Honggang GAO,Ao HE,Zhenghong GAO,Yang NA,Yangping DENG

School of Aeronautics,Northwestern Polytechnical University,Xi'an 710072,China

KEYWORDS Canard rotor/wing aircraft;Dynamics analysis;Flight dynamics;Helicopter;Linearization;Model building

Abstract The aerodynamic layout of the Canard Rotor/Wing(CRW)aircraft in helicopter flight mode differs significantly from that of conventional helicopters.In order to study the flight dynamics characteristics of CRW aircraft in helicopter mode,first,the aerodynamic model of the main rotor system is established based on the blade element theory and wind tunnel test results.The aerodynamic forces and moments of the canard wing,horizontal tail,vertical tail and fuselage are obtained via theoretical analysis and empirical formula.The flight dynamics model of the CRW aircraft in helicopter mode is developed and validated by flight test data.Next,a method of model trimming using an optimization algorithm is proposed.The flight dynamics characteristics of the CRW are investigated by the method of linearized small perturbations via Simulink.The trim results are consistent with the conventional helicopter characteristics,and the results show that with increasing forward flight speed,the canard wing and horizontal tail can provide considerable lift,which reflects the unique characteristics of the CRW aircraft.Finally,mode analysis is implemented for the linearized CRW in helicopter mode.The results demonstrate that the stability of majority modes increases with increasing flight speed.However,one mode that diverges monotonously,and the reason is that the CRW helicopter mode has a large vertical tail compared to the conventional helicopter.The results of the dynamic analysis provide optimization guidance and reference for the overall design of the CRW aircraft in helicopter mode,and the model developed can be used for control system design.

1.Introduction

A Canard Rotor/Wing(CRW)aircraft is a high-speed helicopter which combines the hover and low-speed flight characteristics of a helicopter with the high-speed cruise characteristics of a fixed-wing aircraft.It is promising for both military and civilian fields.CRW has three flight modes:hover and low-speed forward flight in helicopter mode,cruise and high-speed flight in fixed-wing mode,and the conversion mode between these two modes.

The concept of CRW aircraft was proposed by the United States in the early 1990s as they carried out the demonstration and flight test of a CRW aircraft.Bass et al.1studied the aerodynamic characteristics of a CRW airplane in the helicopter mode and fixed-wing mode via a wind tunnel test.Pandya and Aftosmis2investigated the aerodynamic load of CRW aircraft through non-viscous numerical simulations to understand the flight characteristics of CRW aircraft during conversion from helicopter to fixed-wing.Kong and Park3examined the transient performance of the CRW aircraft propulsion system during flight mode conversion on Boeing's tip-jet CRW aircraft(Fig.1)with an H-tailed aerodynamic layout.In China,the research of CRW aircraft mainly focuses on a series of basic research4-10and flight test verification by Gao's team of Northwestern Polytechnical University.Among them,Deng et al.7and He et al.8studied the aerodynamic interference between the main rotor and fixed-wings based on wind tunnel tests.Sun et al.4,5explored the aerodynamic characteristics of the aerofoil of main rotor using CFD;their team completed the flight tests for all flight modes of the CRW aircraft.

A flight dynamics model can be used to evaluate the performance of an aircraft's overall design and guide the optimization of the overall design with the evaluation results; in addition,it can be used to design the flight control system.Model-based advanced control methods,in particular,have higher requirements for the flight dynamics model.Therefore,the study of flight dynamics characteristics is especially important for both the overall design and the flight control system design.

Hover and low-speed flight are the main applications of the helicopter flight mode for CRW aircraft.The dynamic characteristics of the CRW in hover were studied by means of flight identification in Ref.10;however,the method is difficult to be applied in low-speed flight because the frequency sweep signal needs to be added onto a trim state,and it is difficult to operate a CRW aircraft manually at the trim state in low-speed flight.Moreover,the method described in Ref.10can only be used to obtain a linear model.This paper focuses on the CRW aircraft(Figs.2 and 3)developed by the author's group,and uses the method of wind tunnel test,theoretical analysis,and empirical formula to study the flight dynamics characteristics of the aircraft in hover and low-speed flight.Since the CRW in Fig.2 can fly in fixed-wing mode when the speed is greater than 17 m/s,and the helicopter mode only needs to perform hover and low-speed flight,this paper chooses the speed range from 0 to 20 m/s to study the helicopter mode of CRW.

