Modelling and Analysis of a Coaxial

Naman Rawal Abhishek Graduate Student Assistant Professor [email protected] [email protected] Department of Aerospace Engineering Indian Institute of Technology Kanpur Kanpur, India

ABSTRACT This paper proposes a new coaxial tiltrotor design and discusses the developmentof equationsof motion for its analysis in airplane mode, mode, and transition mode flights. The proposed design is characterized by a coaxial prop- rotor system that is capable of convertingfrom a lifter to a thruster between vertical and forward flight conditions. Half of the wing is also tilted along with the rotor, while the remaining out-board half remains in horizontal position at all times. The rotor dynamics is modeled using rigid blades with only flap degree of freedom. Inflow is estimated using Drees’ model. This paper is divided into two parts. First part of the paper compares and highlights the performance improvementof the proposed vehicle concept over that of the conventionalcoaxial helicopter. Second part analyses the problem of transition of the vehicle from helicopter mode to the fixed wing mode in a quasi-steady manner. Proposed tiltrotor configuration offers significant improvements over helicopter configuration with dramatic improvements in maximum cruise velocity, range, endurance and rate of climb. The quasi-steady transition analysis shows that the proposed design can transition from helicopter mode to aircraft mode successfully at wide range of forward speeds.

INTRODUCTION velopment of a real-time flight simulation model for the XV- 15 (Ref. 3). Eight subsequent simulation periods provided Conventional continue to be significantly slower, major contributions in the areas of control concepts, cockpit noisier and suffer from higher vibration than their fixed wing configuration, handling qualities, pilot workload, failure ef- counterparts. They also have lower range and endurance for fects and recovery procedures, and flight boundary problems similar all up weights. Fixed wing aircraft seems to be bet- and recovery procedures. The fidelity of the simulation also ter in every possible aspect, except that they can’t hover or do made it a valuable pilot training aid, as well as a suitable tool Vertical Take-off and Landing (VTOL). As helicopters pro- for military and civil mission evaluations. Recent simulation gressed past the World War II, researchers started running periods have provided valuable design data for refinement of into expected speed limitations inherent to all that automatic flight control systems. Throughout the program, fi- generate all of its lift and thrust from a rotor in edgewise delity has been a prime issue and has resulted in unique data flight. In addition to a push for higher cruise speeds there and methods for fidelity evaluation which are presented and was requirementfor greater range. There were several ways to discussed in (Ref. 4). achieve improvedspeed and range and they all have their mer- As discussed in (Ref. 5), even though the XV-3 was gener- its. Therefore, in 1950s and 1960s there was a significant im- ally the best behaved, it had several notable issues. During petus on the development of novel hybrid air vehicles, such as low speed flight, weak lateral-directional dynamic stability , compound helicopters etc., which could bridge the and longitudinal and directional controllability were experi- gap between aircrafts and helicopters. Most of these designs enced (Ref. 6). These issues were attributed to rotor wash were complicated and could not be realized with the state of at low altitudes. During cruise, lateral-directional (dutch-roll) technology available during 1950s and 60s. and longitudinal (short-period)damping was reduced as speed During 1970s work began on the XV-15 (Refs. 1,2) which was increased (Ref. 7). This was found to be due to large in- was a 13,000 lb (6000 kg) tiltrotor. Each rotor had three 12.5 plane rotor forces created from blade flapping resulting from foot radius blades mounted on a gimbaled hub. A large mod- aircraft angular rates. In spite of these issues, the tiltrotor con- eling effort was undertaken to accompany development and figuration proved to be effective and transition between heli- flight test of the aircraft. This modeling effort included de- copter and airplane mode was shown to be safe, paving the way for additional research efforts for this configuration. Presented at the 4th Asian/Australian Rotorcraft Forum, IISc, India, November 16–18, 2015. Copyright c 2015 by the An in-depth NASA investigation examined several types Asian/Australian Rotorcraft Forum. All rights reserved. of rotorcraft for large civil transport applications, and con- 1 cluded that the tiltrotor had the best potential to meet the de- APPROACH sired technology goals (Ref. 8). Goals were includedforhover and cruise efficiency, empty weight fraction, and noise. The This paper discusses the modelling, performance study and tiltrotor also presented the lowest developmental risk of the transition analysis of a hybrid tiltrotor/ vehicle, which configurations analysed. One of the four highest risk areas consists of a tilting coaxial rotor system (see Fig. 1). In ad- identified by the investigation was the need for broad spec- dition, the portion of the wing in the downwash of the rotors trum active control, including flight control systems, rotor also tilt along with the rotor system. This arragnement allows load limiting, and vibration and noise. Some general infor- the vehicle to take-off and land and transition into a fixed wing mation on the stability and control of can be found aircraft by tilting the rotor and half-wing combination. in the Generic Tiltrotor Simulation (GTRS) documentation of the XV-15 modeling as documented by Sam Ferguson of Sys- tems Technology, Inc. (Refs. 9, 10). Even though a lot of research has been done in tiltrotors, mainly XV-15, a coaxial tiltrotor has never been realised and is still in experimental stage. Moreover, literature that de- scribes the basics of coaxial tiltrotor aeromechanics using the basic Euler equations, flapping equations of motions, and ba- sic helicopter and airplane theory is difficult to find. A mono tiltrotor design has been proposed by the Baldwin Technology Company as an innovative VTOL concept that integrates a tilt- (a) Hover mode (b) Airplane mode ing coaxial rotor, an aerodynamically deployed folding wing, Fig. 1. Proposed model for a coaxial tiltrotor and an efficient cargo handling system (Ref. 11). It was found that the MTR concept, if practically realized, offers unprece- The hover performance analysis is carried out using Blade dented performance capability in terms of payload, range, and Element Momentum Theory (BEMT). The forward flight mission versatility. analysis is performed using a coaxial rotor dynamics analysis This paper proposes and analyses a new design for coaxial in which the blades are modelled as rigid with hinge offset and tiltrotor which is different and possibly simpler than that stud- root spring with only flap degree of freedom. Aerodynamic ied in (Ref. 11). Current design with coaxial rotor promises to loads are estimated using Blade Element Theory (BET) cou- offer several advantages over conventional tandem rotor tilt- pled to Drees inflow model. The trim analysis in systems (Ref. 12): mode and airplane mode is carried out by Newton Raphson approach. • lower actuator forces and moments for tilting the rotors The transition analysis is carried out in a quasi-steady man- as coaxial rotors tend to have significantly lower gyro- ner and the ability of the proposed design to perform success- scopic loads compared to single rotors during transition. ful transition from helicopter mode to aircraft mode is studied at different forward speeds. The objective is to develop the • tilting of the wing portion below the rotor minimizes the methodology for analysis to provide systematic performance hover losses due to aerodynamic download and interfer- comparison with an equivalent pure helicopter (coaxial con- ence of the rotor wake and the wing. figuration). Impact of the proposed design changes on the key performance parameters such as speed, range, endurance etc. • while a fully tilting wing is ideal to avoid losses, a half are discussed in detail. The detailed derivation of the equa- tiltwing reduces actuator forces and moments compared tions for hover, forward flight trim and performance analysis to a full tiltwing. and transition analysis is presented in Appendix A.

