Yi LIU,Chngchun XIE

aBeijing Institute of Electrical and Mechanical Engineering,Beijing 100074,China

bSchool of Aeronautic Science and Engineering,Beihang University,Beijing 100083,China

KEYWORDS Aeroservoelasticity;Flexible aircraft;Geometric nonlinearity;Nonlinear coupled dynamics;Stability control

AbstractA unified theoretical aeroservoelastic stability analysis framework for fiexible aircraft is established in this paper.This linearized state space model for stability analysis is based on nonlinear coupled dynamic equations,in which rigid and elastic motions of aircraft are both considered.The common body coordinate system is utilized as the reference frame in the deduction of dynamic equations,and significant deformations of fiexible aircraft are also fully concerned without any excessive assumptions.Therefore,the obtained nonlinear coupled dynamic models can well reflect the special dynamic coupling mechanics of fiexible aircraft.For aeroservoelastic stability analysis,the coupled dynamic equations are linearized around the nonlinear equilibrium state and together with a control system model to establish a state space model in the time domain.The methodology in this paper can be easily integrated into the industrial design process and complex structures.Numerical results for a complex fiexible aircraft indicate the necessity to consider the nonlinear coupled dynamics and large deformation when dealing with aeroservoelastic stability for fiexible aircraft.

1.Introduction

Aeroelasticity and fiight dynamics are two factors that signi ficantly affect an aircraft’s fiight envelope.For a very fiexible aircraft,these two problems become more complicated and important in the design stage,since the fiexibility of such an aircraft often leads to significant structural deformation and presents notable geometrically nonlinear characteristics.1,2Thus the linear theories and methods used in traditional aeroelasticity and fiight dynamics analysis are not relevant.3,4Furthermore,the extreme length and low stiffness of the fiexible wings5result in natural vibration frequencies in the order of the fiight dynamics so that the aircraft experiences instability characterized by the interaction between the vehicle fiight dynamics and the structural vibrations.6This is a typical coupleddynamic problem betweenflightdynamicsand aeroelasticity,which makes those two subjects cannot be treated unrelated anymore.Therefore,unified coupled dynamic modeling considering fiight dynamic and aeroelastic features is desperately demanded for fiexible aircraft when dealing with aeroelastic and fiight dynamic problems,especially a model taking nonlinear large deformation into account.Nowadays,aeroservoelasticity has become an inevitable analysis process with the control system becoming more and more important.For fiexible aircraft,the effect of a control system acting on the coupled nonlinear dynamics is a new critical problem worth studying,so accurate coupled dynamic modeling concerning about aeroelastic and fiight dynamic features,and a unified aeroservoelastic analysis framework need to be studied.Stability characteristics under the composite influence of the control system and coupled dynamics are also urgent to be figured out.

Early work addressing fiexible structural coupled fiight dynamics and aeroelasticity was performed by Waszak and Schmidt.7The influences of a fiexible structure on fiight dynamics and frequency response of different modal were studied in their work.They found that with an increase of the structural fiexibility,the difference between the rigid modal frequency and the elastic modal frequency tended to get blurry.Soon after that,Van Schoor’s research8also demonstrated the critical importance of considering the aircraft structural dynamics when analyzing the aircraft fiight dynamics offlexible aircraft.Upon their innovative work,a study of coupled fiight dynamics and aeroelasticity was executed,and many researchers suggested that those two disciplines should be analyzed in a unified framework to provide an integrated and comprehensive analysis.9,10Linear cases were studied generally later on,where the structural deformation was small and the change of aircraft configurations was not significant.11For very fiexible aircraft,flight dynamics and aeroelasticity present significantly nonlinear characteristics because of their large elastic structural deflection and the strong coupling between aerodynamics and structural dynamics.Nonlinear analysis methods are strongly recommended.More recently,many American researchers1,2,12focused on analyzing the nonlinear aeroelasticity and fiight dynamics of fiexible aircraft.Their works13,14revealed a significant difference between the shortperiod and phugoid modes of a very fiexible aircraft compared to those of rigid-body,linear aeroelastic cases.In addition,tailless airplanes15with relatively low rigid-body pitch inertias can have relatively high short-period frequencies that interact with elastic deformations,16resulting in a dynamic instability which is known as body-freedom fiutter.6That is a typical instability phenomenon about the fiight dynamics and aeroelasticity coupling.However,most of the nonlinear studies were based on nonlinear beam17modeling methods,which is not convenient for modeling and analyzing in the industry.Multi-body dynamic modeling18and modeling based on quasi-coordinates19can solve the nonlinear coupling problem to a certain extent,but it is not convenient to be applied on all kinds of very fiexible aircraft,especially on complex aircraft,because of the complicated form of equations.A more general applicable coupled nonlinear stability analysis of veryflexible aircraft for engineering application still needs further study.

