Simulation on the dynamic stability derivatives of battle-structuredamaged aircrafts

Bai-gang Mi

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

Keywords:Flying wing Fragment damage Continuous rod damage Combined dynamic derivative Computational fluid dynamics(CFD)

ABSTRACT Accurately evaluating the aerodynamic performance of a battle-structure-damaged aircraft is essential to enable the pilot to optimize the flight control strategy.Based on CFD and rigid dynamic mesh techniques,a numerical method is developed to calculate the longitudinal and longitudinal-lateral coupling forces and moments with small amplitude sinusoidal pitch oscillation,and the corresponding dynamic derivatives of two fragment-structure-damaged and two continuous-rod-damaged models modified from the SACCON UAV.The results indicate that,at the reference point set in this paper,additional positive damping is generated in fragment-damaged configurations;thus,the absolute values of the negative pitch dynamic derivative increase.The missing wingtip induces negative pitch damping on the aircraft and decreases the value of the pitch dynamic derivative.The missing middle wing causes a noticeable increase in the absolute value of the pitch dynamic derivative;the missing parts on the right wing cause the aircraft to roll to the right side in the dynamic process,and the pitch-roll coupling cross dynamic derivatives are positive.Moreover,the values of these derivatives increase as the damaged area on the right wing increases,and an optimal case with the smallest cross dynamic derivative can be found to help improve the survivability of damaged aircraft.

1.Introduction

The continuous development of aviation science and technology has had a profound impact on modern warfare,and the increasing complexity of the battlefield also tests the survivability of all kinds of flight vehicles[1,2].Various battled-damaged modes may appear on these aircrafts.For example,the fuel combustion explosion may lead to aerial disintegration and this serious damage to aircraft even cannot be rescued.However,if only some components of the body are attacked and damaged,it is possible for the pilot to control the aircraft by using the other undamaged parts.In this case,accurately evaluating the aerodynamic performance of the damaged-aircraft is significantly important as the reason that it directly determines the flight quality and further affects the flight control strategy[3,4].

For general flight vehicles,most of the aerodynamic forces are generated from various lifting surfaces(including wings and rudders).Once they are attacked and damaged,the whole aerodynamic performance and the flight quality of the aircraft will change,which will further threaten the aviation safety.Therefore,it is very important to accurately obtain the aerodynamic characteristics of the lifting-surface-damaged aircraft to support the evaluation of its flight capability[5,6].Different types and degrees of penetration damage will occur on the wing depending on the different warheads used.Many research institutes and individuals have carried out a series of aerodynamic characteristic tests and computational analysis of battle damage configurations to evaluate their flight performance[7].Irwin[8]designed a series of wind tunnel tests to investigate the difference of aerodynamic effects between the solid and hollow hole’s sidewall surface to evaluate the low speed performance of the damaged aircraft to support the landing process.It was concluded that the detrimental effects of both gunfire and missile damage were greater for wings with solid construction.Djellal[9]proposed two experimental studies to evaluate the influence of simulated gunfire damage on the performance of a typical aircraft model with holes in the wing.The results indicated that the damage caused significant aerodynamic coefficient degradation related to the diameter of the holes and their spanwise and chordwise locations.Render[10,11]carried out several wind tunnel tests on an airfoil with an axis-varied hole.These wind tunnel tests indicated that quarter-and half-chord locations were more sensitive to damage than leading and trailing edges[12,13].Some more complicated damage models were considered by Etemadi[14],he used both experimental and CFD methods to assess the change in the aerodynamic characteristics of triangular and star-shaped damaged airfoils with repair patches;the results demonstrated the importance of the shape factor in evaluating the performance of battle-damaged vehicles.These efforts revealed the mechanism of the structure damage effect on the aerodynamic forces and moments of the aircraft,however,both of these researches were also just connected with the static aerodynamics,which greatly reduced the practicality of the related conclusions on the damaged aircraft.Frink[15]presented a computational study to assess the utility of NASA CFD codes for capturing the degradation in static stability and aerodynamic performance of a general transport model undergoing progressive losses to the wing,vertical tail,and horizontal tail components.The study demonstrated the high level of accuracy of CFD methods in evaluating the aerodynamics of battle-damaged aircraft and firstly calculated its static stability,which took the aerodynamic performance simulation on damaged aircraft a step further.Based on the aerodynamic data from wind tunnel test or calculation,some researchers proposed studies on simulating the flight dynamics of the damaged aircraft.Nesrin[16]represented an approach to predict the flight dynamics and static derivatives of structurally damaged transport aircraft(spanwise full loss damaged model of C-17 aircraft).He derived the stability and control derivatives from basic principles and theoretical aerodynamics.Jeffery[17]used the vortex lattice method to calculate the aerodynamic forces and generate a reduced order model to evaluate the damage influence on the flight quality of the aircraft.Jong[18]constructed a 6-DOF model for a damaged asymmetric aircraft,and the fight dynamic mode analysis indicated that very different qualities were obtained for damaged and undamaged configurations.Wen[19]analyzed the aerodynamic characteristics of a structurally damaged aircraft by using trim,linearization,stick-fixed response and disturbance simulation,through which the remaining flight performance was calculated.The results indicated that the damage caused an offset of the center of gravity and pitch-roll,pitch-yaw coupling.Because of the randomness of gunfire and missile attacks,different damage directions should be analyzed.These jobs further increased the practicability of the research on damaged aircraft,however,few dynamic data to model the aircraft was presented and the accuracy of the results needed to be further improved.