• the remaining fixed half wings is expected to help dur- PERFORMANCE ANALYSIS IN HOVER ing the transition by contributing to lift generation at low forward speeds. process. Numerical studies are done by modifying the XV-15 aircraft parameters for a coaxial tiltrotor design proposed in this pa- per. The modified set of parameters were then used as input The trim analysis for coaxial rotor system is first developed for a quasi-steady simulation. The complete set of data is pre- and validated with Harrington rotor’s test data. The trim anal- sented in Appendix B and are taken from (Refs. 2, 3). Table ysis for coaxial rotor system is then refined to model coaxial 1 summarises the input parameters used for this simulation. tiltrotor configuration. The performance benefits in terms of The following items should be noted prior to this discussion: range, endurance, rate of climb etc. of proposed coaxial tiltro- tor design over corresponding coaxial helicopter are system- atically studied. A quasi-steady analysis of transition of the • The center of gravity is assumed to be constant and does tiltrotor from helicopter mode to aircraft mode is carried out. not move with the movement of the nacelle. 2 • All thetrims wererun at 6000kg grossweight, a0◦ flight path angle and turn rate equal to 0◦/s. 0.01 ) P • Current analysis does not include modelling of stall char- 0.008 acteristics of either rotary wing or fixed wing. 0.006 • The performance analysis provides results for the mini-

mum power consumption of the rotors under a given set 0.004 of conditions, and so provides a datum against which the Power Coefficient (C efficiency of a real rotor can be measured. 0.002 Present Simulation Harrington’s Experiment

0 No. of blades per rotor 3 0 0.2 0.4 0.6 0.8 1 Thrust Coefficient (C ) −3 No. of rotors 2 T x 10 Rotor solidity 0.089 No. of engines 1 Fig. 2. CT vs CP prediction for Harrington Rotor 1 at 450 Max. take-off weight(kg) 6000 RPM and σ = 0.054 Rotor Diameter(m) 8 Wingspan(m) 10 due to fuselage is unavoidable but design adjustments can be Max. power available(kW) 1800 made such as a tiltwing to avoid vertical drag due to wing. Fuel weight(kg) 676 In the simplest form the vertical drag can be accounted for Wing aspect ratio 6.25 by assuming an equivalent drag area fν or a drag coefficient Specific fuel consumption (kg/Watt-s) 3.8 × 10−4 CDv based on a reference area, say Sre f . This means that the extra rotor thrust to overcome this drag will be Table 1. Key design inputs for coaxial tiltrotor simulation

1 2 ∆T = Dv = ρν¯ fnu (1) Validation of Hover Analysis 2

The modelling process involved developing a trim routine for where ν¯ is the average velocity of the rotor slipstream. a generic coaxial rotor and then modifying the math model Using the Simple Momentum Theory, an expression for to incorporate the effects of a tilting mast. The hover per- calculating the net rotor power requirement is given by formance prediction for a regular coaxial rotorcraft using the (Ref. 14): BEMT is validated against the experimental results obtained by Harrington for nominally full scale rotor systems (Ref. 13). The results comparing the current predictions with experi- W Vc W mental data for Harrington Rotor 1 with 25ft diameter and P = + + P0 (2) fν   solidity of 0.054 for coaxial rotor with untwisted blades anda 1 − 2 v2ρA fν A u A taper ratio of approximately3:1 are shown in Fig. 2. Predicted  u   t   results show excellent correlation. Using the above equation, the download penalty is calcu- The coupled trim code for coaxial rotor has also been de- lated for three different configurations based on the area of veloped and validated and the combination of the two analysis wing that is tilted along with the rotor system and shown in would be used to carry out performance evaluation and tran- Table 2. These three configurations are: sition analysis of the novel coaxial tiltrotor/tiltwing vehicle.

1. No wing area under tilt Hovering Performance Comparison 2. Half wing area under tilt An important feature of the proposed coaxial tiltrotor is the half tiltwing design. It is normally assumed that the total 3. Full wing area under tilt thrust, T, required by the rotor system is equal to the weight of the helicopter, W. However, there is usuaiiy an extra incre- ment in power required because of the download or vertical It can be intuitively stated that the actuator force required drag, Dv, on the helicopter fuselage that results from the ac- to tilt a wing is directly proportional to the wing area under tion of the rotor slipstream velocity. Typically, the vertical tilt. On the other hand Table 2 shows an inverse relation in download on the fuselage can be up to 5% of the gross takeoff power required to hover with area of wing under tilt. Thus, weight, but it can be much higher for rotorcraft designs such qualitatively, it can be stated that a half tiltwing would provide as compounds or tilt-rotors that have large wings situated in a trade-off between actuator forces and download penalty and the downwash field below the rotor. The download penalty is therefore incorporated in the current hybrid tiltrotor design. 3 Vehicle Configuration Power Required in Hover No Tiltwing 2245 kW 0.07 Half Tiltwing 1546 kW 0.06 λ Full Tiltwing 1165 kW u λ l Table 2. Comparison of power required in hover for dif- 0.05 ferent tiltwing configurations λ 0.04

COAXIAL TILTROTOR SIMULATION Inflow, 0.03 The simulation was designed to incorporate the elements of basic rotary wing dynamics along with reasonable assump- 0.02 tions while maintaining the accuracy of the model. The per- formance analysis involves running the simulation for oper- 0.01 0 20 40 60 80 100 ational extremities of the proposed vehicle and obtaining the True Airspeed (in m/s) trim results. A qualitative and quantitative analysis is then carried out to predict the performance advantage of one oper- Fig. 3. Non-dimensional inflow in helicopter mode( w/o ational mode over another. wings) The most important factor in the development of vehicles that can operate in multiple configurations/orientations is the 1 feasibility of transition. To investigate the proposed design’s λ 0.9 u transition capabilities, a quasi-static analysis is carried out for λ a prescribed transition corridor. The results of this analysis l then throws light on the feasibility and nature of transition. 0.8 λ 0.7 Trim Analysis Inflow, 0.6 The two operational extremities of the proposed design are 0.5 helicopter mode (w/o wings) and airplane mode (with wings) for which the performance analysis is carried out. The vehicle 0.4 configuration transitions from helicopter mode (w/o wings) ◦ ◦ 80 100 120 140 160 180 200 (βM = 0 ) to airplane mode (with wings) (βM = 90 ) where True Airspeed (in m/s) βM is the angle between the coaxial rotor mast with the verti- cal axis of the aircraft. The analysis was carried out for speeds from 0 m/s to 105 m/s (378 km/hr) and from 90 m/s (324 Fig. 4. Non-dimensional inflow in airplane mode (with km/hr) to 200 m/s (720 km/hr). wings) Figures 3 and 4 shows trim results for non dimensional In airplane mode (with wings), the cyclic pitch require- rotor inflow on upper and lower rotor. In helicopter mode (w/o ments are very small compared to the collective pitch require- wings), the upper rotor experiences a lesser inflow compared ments. This result seem intuitive as the vehicle is moving to the lower rotor. This can be attributed to the fact that the in the axial direction. The collective power requirement for lower rotor operates in the vena contracta of the upper rotor. both rotors are almost equal. The difference between the two In airplane mode (with wings), the inflow velocities seen by collectives is negligible compared to the high requirements in both the rotors are almost equal for a given flight condition as forward flight. the difference due to interference is very small compared to The collective pitch requirement shown in the results may the high inflow velocities as it operates at higher speeds. appear absurd as they range from about 0◦ to 45◦ which is Figures 5 and 6 the control angle requirements to trim impossible to achieve without encountering stall. A further the vehicle in a given flight condition. In helicopter mode investigation of local blade element reveals the effective an- (w/o wings), the collective pitch requirement for lower rotor gle of attack at the blade element. The results are shown in is slightly higher than the upper rotor which is again a con- Figures 8, 7 for extremities i.e. helicopter mode (w/o wings) sequence of the lower rotor operating under the influence of and airplane mode (with wings). It is clear from the plots that the wake of the upper rotor. At hover, the cyclic pitch re- the blade root might be under stall but for most part, the ef- quirements are zero but as the vehicle gains forward speed, fective angle of attack is within reasonable limits. the rotors tilts backwards as the longitudinal cyclic pitch re- The attitude of the vehicle in helicopter mode (w/o wings) quirement increases in the negative direction. Lateral cyclic is shown in Fig. 9. Pitch requirements in helicopter mode pitch requirementvariation is very small as the rotor mast tilts (w/o wings) shows a decrease which means a nose down to counter the unbalanced side slip forces and roll moments. movement which when compared to the longitudinal cyclic 4 20 20

10 15

0 10

° θ β = 0 , V = 0 m/s −10 0 5 M u (in degrees) e

θ α 0 l θ 0 −20 1c Control Angles (in degrees) θ 1s −30 −5 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 True Airspeed (in m/s) Non−dimensional radial position r/R

◦ Fig. 5. Control angles in helicopter mode (w/o wings) Fig. 7. Effective angle of attack at the blade for βM = 0 at V = 0 m/s 90 θ 20 80 0 u θ 70 0 l 18 θ 60 1c θ 50 1s 16 40 30 14 ° (in degrees) β e = 90 , V = 180 m/s M