Aeroservoelasticity,studying the interaction between an aeroelastic system and a control system,is a compulsory work in aircraft design.Similarly,linear cases were firstly studied when structural deformation was small and the difference between the rigid motion frequency and the elastic vibration frequency was obvious.Among that,aerodynamic stability derivative correction methods have been widely used and even embedded in commercial software.For fiexible aircraft,aeroservoelastic analysis seems more significant,since the nonlinear coupled dynamics itself is complicated,and the interaction of the elastic vibration mode on fiight dynamics may seriously affect the aeroservoelastic stability.Many researchers tried to combine fiexible dynamic equations with traditional control methods to investigate the special aeroservoelastic features of fiexible aircraft.Meirovitch and Tuzcu20established a unified maneuver fiight theory of elastic aircraft,used a linear control system to simulate in the time domain,and studied dynamic characteristics.Static output feedback control was added on a nonlinear aeroelastic system by Patil and Hodges,21and the characteristics of fiutter and response under gust for a slender wing were analyzed.Palacios et al.22used an intrinsic beam and nonlinear modal method combined with a linear control strategy to study the closed-loop response of cantilever wing and full fiexible aircraft under discrete gust.Raghavan et al.23utilized a reduced-order model based on mean axis coordinates to eliminate the coupling between the rigid mode and the elastic mode and make a very fiexible fiying wing fiy straight and circle with a certain height and speed under gust turbulence via nonlinear control.Joshi24combined the shortperiod mode and structural elastic modes to study the structure-control interactions from the structural response aspect(open-and closed-loop responses).Haghighat et al.25developed a unified dynamic framework and an improved model-predictive control formulation to investigate the gust load alleviation for a highly fiexible aircraft.These works above are providing great references for later studies,but all have their own application imitations in certain ways or objections.However,the interaction between a control system andflexible aircraft is various and complicated due to nonlinear coupling dynamics,so more generally applicable aeroservoelastic stability modeling for fiexible aircraft needs further study,especially for complex fiexible aircraft,which has a significant reference value for engineering application.

The first objective of this paper is to establish a coupled nonlinear dynamic model for fiexible aircraft based on common body coordinates,which can not only deal with coupling effects between those two disciplines(flight dynamics and aeroelasticity),but also fully take large deformation(structural geometrical nonlinearity)into account.This develops a generally applicable coupled dynamic model for various kinds offlexible aircraft that is convenient for researchers to investigate and understand the special coupled characteristics betweenflight dynamics and aeroelasticity of fiexible aircraft.Then,a unified aeroservoelastic stability analysis framework in a state space form is established by linearizing coupled dynamic equations around the nonlinear equilibrium state under a smalldisturbance hypothesis and combining the control system model in the time domain.The whole analysis framework in this paper follows traditional analysis ways and makes a step further by building more generally applied nonlinear coupled dynamic equations for fiexible aircraft.Therefore,it can be easily applied in the industry and convenient for researchers to investigate and understand the special coupled stability characteristics between fiight dynamics and aeroelasticity under control for fiexible aircraft.