Fig.1 CRW aircraft of Boeing.

Fig.2 Our CRW aircraft.

The CRW in Fig.2 has three flight modes:(A)Helicopter mode,which operates in the same way as a conventional single rotor with tail rotor helicopter.The power of hover and forward flight is derived from the main rotor.Tail rotor is used to balance the anti-torque generated by the main rotor.The propeller at the head of the CRW does not work in this mode.(B)Fixed-wing mode,which is similar to a threesurface aircraft.At this mode,the main rotor is locked as a fixed wing surface,and the forward flying power comes from the propeller at the head of the CRW.(C)Conversion mode,which is a process of switching from helicopter mode to fixed-wing mode.During the conversion process,the lift provided by the main rotor decreases,and the power of the propeller at the head of the CRW increases gradually.The main rotor and the propeller both work in this mode.The purpose of this paper is to study the flight dynamics characteristics of CRW in helicopter mode,so the propeller is not modelled in the study.

To consider both the fixed-wing mode and helicopter mode,the CRW aircraft has a unique aerodynamic layout,including(A)a small aspect-ratio main rotor with symmetrical aerofoil on the leading and trailing edges;and(B)a three-surface aerodynamic layout that consists of a canard wing,rotary wing and tail wing.To analyse the distinct aerodynamic characteristics of the CRW in comparison with those of a traditional helicopter,this study develops the aerodynamic model of the trapezoidal rotary wing model in the helicopter mode based on the blade element theory,11-13in combination with the results of the wind tunnel test of an elliptical aerofoil with obtuse trailing edge.The tail rotor model is also obtained by the blade element theory.The aerodynamic models of the canard wing,horizontal tail and vertical tail are obtained by theoretical analysis,and the aerodynamic interference model of the main rotor to the fuselage is considered according to the empirical formula;next,the abovementioned rotor model and the aerodynamic models of the fixed wings and fuselage are integrated to construct the flight dynamics model of the CRW.The model is validated by flight test data.Based on the above results,the trim and stability characteristics of the CRW in hover and low-speed forward flight are studied,which lays a foundation for the subsequent overall optimization design and flight control system design.

Fig.3 Three-view diagram of CRW aircraft.

2.Flight dynamics model of CRW aircraft in hover and lowspeed flight

The CRW studied in this paper is a motor-driven single-rotor helicopter with a tail rotor.The main parameters are listed in Table 1.

To formulate the motion equations of CRW,first,we define a body reference frame.This frame is fixed to the airplane with its origin O at the centre of gravity.The x axis is parallel to the geometrical horizontal fuselage datum,y axis is aligned to the starboard and z axis is directed‘‘downwards'”.According to Newton's law and the momentum moment theorem, the dynamic equations of the aircraft under the body frame14-16can be written as follows:

where m is the aircraft mass.u,v and w are the components of the flight velocity on the x,y and z axes of the body axis system.q,p and r are the pitch rate,roll rate and yaw rate,respectively,Ix,Iyand Izare the moments of inertia of the airplane.Ixzis the product of inertia.Fx,Fyand Fzare the resultant forces on the x,y,and z axes,respectively.Mx,Myand Mzare the resultant torques of the x,y and z axes.

Table 1 Main parameters of the CRW aircraft.

The resultant forces and moments acting on the centre of gravity of the aircraft are

where the subscripts denote the following:main rotor(mr),canard wing(cw),horizontal tail(ht),vertical tail(vt),fuselage(f)and tail rotor(tr).

The angular motion equation is

where ?,φ and ψ are the pitch angle,roll angle and yaw angle,respectively.

Eqs.(1),(2),and(5)collectively represent the flight dynamics model employed in this study.

2.1.Aerodynamic force and moment of main rotor system

The main rotor hub of the CRW aircraft is in the form of a seesaw with constraints;thus,hub moment exists in the configuration.The axis system of the main rotor hub is defined as follows:the origin is at the centre of the main rotor hub;the z axis is directed downwards along the rotation axis of the main rotor;the y axis is aligned to the starboard,vertically to the rotation axis;and the x axis is set according to the right-hand law.Thus,the aerodynamic forces and moments generated by the main rotor relative to the centre of gravity of the CRW are

Based on the blade element theory and wind tunnel test results,the aerodynamic force and moment of rotor system are calculated in the following sections.