20 α

Control Angles (in degrees) 10 12 0

80 100 120 140 160 180 200 10 True Airspeed (in m/s) 0 0.2 0.4 0.6 0.8 1 Non−dimensional radial position r/R Fig. 6. Control angles in airplane mode (with wings) ◦ Fig. 8. Effective angle of attack at the blade for βM = 90 pitch requirement in helicopter mode depicts that the rotor at V = 180m/s mast tilts backward as the vehicle pitches in nose down di- The induced power is generated due to the drag induced rection. The roll requirement is very negligible in helicopter as a consequence of development of lift at the blade element. mode (w/o wings). In airplane mode (with wings), the roll Climb Power is the power required to climb. Parasite Power and pitch requirements are shown in Fig. 10. is required to overcome the drag of the helicopter. Profile power is needed to turn the rotor in air. Since the vehicle Performance Prediction is in forward flight, the rotor disk is always under climb and thus climb power contributes significantly to the total power To analyse the performance of the coaxial tiltrotor, several requirement. Parasite power is accounted for using the para- performance parameters are evaluated and compared for he- site drag which is calculated in terms of equivalent flat plate licopter mode (w/o wings) against the airplane mode (with area. For a tiltwing configuration, this flat plate area will be wings). These performance parameters include calculation of a function of the mast angle but is assumed to be constant for Power Requirements, Rate of Climb, Endurance and Range. this analysis. Further, the advantages of a half tiltwing design are investi- gated using Simple Momentum Theory. The power curves for both helicopter mode (w/o wings) and airplane mode (with wings) are shown in Fig. 11. The With all the power transmitted to the main rotor system following inferences are drawn from the results obtained. through the shaft, we have P = ΩQ. It can be shown that the nondimensional power coefficient is equal to the nondimen- • The power requirements for helicopter mode (w/o wings) sional torque coefficient. The total power can be written as a reaches the power available for much lower airspeed than combination of four components as: when the vehicle is in the airplane mode (with wings). This is a reflection of the clear airspeed advantage that P = Pi + Pc + Pp + P0 (3) the airplane possesses over the conventional helicopter. 5 5 2500 pitch Helicopter Mode roll Airplane Mode 2000 Power Available

0 1500

1000

−5 Power (in kW)

500 Attitude/ Euler Angles (in degrees) −10 0 0 20 40 60 80 100 0 50 100 150 200 250 True Airspeed (in m/s) True Airspeed (in m/s)

Fig. 9. Attitude/Euler angles in helicopter mode (w/o Fig. 11. Predicted rotor power vs. true airspeed wings) WTO−Wf V dW R = − ∞ (4) 6 cP ZWTO pitch 4 The total distance covered during a range flight is the in- roll tegral of this expression and is shown in Fig. 12 for both he- 2 licopter (w/o wings) and airplane mode (with wings). The predictions made from the results are itemized below. 0 • The maximum range of the vehicle increases by 100% in −2 airplane mode (with wings) than in helicopter mode (w/o wings). The maximum range is predicted to exceed 1600 −4 kms in airplane mode (with wings) compared to 800 kms

Attitude/ Euler Angles (in degrees) in helicopter mode. −6 80 100 120 140 160 180 200 • The vehicle is capableof flying longer distances at higher True Airspeed (in m/s) speeds in airplane mode (with wings) than in helicopter mode (w/o wings). Fig. 10. Attitude/Euler angles in airplane mode (with wings) • Upto 400% increase is predicted in range when flying in • The maximum cruise speed in helicopter mode is pre- airplane mode (with wings) at 100 m/s (360 km/hr) as dicted to exceed 100 m/s (360 km/hr) whereas in airplane compared to flying in helicopter mode (w/o wings) at the mode (with wings), the vehicle can theoretically achieve same speed. speeds exceeding 200 m/s (720 km/hr).

• Upto 75% reduction in power requirements is predicted Endurance while flying in airplane mode (with wings) as compared to helicopter mode (w/o wings). The maximum reduc- Total time an airplane stays in the air on a full tank of fuel for tion in power requirements can be achieved while flying a given flight condition is defined as the endurance of an air- at airspeeds close to 100 m/s (360 km/hr) than in heli- craft. The generalised endurance parameters are specific fuel copter mode (w/o wings) at the same speed. consumption, aircraft weight, engine power, and fuel weight. Endurance can be calculated using the following equation.

Range of Flight WTO−Wf dW E = − (5) cP ZWTO Range of flight is defined as the total ground distance tra- versed on a full tank of fuel. The weight of fuel consumed The total time an airplane can fly is the integral of this per unit power per unit time or Specific Fuel Consumption(c) expression and is shown in Fig. 13 for both helicopter and dictates the range of an aircraft. The generalised range equa- airplane mode (with wings). The inferences drawn from the tion for propeller-driven airplane is given by: results are presented below. 6 The rate of climb(ROC) capability vs. airspeed is shown 2000 in Fig.14 for both flight modes. As a result of much lower Helicopter Mode Airplane Mode power requirements in airplane cruise, the rate of climb capa- bility far exceeds that of in helicopter mode (w/o wings) . The 1500 power available can be judged by comparing it with the en- gine power installed in XV-15 which consists of two engines of about 1250 kW whereas for a coaxial rotor the power avail- 1000 able is assumed to be 1800 kW. Following inferences can be drawn from the results for climb capabilities: Range (in km) 500 • The maximum rate of climb of the vehicle is predicted to exceed 20 m/s (72 km/hr). • The vehicle can climb faster as higher airspeeds in air- 0 0 50 100 150 200 250 plane mode (with wings) as comparedto helicopter mode True Airspeed (in m/s) (w/o wings) .

Fig. 12. Predicted range vs. true airspeed • Upto 500% increase is predicted in rate of climb when the vehicle is flying in airplane mode (with wings) at an • The maximum endurance of the vehicle is predicted to airspeed of 100 m/s (360 km/hr) than in helicopter mode exceed 4.5 hours. (w/o wings) . • Upto 450% increase in endurance is predicted in airplane mode (with wings) than in helicopter mode (w/o wings). 30 The maximum improvement can be achieved by flying at Helicopter Mode 100 m/s (360 km/hr). Airplane Mode 25 • For the same endurance, the vehicle is now capable of flying two times faster in airplane mode (with wings) 20 compared to helicopter mode (w/o wings). 15

10 5 Rate of Climb (in m/s) 5 4 Helicopter Mode Airplane Mode 0 0 50 100 150 200 250 3 True Airspeed (in m/s)

Fig. 14. Predicted rate of climb capability vs. true airspeed 2 Endurance (in hrs) 1 TRANSITION ANALYSIS During transition, the mast angle changes the aircraft config- ◦ ◦ ◦ 0 uration as it varies from 0 to 90 where βM = 0 corresponds 0 50 100 150 200 250 to the helicopter mode (w/o wings) when the rotor plane is True Airspeed (in m/s) parallel to the horizontal axis. The mast tilts forward with a step size of 2.5◦ until the rotor plane is perpendicular to the Fig. 13. Predicted endurance vs. true airspeed ◦ horizontal axis. The operating speed range for βM = 0 is 0 m/s to 50 m/s with a step size of 2.5 m/s. With every sub- sequent step increment in mast angle, each velocity step gets Rate of Climb ◦ incremented by 2.5 m/s. i.e., at βM = 2.5 , the velocities span ◦ Another important parameter in the assessment of aircraft per- from 2.5 m/s to 52.5 m/s. Likewise, for βM = 90 , the velocity formance is climb capability, particularly the rate of climb. range is 90 m/s to 140 m/s. This is the prescribed transition The maximum rate of climb is characterized by the ratio of corridor for a quasi-steady transition analysis. the excess power to the aircraft gross weight as given by For better visualization, the transition results are plotted with a colour gradient. A range of colours are generated cor- Pavail − Preq responding to trim results for every value of βM. The colour Vcmax = (6) field is indicated using a colour bar. WTO 7 Inflow Velocities 0.7 90 Figures 15 and 16 shows the variation of inflow on lower and 80 0.6 upper rotor with respect to forward speed. Note that ’inflow’ ) 70 u here refers to the flow velocity perpendicular to the rotor disk. λ 0.5 As the rotor disk tilts forward, more and more of the resultant 60 flow becomes perpendicular to the rotor disks or parallel to 0.4 50 the mast. 0.3 40 For βM close to zero, the inflow ratio dips with increase 30 in airspeed before rising again. This is consistent with con- 0.2 ventional rotor system results. For mast angles greater than Inflow upper rotor ( 20 ◦ 0.1 20 , the inflow ratio increases with increase in airspeed. This 10 seems intuitive as the rotor disk is now more aligned with the 0 0 oncoming flow. 0 50 100 150 True Airspeed (in m/s) Figure 17 shows the variation of flow parallel to the rotor disks. For a conventional helicopter, flow parallel to the rotor Fig. 16. Non-dimensional inflow in lower rotor disks is a measure of the forward velocity of the vehicle which is why this quantity is referred to as advance ratio, denoted by 0.35 90 µ. However in this analysis, advance ratio is the flow velocity )