Previous studies have shown that the static aerodynamic performance of an aircraft with different levels of damage varies greatly.In fact,the most important task for the damaged aircraft is to control and balance its body by using the remaining rudder surfaces and the engine power to ensure a basic safe flight and landing.Therefore,it is necessary to accurately evaluate the stability(especially the dynamic stability)of the damaged aircraft to optimize the flight control strategy.The current investigations on the dynamic stability mainly focus on calculating dynamic derivatives of the undamaged aircrafts with time-domain[20-22]or frequency-domain[23,24]CFD methods.These calculations have been widely validated by using various standard models.Mi has also done much work in this field and develops several CFD simulating methods to identify both the combined and single dynamic derivatives to support the further analysis[25].However,the dynamic aerodynamic characteristics of the undamaged and battled-damaged aircrafts are different,and we need to propose new studies on the dynamic aerodynamic performance of the damaged aircraft.

In this paper,the static and dynamic aerodynamic characteristics of a damaged aircraft are systematically analyzed to guide its flight control strategies.Combining the CFD and dynamic grid techniques,we mainly focus on evaluating the dynamic aerodynamic performance of damaged configurations attacked by different warheads,particularly when there has been local cutting damage to the wing caused by a continuous rod warhead or local perforation damage caused by a fragment warhead.Furthermore,we develop an approach to identify the dynamic stability derivatives for analyzing the dynamic performance of a damaged aircraft.

2.Battle-damaged model and computational grids

2.1.Battle-damaged model

The model used in this study is an enlarged version of a wind tunnel test configuration named SACCON(Stability and Control Configuration)[26-28].The original SACCON model is a typical flying wing UAV configuration and is designed for research on dynamic stability and control system analysis.The UAV model has a lambda wing platform with a leading-edge sweep angle of 53°.The root chord iscr=1.0606 m,and the wingspan isb=1.538 m.The mean aerodynamic chord is 0.479 m,and the corresponding wing area issref=0.77 m2.The moment reference points(MRPs)are located 0.855414 m from the head of the wing.The MRP is also the point of rotation(POR)for dynamic tests.Figs.1 and 2 show the basic model of SACCON UAV.

The SACCON model can be represented by three characteristic components:the fuselage,the wing-body junction,and the wingtip.Three different airfoils are used for designing the fuselage,while the wing-body junction part is directly constructed by extending an airfoil.To reduce the radar signal,the leading edge of the wing is set parallel to the trailing edge,and the wingtip trailing edge section is also parallel to the trailing edge of the fuselage.The section on the outer wing is twisted 5°around the leading edge to improve the aerodynamic performance at a large angle of attack.

Fig.1.Original SACCON UAV geometry.

Fig.2.Different views of SACCON model.

The original SACCON model is enlarged 10 times to fit the size of the typical UAV used in the army;moreover,this model replicates different forms of damage that are inflicted by different warheads.Generally,there are three categories of battle damage to an aircraft caused by gunfire or a missile:fragment damage,discrete rod damage and continuous rod damage.Typical fragment damage usually manifests as the perforation of aircraft components.The effect of discrete rod damage is similar to that of fragment damage,except for the damage shape on the aircraft.By comparison,continuous rod damage has a larger killing range and can often cut an entire part of an aircraft.In this study,we analyzed the effect of typical fragment damage and continuous rod damage on an aircraft.Two different types of fragment damage on the right wing are considered.One model is damaged to form a single large hole with a radius of 0.4 m,and the other model is damaged to form nine small holes distributed equidistantly along the line parallel to the leading edge of the wing.The radius of the small holes is 0.1 m.These parameters are abstracted from our several tests and may be not well agreed to the actual impact,but they can help to explain the problems.Continuous rod damage is also investigated with two models:one with loss of the wingtip and one with loss of the middle wing.Figs.3 and 4 show the battle-damaged models and their geometric dimensions.The wing areas of the undamaged and damaged models are compared in Table 1.Please note that the actual wing areas shown in Table 1 are used to calculate the aerodynamic coefficients,and the reference length is set toc=4.79 m as the reason that the feature length of the middle wing section is retained in all the undamaged and damaged models.

