Ma Dongli,Wang Shaoqi,Yang Muqing,Dong Yongpan

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

Dynamic simulation of aerial towed decoy system based on tension recurrence algorithm

Ma Dongli,Wang Shaoqi,Yang Muqing*,Dong Yongpan

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

Dynamic model of aerial towed decoy system is established and simulations are performed to research the dynamic characteristics of the system.Firstly,Kinetic equations based on spinor are built,where the cable is discretized into a number of rigid segments while the decoy is modeled as a rigid body hinged on the cable.Then tension recurrence algorithm is developed to improve computational efficiency,which makes it possible to predict the dynamic response of aerial towed decoy system rapidly and accurately.Subsequently,the efficiency and validity of this algorithm are verified by comparison with Kane’s function and further validated by wind tunnel tests.Simulation results indicate that the distance between the towing point and the decoy’s center of gravity is suggested to be 5%–20%of the length of decoy body to ensure the stability of system.In up-risen maneuver process,the value of angular velocity is recommended to be less than 0.10 rad/s to protect the cable from the aircraft exhaust jet.During the turning movement of aircraft,the cable’s extent of stretching outwards is proportional to the aircraft’s angular velocity.Meanwhile,the decoy,aircraft and missile form a triangle,which promotes the decoy’s performance.

1.Introduction

With precision guided weapons equipped in modern combat operations,a serious threat is posed to the safety of aircraft.To protect the aircraft from radar guided missile so as to improve the survivability of aircraft on the battlefield,the towed decoy system came into being.In the system,the decoy is towed by flexible cable attached upon the aircraft to follow the motion of aircraft.As a result,hostile radar is deceived so that the missile will track down the decoy instead.Due to its excellent combat performance,the towed decoy system has now become an important airborne weapon.Aircraft including F-16,B-1 B and F/A-18 have been successively equipped with towed decoy system.1–3

Dynamic complexity of towed decoy systems manifesting as the concussion,even the instability of the decoy as well as the relaxation and tension of the cable,affects their final working status and combat effectiveness.Severe dynamic problems have already been observed in flight trails.To predict whether the aerial towed decoy system will function well or not,dynamic characteristics of the system during aircraft maneuvering motion have to be studied.Requirements listed below have to be met during application.4

(1)The posture of decoy should be maintained stable.

(2)Collision between decoy and aircraft should be avoided.

(3)The tension of cable should be limited within the allowable value.

(4)The cable should not be blown by the hot exhaust jet of aircraft.

Throughout the literatures dealing with the towed decoy system,radar radiation characteristics and interference mechanism are mainly concerned,while the dynamic problems are hardly involved.But for other towing systems such as low frequency antenna and towing target,5–23many researches can be found in the past few decades.

Assuming that the tangential air drag force and steady state cable extension are negligibly small,Zhu et al.5,6investigated the steady state response and stability of ballooning strings in air as well as the dynamic response of a circularly towed cable-body system.Modeling the towed cable-body as a variable-length rigid-cable pendulum,Doty7simulated the trajectory of the towed body during release without considering the drag on the cable.After that,Djerassi and Bamberger8proposed an order-n algorithm for the simulation of the motion of cable during deployment from two moving platforms.Henderson et al.9pointed out that active control of the target has greater effect than passive modifications by performing a fully three-dimensional simulation of the dynamic behavior of towed cable-bodysystem.Quisenberryand Arena10,11analyzed the discrete cable model and pointed that the thin rod model was more efficient than discrete lumped mass model with the same number of segments.Williams et al.12–17modeled the cable using dynamics of multi-body systems method,and then studied the stability of cable in cross wind and in circular path.Besides,his research also focused on the constrained path-planning,optimal control strategy and influence of related parameters.Koh and Rong18studied the analytical formulation and numerical strategy based on an iterative finite difference scheme,then analyzed the largedisplacement low-tension cable motion.

With the development of computer technology,the finite element method has been applied to towed cable system.Zhu and Meguid19investigated the effect of pertinent parameters and stability of the generalized model for aerial refueling by using the finite element method with three-nodded,curved beam element.Sun et al.20modeled and analyzed the cable using a nodal position finite element method,which calculates the position of cable directly instead of the displacement.By discretizing the cable and employing static equilibrium of each element,Maixner and McDaniel21proposed a simple and robust solution methodology which can provide preliminary calculations for cable in steady,non-uniform flow.Recently,Xing and Zheng22studied the deploying process modeling and attitude control for a satellite with a large deployable antenna using multi-rigid-body dynamics,hybrid coordinate and substructure.Sun et al.23researched the model validation for the towed cable system described by a lumped mass extensible cable using flight data and then researched optimal trajectory generation for the towed cable system with tension constants using model predictive control.Yan et al.24investigated the dynamic characteristics of towed decoy during release based on Kane’s function and achieved satisfactory results.

