Aeroelastic stability of full-span tiltrotor aircraft model in forward flight

Zhiquan LI,Pinqi XIA

College of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Aeroelastic stability of full-span tiltrotor aircraft model in forward flight

Zhiquan LI,Pinqi XIA*

College of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Aeroelastic stability; Forward flight; Full-span model; Modal analysis; Tiltrotor aircraft

The existing full-span models of the tiltrotor aircraft adopted the rigid blade model without considering the coupling relationship among the elastic blade,wing and fuselage.To overcome the limitations of the existing full-span models and improve the precision of aeroelastic analysis of tiltrotor aircraft in forward flight,the aeroelastic stability analysis model of full-span tiltrotor aircraft in forward flight has been presented in this paper by considering the coupling among elastic blade,wing,fuselage and various components.The analytical model is validated by comparing with the calculation results and experimental data in the existing references.The in fluence of some structural parameters,such as the fuselage degrees of freedom,relative displacement between the hub center and the gravity center,and nacelle length,on the system stability is also investigated.The results show that the fuselage degrees of freedom decrease the critical stability velocity of tiltrotor aircraft,and the variation of the structural parameters has great in fluence on the system stability,and the instability form of system can change between the anti-symmetric and symmetric wing motions of vertical and chordwise bending.

1.Introduction

Tiltrotor aircraft has the flight modes of helicopter and fixed wing turboprop airplane through tilting the rotor system mounted to the nacelle at each wing tip.Compared to the conventional helicopter,tiltrotor aircraft has higher flight speedand severe aeroelastic problems in high-speed forward flight.When the aircraft reaches a certain speed,the hub center may rotate and deviate from the original point,leading severe vibration to limit the forward speed of tiltrotor aircraft.When the link between nacelle and wing is elastic and the link stiffness is low,the system is instable in the form of whirl flutter.When the link between nacelle and wing is rigid or the link stiffness is high,the system is installed in the form of divergent motion of wing tip.1

In the investigation of aeroelastic stability of tiltrotor aircraft in forward flight,Reed established a dynamic model for analyzing the mechanism of whirl flutter of propeller aircraft.2Johnson established a simpli fied semi-span analytical model of tiltrotor in forward flight with nine degrees of freedom,which contained the low order modes of wing and rotor.3Nixon et al.established the elastic wing model by using finite element method4and investigated the in fluence of composite blade with bending and torsion coupling on the aeroelastic response of tiltrotor aircraft.5,6The authors considered the elastic link between wing and nacelle in modeling whirl flutter of tiltrotor aircraft.1Hathaway and Gandhi added the torsional degree of freedom in the stability analysis of tiltrotor in forward flight into the model of rigid blade.7The aeroelastic stability of tiltrotor aircraft in forward flight can be improved by changing structural characteristics of wing,such as the use of composite wing with the coupling bending and torsion,8–11the addition of winglet to the tip of wing,12and changing nacelle shape and mounting position.13In modeling tiltrotor aircraft,the used aerodynamic models had been improved.Kim et al.adopted the Theodorsen unsteady aerodynamic theory.14Johnson extended the unsteady aerodynamic theory used in helicopter into the aerodynamic modeling in tiltrotors.15Yue and Xia established a wake bending dynamic in flow model of the tilting rotor16based on the Peters-He dynamic in flow model17used in helicopters to improve the aerodynamic model of tilting rotor for tiltrotor aircraft.Ye et al.developed a CFD method to simulate aerodynamic interaction among rotor,wing and fuselage for tiltrotor aircraft in helicopter mode.18

It is noted that the above studies were based on the semispan model of tiltrotor aircraft.Tiltrotor aircraft has the zygomorphic rotor/wing/nacelle structures about the longitudinal axis of fuselage,and its each vibration mode has the symmetric and anti-symmetric modal shapes.The coupling of fuselage motions with the modes of rotor and wing motions may have complicated in fluences on aeroelastic stability of tiltrotor aircraft in forward flight.Hence,the existing semi-span models have a certain limitation and inaccuracy for stability analysis of tiltrotor aircraft.Howard presented a full-span model of tiltrotor aircraft with rigid blades in which the in fluence of coupling between the elastic blades,elastic wings and fuselage on the stability was not considered.19