Note that the concept of dynamic derivative is consistent for any configuration,that is to say,the method is applicable even if the damage form and reference area change.In addition,the reference area of the damage model changes,resulting in the corresponding aerodynamic changes,and the final dynamic derivative value also changes,reflecting the change degree of local dynamic stability.By comparing the dynamic derivatives of damaged and lossless models,the change of local dynamic characteristics can be judged to a certain extent,so as to provide accurate input for the evaluation of flight quality.

Note that the concept of dynamic stability derivatives is the same for any configuration,that is to say,the numerical method to calculate this value is applicable for both the undamaged and damaged aircrafts.In addition,the actual wing area varies with the damage forms and affects the coefficients of aerodynamic forces and moments,which will finally change the values of dynamic derivatives.However,this has no influence on the result comparison among undamaged and damaged models as the reason the non-dimensional dynamic derivative reflects the local dynamic stability of the aircraft.We can evaluate the local dynamic characteristics through the values to support the flight quality analysis.

2.2.Grids for calculation

The structured grids were generated using the ANSYS ICEM CFD code.Figs.5 and 6 present the computational grids of the damaged models.A boundary layer is set around the wall to simulate the effect of viscosity,and the near-wall spacing is arranged in such a way thaty+≈1.1 in the wall-adjacent hexahedron cells.To accurately capture the detail flow flied around the damaged holes,specific O-blocks are divided and the mesh density is also adjusted,as shown in Fig.7.

The final numbers of grids for the large-hole damaged model,the small-hole damaged model,the lost-wingtip model,and the lost-middle-wing model have 6 million,7 million,5 million and 4.8 million grid points.

The dynamic aerodynamic performance evaluation of these damaged models is closely related to unsteady motion,which should be simulated by using dynamic mesh.The dynamic motions in this study are mainly harmonic rotations around the center of gravity of the models;therefore,a relatively simple and convenient method called the rigid dynamic mesh technique[29,30]is developed.In a rigid dynamic mesh,all the cells and nodes of the grids will rotate along with the model,as shown in Fig.8.During the dynamic process,the relative positions of the nodes remain unchanged;therefore,it is not necessary to regenerate the computational mesh,and the mesh quality can be guaranteed.Moreover,computational efficiency can be greatly improved because no additional mesh deformation is calculated.

Fig.3.Fragment damage.

Fig.4.Continuous rod damage.

Table 1Wing areas of the undamaged and damaged models.

Fig.5.Surface meshes for fragment damaged model.

3.Modeling the dynamic stability derivatives of a battledamaged aircraft

3.1.Numerical method to simulate dynamic stability derivatives

The unsteady aerodynamic forces or moments of an aircraft affected by a small disturbance can be expressed as

whereCirepresents a longitudinal(Cm)and lateral directional(Cl，Cn)aerodynamic force moment coefficient.

Eq.(1)effectively relates motion parameters to aerodynamic forces or moments,and the derivatives of the aerodynamic forces for these parameters can be obtained by using a Taylor expansion.The derivatives of the forces with respect to the angle of attackα and sideslip angleβ(Ciα，Ciβ)are called static derivatives,which indicate the effects on the aerodynamic force or moment increments caused by a unit angle change and predict whether the aircraft can return to its original balance state after being disturbed.The derivatives with respect to the motion parameters(˙α，˙β，p，q，r)are defined as dynamic derivatives(Ci˙α，Ci˙β，Cip，Ciq，Cir),which indicate whether the aircraft can return to its balanced state after a disturbance.Both the static and the dynamic derivatives are key parameters for flight quality analysis and control system design and can be used to determine the static and dynamic stability of the aircraft[31,32].