µ 80 parallel to the rotor disk in longitudinal direction. The results 0.3 for flow parallel to rotor disk shows an increase for small mast 70 angles. As the mast tilts further forward, the flow becomes 0.25 60 more and more perpendicular to the rotor disk. From the Fig. 17, it can be seen that the flow lines starts decreasing for mast 0.2 50 ◦ angles greater than 45 . 0.15 40 30 0.1 0.7 90 20 0.05 80 10 0.6 Advance ratio w.r.t. rotor disc (

) 70 l 0 0 λ 0.5 0 50 100 150 60 True Airspeed (in m/s) 0.4 50 Fig. 17. Non-dimensional advance ratio on both rotors 0.3 40 30 control sticks can be assigned to the collective pitch require- 0.2 ment and would influence the throttle. Similarly the cyclic Inflow lower rotor ( 20 pitch controls can be assigned to the two pedals and would 0.1 10 influence the orientation of the vehicle.

0 0 Figures 18 and 19 shows the collective pitch requirement 0 50 100 150 for both the rotors. The collective pitch requirement shows an True Airspeed (in m/s) initial decrease which would mean a pull back of the collec- tive stick control at lower mast angles for increasing speed. Fig. 15. Non-dimensional inflow on upper rotor As the mast tilts forward, the control stick is pushed to gain airspeed. For higher mast angles, the collective pitch require- ment decreases relatively. Trim Control Angles Fig. 20, depicts the lateral cyclic pitch control angle. The The proposed vehicle design has four control parameters, two lateral cyclic pitch required to trim the aircraft as the nacelle collective pitch control, one foreach rotorand two cyclic pitch tilts forward decreases initially, followed by a sharp before control, common for both rotors. These four control param- evening out to nearly shows. This follows intuition as the mast eters are depicted in Figures 18, 19, 20, and 21. To control would not require to tilt laterally to maintain flight in airplane the vehicle, the pilot must be able to influence all the control mode (with wings). The lateral cyclic pitch control shows parameters at the same time. Since, there are four control pa- huge variations during transition to balance the roll moments rameters, four control columns are required. Let us assume and side forces generated. that the vehicle has two control sticks located in front of the Fig. 21, depicts the longitudinal cyclic pitch control an- pilot and two pedal controls activated by the pilot’s feet. Two gle. The longitudinal cyclic pitch required to trim the aircraft 8 as the nacelle tilts forward tends to be negative, which would imply a tilt opposite to the direction of motion. This results in 4 90 apparent negative speed stability while converting (aft stick 80 requirement for increasing airspeed). There is a sharp de- 2 70 crease in the rate of change of longitudinal cyclic pitch. The longitudinal cyclic pitch eventually tends to even out to nearly 60 ◦ 0 zero as the rotor mast tilts close to 90 . This follows intuition 50 as the mast would not require to tilt laterally to maintain flight in airplane mode (with wings). 40 (in degrees) −2

1c 30 θ

50 90 −4 20 80 10 40 −6 0 70 0 50 100 150 30 60 True Airspeed (in m/s)

50 20 Fig. 20. Lateral cyclic pitch of both rotorsvs. true airspeed 40 (in degrees)

u 0 10 30 0 90 θ 20 −5 80 0 10 −10 70

−10 0 −15 60 0 50 100 150 True Airspeed (in m/s) −20 50 −25 40 Fig. 18. Collective pitch of upper rotor vs. true airspeed (in degrees) 1s −30 30 θ −35 20

50 90 −40 10

80 −45 0 40 0 50 100 150 70 True Airspeed (in m/s) 30 60 Fig. 21. Longitudinal cyclic pitch of both rotors vs. true 50 20 airspeed 40 speed and the mast tilts further forward, the wing starts con- (in degrees) l 0

θ 10 30 tributing to the lift and the rotors start acting as propellers to 20 provide thrust in forward flight. Also, the pitch requirement 0 decreases as the wing generates higher lift for smaller pitch 10 angles at higher speeds. Thus, the sharp decrease in pitch. −10 0 The roll requirement, shown in Fig. 23. 0 50 100 150 True Airspeed (in m/s) Power in Transition Fig. 19. Collective pitch of lower rotor vs. true airspeed The power consumption over the transition range is shown in Fig. 24. It can be noticed that the power consumption de- Attitude Parameters creases as the rotor mast starts tilting forward upto 15◦ de- noted by the green region. The transition from green to cyan The pitch requirement, shown in Fig. 22, shows an increasing to denotes an increase in the rate of power consumption. This pitch requirement in the early stages of the transition which rate increases upto 40◦ followed by a decrease in the rate of later evens out. At all points of transition, the pitch attitude power consumption denoted by the region in blue to black. is decreasing with increasing airspeed. The increasing pitch requirements with increasing mast angle for smaller mast an- CONCLUSIONS gles can be attributed to lesser contribution of lift in providing lift at lesser speeds. Thus, the vehicle pitches up to allow the This paper discusses the development of the equations of mo- rotors to continue providing lift. As the vehicle gains forward tion for coaxial tiltrotor aircraft covering airplane mode, heli- 9 25 90 3000 90 80 80 20 2500 70 70

15 60 2000 60

50 50 10 1500 40 40

5 30 Power (in kW) 1000 30 Pitch (in degrees) 20 20 0 500 10 10

−5 0 0 0 0 50 100 150 0 50 100 150 True Airspeed (in m/s) True Airspeed (in m/s)

Fig. 22. Pitch attitude vs. true airspeed Fig. 24. Power consumption vs. true airspeed 2. The maximum range of the vehicle increases by 100% 0.5 90 in airplane mode than in helicopter mode (w/o wings). 80 Up to 400% increase is predicted in range when flying 70 in airplane mode at 100 m/s (360 km/hr) as compared to 0 flying in helicopter mode (w/o wings) at the same speed. 60

50 3. The maximum endurance of the vehicle is predicted to −0.5 exceed 4.5 hours and the vehicle can fly up to two times 40 faster for the same endurance. A maximum possible 30 increase of 450% in endurance is predicted in airplane Roll (in degrees) −1 20 mode compared to the highest possible speed in heli- copter mode (w/o wings). 10

−1.5 0 4. A maximum of 500% increase is predicted rate of climb 0 50 100 150 when the vehicle is flying in airplane mode at an airspeed True Airspeed (in m/s) of 100 m/s (360 km/hr) as compared to flying in heli- copter mode (w/o wings) at the same speed. Fig. 23. Roll attitude vs. true airspeed 5. The analysis demonstrates the capability of the coaxial copter mode, and conversion mode flight. Coaxial rotor trim tiltrotor of transitioning from helicopter mode to aircraft and performance analysis is developed and refined for tiltrotor mode at wide range of speeds. The simulated vehicle analysis. Subsequent analysis addresses the performance and could transition from helicopter mode to aircraft mode transition aspects of the coaxial tiltrotor aircraft from the per- for all the airspeeds between 0 m/s to 90 m/s. spective of a quasi-steady trim. Qualitative analysis demon- strated substantial benefit of coaxial tiltrotor design if it were 6. The proposed half tiltwing design is expecter to require to be technically realized. The coaxial tilt rotor was modelled smaller actuators with reduced power requirements when using rigid blades with root spring and hinge offset. Rotor in- compared to a full tiltwing while giving advantage of re- flow was modelled using Drees’ model. Coupled rotor trim duced download penalty. analysis was carried out using Newton-Raphson to perform performance and transition analysis. In general, the proposed coaxial tiltrotor design offers significantly improved perfor- APPENDIX A mance in the aircraft mode when compared with a coaxial he- Governing Equations licopter of same disc loading. The following specific conclu- sions can be drawn from this conceptual study: First, the equations of motion are developed. These equations are then modified for an identical counter-rotating coaxial ro- 1. The maximumcruise speed is predictedto be 100%more tor system. The two set of rotor equations provided the coax- in airplane mode than in helicopter mode. Up to 75% ial rotor loads which are then added to the airframe loads for reduction in power requirements is anticipated while fly- trim analysis. The transition analysis is carried out in quasi- ing in airplane mode compared to helicopter mode (w/o steady manner to demonstrate the feasibility of the vehicle to wings) at higher flight speeds. complete successful transition. 10 Axes Systems

In order to define and transfer forces, moments, and motions for all the moving parts of the vehicle, several sets of coor- dinate systems need to be defined. The coordinate systems and the corresponding transformation matrices are discussed in the following subsections.