Based on the previous research,1a full-span model of tiltrotor aircraft for analysis of aeroelastic stability in forward flight with elastic blades,elastic wings,rigid nacelle and fuselage has been proposed in this paper,to reveal the difference between the semi-span model and full-span model of tiltrotor aircraft,and further study the dynamic instability of tiltrotor aircraft in forward flight.The in fluence of some structural parameters such as the position of hub center and length of nacelle on the aeroelastic stability of tiltrotor aircraft in forward flight is also investigated in this paper.

2.Dynamic modeling of tiltrotor aircraft in forward flight

2.1.Coordinate systems

The full-span model of tiltrotor aircraft in forward flight and coordinate systems are shown in Fig.1.The full-span model consists of two elastic wings fixed at the fuselage,two articulated rotors mounted at the hubs,and two rigid nacelles which can tilt about the mounting point.In Fig.1,the superscripts‘L’and ‘R’represent the coordinate systems at the left and right wing/rotor systems respectively.

To describe the spatial locations and motion relationships among various structural components,the coordinate systems are de fined.The ground coordinate system denoted by subscript‘G’is chosen as the inertial coordinate system.The fuselage coordinate system is denoted by subscript ‘F’,its original point is located at the center of gravity,its IFaxis is backward along the longitudinal axis of the fuselage,its JFaxis is along the wing and points to the tip of wing,and its KFaxis points upward.The fuselage degrees of freedom are de fined by translational displacements(xF,yF,zF),pitch angle αF,roll angle φFand yaw angle ψF.The undeformed wing coordinate system is denoted by subscript‘I’and its original point is located at the mounting point of nacelle.The offsets(xtw,ytw,ztw)between the mounting point of nacelle and the center of gravity are allowed.The wing deformed coordinate system describes the elastic translational deformations(xH,yH,zH)and rotation motions(αH,φH,ψH)at wing tip.The hub coordinate system is denoted by subscript ‘H’.The elastic blade and coordinate systems are shown in Fig.2.The rotating coordinate system denoted by subscript ‘R’is located at the center of hub and is different from the hub coordinate system by the azimuth angle ψ.The rigid flapping coordinate system denoted by subscript ‘U’accounts for the flapping angle β about the JRaxis in rotating coordinate system.The moderate de flection beam theory is adopted to de fine the deformed coordinate system denoted by subscript‘D’and the cross-section coordinate system which are shown in Fig.3.A point P in the rigid flapping coordinate system moves to point P′after elastic de flection(u,v,w).The total blade pitch angle θ1between the deformed elastic axis and the normal direction of blade cross section is de fined as θ1=θ0+φ^,where θ0is the rigid pitch angle due to control pitch and pre-twist,andφ^is the elastic twist angle.The transformation matrices relating the above coordinate systems are given in the Ref.5

2.2.Dynamic modeling of tiltrotor aircraft

The multi-body dynamic equation of full-span tiltrotor aircraft containing fuselage,wings and rotors is derived based on the Hamilton’s generalized principle.For a non-conservative system,the equation is expressed as

where δU is the variation of the elastic strain energy,δT is the variation of kinetic energy,and δW is the work done by nonconservative forces.The contributions to these energy expressions from the blades,wings and fuselage can be summed as

where the subscript‘b’refers to the blade,‘W’the wing,‘F’the fuselage,and Nbthe number of blades.

In this paper,the nacelle is regarded as a coupling concentrated mass with a tilting angle,and mounted at the wing tip.The virtual energy of nacelle is contained in the formulation of wing virtual energy.

2.2.1.Dynamic modeling of elastic blade

According to the coordinate systems shown in Figs.1 and 2 and the kinematic description,the position vector of an arbitrary point on the cross section of the ith blade can be written in the inertial coordinate system as

where A represents the transformation matrix between the relating coordinate systems,h represents the length of nacelle,and[0,η,ξ]is the position vector of an arbitrary point on the cross section in the deformed coordinate system.