Researches on low-frequency antenna and towing target open the door for towing cable system.However,the present methods require large amount of calculation to yield accurate results.This paper presents an efficient way to study the dynamic issues mentioned above.Kinetic equations based on spinor are built,where the cable is discretized into a number of rigid segments and the decoy is modeled as a rigid body hinged on the cable In this instance,the tension of cable at given moment is solved by applying tension recurrence algorithm,which avoids the inversion problem of large-scale mass matrix.In this way,the computing efficiency is significantly improved.It is verified by numerical examples and further validated by wind tunnel tests.In the simulations,relevant parameters,such as the flight conditions are changed to study the characteristics of the system.This paper also investigates dynamic responses of the cable and decoy during the maneuvering motion of aircraft.Hopefully,the qualitative and quantitative conclusions achieved in this research may contribute to the design and application of the aerial towed decoy system.

2.Model of towed decoy system

2.1.Dynamic equations based on spinor theory

The real towed cable is so flexible that its bending stiffness can be neglected.Moreover,with an elastic modulus of 50 GPa,the cable has negligibly small axial elongation during flight but it can be compressed easily.In this case,the model we have to employ is required to illustrate the flexibility,compressibility and little extensibility of the cable.

Accordingly,inthismodel,thetowedcableisdiscretizedinto a certain number of thin rigid rods connected by rotating hinges where axial elongation and bending of each segment are not taken into consideration.The first cable segment is hinged to theaircraftwhilethelastoneishingedtothedecoy.Assumethat themassofeachsegmentisuniformly distributedandthecenter ofgravityislocatedatitsgeometriccenter.Hereweconsiderthe aircraft and decoy as rigid bodies as well.In this way,the towed decoy system is converted into an open-chain multi-body system,which is propitious to model with spinor method.

As shown in Fig.1,the cable segments are numbered as B1to BNin sequence.The aircraft is denoted as B0while the decoy is labeled as BN+1.V is the flight direction of aircraft.For clarity,the point connecting the cable and aircraft is named as‘connection point’while the point linking the cable and decoy is designated as ‘towing point’.

The body-fixed coordinate frame Oixiyiziis established on the gravity center of the ith cable segment(see Fig.2).By definition,the generalized moment of inertia of the ith cable segment measured in Oixiyiziis^J（i）.Velocity inversion spinor is^V（i）while the anti-symmetric matrix of which is~^V（i）.Despite that dual-number is a common expression of spinor,we employ matrix instead,so that all operations have a uniform matrix form in this paper.

Fig.1 Schematic diagram of aerial towed decoy system.

Fig.2 Body-fixed coordinate frames of cable segments and decoy.

where i=1 to N+1;I3×3is a unit matrix of order 3;m（i）is the mass of the ith cable segment;J（i）is the inertia tensor of the ith cable segment on point Oi;V（i）and ω（i）are the centroid velocity and angular velocity column matrix of the ith cable segment measured in Oixiyizi,respectively;and ‘～’is anti-symmetric operation symbol.The anti-symmetric operation of vector a=[ax， ay， az]Tis

The external force spinor on the ith cable segment is

where^Fa（i）is aerodynamic force spinor,^Fg（i）gravity spinor and T（i）=[TxiTyiTzi]Tthe tension column matrix of the ith cable segment measured in Oixiyizi;H（i）=[I3×303×3]Tis used to convert the tension column matrix into spinor form;Φ（i，i+1） is the spinor transformation matrix from Oi+1xi+1yi+1zi+1to Oixiyizi.

where Ψ（i，i+1） isthedirection cosinesmatrix from Oi+1xi+1yi+1zi+1to Oixiyizi;r（i，i+1）is the radius vector from Oi+1to Oi.