2.Theoretical development

The theoretical development can be divided into four parts:fi rstly,nonlinear coupled dynamic equations are founded based on energy methods,and common body coordinates are utilized as the body-reference frame to fully describe rigid/elastic motions and large structural deformations.Secondly,a nonlinear equilibrium state is selected as a benchmark state to linearize the coupled dynamic equations under a smallvibration hypothesis.By introducing the quasi modes of a deformed structure and the non-planar aerodynamic expression,rigid/elastic coupling dynamic state-space modeling for stability analysis is established.Then,control systems for aeroservoelastic analysis are founded with three basic sessions:sensors,actuator,and control laws in the state-space form.At last,aeroservoelastic stability analysis models are established by integrating the dynamic model and the control system according to input/output relations.

2.1.Rigid/elastic motions coupled dynamic equation

Coupled kinematic equations constitute the foundation of the analysis of fiight dynamics and aeroelasticity for fiexible aircraft,which require considerations of rigid motion Degrees of Freedom(DOFs)and elastic vibration DOFs in an integrative way.In this paper,the fiat geodetic reference frame(OXYZ)is utilized as the inertial frame,and the common body coordinate system(oxyz)as the body-reference frame and deduces from energy equations to establish coupled dynamic equations.The origin of the body-reference axes is constrained to an arbitrary pointed position in the structure.For the symmetric longitudinal case discussed later,we define the ox axis along the undeformed airframe pointing towards the back,and the oy axis vertical to the longitudinal symmetric plane of the undeformed aircraft pointing to the right.Once the body-reference frame has been defined,it moves and rotates together with the aircraft in space.Fig.1 shows the relationship and positions of the two coordinate systems.

Fig.1Inertial and body-reference frames.

The position of an arbitrary mass element of an elastic aircraft can be written in terms of its position relative to the local reference system oxyz r and the position of this local reference system relative to the inertial reference frame OXYZ R0,shown in Fig.2.If the body-reference axes oxyz are translating and rotating relative to inertial space with velocity V and angular velocity x and if the position relative to the local reference system can be expressed by the original position r and elastic deformation u(i.e.,R=R0+ r + u),then the absolute velocity of the mass element is

where v0is the velocity of body-reference system,er and eu are the tensor representations of r and u.

Based on the expression of velocity,the kinetic energy of the whole body can be written as

where m is the mass element of the whole aircraft.The obtained kinetic energy expression is complicated and contains many coupled items since it deduced without assumptions and the large structural deformation of very fiexible aircrafts can be considered.The gravity force can be treated as external force,thus the elastic potential energy is only related with elastic deformation u.

As expressed in the body-reference frame,R;V and x are defined asrespectively.The usual Euler anglesare used to define the rotational relationship between the inertial axes and body-reference axes,which is consistent with the analysis of rigid aircraft.The vector x can be written as

Fig.2Position of a mass element under two coordinate frames.

R0;h and u are selected as the generalized coordinates of the system.is corresponding tensor representations.Then define L=T?Ueis the Lagrange function,in which T is the kinetic energy,Ueis the potential energy and the gravity are treated as external force in this derivation,thus we can get

where the right-hand items of Eq.(5)are the generalized forces of corresponding motions.Element energy is expressed by^.QR0;Qhare the generalized force corresponding to the generalized coordinates R0and h.Substitute kinetic energy expression into Eq.(5)then we can obtain the translate equation,rotation equation and elastic equation.

where M is the whole mass of aircraft,Thus the dynamic equation under common body coordinate system for very fiexible aircraft is obtained.

According to the virtual work principle,the generalized force can be expressed as,

The derivations above demonstrate that Fbis the total external force acting on the elastic structure expressed in the body-reference frame and Mbis the total external moment expressed in the body-reference frame.Quis the generalized force on the elastic structure.All the external forces,no matter which ways they are expressed,should include gravity G,aerodynamic force A(aerodynamic force caused both by rigid motion and elastic motion)and the engine thrust T.

In order to express the specific location of the aircraft in the inertialframe,R can be writtenin the inertial frame.Thus,the translate velocity of cg can be expressed by the cg position in space via common translation matrix,where L is the translating matrix between inertial frame and body referenced frame.

In general,the kinetic equations can be expressed as

Introduce the rigid motion mode(Utfor translate motion and Urfor rotation mode)into common body coordinate system.With the specific grid force expressionthe final coupled dynamic equations containing rigid motions and elastic motions are obtained,which is suitable for all the very fiexible aircrafts dynamic modeling.