The velocity vector is determined by taking the time derivative of the position vector Eq.(5)in the inertial coordinate system and is given as

where V1,V2and V3are the three velocity components of V in the undeformed wing coordinate system.

The kinetic energy variation of the ith blade is given by

where ρsis blade density,Abis area of blade section,and R is the blade radius.

Substituting the velocity expression Eq.(6)and the variation of the velocity into Eq.(7),the non-dimensional kinetic energy variation is written as

where m0is the reference section mass per unit length,Ω is the rotor rotational speed,δT0is the contribution associated with the foreshortening effect and the full expression can be seen in Ref.20The subscripts ‘F’and ‘H’refer to the terms associated with the fuselage and hub respectively,and T represents the kinetic energy terms associated with the blade variables.

Under the moderate deflection beam assumption and the formulations in Ref.20,the expression for the variation of non-dimensional strain energy of the ith blade is given by

where U represents the strain energy terms associated with the variables,φ is the elastic torsion of blade,and the termsand()′in Eqs.(8)and(9)are bending curves,such as v′= ?v/?x and

2.2.2.Dynamic modeling of wing and nacelle

The wing/nacelle structure is considered as an elastic beam with the coupling concentrated mass at the wing tip.The nacelle is clamped to the wing.

According to the coordinate systems and kinematic description,the three velocity components of an arbitrary point on the cross section located at x position from the wing root are given by

where the terms with subscript‘W’are de fined similarly to the terms used in blade formulation.

The kinematic energy variation of wing is given by

where ρWrepresents wing density,and mpis the mass of nacelle.

Substituting the velocity components Eq.(10)and the variation of velocity into Eq.(11),the non-dimensional kinetic energy variation is written as

where the above terms are de fined similarly to the terms used in blade formulation.

The formulation process of wing strain energy is similar to that of blade because the wing is also modeled by using the moderate deflection theory.

2.2.3.Dynamic modeling of fuselage

According to the kinematic description of rigid body,the kinematic energy variation of fuselage is given by

where mFis the fuselage mass,and IαF,IφFand IψFare the moments of inertia of fuselage in pitch,roll and yaw directions respectively.

2.3.Virtual work of aerodynamic forces

The virtual work of aerodynamic forces is equal to the product of the aerodynamic forces acting on the structural components and the virtual strain of various degrees of freedom.Deriving the virtual work of blade needs to determine the relative air velocity at the cross section of blade,which is composed of three parts:the velocity caused by blade motion,the velocity caused by fuselage motion and the rotor in flow.Taking the time derivative of the position vector Eq.(5),the velocity caused by blade motion in deformed coordinate system can be obtained by

The velocity caused by fuselage motion in deformed coordinate system is given as

where Vx,Vy,Vzare the flight velocity components of tiltrotor aircraft in ground coordinate system.

The rotor in flow is mainly from the induced in flow which is given in deformed coordinate system as

where λiis the induced velocity produced by the thrust perpendicular to the rotor plane.

The total air velocity at the cross section of blade in deformed coordinate system is determined by

where UR,UTand UPare velocity components in the deformed coordinate system.

According to the quasi-steady aerodynamic model described in Ref.20,the aerodynamic lift,drag and moment components on the cross section of blade in deformed coordinate system can be derived.The aerodynamic components in deformed coordinate system can be transformed into the undeformed coordinate system.The virtual work variation of aerodynamic forces on blade is given by

where the subscript‘b’refers to the aerodynamic forces on the blade,and the superscript‘A’refers to the aerodynamic forces in the undeformed coordinate system.

Similarly,the virtual work variation of aerodynamic forces on the wing is derived as

where the terms with subscript‘W’are de fined similarly to the terms used in blade formulation of aerodynamic forces.

The virtual work of aerodynamic forces on the fuselage includes the work done by the aerodynamic forces on rotor LF1,by the aerodynamic forces on wing LF2and by the aerodynamic forces on aerodynamic con figurations of fuselage LF3.The expressions of these aerodynamic forces and moments and the corresponding transformation matrix can be seen in Ref.5The virtual work done by the aerodynamic forces on the fuselage is given by

where all the forces and moments above are included in LF,and LF=LF1+LF2+LF3.