Accordingly,the Newton-Euler kinetic equation in spinor form is

2.2.Recurrence relations of cable’s tension

There are recurrence relations among cable segments in the towed decoy system.Firstly,the recurrence relations of middle cable segments are derived.Torsional deformation has negligibly little influence on force on the cable segment.Hence,the ith hinge displacement(shown in Fig.3)is expressed as

Recurrence relation of velocity inversion spinor between the ith and(i+1)th cable segment is

Fig.3 Schematic diagram of hinge displacement.

where G（i+1）=[03×3， I3×3]Tis used to convert variables in articulated freedom into spinor form,which is orthogonal to H（i+1）;Φ*（i+1，i）is the transformation matrix of velocity inversion spinor from Oixiyizito Oi+1xi+1yi+1zi+1.

Time differentiation of Eq.(9)yields the acceleration recurrence relation:

After that,substituting Eq.(7)into Eq.(11)and left multiplying both sides of the equation by HT（i+1）,we can eliminate ¨s（i+1）and obtain the following:

HT（i+1）^J-1（i+1）^F（i+1）

Similarly,the recurrence relation of tension of segment connected to the aircraft and segment attached to the towed decoy are derived as Eqs.(16)and(17),respectively.

Consequently,the recurrence relations of sequence{T（1），T（2），...，T（N），T（N+1）}are summarized as follows:

3.Tension recurrence algorithm

As seen from Eq.(20),0= αT（1）+ β is contained implicitly.As a result,once the column matrix β and square matrix α are derived,tension offirst segment will be obtained by T（1）=-α-1β.Substitution of T（1）into Eq.(20)yields all the rest tensions.The recurrence relations are only related to the displacement and velocity of the system,but are irrelevant to the acceleration.Accordingly,matrix β and square matrix α at given moment are available although the acceleration is unknown.

According to the coefficient matrixes Q,A,B and C in Eqs.(15),(18)and(19),we construct the recurrence sequence{q1， q2， ··· qN， qN+1| q ∈ R3}that subjects to relations as follows:

Eq.(21)implicitly contains qN+1= αq1+ β whose null point is q0=-α-1β,i.e.,the tension offirst cable segment T（1）.Steps to solve Newton-Euler kinetic equation of the towed decoy system are posed as follows:

Step 1.Calculate column matrix β.

Setting q1=03×1and substituting it into Eq.(21),we can get¯qN+2.

Step 2.Calculate square matrix α.

Setting kth row of q1as 1 and other rows as 0,then substituting it into Eq.(21),we can get¯qN+2.Thus,the kth column of α can be expressed as Eq.(23).Matrix α is obtained by calculating every αk（k=1，2，3）.

Step 3.Calculate the tension of cable segments.

T（1）can be solved by T（1）=q0=-α-1β.Substitution of T（1）into Eq.(20)yields other tensions{T（2）， T（3）， ···，T（N），T（N+1）}.

Step 4.Calculate the acceleration of the system.

Substitution of the tension of cable segments into Eq.(7)yields the acceleration of towed decoy system.Here we suppose that the artificially given movement of the aircraft will not be affected by the towed decoy.In the following instances,researches will focus on the dynamic response of cable and decoy.

4.Numerical method validations

4.1.Exemplification

Kane’s function is a common method to establish dynamic equations for discrete system.It can be used in complex system with multi-degree offreedom.25Yan et al.investigated the dynamic characteristics of towed decoy during release based on Kane’s function in Ref.24and achieved satisfactory results.Therefore,in this section,the efficiency of tension recurrence algorithm is verified by numerical simulations compared with Kane’s function.

A compound multi-pendulum(shown in Fig.4)is considered.Fig.5 demonstrates the time response of a multipendulum where the rod number is amount to 300.The rods have a uniform mass of 0.5 kg and a length of 0.1 m.The initial conditions at t=0 are φ1= φ2= ···= φ300=-π/3 rad,˙φ1=˙φ2=···=˙φ300=0 rad/s.

Tension recurrence algorithm in this paper has the same results with Kane’s function derived in Ref.24.The results are always consistent with rod number increasing since the dynamics equations and Kane’s function are essentially the same.

Time spent on simulation increases with rod number in the multi-pendulum,as illustrated in Table 1.Note that simulation for compound double pendulum with tension recurrence algorithm only takes 0.05 s while simulation employing Kane’s function takes 0.08 s.The difference is even more significant when rod number is up to 100 in multi-pendulum systems.