It is relatively easy to obtain the static derivative by interpolating the static aerodynamic force or moment at different angles.The dynamic derivative involves the dynamic motion of aircraft,and its identification is much more difficult and costly.The classic method for calculating the dynamic derivative is the small amplitude harmonic oscillation technique[33].This method,which evolved from the wind tunnel test,can be applied to calculate all combined dynamic derivatives in subsonic,transonic and supersonic states.

We take the longitudinal dynamic derivative as an example to introduce the small amplitude oscillation method.Generally,the longitudinal combined dynamic derivativeCm˙α+Cmqcorresponds to the sinusoidally pitching oscillation,as shown in Fig.9.

This oscillation forces the aircraft to pitch around its center of gravity,and the motion is formulated as

The pitch angular velocity and the angle of attack rate have the same expression when the freestream is stable,which is denoted as

Then,the instantaneous pitch moment can be described using the Taylor expansion as

Fig.6.Surface meshes for continuous rod damaged model.

whereΔ˙αand bqare dimensionless parameters of the angle of attack and the pitch angular velocity,respectively.These parameters are expressed as:

The aerodynamic performance is basically linear or weakly nonlinear at small and medium angles of attack,so the higher-order terms can be omitted,and the formula is simplified as:

By combining the motion equations and the reduced frequency formulak=ωc/2V∞,we can finally obtain the dimensionless calculation equation of the longitudinal dynamic derivative:

Fig.7.Grid generation around the damaged holes.

Fig.8.Rigid dynamic mesh.

whereCmωt=2nπcan be obtained by calculating the unsteady periodic flows,andCm0is the balance moment in the initial static state.The method focuses on the instantaneous effect at one single point and is called the single-point identification method.

In general,the flow field of an undamaged aircraft is basically symmetrical.Longitudinal motion will not cause lateral directional forces or moments;this fact is the basis of the flight quality analysis of conventional aircraft.However,this statement does not hold for the battle-damaged aircraft in this study.The geometry of the aircraft is not symmetrical due to the unilateral structural damage;moreover,the aerodynamic forces on the left and right wings are also not symmetrical.The pitch-roll-yaw coupling moments caused by the dynamic motion in the longitudinal direction can also lead to the corresponding coupling dynamic derivative.Since the damaged configuration studied in this paper is a flying wing and the directional damping of this vertical tailless aircraft is small,we mainly focus on the pitch-roll coupling effect.

For an aircraft with unilateral structural damage,the roll moment due to pitch oscillation can be expressed as

By omitting the higher order terms,Eq.(10)can be simplified as:

The pitch-roll coupling combined dynamic derivative can finally be defined as

Note that in contrast with the undamaged configuration,the asymmetric geometry of battle-damaged aircraft leads to several coupling dynamic derivatives,which will lead to much more complicated flight qualities on both the stability and controllability of the aircraft.

3.2.Method validation

First,we use the rigid dynamic mesh technique to simulate the pitch oscillation of the original SACCON model to test the previously developed dynamic derivative identification method.The SSTk-ωturbulence model[34]is used to calculate the unsteady aerodynamic forces and moments during the dynamic motion as its extensive adaptability and high accuracy on the external flow of the aircraft,in which the chosen initial angles of attack are 0°，5°，10°，15°and 20°.The pitch motion is defined asα=α0+1°sin(6πt),and the corresponding reduced frequency isk=ωc/2V∞=0.09.Note that the parameters are determined according to the actual wind tunnel test on dynamic derivative.Usually,a wide range of dynamic parameters are set to identify the dynamic performance of the aircraft during the experiment process,here we only adopt a group of typical data to develop and validate our method.

Figs.10 and 11 present the comparisons of the normal force and pitch moment combined dynamic derivatives(CN˙α+CNqandCm˙α+Cmq)between the CFD results and experimental data obtained from DNW and NASA wind tunnels[16].The results show that both the normal force and the pitch moment dynamic derivative agree well with experimental values in the range of small and medium angles of attack.The nonlinear flow separation at a large angle of attack leads to relatively low calculation accuracy;however,the simulation results are still close to the error band of the experimental data.Therefore,it can be said that the CFD method of identifying dynamic derivatives is reliable.

When the wing is attacked and penetrated to form large or smalls holes by the fragment objects,a large number of separation vortices are generated in these damaged holes.Moreover,the strengths and influences of the vortices will change with the shapes of the holes and the angle of attack,which causes certain interference to the accuracy and stability of the CFD method.In spite of the lack of dynamic derivative test on the damaged models,we can still explain the rationality of the method by using the above cases at high angle of attack.For the original flying wing model,when the angle of attack increases,separated vortices generate from the leading edge of the wing,and their strengths also increases.Fig.12 show the spatial streamlines of the model at high angles of attack(15 and 20°).The complicated flow flied is similar to those of damaged configurations,but the calculated dynamic derivatives in Fig.at these angles of attack are still close to the experimental data,which indicates that the numerical methods is reasonable in calculating the dynamic derivatives with complicated flow fields and can be used to identify the results of damaged configurations.