2.1.1.Calculation of blade element aerodynamic force

First,three axis systems related to the blade element were defined:17the wind coordinate system of the blade element,the rotating coordinate system of the blade element and the fixed coordinate system of the blade element.The coordinate origins are all located at the centre of aerodynamic force of the blade element.The detailed definitions of the three axis systems can be found in Ref.17

In Fig.4,U denotes the local inflow velocity and can be decomposed to UTin the pitch-fixed plane and UPvertical to the pitch-regulated plane.θ is the blade pitch,and φ is the inflow angle.The main rotor of the CRW is a symmetric elliptical aerofoil;thus,the angle of attack of zero lift equals zero.According to Fig.4,the lift and drag acting on the blade element are

where a is the lift curve slope of the aerofoil,ρ is the air density,c is the chord length of the blade element,and CDis the drag coefficient.

Fig.4 Schematic of relationship between blade element inflow and force.

The planform of the blade is trapezoidal,and thus,the chord length of the blade element varies at different spanwise positions,as shown in Fig. 5.Therefore,the relationship between the chord length of the blade element c and the distance from the blade element to the hub centre r must be determined.x1in Fig.5 denotes the blade root cutting coefficient,R is the main rotor radius,and x?r/R is the dimensionless spanwise distance between the blade element and the hub centre.

Subsequently,c can be written as

where c0is the root chord length,and c1is the tip chord length.It is assumed that:subsequently,

As the CRW adopted an obtuse trailing edge aerofoil with symmetric leading and trailing edges,the associated lift-drag characteristics are different from those of conventional aerofoils.To simulate the characteristics of the main rotor,the lift curve slope a in Eq.(8)and drag coefficient CDin Eq.(9)were obtained from a wind tunnel test.18-21The mounting of the test model in the wind tunnel is shown in Fig.6.

Fig.7 is the experimental data of the aerofoil lift coefficient CLwith the Angle Of Attack(AOA)α when the Mach number equal to 0.4.The lift coefficient varies linearly with α within a small range of α.

Fig.8 shows the variation of the aerofoil lift coefficient CDwith α;it is noted that the drag characteristics are considerably different from those of conventional aerofoils.As is required in the following calculations by integration of Eq.(9),the drag coefficient acquired from the wind tunnel test is approximately fitted as

Fig.5 Geometric relation between the blade element and main rotor.

Fig.6 Mounting of the high-speed aerofoil test model in the wind tunnel.

Fig.7 Curve of aerofoil lift coefficient versus the angle of attack.

Fig.8 Curves of the aerofoil drag coefficient versus the angle of attack.

As the blade element α varies along the blade spanwise,substituting Eq.(12)directly into Eq.(9)would add complexity to the aerodynamic integration.Therefore,the average value of blade α is approximately denoted asas defined in Ref.11Consequently,the drag coefficient can be denoted as

where σ is the main rotor solidity,which can be obtained according to Fig.5.

k3is defined as k3= (c1+c0)(1 - x1);subsequently,

The inflow analysis can be found in Ref.11

2.1.2.Calculation of main rotor aerodynamic force and moment

The main rotor aerodynamic force of the CRW varies cyclically.Normally,the aerodynamic force is taken as the average per revolution;thus,the thrust of the main rotor can be written as

The thrust coefficient is defined as

where A is the area of the main rotor disk,N is the number of blades,and Ω is the rotational angular velocity of the main rotor.

Ignoring the high frequency flap and integrating Eq.(15),the following expression can be obtained:

where θ0is the collective pitch,B1is the longitudinal cyclic pitch,pwis roll rate on the wind coordinate system of the blade element,and μ is the forward ratio of the hub.It is observed that Eq.(17)contains an unknown variable inflow ratio λ.To acquire the thrust coefficient CT,the relationship between λ and CTmust be determined.

