performance issues related to collective pitch. T ft/sec.
Depending on the tip speed and the design cruise speed, some benefit can be realized by careful selection of collective pitch, Advance Ratio Advance Ratio 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.5 -4 -2 -6 2 1.5 1.5 0.5 0.5 -4 -0.5 -0.5 Collective Pitch (deg) Collective Pitch (deg) -1 -2 -1 6 -1.5 -1.5 -2 -2 100 150 200 250 300 350 100 150 200 250 300 350 Airspeed (kts) Airspeed (kts) (a) V = 230 ft/sec (a) V = 230 ft/sec T T Advance Ratio Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.8 1 1.2 1.4 1.6 1.8 4 4 -8 -6 3.5 3.5 -4 -2 3 2.5 2.5 -6 1.5 1.5 1 1 0.5 0.5 -4 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -2 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (b) V = 345 ft/sec (b) V = 345 ft/sec T T Advance Ratio Advance Ratio 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 1.4 1.6 -2 3.5 3.5 -10 3 3 2.5 2.5 -8 1.5 1.5 -6 0.5 0.5 -4 0 0 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -2 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 450 100 150 200 250 300 350 400 450 Airspeed (kts) Airspeed (kts) (c) V = 460 ft/sec (c) V = 460 ft/sec T T Fig. 18. Spindle tilt angle for an articulated rotor hinged Fig. 17. Flapping angle for an articulated rotor hinged at at the center of rotation vs. airspeed and collective pitch, the center of rotation vs. airspeed and collective pitch, V T V = 230–460 ft/sec.
= 230–460 ft/sec.
T Advance Ratio Advance Ratio 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.5 2.5 2 2 2 1.5 1.5 1 1 0.5 0.5 -0.5 -0.5 Collective Pitch (deg) Collective Pitch (deg) -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 100 150 200 250 300 350 Airspeed (kts) Airspeed (kts) (a) V = 230 ft/sec (a) V = 230 ft/sec T T Advance Ratio Advance Ratio 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.8 1 1.2 1.4 1.6 1.8 4 4 3.5 3.5 50 100 2.5 2.5 1.5 1.5 1 1 0.5 0.5 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Airspeed (kts) Airspeed (kts) (b) V = 345 ft/sec T (b) V = 345 ft/sec T Advance Ratio Advance Ratio 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 1.4 1.6 3.5 3.5 3 3 2.5 2.5 1.5 1.5 0.5 500 0.5 1000 1500 0 0 Collective Pitch (deg) Collective Pitch (deg) -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 100 150 200 250 300 350 400 450 100 150 200 250 300 350 400 450 Airspeed (kts) Airspeed (kts) (c) V = 460 ft/sec (c) V = 460 ft/sec T T Fig. 19. Tip path plane angle of attack for an articulated Fig. 20. Power required for an articulated rotor hinged at rotor hinged at the center of rotation vs. airspeed and col- the center of rotation vs. airspeed and collective pitch, V T lective pitch, V = 230–460 ft/sec.
T = 230–460 ft/sec.
but adequate performance and controllability is possible over a range of collective pitch settings.
Elastic Blades A generic CAMRAD II elastic blade model was developed to determine what effect elasticity has on stability. Structural dynamic properties for a production blade are preferable, but elastic properties for a high advance ratio rotor were not avail- able. Instead, elastic properties were chosen to approximate what a production blade might have.
The model was intended to be as simple as possible. The blade has no chordwise offsets of center of gravity, tension center, or shear center, and uniform stiffness. The blade fre- quencies were designed based on a hover tip speed of 650 ft/sec. The flap and lag stiffness values were adjusted for a fundamental lag frequency near 1.2/rev and ratio of lag to flap stiffness of 30:1. Three separate torsion stiffness values were selected for comparison. They were chosen to produce funda- mental torsion frequencies of 4.5/rev, 6.5/rev, and 8.5/rev at a 650 ft/sec tip speed.
A fan plot for the elastic blade model is shown in Fig. 21.