2.4.Finite element discretization of blade and wing

Substituting the above derived virtual energy and work expressions into Eq.(1)can obtain the following expression:

To establish the coupling multi-body dynamic model of tiltrotor aircraft,the elastic blade and wing need to be discretized into a number of finite elements.The discretization process can be seen in Ref.20After discretization,the virtual energy Eq.(1)is expressed by

where N is the number of finite elements.Spatially assembling expression Eq.(22)can obtain the dynamic equations of the ith blade and wing.

The discretized dynamic equation of blade is derived in the rotating coordinate system of blade.The dynamic equations of hub,wing and fuselage are derived in the non-rotating coordinate system.Hence,the dynamic equation of blade needs to be transformed into the non-rotating coordinate system so as to obtain the global modes of tiltrotor aircraft in the nonrotating coordinate system.The transformation process can be seen in Ref.20The dynamic equation of blade after transformation is given by

where

where β,ζ and φ represent the flapping,lag and torsion modes of rotor respectively,the superscripts are the order number of rotor modes,and the subscripts ‘0’,‘1c’and ‘1s’represent the collective,longitudinal and lateral modes of rotor respectively.

2.5.Assembling of dynamic equations

2.5.1.Assembling of rotor/nacelle/wing coupled system

As shown in Fig.1,the rotor is coupled with the nacelle/wing system by six degrees of freedom between the hub and wing tip.The transformation relationship between the hub degrees of freedom and the wing tip degrees of freedom is given by

where the subscript‘tip’refers to wing tip.

Transforming the hub degrees of freedom in Eq.(23)into the wing tip degrees of freedom by the transformation relationship Eq.(24),the equations of motion of rotor/hub and the equations of motion of wing can be assembled at the wing tip.

2.5.2.Assembling of full-span coupled system

In this paper,the left-hand coordinate systems are used in the formulation of left-span rotor/nacelle/wing coupled system.According to Ref.19,the transformation matrices established in the right-hand coordinate systems and in the left-hand coordinate systems are consistent.When assembling the dynamic equations of systems in the left and right hands with the dynamic equation of fuselage,the degrees of freedom need to be transformed to the same hand coordinate system.The coordinate systems and attitude angles of fuselage in the right-hand and left-hand coordinate systems have the transformation relationship as

where the superscript‘L’means degrees of freedom of fuselage in the left-hand coordinate system.Using the transformation relationship Eq.(25),the dynamic equation of fuselage and the left and right structural systems can be assembled in the inertial coordinate system,as shown in Fig.4.

In Fig.4,the superscripts‘L’and ‘R’represent the left-and right-span equations respectively,qr,qWand qFrepresent the degrees of freedom of rotor,wing and fuselage respectively,the terms with the subscripts ‘r’,‘W’and ‘F’refer to dynamic equations of rotor,wing and fuselage respectively.

According to Fig.4 and assembling the mass,damping and stiffness matrices,the full-span dynamic equations of tiltrotor aircraft are given as

The diagram of spatial degrees of freedom of tiltrotor aircraft is shown in Fig.5.The elastic blades and wings are discretized into 48 nodes of finite elements,and each node contains six degrees of freedom.

Considering the structural and aerodynamic damping,the dynamic Eq.(26)needs to be transformed into the equations in state space form,which can be seen in Ref.21

3.Model validation

The test model of tiltrotor aircraft in Ref.5is used as the base model.To validate the model by using the test model,the degrees of freedom of left-span rotor/wing system and fuselage in Eq.(26)are removed,and then the full-span model is transformed into a semi-span model without fuselage motions.The variation of modal frequencies of semi-span model with forward speed is shown in Fig.6,and the variations of damping ratios of the modes of wing are shown in Figs.7(a)–(c).The terms β0,βsand βcrepresent the collective,lateral and longitudinal modes of the rotor in out-plane respectively.The terms ζsand ζcrepresent the lateral and longitudinal modes of rotor in in-plane respectively.The terms p,q1and q2represent the firstorder torsion,vertical bending and chord-wise bending modes of wing respectively.The present analysis is validated by comparing the calculated results of wing modes with the model established by Johnson3based on rigid blade and cantilever wing,the model established by Nixon4based on rigid blade and elastic wing,and the experimental results from the full-scale tests of a semi-span con figuration in wind tunnel provided by NASA.3According to the description in Ref.3,the test gave primarily the dynamic characteristics for the wing vertical bending mode.The frequency and damping ratio were determined from the decaying transient motion of wing tip after imparting initial excitation on wing tip,and the data were limited by the tunnel maximum speed of 200 kn/h(advance ratio is about 0.6).