The high-speed flight of CRW is realized by fixed-wing mode.The helicopter mode only performs hover and flight at low-speeds and the CRW has a larger rotor disc load,however,the dynamic inflow model is suitable for the application in the case of a low rotor disc load and medium to large speeds.22The free wake model based on the rotor vortex theory allows the vortex to move with the local airflow,and it can automatically consider the wake distortion caused by the self-induced wake and unsteady motion of the aircraft.Therefore,this model is suitable for use in the unsteady aerodynamic calculation of rotors with hover and low-speed flight characteristics,but the calculation is complex,and there are only a few such relevant applications in engineering.Thus,in this study,it is assumed that the induced airflow of the CRW in hover and low-speed flight is uniform.Based on the momentum theory,12the relationship between λ and CTis

where whis the component of flight speed on the z axis of the hub axis system.Combining Eqs.(17)and(18),the values of λ and CTcan be determined using the Newton algorithm.

Similarly,we can refer to Refs.11-13to obtain the lateral force Sh,backward force Hhand the anti-torque Qmrgenerated by the main rotor.

The main rotor hub of the CRW aircraft is in the form of a seesaw with constraints,and the outreach of flapping hinge is zero.Therefore,the pitch moment and roll moment acting on the hub can be defined as follows:

where Kβis the hub unilateral flap stiffness.a1and b1are the flap backward angle and side angle,respectively,defined with respect to the flap angle β represented by the Fourier coefficient;in this case,the harmonics beyond the second order are ignored.The derivation of the flap equation is not presented in detail in this paper,and interested readers can find it in Refs.23,24

2.2.Aerodynamic force and moment of canard wing,horizontal tail,vertical tail and fuselage

The airflow speed at the horizontal tail is

where ηhtis the loss coefficient of dynamic pressure at the horizontal tail.Because the CRW has a T-shaped tail,the airflow at the horizontal tail is not considerably affected by other components,and ηhttakes 1 here.vx,vyand vzare the components of flight speed on the body axis.xht,yhtand zhtcomprise the coordinate representation of the vector in the body axis from the centre of the aircraft to the aerodynamic centre of the horizontal tail.Considering the installation angle to be φht,the aerodynamic force and moment generated by the horizontal tail relative to the gravity centre of the aircraft are

The aerodynamic force and moment generated by the canard wing can be determined in the same manner as those for the horizontal tail;and the dynamic pressure loss coefficient ηhttakes 1 here.

The solution method of the aerodynamic force and moment of the vertical tail is similar to those of the horizontal tail and can be obtained by simply replacing α in the formula with the sideslip angle β1.

For the research on hover and low-speed flight carried out in this study,the aerodynamic force generated by the fuselage can be ignored.

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2.3.Aerodynamic force and moment of tail rotor system and interference

The tail rotor system does not have an automatic swashplate and only the collective pitch can be adjusted.For the miniature helicopter considered in this study,the flap moment of the tail rotor can be ignored,and only the rotor thrust Ttrand antitorque Qtrwere considered.The calculation method is similar to that of the main motor thrust Tmrand anti-torque Qmrand is not presented herein.The tail rotor adopts the NACA0012 aerofoil.The force and moment acting on the aircraft centre of gravity due to the tail rotor are

As the CRW in this study has a T-shaped tail(Figs.2 and 3),the horizontal tail is much higher than the main rotor,and only the influence of the main rotor on the fuselage is considered during forward flight in the helicopter mode.This influence is equivalent to the vertical washing resistance,which is represented by the vertical weight gain coefficient K.14The interference torque generated by the wake flow is generally obtained by wind tunnel testing,and it is ignored in this study.

where the main rotor disc load isthus,the gravity of the CRW is G=Kmg.

3.Flight test validation of dynamics model

To test the applicability of the flight dynamics model of the CRW in helicopter mode developed above with flight data,special flight tests should be designed.First,the CRW is maintained in a hovering state at a certain height;next,a frequency sweep signal is assigned to the pitch channel,roll channel,yaw channel and altitude channel,sequentially.As a result,the aircraft will fly at a low-speed toward the front,back,left and right along the hovering point.As the spectrum of the sweep data is uniform and the aircraft is fully motivated,the flight data can be used to verify the aerodynamic model.

Using the control input of the flight test as the model input,the simulation response of the model is compared with the flight data to verify the accuracy of the model.The time domain verification results for triaxial angular motion and line motion are as shown in Fig.9.