Before the calculation,we should evaluate the mesh convergence.Although different damaged models generate different amount of computational grids,they share the similar refined strategy and grid density in boundary layer.The grid of the undamaged model is used to testify the mesh convergence by calculating the dynamic stability derivatives,and the total nodes are 10 million,8 million,6 million,and 3 million.The calculated dynamic derivatives are listed in Table 2.The results calculated with 8 million grids are very close to those of 10 million,which indicates that it is adequate to obtain relatively accurate results by using 8 million girds.Therefore,the subsequent mesh generations for both the undamaged and damaged models will adopt this strategy.

4.CFD modeling

4.1.Static aerodynamic calculation

First,the static aerodynamic performances of an undamaged model and of four damaged models are simulated by using the CFD method to clarify the differences in aerodynamic forces and moments among these models.The simulation conditions are listed in Table 3.

Fig.13 shows the static aerodynamic forces and moments of these configurations.In general,the change values of the static aerodynamic results between the undamaged and damaged models should be presented to clearly show the effect of the damaged forms on the performance of the aircraft.However,this is not the focus of the article,and we directly give the coefficients of all the models.The lift coefficients of the fragment-and continuous-rod-damaged models are basically the same as those of the undamaged model,however,these lift coefficients differ slightly when the angle of attack is large.In that case,the flow fields of models with a larger loss on the lift surface are more sensitive than those of other models.Fig.14 shows the surface streamlines of these models at high angle of attack(α0=10°),where we can clearly see that the separation area of the model lost middle wing is relatively large than those of other models,therefore its lift coefficient is slightly smaller.

Compared to the undamaged model,the fragment-damaged model has relatively large drag coefficients,whereas the continuous-rod-damaged models have relatively drag coefficients.Due to the increase in the wetted area of the wing body and the flow separations in the small holes(Figs.15 and 16),both the friction(related to wetted area)and the pressure drag(related to flow separations)of the distributed-small-hole damaged model increase.Therefore,the drag of this model is the largest among all the configurations at the same angle of attack.The drag coefficient of the large-hole damaged model is slightly larger than that of the undamaged model at small and medium angles of attack,but the drag increment is weakened at a large angle of attack because of the increase in airflow penetrability.The area of the wingtip-loss model decreases;thus,the friction drag is correspondingly reduced.Similar trends are found in the middle-wing-damaged model.However,the wingtip vortex effect of this model is greatly extended and can even affect the left wing,leading to much more lift-induced drag,and the total drag of the middle-wing-loss model is larger than that of the wingtip-loss model.

Fig.9.Small amplitude sinusoidally pitching oscillation.

Fig.10.Normal force combined dynamic derivatives.

Fig.11.Pitch moment combined dynamic derivatives.

The pitch moment is closely related to the lift distribution.For the damaged models,the difference in the lift surface loss area before and after the moment reference point(MRP)may cause the pitch moment to have different characteristics in different damaged models,as shown in Fig.17.The large hole is located at the front of the MRP and decreases the partial lift,which leads to an additional pitch-down moment compared to the undamaged model.For the distributed small-hole damaged model,the total lift surface loss is still revealed in the front area.However,this loss area is smaller than that in the large-hole damaged model,and the additional pitch-down moment is also relatively small.When the aircraft is attacked and loses the right wingtip,the surface area decreases more behind the MRP than in the front area;however,the part of the front loss located in the high-pressure-gradient region may also generate considerable lift,so the net effect on pitch moment in this model is nearly the same as that in an undamaged aircraft.For the middle-wing-loss model,the lift substantially decreases in front of the MRP,which results in an obvious pitch-down moment.The vortex generated from the“new-wingtip”of the wingtip-lost model decreases the local angle of attack of this region,which will lead to a rolling moment for the damaged aircraft,which will be much more apparent on the middle-wing-lost model as that the generated vortex is closer to the symmetry of the aircraft.