2.2.Rigid/elastic coupled state space modeling and stability analysis

Considering thegeometrically nonlinearlargestructural deformation,the nonlinear trimmed state is selected as a benchmark state for stability analysis.Then the existent large deflection under trim state can be expressed as solidified deformation r in common body coordinates to represent the various deformed configurations,in which the cg and moment of inertia may be changed under different deformation.Under the large deformed configuration,the small vibration hypothesis around equilibrium states can be adopted for stability analysis and expressed with quasi-modes.The quasi-mode is obtained at the linearized state around the nonlinear equilibrium states,in which the nonlinear updated tangential stiffness and mass matrix are concerned due to structural geometric nonlinearity.After the linearization,the small vibration can be expressed with modes just like in the linear case,which can be written as:

With the expression of quasi-modes,the Eq.(14)can be rewritten in matrix form as Eq.(15)

where

Select the straight fiight state as benchmark to linearize Eq.(15)by small disturbance assumption,thenThe linearized dynamic equatio

ns are,

where

Although the smalldisturbance hypothesis has been adopted,there are still many coupling items in Eq.(16).Since the large deformation under nonlinear equilibrium state has been solidified,the aircraft can be treated as a complete new configuration.In the equations deduction above,the origin of body referenced frame is an arbitrary point,for stability analysis now the origin of body referenced frame is pointed and transferred to the cg of current nonlinear deformed configuration.Thus,rGequals to zero and the rigid motion modal functions are introduced to simplify the equation,the translational motion wtand rotational motion wrcan be expressed as

From the deduction above we can see that when the origin of body reference frame is transferred to the cg of deformed nonlinear equilibrium configuration,the dynamic equation can be greatly simplified and the inertial coupling can be eliminated.Attention should be paid that all the parameters in Eq.(18)are described in the nonlinear deformed body reference frame,x is the rigid rotational velocity about the cg of deformed configuration,F,M,Q are the total force,total moment and generalized force under the deformed body reference frame.

As for the unsteady aerodynamic modeling,nonplanar doublet lattice method and rational function fitting strategy are utilized as routine in aeroelastic analysis.So here gives the final aerodynamic model results directly.

where q1is the dynamic pressure of inflow,USand qSare the modal matrix and modal coordinates of structural motion under deformed configuration.fA(t ) is the generalized aerodynamic force obtained by non-planar doublet lattice method.ASSnare the generalized aerodynamic force influence matrix caused by the aircraft’s motion,including rigid motions and elastic motions.ASCnare the generalized aerodynamic force influence matrix caused by the control surface rotations.ASSn,ASCnand Dxa(s ) are the fitting coefficient matrices obtained by rational function fitting strategy based on Minimum State(MS)method.L=cref=2 is half of the referenced chord length,V1is the fiight velocity.

Substitute Eq.(19)into Eq.(18)and rewrite the equations into matrix form,then we can get

Mee,Bee,Keeare the generalized modal mass matrix,damping matrix and stiffness matrix under nonlinear equilibrium state.ncis the control variable number.nsis the Laplace variable number.nris the additional aerodynamic roots number.

Converting Eq.(20)into state space form,we can get Eq.(21)

where

By analyzing the eigenvalues and eigenvectors of Eigen matrix Aaeunder different fiight velocity,the stability characteristics can be obtained.According to the stability criterion,if the real parts of all eigenvalues are negative,the system is stable;if the system contains the positive real parts of the eigenvalues,the system is unstable;and if the system has an imaginary eigenvalue and other eigenvalues have negative real parts,the system is critical stable.

The rigid/elastic coupling dynamic state space modeling above is derived from the large nonlinear deformation of elastic aircraft based on common body coordinate frame.It well meets the demand of rigid/elastic coupled stability analysis under large deformation for very fiexible aircraft.