Comparisons of the flight test results and the model simulation shown in Fig.9 indicate that the predictive results agree well with the test results.This conformity demonstrates the high reliability of the developed model,and it can be used in flight dynamic analysis and control design.

Fig.9 Comparisons of the model prediction and flight test results.

4.Trimming and stability analysis

With the increase of forward flying speed of CRW helicopter mode,not only the main rotor produces lift,but also the canard wing and horizontal tail will produce lift.Therefore,for low-speed forward flight,the trim and stability characteristics of helicopter mode were investigated.

The CRW aircraft has two redundant control systems including the main rotor and tail rotor in the helicopter mode,and the canard wing and horizontal tail control surfaces.Due to the low control efficiency of the canard wing and horizontal tail surfaces in hover and low-speed forward flight,their control functions were ignored.Only the control input of the helicopter is considered,the unknown variables under the trim condition of the CRW dynamic model in the helicopter mode consisting of Eqs.(1),(2)and(5)contain four control inputs:collective pitch θ0,longitudinal cyclic pitch B1,lateral cyclic pitch A1,tail pitch,and two attitude angles,namely,the pitch angle ? and roll angle φ.Other state variables include u,v and w as the given trim condition;q,p and r are assigned a value of zero.Next,trimming and linearized small perturbations at the trim point were introduced for the CRW in hover and low-speed forward flight.

4.1.Trimming and linearized small perturbation

Let X=[FxFyFzMxMyMz]T; the trim cost function is designed as follows:

where weight is the weight designed based on the contributions of different variables to trimming.Subsequently,the trim process is performed to determine the state variables and control variables such that the cost function tends to zero.θ0,B1,A1,and φ are the variable parameters;u,v and w represent the given trim speed;q,p,r,and ψ are taken as zeros;and Eq.(27)is the objective function used to acquire the trim value.

The detailed algorithm of the linearized small perturbation is shown in Fig.10.The dynamics model for the CRW in the helicopter mode was developed using Simulink.Suppose x0is the state variable of a trim point;a small perturbation Δx is superposed with x0,and the Simulink program is run to determine the small perturbation-generated force and moment Δy′.Δy′/Δx is used to approximate the gradient at x0.It is shown that the smaller Δx is,the closer Δy′/Δx is to Δy/Δx;therefore,for the linearized small perturbation,an extremely small value of perturbation was chosen.

To introduce a linearized small perturbation,the Simulink model was processed as shown in Fig.11.Where H is flight altitude.As shown in Fig.11,first,the force and moment transmission of the nonlinear model was cut off to ensure that the model did not reach a new state different from the previous trim state when a small perturbation was added to the state variable.Next,the trim values were assigned to the corresponding variables,and small perturbations were imposed to the state variables one-by-one independently. We ran the model to obtain the increments in the force and moment,generated by the small perturbations. Finally, the force and moment increments were divided by the perturbation.The derivatives of the force and moment with respect to the state variables can be determined at this trim state.Next,the eigenvalues of the system at this trim state were calculated.

Fig.10 Linearized small perturbation.

4.2.Results and analysis

4.2.1.Trimming of hover and low-speed forward flight

The trimming of the CRW in hover and low-speed forward flight was performed using the method proposed in the paper.First,it was assumed that the CRW was flying at a fixed height,and the altitude was selected as the local altitude of 440 m.Second,the states pertaining to 0,5,10,15,and 20 m/s were selected for trimming.The corresponding results are shown in Fig.12.

The trimming results of the model established this paper in hover were verified by flight data.The results are presented in Table 2.As it is difficult to realize straight-line flight at a fixed height in the case of the model trim conditions under manual manipulation,only the trim results of hover were compared with the flight data in this study.

Table 2 shows that most flight data in hover are remarkably close to the corresponding trim results obtained using the model.Among them,the result of the deviation of tail pitch trimming between the flight data and model data is slightly larger than that of the others.The probable reason for this may be that the function of tail rotor is to balance the anti-torque generated by the drag of the main rotor rotation;however,the airflow around the main rotor is complex and the drag cannot be calculated accurately.