As the flow fields are basically symmetrical when the angle of attack ranges from 0 to 12°,the roll moments of the undamaged model are nearly 0.As the area loss on the right wing increases,the coupling roll moment also increases.The smallest area loss leads to the smallest increase in roll moment for the distributed-small-hole damaged model,followed by the large-hole damaged model.The coupling roll moment increase is not large and changes slowly with respect to the angle of attack.The continuous rod damage cuts part of the right wing,especially in the case of middle wing loss.In this case,the coupling roll moment increases and changes significantly with the angle of attack.

4.2.Dynamic aerodynamic simulation

Based on the static CFD solutions of the undamaged and damaged models,the rigid dynamic mesh technique is developed to simulate unsteady small amplitude oscillation,and the combined dynamic derivatives are identified with the established methods.The dynamic sinusoidal motion is defined asα=α0+αmsin(ωt)=α0+1°sin(3.8594t),and the corresponding reduced frequency isk=ωc/2V∞=0.05.The other computational parameters are the same as those in static calculations:Ma=0.6,H=6 km,and the reference chords for these damaged models are bothc=4.79 m.

4.2.1.Pitch combined dynamic derivative

The hysteresis loops of the pitch moment coefficient of the fragment-and continuous-rod-damaged models are presented in Fig.18.As the instantaneous angle of attack increases,the circumferential direction of all the hysteresis curves is counterclockwise,which indicates that damping characteristics are obtained for the pitch moment at these attack angles.Due to the different types and sizes of the missing parts in these models,the azimuths and sizes of the hysteresis loops are also different,causing different amounts of longitudinal damping.

The identified combined dynamic derivatives of the pitch moment coefficientsCm˙α+Cmqof the models are shown in Fig.19.The difference inCm˙α+Cmqbetween the damaged and undamaged aircraft is mainly caused by the differences in the unsteady aerodynamic forces before and after the rotating point(the MRP in this study),which is related to the missing area of the lift surface and is affected by the pitch moment generated by the forces on the missing surface.

Fig.12.Spatial streamlines of SACCON model at high angles of attack.

Table 2Dynamic derivatives of different amount of grids atα=5°.

Table 3Static calculation conditions for undamaged and damaged models.

For the large-hole damaged model,the missing part is located at the front of the pitch axis.Compared to the undamaged aircraft,the damaged aircraft has reduced lift in the front part,which causes a pitch-down moment and provides additional damping to the system,helping it to resist its dynamic motion during the pitching-up and pitching-down processes.Therefore,the signs of the combined dynamic derivativesCm˙α+Cmqare consistent with those of the undamaged aircraft,but the absolute values of these derivatives increase.When the damage is caused by distributed holes on the right wing,8 holes are located before the pitch axis,and only 1 hole is located behind the pitch axis in the low-lift region.The total effect is reflected in the lift loss of the front area,and an additional pitch down moment is obtained to resist the dynamic motion.The sign ofCm˙α+Cmqis still negative,and the absolute value is larger than that in the undamaged model.However,the equivalent missing lift surface is smaller in the distributed-hole damaged model than in the large-hole damaged model,and the absolute values ofCm˙α+Cmqare also smaller for the distributed-hole damaged model.The local structure loss in the fragmentdamaged model changes the local characteristics of the overall flow,especially at large angles of attack ranging from 8 to 12°,and the dynamic derivative changes slight with the angle of attack.However,the large structure missing in the large-hole damaged model changes the development of dynamic derivatives with the angle of attack.If the angle of attack ranges from 0 to 4°,which is unfavorable for aircraft control,the blue curve shape of the dynamic derivatives is quite different from the others,as shown in Fig.19.

Continuous rod damage may result in the partial or entire loss of the right wing and have much more influence on aerodynamic performance than fragment damage.On the one hand,the lift surface is substantially missing;on the other hand,the action region of the wingtip vortex will also change in response to the loss of a component,as shown in Fig.20.When the wingtip is damaged,although more of the lift surface behind the pitch axis is missing before the pitch axis,the front missing part contains the leadingedge suction zone,so there is less of a difference in lift loss before and behind the axis.However,the total lift is still reduced,leading to a pitch up moment.Therefore,the absolute values of the negative dynamic derivative are smaller than those in the undamaged model.At a high angle of attack,the lift loss difference is further reduced due to the increased effect of the wingtip on the leading edge of the middle wing,and the additional pitch-up moment is reduced.At this time,the dynamic derivativeCm˙α+Cmqis still negative and changes more gently than that in the undamaged model.For continuous rod damage without a middle wing,the dynamic aerodynamic force changes much more dramatically.The lift surface before the pitch axis is substantially missing,and the substantial loss in lift in this region generates a pitch-down moment to resist the dynamic oscillation.Thus,the absolute value ofCm˙α+Cmqobviously increases.When the initial angle of attack increases to 12°,the wingtip affects the inner wing section and further increases the additional pitch-down moment,which causes a more apparent change in the dynamic derivative with the angle of attack.