2.3.Control systems modeling

For a conventional aircraft,the control systems often constitute with three basic parts:sensors,an actuator,and control laws.In this paper,the state space model is utilized for stability analysis,so the control systems are also established with a state space form in the time domain.Attention should be paid to that since the structural deformation is large,the transformation between local coordinates and global coordinates should be concerned.Besides the transformation between two coordinates systems,control systems modeling in a traditional linear case can be adopted for fiexible aircraft,so control systems modeling is introduced briefly below.

2.3.1.Sensors modeling

Accelerometer and angular velocity meter are the common used sensors on the aircraft.When the structural deformation is large,the directions of motion that the sensor obtained may be different from the original settings,thus the transformations between local coordinates and global coordinates is very important.For example,when the bending deformation of wing is significant,the direction of accelerometer on wingtip is vertical in local coordinate but not vertical in global coordinate.

The typical relationship between in and out of a single sensor can be expressed with the transfer function below,

where n and x are the damping ratio and cut-off frequency of sensors.Fortimedomain modeling,are selected as state variables and ys=xsas output variable.Thus the sensors state space modeling can be written as

2.3.2.Actuator modeling

For actuator modeling,second order servo and first order motive servo are utilized,ignoring the nonlinear effects.Then the transfer function of actuator is

where a is the time constant,uais the command input of actuator.All these parameters can be obtained by physical equations of actuator or fitted via frequency response curves.Similarly,when the deformation is significant,the direction of rudder deflection should be defined in local coordinate.Selectas state variables andas output variable,the state space model for single actuator can be obtained,

2.3.3.Control laws modeling

Typical fiight control laws systems constitute with the filter,integral,delay and gain system etc.Similar with actuator and sensors,the control laws system often expressed with serial-parallel combination of sub-systems.So the control laws system can be modeled in state space form via appropriate state variables selection as follows,

Considering the control laws may vary with different aircraft configuration and design approach,there is no unified mathematical model for control laws and only can be defined by the specific control form and parameters.

2.4.Aeroservoelastic model for fiexible aircraft

Aeroservoelasticity mainly investigatesthe stability and response characteristics of aircrafts under control.An aeroservoelastic system often consists of four parts as shown in Fig.3.According to the system model,the relation between the input and output can be expressed as

Make the series-wound actuator and fiexible aircraft as open loop,the state space model can be written as:

Now considering the whole closed loop aeroservoelastic system,selectas state variable and y=ysas output variable,the state space model for whole system is:

Attention should be paid on that Aaein the eigenmatrix above is related to the dynamic pressure.So when the air density keeps unchanged,the eigenvalues of closed loop system vary with the increase of the fiight speed,which makes the root locus in the complex plane.Thus the stability characteristics can be investigated:whether there is any root locus cross the imaginary axis.The mode,whose responding eigenvalue cross the imaginary axis,is the key mode.The speed and frequency of the crossing point are the critical speed and frequency in stability analysis in time domain.Combined with eigenvalue and eigenvector,the unstable modes can be expressed,which helps to distinguish and understand the phenomena and mechanism of instability.

Fig.3Aeroservoelastic model in the time domain.

3.Computational example

In this section,a complex aircraft model is selected as an example to execute and verify nonlinear dynamic modeling and aeroservoelastic stability analysis methods.The thinking and logic of the presented computational example are as follows.A computational model is introduced at first.Since the stability analysis is based on the nonlinear equilibrium state,a static aeroelastic analysis is introduced.Because of the space limit,only the analysis results are presented;the specific analysis is based on the quasi-steady assumption of Eq.(13),and the analysis fiow and example can be found in Ref.4.Then a fiutter analysis is executed both in a traditional way in the frequency domain and in a state space way in the time domain to verify the accuracy of structural modeling and aerodynamic modeling.After that,the rigid/elastic coupled stability in an open loop and the coupled aeroservoelastic stability in a closed loop are analyzed and compared to demonstrate the special characteristics of aeroservoelasticity for fiexible aircraft.

This computational example can test the feasibility of theoretical modeling and analysis methods established in this paper in engineering application and investigate the geometrically nonlinear aeroservoelastic characteristics of complex fiexible aircraft,providing great reference and technical support forflexible aircraft design.