Gravity Coordinate System

The gravity axes system is shown in Fig. 25. It is fixed to the aircraft centre of gravity when it is stationary on the ground. The z-axis points vertically downwards in the direc- tion of gravity or the center of the Earth. The x-axis points North and the y-axis points East. Fig. 26. Body-fixed coordinate system

The nacelle axes system shown in Fig. 27 is defined with respect to the body axes system with the help of mast ◦ angle(βM). When βM = 0 , the vehicle is in helicopter mode. In this configuration, the x-axis runs parallel to the x-axis of the body frame, the y-axis runs out the right wing, and the z- axis is perpendicular to the x and y axes directed downward. This system is centered at the nacelle axis of rotation. It is fixed to the nacelle and rotates with it about the y-axis or na- celle axis of rotation).

Fig. 25. Gravity coordinate system

Body Coordinate System

The body axes system is defined with respect to the grav- ity axes system with the help of euler angles which determine the attitude of the aircraft. This system is fixed to the air- craft at the CG and rotates with it.As shown in Fig. 26 the x-axis is directed out of the nose of the aircraft and is paral- lel to the longitudinal axis, the y-axis is directed out of the right wing and the z-axis, perpendicular to the x and y axes, Fig. 27. Nacelle-fixed coordinate system directed downward. The roll, pitch and yaw then refers to the rotation about x-axis, y-axis and z-axis, respectively. TG→B denotes the rotation matrix defined by the Euler Angles. iˆB cosβM 0 −sinβM iˆNAC jˆB = 0 1 0 jˆNAC (8) iˆ iˆ ˆ   ˆ  B G kB sinβM 0 cosβM kNAC jˆ = T jˆ (7)  B G→B  G   kˆ kˆ      B   G Non-rotating Hub Coordinate System where     The non-rotating hub axes system is centered at the rotor ◦ cθcψ cθsψ −sθ hub as shown in Fig. 28. In helicopter mode, i.e. βM = 0 , the x-axis runs parallel to the x-axis of the body frame, however TG→B = sφsθcψ − cφsψ sφsθsψ + cφcψ sφcθ cφsθcψ + sφsψ cφsθsψ − sφcψ cφcθ it is directed aft. The y-axis runs parallel to the body y-axis and the z-axis is directed upwards. The non-rotating hub axis   system is fixed to the hub, therefore when the nacelle rotates, Nacelle Coordinate System the axis rotates with it. 11 Fig. 28. Non-rotating hub-fixed coordinate system Fig. 29. Rotating blade-fixed coordinate system ˆ ˆ iNAC −10 0 iNR parallel to the blade, directed out of the blade. When βk = 0, jˆNAC = 0 1 0 jˆNR (9) the z-axis is directed downward and the y-axis is then perpen- ˆ   ˆ  kNAC 0 0 −1 kNR dicular to the x and z axes, directed opposite to the direction   of motion.     iˆB −cosβM 0 sinβM iˆNR jˆ jˆ B = 0 1 0 NR (10) iˆ iˆ′ ˆ   ˆ  ROT cosβk −sinβk 0 kB −sinβM 0 −cosβM kNR ′     jˆROT = 0 1 0 jˆ (13)   ˆ    ˆ′     kROT  sinβk 0 cosβk k  Rotating Hub Coordinate System       Momentum Theory for Coaxial Rotors The rotating hub axes system, shown in Fig. 29 is fixed to the center of the rotor and rotates with it. It is defined with The Momentum Theory or Simple Momentum Theory pro- respect to the non-rotating hub axes system with the help of th vides the most parsimonious flow model to study the hovering the azimuthal angle of rotation of the k blade (ψk) . When performance of a coaxial rotor system. The main advantage ψk = 0, the rotor blade passes over the tail of the aircraft. of the Simple Momentum Theory, however, is that it provides At this point, the x-axis is parallel to the x-axis of the body results for the minimum power consumption of the rotors un- however, it is directed aft. The y-axis runs parallel to the body der a given set of conditions, and so provides a datum against y-axis. The z-axis is perpendicular to the x and y axes and is which the efficiency of a real rotor can be measured. directed upward. There are four primary ways of modelling the inflow of coaxial rotors:

iˆNR cosψk −sinψk 0 iˆROT jˆNR = sinψk cosψk 0 jˆROT (11) 1. The two rotors rotating in the same plane (practically, ˆ   ˆ  kNR 0 0 1 kROT  they would be very near to each other with minimal rotor   spacing) and operated at equal thrusts.     2. The two rotors in the same plane of rotation and operated iˆNAC −cosψk sinψk 0 iˆROT at equal and opposite torques, i.e., at a torque balance. jˆNAC = sinψk cosψk 0 jˆROT (12)      kˆ 0 0 −1 kˆ  NAC  ROT  3. The two rotors rotating at equal thrusts, with the lower       rotor operating in the fully developed slipstream of the Blade Coordinate System The blade axes system, denoted upper rotor. by a prime, flaps with the blade. It is centered at the blade 4. Two rotors rotating at equal and opposite torques, with hinge point and defined with respect to the rotating hub axes the lower rotor in the fully developed slipstream (i.e., the th system by the flap angle of the k blade (βk). The x-axis runs vena contracta) of the upper rotor. 12 where

−cosβM 0 sinβM TNR→B = 0 1 0   −sinβM 0 −cosβM  

The velocity vector V~NR can be written as ~ ˆ ˆ ˆ VNR = VNRx iNR +VNRy jNR +VNRz kNR (16)

We now define the non dimensional forward speed or ad- vance ratio on the rotor disk as V µ = NRx (17) ΩR

Inflow on Upper Rotor

The inflow on the upper rotor consists of two velocities, i.e., inflow due to vehicle velocity and inflow due to induced velocity. Following the momentum theory in hover, let us as- sume that the induced velocity at the upper rotor disk is νu, and in the far wake, wu = 2νu. The induced velocities are di- rected parallel and opposite to the direction of the thrust vec- tor, Tu. We define the non dimensional induced velocity on Fig. 30. Flow model showing the coaxial rotor system the upper rotor as νu with lower rotor in the vena contracta of the upper ro- λiu = (18) tor (Ref. 14) ΩR Thus, the total non-dimensional inflow on the upper rotor These cases are discussed in detail along with deriva- can be written as tions of the induced power requirements and thrust sharing VNR ν λ = z + u (19) in (Refs. 14, 15). Practically, there is always a finite spacing u ΩR ΩR between the two rotors in a coaxial system to avoid blade col- lisions between the two rotors. Also, the two rotors will gen- Mass flow rate through the upper rotor is given by erally operate at equal and opposite torques to provide zero 2 2 net torque to the helicopterwhen it is operatingin steady flight m˙ u = ρA (λuΩR) + µΩR) (20) conditions. Therefore, for all the cases mentioned above, Case q 4 is of primary practical importance, and is employed in the By momentum conservation present analysis to calculate inflow through coaxial rotors. ˙ 2 2 2 Figure 30 shows the flow field of a coaxial rotor with the Tu = (m)u2νu = 2ρA(ΩR) λiu λu + µ (21) lower rotor operating in the vena contracta of the upper rotor. q Thrusts shared by upper and lower rotors are represented by Non-dimensionalzing thrust on the upper rotor, the thrust Tu and Tℓ, respectively. Because, the upper rotor is not directly coefficient is given by affected by the wake of the lower rotor, it can be treated as a single rotor that is connected to the lower rotor only through 2 2 CTu = 2λiu λu + µ (22) the need for a torque balance. q For a vehicle operating at forward speed V at a certain Rearranging equation 22, the induced inflow on the upper flight path angle(θFP = 0 in this case) from the horizontal rotor can be written as, plane in gravity axes system, the velocity can be written as CTu λi = (23) u 2 λ 2 µ2 V~G = −V cosθFPiˆG +V sinθFPkˆG (14) u + p Using equation 23 and equation 19, the inflow equation for Using equations 7 and 10, we can write the flow velocity the upper rotor can be written as on the rotor disk as