It can be seen in Fig.6 that after the forward speed is over 0.8ΩR,there are great frequency discrepancies of rotor flapping modes between the present model and the Nixon’s model because of the more significant in fluence of the pitch- flap coupling coefficient on the flapping frequencies of rigid blade than those of elastic blade.Hence,the present model based on elastic blade in this paper is more accurate than the Nixon’s model based on rigid blade.The wing modal frequencies calculated in this paper agree with the results of the Nixon’s model and experimental data.It can be seen in Fig.7 that the obvious discrepancies among all the calculated and experimental results of modal damping ratios exist.There are two main reasons for the obvious discrepancies:(A)the analytical models adopt the simplified wing structural model based on the cantilever beam theory,which could not fully represent the real wing structure used in experiments;(B)the quasi-steady aerodynamic models were used in all the theoretical models without considering the unsteady aerodynamic and nonlinear problems.However,the changing trend of all the curves in Fig.7 is consistent.It is needed to establish more accurate analytical model for reducing the discrepancies.

To illustrate the in fluence of the fuselage motion on the stability of semi-span model,the degrees of freedom of left-span rotor/wing system in Eq.(26)are removed,and the full-span model is transformed into a semi-span model containing the fuselage motion.The variations of modal frequencies and damping ratios with and without fuselage motion are shown in Figs.8 and 9 respectively.It can be seen in Fig.8 that the fuselage motion has no evident in fluence on the modes of rotor but has great in fluence on the modes of wing.When considering the motion of fuselage,the coupling between fuselage and wing increases the modal frequencies of wing.Studies indicate that the frequencies of wing decrease with the increase of the weight of fuselage,and the frequencies of wing with and without fuselage motion are consistent when the fuselage mass is set to be in finite.It can be seen in Fig.9 that the in flow caused by fuselage motions has little effect on the modal damping ratios of rotor.The damping of wing decreases when considering the fuselage motions,which can be explained by the fact that the increase of modal frequencies of wing indicates the increase of stiffness of wing,resulting in the decrease of elastic deformations of wing.When considering the fuselage motions,the instability critical forward speed of semi-span model decreases to about 1.28ΩR.

4.Stability analysis of full-span model

When considering the fuselage motions,the rotor/nacelle/wing systems on both sides of tiltrotor aircraft are not two separate systems.Each global mode of full-span model of tiltrotor aircraft has two symmetric and anti-symmetric modal shapes.The variation of modal frequencies and damping ratios of full-span model with forward speed are shown in Figs.10 and 11 respectively.Since the anti-symmetric modes of wing are coupled with the fuselage motions,the anti-symmetric modal frequencies of wing q1and q2are higher than the symmetric modal frequencies of wing q1and q2,and the symmetric modal damping ratios of wing q1and q2are higher than the anti-symmetric modal damping ratios of wing q1and q2due to the effects of fuselage motions.The symmetric and antisymmetric modal frequencies of rotor are consistent basically,but the symmetric and anti-symmetric modal damping ratios of rotor have some differences.The anti-symmetric q1mode is instable when the forward speed of tiltrotor aircraft is about 1.28ΩR.