3.1.Model

The Finite Element Model(FEM)and aerodynamic models of the computational example are shown in Figs.4 and 5,respectively.This fiexible aircraft has a similar configuration to GlobalHawk and consistsofhigh-aspect-ratio wings,a comparatively rigid fuselage,and a vee tail.There is one aileron at the trailing edge in the middle of each wing and two elevators at each empennage.The whole wingspan is 15.3 m,and the aspect-ratio is 17.The aircraft weighs 2500 kg with a fuselage of 7.6 m.The FEM model was constructed with MSC.Nastran software.It consists of 49872 grids,14298 beams elements,51886 plate shell elements,and 2132 concentrate mass elements.The designed critical fiight speed of cruise state is 180 m/s.

3.2.Static aeroelastic analysis

A non-planar vortex lattice method and geometrically nonlinear structural analysis are utilized to execute the nonlinear static aeroelastic analysis,in which the fiight straight cruise state is selected.

Fig.4FEM model of the computational example.

Fig.5Aerodynamic model of the computational example.

Fig.6Nonlinear static deformation.

Fig.7Initial and deformed aerodynamic models.

When the deformation is significant,the structural stiffness varies with the deformed configuration and present notable geometric nonlinearity.In nonlinear aeroelastic analysis,the following force effect and non-planar aerodynamic modeling are concerned.The structural stiffness is updated in nonlinear computation,and thus the real force condition and deflection can be expressed.The obtained nonlinear structural deflection is shown in Fig.6,and the maximum deformation at the wingtip is 350 mm,which is about 5%of the semi span.A comparison between initial and deformed aerodynamic models is presented in Fig.7,indicating that the non-planar aerodynamic effect is inevitable.Although the nonlinear deformation is not that big,the influence of deformation on structural dynamics is still obvious,which will be discussed in the following section.

3.3.Flutter analysis

3.3.1.Stability analysis in the frequency domain

In order to compare with the coupled dynamic stability analysis and provide a preliminary reference,a fiutter analysis is executed with first 8 symmetric elastic modes(the detailed modes information is shown in Table 1)that are shown in Fig.8.As is well known,a linear fiutter has no relation with deformationand force condition.Here,the planar doublet lattice model is combined with the pk method to obtain linear fiutter characteristics.

Table 1Elastic modes information.

The linear fiutter analysis is unrelated with loads and deformations.The commonly used pk method is adopted to execute the linear fiutter analysis,and results are presented in Fig.9.Because of the unsteady aerodynamics effects,the mode frequency and mode damping vary with an increase of the fiight speed.When the mode damping becomes positive(i.e.,the g-curve across the x=0 axis)the fiutter occurs,and the critical fiutter speed is obtained.The linear fiutter analysis results indicate that the Mode 7(2nd horizontal wing bend mode)is the key mode,resulting in the critical fiutter speed at 320 m/s.The nonlinear fiutter analysis is related with structural load condition and deformation,so in nonlinear analysis,the quasi-modes,which are linearized around the nonlinear equilibrium states,are combined with the non-planar doublet lattice method to execute the fiutter analysis.Nonlinear analysis results are shown in Fig.10 and the physical meaning of thefigure is similar to linear results that have been illustrated above.Compared with linear analysis results,the elastic deformation changes the structural stiffness and dynamic characteristics,and the nonlinear fiutter key mode switches to the 1st horizontal mode,decreasing the critical fiutter speed to 220 m/s.Since the nonlinear analysis considers that the dynamic characteristics vary under different deformed configurations and force conditions,the real critical fiutter boundary and key mode can be reflected in the nonlinear analysis.Thus the potential safety risk can be avoided,so the nonlinear aeroelastic analysis is necessary and inevitable for engineering aircraft design.

Fig.8Symmetric elastic modes.

Fig.9Linear fiutter results obtained by the Pk method.

Fig.10Nonlinear fiutter results obtained by the Pk method.