VNRz CTu −1 λu = + (24) V~ T T V~ (15) 2 2 NR = NR→B G→B G ΩR 2 λu + µ     13 p Inflow on Lower Rotor Thus, the inflow equation on the lower rotor then becomes

The lower rotor is identical to the upper rotor rotating in VNRz the opposite direction. The wake from the upper rotor now λ = λi + λi + (32) ℓ ℓ u ΩR strikes the lower rotor parallel and opposite to the thrust vec- tor Tℓ with velocity wu. At infinity, wu = 2νu and the wake A Blade Element Theory area contracts from A to . Since, the lower rotor is not at 2 A A detailed and systematic derivation of blade element theory infinity, we assume a contraction factor fc = , where Ac Ac is discussed in (Refs. 14, 16). The rotor blade is idealized as is the area of the wake from the upper rotor that strikes the a rigid beam element undergoing only flapping motion. The th lower rotor. The inflow field on the lower rotor now consists position vector of any point P on the k blade, of an Nb bladed of two regions, i.e., the inner inflow field of area Ac which ex- rotor system, in the deformed state can be written as periences wake from the upper rotor and the outer inflow field which is unaffected by the wake of the upper rotor. r~ = riˆ′ = r cosβ iˆ + r sinβ kˆ (33) From continuity,the velocity of the wake of the upper rotor p k ROT k ROT is fcνu. We assume that the induced inflow on the lower rotor The angular velocity of the rotor blade is taken as ω~ = is νℓ. ΩkˆROT . It is assumed that the angular velocity of the rotor is a The non dimensional inflow on the inner region can be dω~ constant, and hence, the angular acceleration = 0. written as dt VNR ν f ν λ = z + ℓ + c u (25) ℓinner ΩR ΩR ΩR The absolute velocity of point P has two components: one due to rotation and another due to the flapping motion in the and on the outer region can be written as rotation frame. The absolute velocity of point P is given by

VNRz νℓ λℓouter = + (26) ΩR ΩR dr~p v~ = + ω~ × ~r (34) p dt p The non dimensional induced velocity on the lower rotor  rel can be defined as νℓ λiℓ = (27) ΩR dβk v~p = −r sinβk iˆROT Total Mass flow throughthe lower rotor can then be written dt as + Ωr cosβk jˆROT (35) dβ m˙ ρA λ ΩR 2 V 2 r cosβ k kˆ ℓ = c ( ℓouter ) + NRx + k ROT (28) dt q 2 2 + ρ(A − Ac) (λℓinner ΩR) +VNR x The absolute acceleration of point P is given by q By conservation of momentum, the thrust of the lower ro- tor can be expressed as, 2 d r~p a~p = Tℓ = m˙ℓwℓ − ρAc( fcνu)( fcνu) (29) dt2  rel dω~ dr~p (36) where wℓ is the velocity of the wake of the lower rotor far + × ~r + 2ω~ × dt p dt downstream. The work done by the lower rotor can be written  rel as + ω~ × (ω~ × ~rp) mw˙ 2 ρA ( f ν )( f ν )2 T (ν + ν )= ℓ − c c u c u (30) ℓ u ℓ 2 2 Eliminating w from equation 29 and 30, and substituting 2 ℓ dβk m˙ from eq. 28, we get the following expression for induced a~p = −r cosβk ℓ dt inflow on the lower rotor.   d2β −r sinβ k − Ω2r cosβ iˆ k dt2 k ROT  (37) 2 2 dβk 1 fc(CT + fcλ ) + −2Ωr sinβ jˆ  ℓ iu 2 3  k dt ROT λiℓ = − fc λiu 2CT (λ + ( f − 1)λ )2 + µ2 (31)   ℓ ℓ c iu 2 2  2 2  dβk d βk  + (λℓ − λiu ) + µ  + −r sinβ + r cosβ kˆ  p  k dt k dt2 ROT     ! − λiu p 14 Flapping Equation of Motion V~h = µΩRcosψkiˆROT The flapping equationof motion for a blade is derivedby mak- − µΩRsinψ jˆ (44) ing the total moment about the flap hinge at the root equal to k ROT zero. − λΩRkˆROT

Resolving these velocity components in the blade-fixed (M¯ ) + (M¯ ) = 0 (38) I flap ext flap system, we have

The inertia moment about the root is given as ′ V~h = (µΩRcosψk cosβk − λΩRsinβk)iˆ ′ R − µΩRsinψk jˆ (45) M¯ = r~ × (−ρdra~ ) (39) I p p − (µΩRcosψ sinβ + λΩRcosβ )kˆ′ Z0 k k k Integrating the term over the length of the blade, the inertia Invoking a small angle assumption, equation 45 can be moment in flap motion can be expressed as written as

2 d βk ′ ¯ V~h = (µΩRcosψk − λΩRβk)iˆ (MI ) flap = Ib 2 dt ′ (40) − µΩRsinψk jˆ (46) Sβ kβ 2 + 1 + e + Ω Ib sinβk cosβk ˆ′ I I Ω2 − (µΩRcosψkβk + λΩR)k  b b 

R 2 The relative air velocity at the blade cross section at radial where Ib = 0 ρr dr is the mass moment of inertia of the blade about the flap hinge at the center of the hub. station r from the hub center is given as R Invoking small angle assumption for flap angle βk, the in- ~Vrel = V~h − v~p (47) ertia moment in the flap can be simplified as Substituting equations 43 and 46, we get

2 ˆ′ d βk Sβ kβ 2 ~Vrel = (µΩRcosψk − λΩRβk)i (M¯ I ) flap = Ib + 1 + e + Ω Ibβk (41) dt2 I I Ω2 ˆ′  b b  − (µΩRsinψk + Ωr) j (48) ∗ − (µΩRcosψ β + λΩR − Ωrβ )kˆ′ Sβ kβ 2 k k k We define, 1 + e + 2 = νβ , which allows the in- Ib IbΩ ertia moment to be expressed as  The velocity components can then be expressed as Tangential velocity component at any radial location r: d2β (M¯ ) = I k + ν2Ω2I β (42) I flap b dt2 β b k r U = Ωr + µΩRsinψ = ΩR + µ sinψ (49) T k R k   The reduced expression, assuming βk to be small, for the velocity of point P, in the primed blade axes system, can be Radial velocity component along the blade: written as UR = µΩRcosψk − λΩRβk (50) ′ dβk ′ ′ ˙ ′ v~p = Ωr jˆ + r kˆ = Ωr jˆ + rβkkˆ (43) dt The normal velocity component at r:

For the sake of consistency and convenience, the time UP = λΩR + Ωrβk + βkµΩRcosψk (51) ∗ dβk derivative of flap βk is non-dimensionalized as βk = Ω = dΩt The resultant velocity of the oncoming flow can be written, dβk Ω = Ωβ˙ where ψ is denoted as nondimensional time Ωt. by assuming UP < UT , as dψ k We follow from the equations derived in earlier, that the 2 2 U = UT +UP ≈ UT (52) flow velocity over the rotor disk consists of a horizontal com- q ponent or advance ratio(µ), and a normal componentor inflow The inflow angle is given by ratio(λ). The relative air velocity at the blade section due to the motion of the helicopter and the total induced flow can be UP written as components along the kth blade axes system as tanφ = (53) UT 15 and for small angles The velocity components can be written in non dimen- sional form(from equations 49 and 51) as U φ ≈ P (54) UT UT r u = = + µ sinψ (68) T ΩR R k The sectional aerodynamic lift and drag acting on the air- ∗ foil are given as UP r uP = = λ + βk + βkµ cosψk (69) 1 2 ΩR R L = ρU CCl (55) 2 UP Substituting Cl = aαe and αe = θ − φ = θ − in equa- UT 1 tions 52 and 55, the aerodynamiclift per unit span of the blade D = ρU 2CC (56) 2 d can be written as Resolving these forces along the normal and in-plane di- The blade forces per unit span of the blade are rections of the blade fixed frame, 1 F = − ρCa(ΩR)2[u2 θ − u u ]β (70) iROT 2 T P T k