Fig. 21. Frequencies of the CAMRAD II elastic blade mod- The operating speeds and the speed at which the frequencies els with 4.5–6.5/rev torsion frequencies.
were set are shown by solid lines at 230, 345, 460, and 650 ft/sec. The solid symbols are flap and lag modes for the 4.5/rev torsion frequency. The flap and lag modes for the 6.5/rev and 8.5/rev torsion frequencies were nearly the same to the resolution of the plot, so they were not duplicated. The one exception is some interaction between the 4.5/rev torsion mode and the first elastic flap mode which is not present for the other two torsion frequencies. The modes for the three tor- sion frequencies are plotted on the same graph with open sym- bols, but it is important to realize that only one of the torsion modes is present in each model. Since the torsion frequen- cies are less dependent on RPM, the per rev frequencies at the operating speeds of 230–460 ft/sec are higher than 4.5/rev, 6.5/rev and 8.5/rev.
The stability of the elastic blades is shown in Figs. 22–24.
Four modes were used in the elastic blade analysis, one each of teeter, elastic flap, lag, and torsion. The rigid blade teeter- ing mode damping (the only degree of freedom for the rigid blade model) is also shown on the plots for comparison. For these results, the models were trimmed to zero power by tilt- # !" ing the shaft. The lowest tip speed of 230 ft/sec was chosen to !" eliminate the effects of compressibility. Once the trim condi- tion was satisfied, Floquet theory was used to calculate system eigenvalues. The modes were identified by matching the fre- quency and damping to form continuous curves. The damp- ing level shown is the real part of the eigenvalue, so negative Fig. 22. Stability of elastic teetering rotor at tip speed V T numbers are stable, positive numbers unstable.
= 230 ft/sec and torsion frequencies of 4.5/rev.
A hard stability boundary is evident near an advance ratio of 1.5 in Figs. 22–24. Although it appears from the plots that different modes become unstable, but at this boundary several modes become unstable at once. The modes become unstable very rapidly, so it is difficult to obtain a periodic solution and the damping levels (both stable and unstable) in this region are very sensitive to small changes in the trim state. Regardless of the damping levels, it is clear that the rigid blade shows no sign of instability while the elastic blades are clearly unsta- ble and the stability boundary does not depend on the torsion frequency.
The elastic stability is very different from the rigid blade stability in Figs. 8 and 9. For the rigid blade, the rotor is sta- ble to an advance ratio of 3, but for the elastic blade, there is a sharp stability boundary at an advance ratio of about 1.5.
This reinforces the importance of elastic blade properties and shows that even for a teetering rotor, if the blades are not suf- ficiently stiff, an instability will occur.
# !" The rotor thrust and power for these rotor models are shown in Figs. 25 and 26. These show that although there !" is a large difference in stability, there is almost no difference in performance. In Fig. 25, the lift for the 4.5/rev torsion frequency appears to deviate significantly from the other fre- quencies and the rigid blade. The approximately 200 lb of difference in lift represents only about 5% of the vehicle gross Fig. 23. Stability of elastic teetering rotor at tip speed V T weight, so the deviation is actually small. When the torsion = 230 ft/sec and torsion frequencies of 6.5/rev.
frequency is raised to 6.5/rev, the lift is nearly converged to the rigid blade result. The rotor power is dominated by pro- file power, so this deviation is almost imperceptible in Fig. 26.
These results suggest that stability boundary is not caused by changes in the trim state resulting from elastic deflections, but is very sensitive to elastic stiffness.
Conclusions The stability and control of rotors at high advance ratio ap- plicable to a slowed-rotor compound helicopter have been in- vestigated. A simple linear model, rigid blade CAMRAD II models, and an elastic blade CAMRAD II model were devel- oped. The following conclusions are made: 1. The simplified flapping blade analysis suggested that a teetering rotor was the most stable hub configuration.