The combined dynamic derivative calculation of the pitch moment indicates that the differences among the four damaged models are mainly caused by the aerodynamic change before and after the pitch axis due to the lack of an effective lift surface.For the configuration in this study,both types of fragment damage increase the pitch damping of the aircraft,whereas different effects are observed in the cases of wingtip and middle wing damage.A missing wingtip leads to negative damping,whereas a missing middle wing increases the damping effect and causes complicated changes in the dynamic derivative as the angle of attack increases.

4.2.2.Pitch-roll combined dynamic derivative

Fig.13.Static aerodynamic forces and moments of undamaged and damaged models.

Fig.14.Surface streamlines of models at high angle of attack(α0=10°).

Fig.15.Local flow field of distributed small holes damaged model.

Fig.16.Local flow field of large hole damaged model.

Fig.17.Area loss before and after the MRP.

The unsteady roll moment coefficient obtained by pitch harmonic oscillation is shown in Fig.21.As the initial roll moments of the undamaged model in the calculated range of angle of attack are all 0 and the instantaneous flow fields are symmetrical during dynamic motion,the pitch-roll moment for this model is basically 0.The directions of the coupling pitch-roll moment hysteresis loops of the damaged models are basically clockwise,which indicates that the instantaneous roll moment produced by pitch harmonic motion when the ring wing of a UAV is damaged has a negative damping effect on the lateral direction of the aircraft.The increase in the missing area from the small holes to the large holes in the fragment-damaged models and from the wingtip to the middle wing in the continuous-rod-damaged models also causes an increase in the areas of the hysteresis loops of the roll moment,which makes the dynamic coupling characteristic between the longitudinal and lateral directions increasingly apparent.

Fig.18.Unsteady pitch moment coefficient at different initial angles of attack.

Fig.19.Combined dynamic derivative of pitch moment.

We use the established single point method to identify the pitch-roll cross dynamic derivativeCl˙α+Clq;the results are shown in Fig.22.For the undamaged model,the dynamic characteristics of the flow fields for attack angles between 0 and 12°are basically symmetrical,which means that the dynamic pitch motion does not cause an obvious difference between the left and right flow fields around the UAV.Thus,there is no longitudinal interference on the lateral performance,so the dynamic roll moment and pitch-roll cross dynamic derivative are all 0 during the process.When the right wing of the UAV is damaged,the asymmetric geometry of the model leads to a roll moment even in initially static cases,which will make the aircraft roll to the right side around the axis.When pitch oscillation occurs,the instantaneous angle of attack increases during the pitch-up process,and the increased lift on the left wing causes a large instantaneous right-roll moment.In pitch-down motion,the angle of attack decreases,and the lift on the right wing decreases much more than that on the left wing;thus,the relative roll moment will still make the aircraft roll to the right side.Therefore,the coupling roll moment generated by the damaged model makes the aircraft roll to the right side during the whole pitch harmonic motion,which has a negative damping effect on the lateral performance of the whole system and makes the pitch-roll cross dynamic derivativeCl˙α+Clqpositive.

Fig.20.Spatial streamlines around the actual wingtip of continuous rod damaged models.

With the increase in the initial angle of attack,the pitch-roll cross dynamic derivatives of all the damaged models first decrease and then increase.A minimum value can be found at a certain angle of attack.The reason for this phenomenon is closely related to the interference of the local damaged part on the whole dynamic flow field.This finding also indicates that although the additional cross-coupling moment caused by the dynamic pitch oscillation after battle damage brings a series of complicated problems to the aircraft,an optimal control strategy can still be found to help the pilot to balance the damaged aircraft in this weak coupling case of the dynamic derivatives.

Fig.23 shows how the pitch-roll combined dynamic derivatives change with the missing area on the wing surface for different initial angles of attack.The results indicate that the relationship between the value of the pitch-roll cross dynamic derivative and the missing area is not linear;because we have not considered the effect of the lift distribution characteristic in this figure.However,the results shown in this figure still demonstrate that a large missing area on the wing surface can lead to a large cross dynamic derivative,which is consistent with the actual phenomenon.