Fk′ = Lcosφ − Dsinφ (57) 1 C F = ρCa(ΩR)2[u u θ − u2 + d u2 ] (71) jROT 2 T P P a T

Fj′ = −(Lsinφ + Dcosφ) (58) 1 F = ρCa(ΩR)2[u2 θ − u u ] (72) kROT 2 T P T The components of these distributed aerodynamic loads in the rotating blade frame, after neglecting radial drag effects, The external flap moment due to the distributed aerody- can be given as namic lift actig on the blade can be written, using equations 57 and 65, as

F −F ′ sinβ (59) iROT = k k R ˆ′ ˆ′ (Mext ) flap = (ri × Fk′ k )dr Ze R (73) FjROT = Fj′ (60) = (−(r − e)Fk′ )dr Ze Using equations 62, 38 and 41, we get the flap equation as ′ FkROT = Fk cosβk (61) 2 R d βk 2 2 Invoking small angle approximation and assuming L>D, Ib 2 + νβ Ω βk = (r − e)FkROT dr (74) dt e the aerodynamic force components can be approximated as   Z where νβ is the rotating natural frequency of flap dynamics Non-dimensionalising the time derivative term on the LHS Fk ≈ Fk′ = L (62) ROT d2β ∗∗ as k = Ω2β and the integral on the RHS with respect to dt2 k r F ≈−(Lφ + D) (63) the rotor radius R as x = . The flap equation can be written jROT R in symbolic form as

Fi = −F ′ βk (64) ∗∗ ROT i 2 ¯ βk + νβ βk = γMflap (75) The aerodynamic root moment can be obtained as ρaCR4 Here, γ = , the aerodynamic flap moment is then, M~ A =~r × ~F = (r cosβkiˆROT + r sinβkkˆROT ) × ~F (65) Ib 1 R In component form, ¯ Mflap = 2 4 (r − e)FkROT ρaCΩ R e Z (76) ~ ˆ 1 MA = −FjROT r sinβkiROT 1 e 2 = (x − )(u θ − uPuT )dx ˆ 2 e R T + (FiROT r sinβk − r cosβkFkROT ) jROT (66) Z R ˆ + r cosβkFjROT kROT This is a simplified flap equation for a blade with an hinge offset (e) and a flap spring of stiffness kβ . This equation The blade pitch angle consists of the pilot inputand a linear is solved numerically using Newmark’s Algorithm to obtain geometric twist of the blade a steady-state solution while assuming that the aerodynamic loads lag the flap response. The solution obtained is then used θ = θo + θ1c cosψk + θ1s sinψk (67) to calculate rotor hub forces. 16 Drees Inflow 1 R dF C jROT dr s j′ = 2 During the transition from hover into level forward flight, that ρA(ΩR) e dr Z (80) is, within the range 0.0 ≤ µ ≤ 0.1, the induced velocity in 1 aC 2 Cd 2 = (uPuT θ − u + u )dx the plane of the rotor is the most uniform, it being strongly 2πR e P a T affected by the presence of discrete tip vortices that sweep Z R downstream near the rotor plane. Following Glauert’s result for longitudinal inflow in high speed forward flight and con- R 1 dFkROT sidering a longitudinal and a lateral variation in the inflow, the Cs = dr k′ ρA(ΩR)2 dr induced inflow ratio can be written as Ze 1 (81) aC 2 = (u θ − uPuT )dx 2πR e T Z R λi = λo(1 + kxr cosψ + kyr sinψ) (77)

Here kx and ky can be viewed as weighting factors and rep- 1 R dM C iROT dr (82) resent the deviation of the inflow from the uniform value pre- ni′ = 2 ρA(ΩR) e dr dicted by the simple momentum theory. A parsimonious lin- Z ear inflow model frequently employed in basic rotor analysis is attributed to Drees(1949). In this model, the coefficients of R 1 dFkROT the linear part of the inflow are given by : Cn = (r − e) dr j′ ρA(ΩR)2 dr Ze 1 (83) 2 aC e 2 4 1 − cosχ − 1.8µ = (1 − )(u θ − uPuT )dx k = and k = −2µ (78) 2πR e R T x 3 sin χ y Z R  

−1 µx R where the wake skew angle or χ = tan . 1 dFjROT µ λ Cn ′ = (r − e) dr z + i k ρA(ΩR)2 dr   Ze 1 (84) aC e 2 2 Cd = (1 − )(uPuT θ − u + u )dx Rotor Load Calculations 2πR e R P T a Z R

The distribution of aerodynamic and centrifugal forces along Root Loads in Rotating Hub Axes System The root loads in the span, and the structural dynamics of the blade in response rotating hub axes system are obtained by simply transfering to these forces create shear loads and bending loads at the the root loads to the hub. For a non-zero offset, the rotating blade root. hub loads are: For a zero hinge offset, the blade root is at the center of rotation. For a non-zero hinge offset, it is at a distance e out- board from the center of rotation. By ’loads’ we mean ’reac- CF = Cs CM = Cn (85) iROT i′ iROT i′ tion’ forces generated by the net balance of all forces acting CF = Cs CM = Cs + eCs (86) jROT j′ jROT j′ k′ over the blade span. Let si′ , s j′ , and sk′ be the three shear ′ ′ ′ CF = Cs CM = −Cs − eCs (87) loads, in-plane, radial, and vertical. Let ni , n j , and nk be the kROT k′ kROT k′ j′ bending loads, flap bending moment, torsion moment (posi- tive for leading edge up), and chord bending moment (positive The rotating frame hub loads for a counter rotating rotor in lag direction). They occur at the blade root, rotate with the can be written as blade, and vary with the azimuth angle. Thus they are called the rotating root loads or root reactions. Since the forces and moments are calculated by integrating over a complete rota- CF = Cs CM = Cn (88) tion, the centrifugal and trigonometric terms that integrate to iROT i′ iROT i′ CF = −Cs CM = Cs + eCs (89) zero over the limits 0 to 2π are ignored in the formulation. jROT j′ jROT j′ k′ The root forces and moments are calculated in the non dimen- CFk = Cs ′ CMk = Cs ′ + eCs ′ (90) sional form in the blade axes system are given as follows: ROT k ROT k j Loads in Non-rotating Hub Axes System Root loads in the non-rotating hub axes system can be obtained by simply re- R 1 dFiROT Cs = − dr solving them in the NR frame for each blade and adding them i′ ρA(ΩR)2 dr Ze together. Let m = 1,2,...Nb be the blade number. ψm be the 1 (79) aC 2 azimuthal location of each blade m. The non-dimensional = −βk(u θ − uPuT )dx 2πR e T loads can be expressed as, Z R 17 where ~ru denote the position vector of the hub centre of Nb rotor 1 or upper rotor. Similiarly, for lower rotor, the moments CF = (CF cosψm +CF sinψm) (91) iNR ∑ iROT jROT can be written as m=1 Nb ~ ~ ~ CF = (CF sinψm −CF cosψm) (92) MBℓ = TNR→B MNRℓ +~rℓ × FBℓ (108) jNR ∑ iROT jROT m=1   Nb Airframe Loads and Moments CF = CF (93) kNR ∑ kROT m=1 The trim equations for the aircraft depend on the forces and Nb moments of each component. The contributions are CM = (CM cosψm +CM sinψm) (94) iNR ∑ iROT jROT discussed earlier. The methodology used to calculate the air- m=1 frame forces and moments is to determine the lift and drag of Nb CM = (CM sinψm −CM cosψm) (95) each component, and then, using the relative position on the jNR ∑ iROT jROT m=1 aircraft, determine the associated body axis forces and mo- Nb ments. By definition : CM = CM (96) kNR ∑ kROT m=1 1 • Dynamic Pressure: q = ρV 2 (97) 2 1 • Lift: L = ρV 2AC The assumption here is that all blades have identical root 2 L loads, only shifted in phase. In case the blades are similar, 1 2 the hub loads transmit all the harmonics. Such is the case of • Drag: D = ρV ACD damaged or dissimilar rotors. The periodically varying loads 2 over one complete rotation of the rotor blade are The airframe components used for this analysis were the wing, fuselage, horizontal tail, and vertical tail. For the pur- pose of this analysis, the lift and drag from the nacelles were Rotor Drag or H = ρA(ΩR)2 ×C (98) ignored. Assumptions made to develop these equations are FiNR listed below: Side Force or Y = ρA(ΩR)2 ×C (99) FjNR 2 Thrust or T = ρA(ΩR) ×CF (100) L D kNR • C = and C = . L qA D qA Roll Moment or M = ρA(ΩR)3 ×C (101) x MiNR • C is linear, therefore, C = C α. Pitch Moment or M = ρA(ΩR)3 ×C (102) L ℓ Lα y MjNR 3 • CM is linear, therefore, CM = CM α +CM . Torque or Q = ρA(ΩR) ×CM (103) α 0 kNR 2 (104) • Drag Coefficient CD = CD0 + kCℓ . • Wing has a constant airfoil section. To obtain the steady state components, the loads are aver- aged out over one complete rotation. This provides us the hub • Proprotor effects on the airflow over the wing and other loads used to trim the aircraft. aircraft components are negligible. • The small angle approximation was made for the angle Loads in Body Axes System In order to get the contribution of of attack and sideslip angles. these loads in maintaining equilibrium, they need to be trans- ferred to the body axes system. The forces can be easily trans- ferred using the rotation matrices in the following way. Wing