The articulated rotor was unstable above an advance ra- tio of about 2.2 but could be stabilized to higher speed # !" with δ . The gimbaled rotor was unstable above advance !" ratios of about 2 and was not stabilized by δ .
2. Damping predicted by the simplified analysis and a rigid blade CAMRAD II model were similar outside regions of rotor stall. Trimming the CAMRAD II model to an Fig. 24. Stability of elastic teetering rotor at tip speed V T autorotation condition did not influence the stability.
= 230 ft/sec and torsion frequencies of 8.5/rev.
3. Autorotation can be maintained at two distinct shaft an- gles for the same collective pitch setting. There is a size- able difference in lift between the two trim conditions.
4. The optimum collective pitch for the four hub configurations—teetering, articulated with 0% and 5% "# hinge offset, and rigid—was found to be around 0–1 deg to minimize control input and flapping. There was no collective pitch restriction on power for the collective pitch ranges considered.
5. Rotor power required was only increased slightly by in- creasing the tip speed from 230 to 345 ft/sec, but a large increase was seen increasing from 345 to 460 ft/sec.
6. Blade elasticity was found to drastically reduce the rotor stability. For the particular blade stiffnesses considered, a sharp boundary was predicted near an advance ratio of 1.5. The blade elasticity did not significantly affect the rotor performance.
ν θ ν θ References ν θ Hohenemser, K., “A Type of Lifting Rotor with Inher- ent Stability,” Journal of the Aeronautical Sciences , Vol. 17, September 1950.
! Hohenemser, K., “Remarks on the Unloaded Rotor Type Fig. 25. Comparison of rotor lift for elastic and rigid blade of Convertiplane,” Proceedings of the American Helicopter teetering rotors, V = 230 ft/sec and elastic torsion fre- T Society 11th Annual Forum, Washington, DC, April 1955.
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Hohenemser, K. H., “Some Aerodynamic and Dynamic Problems of the Compound Rotary-Fixed Wing Aircraft,” Proceedings of the American Helicopter Society 8th Annual Forum, Washington, DC, May 1952.
!" Hohenemser, K. H., “Aerodynamic Aspects of the Un- ν loaded Rotor Convertible Helicopter,” Journal of the Amer- θ ν ican Helicopter Society , Vol. 2, (1), January 1957.
θ ν θ Hickey, D. H., “Full-Scale Wind-Tunnel Tests of the Lon- gitudinal Stability and Control Characteristics of the XV-1 Convertiplane in the Autorotating Flight Range,” NACA RM A55K21a, Ames Aeronautical Laboratory, May 1956.
Marks, M. D., “Flight Test Development of the XV-1 Convertiplane,” Journal of the American Helicopter Society , Vol. 2, (1), January 1957.
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Sissingh, G. J., “Dynamics of Rotors Operating at High Advance Ratios,” Journal of the American Helicopter Society , Vol. 13, (3), July 1968.
Fig. 26. Comparison of rotor power for elastic and rigid Peters, D. A. and Hohenemser, K. H., “Application of Flo- blade teetering rotors, V = 230 ft/sec and elastic torsion T quet Transition Matrix to Problems of Lifting Rotor Stability,” frequencies of 6.5–8.5/rev and rigid.
Journal of the American Helicopter Society , Vol. 16, (2), April 1971.
Carter Jr., J., “CarterCopter—A High Technology Gyro- plane,” Proceedings of the American Helicopter Society Ver- tical Lift Aircraft Design Conference, San Francisco, CA, Jan- uary 2000.
Johnson, W., “Rotorcraft Aeromechanics Applications of a Comprehensive Analysis,” Heli Japan 98: AHS Interna- tional Meeting on Advanced Rotorcraft Technology and Dis- aster Relief, Nagarafukumitsu, Gifu, Japan, April 1998.
Floros, M. W. and Johnson, W., “Stability Analysis of the Slowed-Rotor Compound Helicopter Configuration,” Pro- ceedings of the American Helicopter Society 60th Annual Fo- rum, Baltimore, MD, June 2004.