3.3.2.Stability analysis in the time domain

Also select the first 8 elastic symmetric modes to establish an aircraft state space model to analyze the aeroelastic stability characteristics and compare with the fiutter analysis results.Those two methods consider the same problem in two different ways,among which the traditional fiutter analysis is more commonly used and accepted.Therefore,the consistence of two analysis results can demonstrate the feasibility of state space modeling established in this paper for complex aircraft in engineering application.Both the linear and nonlinear analyses are executed,in which the nonlinear analysis considers the geometric nonlinearity due to structural deformation and the non-planar aerodynamic effect,while the linear analysis does not.

Fig.11Aeroelastic stability analysis results.

Fig.11(a)presents the linear state space modeling stability analysis result,in which the arrows show the developing trend of root locus.Similar to the linear fiutter analysis results,the 2nd horizontal wing bend mode becomes the key mode and results in the fiutter speed at 320 m/s.Nonlinear analysis results are shown in Fig.11(b)and consistent with the fiutter results that the nonlinear fiutter speed decreases to 210 m/s and the key mode switches to the 1st horizontal bend mode.The good agreement between fiutter results and state space model analysis results proves that the state space stability analysis model is quite suitable for complex aircraft and can be applied on dynamic coupled stability analysis considering rigid motion modes.

3.4.Rigid/elastic motion coupled dynamic stability analysis

In previous section,the applicability of the state space model has been verified.Now take three longitudinal rigid motion modes into consideration to establish a dynamic coupled state space model and analyze rigid/elastic coupled stability characteristics.Likewise,both linear and nonlinear analyses are considered.

Linear analysis results are presented in Fig.12(a).Among the computational speed range,two rigid motion modes are always at low damping states,and no instability occurs.The linear rigid/elastic coupling instability is mainly related with elastic modes and similar to fiutter results:the 2nd horizontal wing bend becomes unstable and results in the critical speed at 320 m/s.Linear analysis results indicate that the aircraft has no serious rigid/elastic coupled stability problem but aeroelastic instability.Fig.12(b)gives nonlinear rigid/elastic coupled stability analysis results.When taking the rigid motion into consideration,the coupled critical speed decreases a little compared with nonlinear fiutter results,but the key mode also switches to a lower-order horizontal bend mode.The consistence between dynamic coupled stability analysis and nonlinearflutter analysis results illustrates that since the longitudinal moment of inertia of this computational example with a regular configuration is not prominent like a fiying wing configuration,the nonlinear rigid/elastic coupling problems is not significant,especially reflected in rigid motion.

Fig.12Coupled dynamic stability analysis.

3.5.Aeroservoelastic stability analysis

Based on the rigid/elastic coupled stability analysis,a servo system is added including an actuator,longitudinal pitching angle and angular velocity sensors,and control laws.The state space model of each part is established and combined together to make a closed-loop model and investigate the aeroservoelastic characteristics of complex fiexible aircraft.

3.5.1.Closed-loop stability analysis

According to the actuator modeling principles introduced before,establish an elevator actuator state space model with the detailed information in Fig.13.cmd is the pilot command.A pitching angle sensor(h)and an angular velocity sensor(wz)are placed 1000 mm ahead of the cg in the fuselage.The sensor model and control laws used in this paper are shown in Fig.14.Combined with the rigid/elastic coupled dynamic model offlexible aircraft,an aeroservoelastic model is established.Eigenvalueanalysistechniquesareadopted to analyze closed-loop stability characteristics.

Fig.15(a)gives the root locus of linear closed-loop stability analysis,in which the critical speed decreases from 320 m/s to 290 m/s compared with open-loop analysis results,and the unstable key mode also changes.In closed-loop stability analysis,the fuselage bending modes are collected by the sensor that is placed on the fuselage and transferred to the control law in the aircraft.Thus the elevator reacts according to the order including the fuselage bending mode’s interaction with rigid motion modes and makes the aircraft unstable,so aeroservoelastic instability occurs.

Fig.13Actuator model at empennage.

Fig.14Sensors and control laws model.

Fig.15Closed-loop stability analysis results.