The equations for lift and drag developed for the wing are F~B = TNR→B F~NR (105) u u as follows: Similarly,   1 2 F~ T F~ Lw = ρV Aw[CLαw (αw − α0L)] (109) Bℓ = NR→B NRℓ (106) 2 The moments in body frameare  a sum of the hub moments and moments produced by the hub forces at the CG of the 1 2 Dw = ρV AwCDw (110) aircraft. We can write the moments due to the upper rotor as 2 where 1 M~ = T M~ +~r × ~F (107) kw = Bu NR→B NRu u Bu πewARw   18 The equations for lift and drag developedfor the horizontal C = C + k C2 Dw D0 w Lw tail are as follows:

The wing forces in gravity frame for an airplane at a given 1 2 LHT = ρV AHT [CLαHT (αHT − α0L)] (119) flight path angle of can be written as: 2

1 2 −Dw cosθFP − Lw sinθFP DHT = ρV A f CD (120) ~ 2 HT FGw = 0 (111) −L cosθ + D sinθ  where w FP w FP 1   kHT = And in the body frame as πeHT ARHT 2 CD = CD + k C ~ ~ HT 0 f LHT FBw = TG→B FGw (112)   The horizontal tail forces in gravity frame for an airplane The moment due to wing forces can be evaluated as at a given flight path angle of can be written as:

~ ~ MBw =~rw × FBw (113) −DHT cosθFP − LHT sinθFP F~G = 0 (121) where~rw is the position vector of the line of aerodynamic HT   center of the wing from the aircraft CG. −LHT cosθFP + DHT sinθFP   Fuselage And in the body frame as F~ = T F~ (122) The equations for lift and drag developed for the fuselage BHT G→B GHT are as follows:   The moment due to horizontal tail forces can be evaluated 1 L = ρV 2A [C (α − α )] (114) as f 2 f Lα f f 0L ~ ~ MBHT =~rHT × FBHT (123) 1 D = ρV 2A C (115) f 2 f D f where~rHT is the position vector of the line of aerodynamic center of the horizontal tail from the aircraft CG. where 1 k f = Vertical Tail πe f AR f C C k C2 D f = D0 + f L f The equations for lift and drag developed for the vertical tail are as follows: The fuselage forces in gravity frame for an airplane at a 1 L = ρV 2A [C (α − α )] (124) given flight path angle of can be written as: VT 2 VT LαVT V T 0L

D L − f cosθFP − f sinθFP 1 2 DVT = ρV A f CD (125) F~G = 0 (116) VT f   2 −L f cosθFP + D f sinθFP where   1 And in the body frame as kVT = πeVT ARVT 2 ~ ~ CD = CD + k f C FB f = TG→B FG f (117) VT 0 LVT The vertical tail forces in gravity frame for an airplane at a   The moment due to fuselage forces can be evaluated as given flight path angle of can be written as:

−D cosθ − L sinθ M~ r ~F VT FP VT FP B f =~ f × B f (118) ~ FGVT = 0 (126) −L cosθ + D sinθ  where~r is the position vector of the line of aerodynamic VT FP VT FP f   center of the fuselage from the aircraft CG. And in the body frame as ~ ~ Horizontal Tail FBVT = TG→B FGVT (127) 19   The moment due to vertical tail forces can be evaluated as

Table 4. Fuselage Parameters M~ =~r × ~F (128) BVT VT BVT Parameter Description Value Units 2 where~rVT is the position vector of the line of aerodynamic Flat Plate Drag, f 1.0 m center of the vertical tail from the aircraft CG. Lift Curve Slope, a f 0.286 /rad

Zero lift angle of attack, α0L f -0.14 rad

Equilibrium Equations Zero AOA Moment Coeffficient, CM0 f -0.0096 kg-m

Moment Coeffience vs. AOA, CMα f 1.145 /rad The equations of equilibrium can be derived by simply adding all the forces and moments in the body frame. The resultant forces and moments in the body frame can be written as a sum of all the components of the aircraft. Table 5. Wing Parameters Parameter Description Value Units ~ ~ ~ ~ ~ ~ ~ ~ A m2 FB = FBu + FBℓ + FBw + FB f + FBHT + FBVT +W = 0 (129) Wing Area, w 16.0 Span, bw 10 m Aspect Ratio, AR 6.25 – Incidence angle, iw 0 rad ~ ~ ~ ~ ~ ~ ~ MB = MBu + MBℓ + MBw + MB f + MBHT + MBVT = 0 (130) Lift Curve Slope, aw 5.31 /rad Zero-Lift Angle of attack, α0L -0.07 rad

The components of forces and moments in body frame to- Profile Drag Coefficient, Cd0w 0.017 – gether are the six equilibrium equations which are used for Wing chord, cw 1.6 m trim analysis using standard Newton-Raphson technique. Wing efficiency factor, ew 0.9 0 Taper Ratio, λ 1 – APPENDIX B Moment coefficient vs. AOA, CMαw 0 rad

Table 3. Rotor Parameters Parameter Description Value Units Table 6. Vertical Tail Parameters Radius, R 4.0 m Parameter Description Symbol Value Units Rotor Speed, Ω 589 RPM Area A 2.4 m2 No. of blades, N 3 – VT b Span b 2.4 m Chord, C 0.4 m VT Aspect Ratio AR 2.4 – Taper ratio, TR 1 – VT Lift Curve Slope a 3.06 /rad Lift Curve Slope, a 5.73 /rad VT Zero-Lift Angle of attack α 0 rad Profile Drag Coefficient, C 0.01 – 0LVT d Profile Drag Coefficient C 0.0017 – Blade Flapping Inertia, I 140 kg − m2 d0VT b Oswald’s efficiency factor e 1 – Flapping Spring Constant, K 1800 kg-m/rad VT β Distance from aircraft CG X 14.5 m Hinge Offset, e/R 0.15 – VT Twist, θtw -41 deg Twist at hub, θtw0 40 deg Rotor(upper) height from CG, h1 2 m Rotor(lower) height from CG, h2 1.5 m Table 7. Horizontal Tail Parameters X distance of mast from CG, Xcg 0 m Parameter Description Symbol Value Units 2 Y distance of mast from CG, Ycg 0 m Area AHT 4.5 m ˙ Rate of Nacelle Movement, βM 0 rad/s Span bHT 4 m Aspect Ratio AR 3.55 – Incidence angle iHT 0 rad Lift Curve Slope aHT 4.03 /rad Zero-Lift Angle of attack α0L 0 rad

Profile Drag Coefficient Cd0HT 0.0088 – Efficiency factor eHT 0.8 0 Distance from aircraft CG XHT 14.2 m

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