The calculation of the pitch-roll cross dynamic derivative of the damaged models shows that,due to the missing area of the lift surface on the right wing,the lift on the left and right wings is not balanced,causing an instantaneous roll moment;this phenomenon causes the aircraft to roll to the right side during pitch oscillation.The total action on the dynamic system is negative damping,therefore,the values of the pitch-roll dynamic derivatives are positive.A large damaged area on the aircraft can lead to a large value of the cross derivative,but the relation between the two is not linear.The pitch-roll dynamic derivatives decrease first and then increase,and a case with the minimum cross derivative can be found to help optimize the flight control system when battle damage occurs.

4.2.3.Influences discussion of POR and MRP on the dynamic derivatives

Fig.22.Pitch-roll moment cross dynamic derivative.

The values of dynamic derivatives are closely related to the reference point.Generally,both the moment reference point(MRP)and the point of rotation(POR)are set as the center of gravity,which should be as stable as possible in actual flight,especially for the flying wing configuration.In principle,the dynamic derivative should be evaluated according to the new center of gravity of the damaged models,however,it is very difficult to timely obtain these points and we still set the reference points of all the models as the original center of gravity of the undamaged configuration,which is also convenient for the further analysis and application.

We use two new reference points to further discuss their influence on the dynamic derivatives.Fig.24 shows the three different reference points P0,P1 and P2,in which P0 is the center of gravity of the original model used in previous study,P1 and P2 are added points for comparison.Figs.25 and 26 show the dynamic derivatives of the undamaged and damaged models.The results indicate that if the position of reference point is changed,the unsteady aerodynamic forces are also changed due to the effective lifting surfaces before and after the point,which will make some new changing rules on the combined dynamic derivatives of pitch moment between the undamaged and damaged models,but it has no influence on the pitch-roll moment cross dynamic derivatives as the reason that the point is always located on the symmetry plane.

Fig.21.Unsteady roll moment coefficient caused by the pitch oscillation.

Fig.23.Pitch-roll moment cross dynamic derivative changing with damaged area of wing surface.

In a word,although the results of the pitch and pitch-roll dynamic derivatives are different with reference points,the numerical simulation and analysis method developed in this paper are applicable to all cases.

5.Conclusion

CFD simulations are conducted on the modified SACCON UAV configuration with four damage modes:two fragment-damaged configurations,one with a large hole on the right wing and one with nine distributed small holes on the right wing,and two continuous-rod-damaged models,one formed by cutting the wingtip and one formed by cutting the middle wing.First,the static aerodynamic forces and moments of the undamaged and damaged models have been calculated and compared.On this basis,the dynamic pitch and pitch-roll coupling moments of these models with unsteady pitch oscillation have been further computed by using the rigid dynamic mesh technique.A single point method is established to identify the pitch and pitch-roll coupling cross dynamic derivatives.

Whether caused by fragmentation or continuous rod damage,the partial loss of the lift surface on the right wing of the aircraft causes geometric asymmetry and further leads to an asymmetric flow field,which generates a coupling pitch-roll moment on the aircraft.In the static state,the lift characteristics of the undamaged and damaged models are nearly the same,but the drag varies greatly because of the different wetted areas of the configuration and the effect of the wingtip vortex.The pitch moment is closely related to the lift distribution before and behind the MRP,and the coupling pitch roll moment increases with the increase in missing area on the right wing.

Fig.24.Different reference points for comparison(P0:the center of gravity of the original fly wing model;P1,P2:new added points).

Fig.26.Pitch-roll cross dynamic derivatives at different reference points.

The method presented in this paper is also applicable to the identification of other cross-dynamic derivatives in multi-axis coupling cases.However,the battle damage situation for an aircraft can be much more complicated,and the dynamic aerodynamic performance and dynamic stability may differ greatly from those of an undamaged aircraft.In addition,limited by the space of the article,the selected parameters of the holes on the damaged models are relatively simple.Actually,both the shape,size,position and direction of the hole,and the flight speed,altitude and other factors have significant impact on the dynamic aerodynamic performances of the aircraft.Therefore,the job in this article is still relatively preliminary.We need to propose a further research on the effects of these factors and analyze the physical mechanism,and develop the validation of the CFD simulations through the comparison with wind tunnel test.

Declaration of competing interest

The authors declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work,there is no professional or other personal interest of any nature or kind in any products,service and/or company that could be construed as influencing the position presented in,or the review of,the manuscript entitled.

Acknowledgements

The authors would like to acknowledge the support of National Natural Science Foundation of China(Grant No.11672236)and Project funded by China Postdoctoral Science Foundation(Grant No.2018M641381).