Fig.15(b)presents nonlinear closed-loop stability analysis results.The critical speed decreases form 207 m/s to 150 m/s compared with the nonlinear open-loop analysis.The critical speed decrease is much more obvious than that in the linear case.This is caused by the structural geometric nonlinearity.The structural deflection has an influence on the structural dynamics.The quasi-modes used in the nonlinear stability analysis are quite different form the linear modes.This can be identified from Table 2 that the frequency of quasi-modes in nonlinear analysis declines,especially the frequency of the 1st fuselage bend mode,which declines from 13.73 Hz to 12.35 Hz.Therefore,in nonlinear closed-loop stability analysis,the fuselage bending modes are easier to be collected and participate in the control system.Additionally,the nonplanar aerodynamic effect may also worsen the stability characteristics of fiexible aircraft.

This computational example indicates that complex fiexible aircraft may have aeroservoelastic stability problems.The structural geometric nonlinearity due to large deflection may make the frequency of the fuselage bending mode decline and decrease the critical speed in closed-loop stability analysis.This presents convincing demonstration that closed-loop aeroservoelastic stability analysis has significant meanings for complex fiexible aircraft design in engineering.

3.5.2.Modified closed-loop analysis with filters

In last section,linear and nonlinear closed-loop stability analyses are completed,and the nonlinear closed-loop criticalspeed decreases a lot and far beyond the envelop limitation of 180 m/s.In order to increase the critical speed,reasonable structural filters should be added to the control system according to the nonlinear dynamic coupled characteristics of fiexible aircraft.According to the computational model,structural filters are designed and added in the control system as shown in Fig.16,and the stability characteristics with filters are analyzed in both linear and nonlinear cases.

Table 2Comparison between linear and nonlinear modes.

Fig.16Control system with structural filters.

Fig.17Closed-loop stability analysis results with structural filters.

Fig.17 presents stability analysis results with structuralfilters.The linear critical speed increases from 290 m/s to 302 m/s,and the nonlinear critical speed increases from 150 m/s to 200 m/s.This demonstrates that the closed-loop stability critical speed can be improved via reasonable structuralfilters design for complex fiexible aircraft,especially for nonlinear dynamic coupled stability analysis.

4.Conclusions

A theoretical nonlinear coupled dynamic model without any overfull hypothesis for fiexible aircraft has been established in this paper,which can be commonly used for both fiight dynamic problems and aeroelastic problems.For easy implementation and intuitionistic understandings,the smallturbulence hypothesis is adopted for stability analysis.By transferring the center of the common body coordinate system to the center of gravity under a nonlinear large deformed equilibrium configuration,the inertia coupling can be eliminated for rigid/elastic coupled stability analysis.In order to meet the demand of modern aircraft design,a control system is considered in stability modeling,and thus a unified aeroservoelastic stability analysis framework for fiexible aircraft are founded in the time domain.A complex aircraft model is selected as an example to illustrate special rigid/elastic coupled stability characteristics and aeroservoelastic stability characteristics.Moreover,the adaptability in engineering application of theoretical analysis methods established in this paper is verified via the computational example.The computational example shows that geometrics nonlinearity caused by large structural deflections may have big influences on coupled aeroelastic stability and aeroservoelastic stability characteristics as follows:

(1)Because of large structural deformations,the quasimode frequency declines and changes the fiutter coupling form.Thus the nonlinear fiutter critical speed is much lower than the linear fiutter critical speed.

(2)In the computational example of this paper,the control system may worsen the stability characteristics for fiexible aircraft and cause aeroservoelastic problems under certain control laws,especially in nonlinear analysis.A modified control system with suitable structural filters can depress the sensitivity to elastic vibration and improve the instability critical speed.

The theoretical method of nonlinear aeroservoelastic stability analysis established in this paper is generally suitable for complex fiexible aircraft,which has a significant meaning in engineering application.Traditional linear aeroelastic analysis cannot describe aeroelastic mechanics and predict the nonlinear phenomena of fiexible aircraft.Nonlinear dynamic coupled stability and nonlinear aeroservoelastic stability analyses can provide reliable fiight boundaries and become inevitable.Future work will focus on the further application and validation of nonlinear coupled dynamic modeling.

Acknowledgment

This work was supported by the National Key Research and Development Program of China(No.2016YFB0200703).