In this paper,we change the position of hub center relative to the gravity center of tiltrotor aircraft by changing the mounting position of wing root,mounting position of nacelle and the length of nacelle,and then investigate the in fluences of position change of hub center on the aeroelastic stability of tiltrotor aircraft in forward flight.The coordinate system of undeformed wing is shown in Fig.12.There is an offset(xtw,ytw,ztw)between the mounting point P of nacelle and the gravity center of aircraft OF.The studies have indicated that the source of the instability of tiltrotor aircraft in forward flight is the pitch moment generated by the upward force Fxon rotor disc which can produce the vertical deformation of wing through the coupling between the torsion and vertical deformation modes of wing.1Meanwhile,the aerodynamic force Fq1on wing can enhance the stability for the system,and Fq2decreases the stability.The instability of tiltrotor aircraft in forward flight mainly occurs in wing modes.In this paper,the instability critical forward speeds for the symmetric and anti-symmetric modes of vertical deformation q1of wing and the symmetric mode of chord-wise deformation q2of wing at different positions of hub center are calculated to investigate the changing relationships between the system stability and the position of hub center.

The longitudinal position of the hub center xtwcan be changed by changing the mounting point of wing root.The changing range is 0.5R before and after the center of gravity.The instability critical speed with the variation of the longitudinal position of hub center is calculated as shown in Fig.13,in which Vfis the flutter speed.As the hub center moves backward,the pitch moment produced by the aerodynamic force Fxabout the center of gravity decreases,and the instability critical speed of symmetric and anti-symmetric vertical bending modes q1of wing increases gradually.The instability critical speed of symmetric chord-wise bending mode q2of wing decreases firstly.When the hub center moves behind the gravity center,the moment directions of aerodynamic forces Fyand Fq2are opposite,and the instability critical speed of symmetric chord-wise bending mode q2of wing increases as the hub center moves backward.

Fig.14 shows the variation of instability critical speed with wing length ytwfrom 1.1R to 2R.With the increase of wing length,the moments produced by aerodynamic forces Fq1and Fq2about the center of gravity increase,the instability critical speed of wing in vertical bending increases,and the instability critical speed of wing in chord-wise bending decreases.The instability form of the system changes gradually from the divergent motions of anti-symmetric mode of q1to the divergent motions of symmetric mode of q2.

Fig.15 shows the variation of instability critical speed with the change of vertical mounting position ztwof nacelle.The vertical position of nacelle varies from 0.5R below the center of gravity to 0.2R above the center of gravity.The upward movement of nacelle has signi ficant in fluences on the stability of symmetric mode of q1.When the mounting position of nacelle moves upward,the instability form changes from divergent motion of symmetric mode of q1to anti-symmetric mode of q1,and the instability critical speed keeps relatively a constant.

Fig.16 shows the variation of instability critical speed with the length of nacelle from 0.1R to R.Increasing the length of nacelle increases not only the inertia about the tilting axis and the center of gravity,but also the moments of aerodynamic forces about the tilting axis and the center of gravity.Therefore,the instability critical speed of each mode decreases with the increase of the length of nacelle,and the instability form is mainly the divergent motion of anti-symmetric mode of q1.

5.Conclusions

In this paper,a full-span model including the elastic blade,elastic wing,nacelle and fuselage for aeroelastic stability analysis of tiltrotor aircraft in forward flight has been proposed.The in fluences of the fuselage degrees of freedom and the position of hub center on the stability of system have been investigated.The results indicate that

(1)When the elastic blade model is used to analyze the stability of tiltrotor aircraft in forward flight,the instability critical flight speed of tiltrotor aircraft is higher than that when the rigid blade model is used.

(2)When the fuselage motions are considered,the fuselage motions are coupled with the anti-symmetric mode of wing,resulting in higher anti-symmetric modal frequencies of wing than the symmetric modal frequencies of wing,and the instability critical flight speed of tiltrotor aircraft is lower than that when the fuselage motions are not considered.

(3)In full-span model,the anti-symmetric modal frequencies of wing are higher than the symmetric modal frequencies of wing.

(4)In full-span model,the position of hub center and the center of gravity can affect the instability critical speed of tiltrotor aircraft in forward flight,and the instability form of the system can change between the antisymmetric and symmetric wing motions of vertical and chord-wise bending.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(No.11572150).

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5 September 2016;revised 15 December 2016;accepted 26 July 2017

Available online 16 October 2017

?2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.Thisis anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail address:xiapq@nuaa.edu.cn(P.XIA).

Peer review under responsibility of Editorial Committee of CJA.