Fig.12 shows that:(A)the collective pitch of main rotor and tail rotor pitch decrease with the increase of forward flight speed,whereas the power required by the main rotor and tail rotor decreases;this is in agreement with the flight characteristics of conventional helicopter.(B)The pitch angle of the aircraft while hovering is negative as the centre of gravity of CRW is in front of the main rotor axis.The aircraft pitch angle decreases with the increasing flight speed and this is in agreement with the flight characteristics of conventional helicopter.The roll angle,mainly to generate the lateral component of the main rotor thrust for balancing the thrust generated by the tail rotor,also decreases with increasing speed.As with increasing flight speed,the power required by the main rotor decreases;thus,the values of the main rotor anti-torque,tail rotor thrust and roll angle trim decrease.The result is logical.In addition,Fig.12 also shows that unlike a conventional helicopter,a larger canard wing and horizontal tail can generate considerable lift for the CRW aircraft during forward flight in the helicopter mode.The lift generated by the canard wing and horizontal tail is nearly equal to half of the lift generated by the main rotor.

4.2.2.Stability analysis of helicopter mode in hover and lowspeed flight

A linearized small perturbation was implemented for the trim points of helicopter mode in hover and low-speed forward flight.Eigenvalues were solved for the state space model,and the variation of the eigenvalues with flight speeds is shown in Fig.13.

It is seen from Fig.13 that with increasing flight speed in the helicopter mode,most eigenvalues move toward the left of the real axis,i.e.,the stability of these modes increases.However,there is a pair of minor eigenvalues that appears in the form of conjugate complex roots while hovering,which separates to two real-roots in forward flight: One moves toward the direction of the positive real axis and exhibits unstable monotonous divergence.To determine the reason for this,the eigenvector of this positive real root are calculated.Considering the state point V=10 m/s as an example,the positive real root is 0.148 and the corresponding eigenvector is[-0.12 0.60 0.02 0 0 0.13 0 0 0.78]T.

Fig.11 Simulink model process for linearized small perturbation.

Fig.12 Trim results of the CRW aircraft in hover and low-speed flight mode.

Comparing this with the state variable X=[u v w p q r φ ? ψ]Tchosen for the linearized small perturbation,it is noted that this positive real root is dominated by the lateral velocity v and yaw angle ψ.With increasing flight speed,the stability of this mode degrades.

Table 2 Comparison of flight data and model trimming results in hover.

Fig.13 Variation of eigenvalues with flight speeds.

The reason of appearing the positive real root is also analyzed according to the configuration of CRW helicopter mode.It is found that the eigenvalue of the instability is dominated by the derivative of yaw moment versus lateral velocity Nv,and the Nvof CRW helicopter mode in forward flight is mainly provided by tail rotor and vertical tail.The mode is similar to the spiral mode in fixed-wing aircraft.When the CRW flies forward in helicopter mode,its large vertical tail will weaken the stability of the spiral mode.

5.Conclusions

(1)The good consistency between the simulation and flight test proves that the modeling method in this paper is feasible and the flight dynamics model of CRW in helicopter mode established in the study is reliable.The model can be used for flight dynamic analysis and control law design.

(2)For multi-channel coupling of CRW helicopter mode,the traditional trim method is very complicated and requires a large amount of computation.According to the principle of trim,the method of using optimization algorithm to trim the model is put forward.The trim results of hover and low-speed forward flight of CRW helicopter mode agree well with the flight characteristics of conventional helicopter,and they also reflect the unique characteristics of the CRW aircraft.The trim results can provide guidance for the overall optimization design of the helicopter mode of the CRW aircraft.

(3)A method of linearized small perturbation using a simulation tool was proposed,and the variation trend of the eigenvalues with the flight velocity in helicopter mode was analysed based on the linearized results.The results provide deeper understanding pertaining to the CRW aircraft in helicopter mode and can be used as a guide for the overall design and control system design.

In all,the work described in this paper provides the process and method for the study on the flight dynamics modelling for CRW aircraft in helicopter mode.The trimming method proposed in the study is also applicable to other aircraft,which provides a simple and convenient method for aircraft trimming.At the same time,the small perturbation linearization method described herein can also prove to be an option for future scholars to perform small perturbation linearization.The results lay a foundation for subsequent overall optimization design and flight control system design.