Tension recurrence algorithm avoids the inversion problem of large-scale mass matrix.Accordingly,the computation complexity for calculating the towed decoy system is lower than Kane’s function.By adopting this algorithm,the computing speed significantly increases.In addition,since the tension recurrence algorithm is easy to program,the demand of rapid analysis for engineering is satisfied.

Fig.4 Schematic diagram of compound multi-pendulum.

Fig.5 Time response of multi-pendulum where the total number of rods is 300.

Table 1 Time spent on simulation of multi-pendulum system.

4.2.Wind tunnel tests

In order to further verify the numerical method and observe the motion of towed decoy system,wind tunnel tests are conducted under some specific situations.The experiments are performed in the 3.5 m single-loop low-speed wind tunnel at Aviation Industry Corporation of China(AVIC)Aerodynamics Research Institute in Harbin.The main test section is nominally 3.5 m×2.5 m in cross-section and 5.5 m in length.The maximum wind velocity is 73 m/s,with an average turbulence intensity of 0.1675%.

Fig.6 Aircraft model in wind tunnel.

Table 2 Parameters of towed cable in wind tunnel tests.

Table 3 Parameters of towed decoy in wind tunnel tests.

As shown in Fig.6,the aircraft model is clamped on the tunnel wall with an adjustable angle of attack,where the towed cable system is joined on the ventral position of aircraft.The cable is made of nylon and the decoy is made of aluminum alloy.Tables 2 and 3 show the physical parameters of cable and decoy,respectively.The position and altitude of decoy are observed by three high-speed cameras with a sampling frequency of 1000 s-1which are placed outside of the wind tunnel as illustrated in Fig.7.Based on binocular vision theory,the camera is applied with synchronous control method to reconstitute the objects accurately.A high-precision tension sensor is installed at aircraft model to measure the tension of cable at connection point.

The experiments are operated in constant flow with speeds of 30 m/s,40 m/s and 50 m/s,respectively.For comparison,simulations under the same conditions are also performed.Fig.8 shows the shape of cable and altitude of decoy at a wind speed of 40 m/s.We can see that the simulation results are consistent with wind tunnel tests.Results also indicate that the number of segments barely exerts any effect on the shape of cable in steady flow.

Fig.7 Synchronous observation using three high-speed cameras.

The dynamic process of towed decoy system is obtained as the speed of wind uniformly increases from 30 m/s to 50 m/s at an acceleration rate of 2 m/s2.The tension of the cable and pitching angle of the decoy during this process are demonstrated in Fig.9 where the start time of acceleration is set as t=0 s.

Results indicate that the number of segments in thin rod model has a great effect on the accuracy of simulation during non-stationary motion.The fidelity of simplified physical model is higher with more segments.It also turns out that the simulation results are basically consistent with wind tunnel tests when the segment number of cable amounts to 300.

5.Numerical simulations

5.1.Input data

Numerical simulation requires complete mass,inertia and aerodynamic data of the cable and decoy to yield correct results.The towed decoy system data are divided into two parts:physical data(weight,geometry and moments of inertia)and aerodynamic data(force and moment coefficients),which are presented below.

5.1.1.Physical data

The relevant physical parameters of the cable and decoy are listed in Tables 4 and 5,respectively.

5.1.2.Aerodynamic data

(1)Aerodynamic force of cable

The real towed cable is so flexible that it is difficult to predict the aerodynamic force on it accurately.So in our case,the engineering empirical formulas proposed by Jaremenko26are employed to calculate the aerodynamic force on the cable segments.

As demonstrated in Fig.10,the aerodynamic force on the ith cable segment is calculated by

where ρ is air density,dithe diameter of the ith cable segment and lithe length of the ith cable segment;Ct,Cnand Cbstand for the cable’ s coefficient of drag in the tangent,normal and binormal directions in the corresponding Reynolds number and Mach number,respectively;Vtai,Vnaiand Vbairepresent airspeed component in the tangent,normal and binormal direction,respectively.

Fig.8 Shape of cable and altitude of decoy at a wind speed of 40 m/s.

(2)Aerodynamic force of decoy

In the airflow coordinates,decoy’s aerodynamic lift La,drag Daand pitching moment Maare calculated by

where Vdis the airspeed of decoy;Sdis the reference area of aerodynamic coefficients,which is the cross-sectional area of decoy;CL,CDand Cmrepresent aerodynamic coefficients of lift,drag,and pitching moment,respectively;The pitching moment coefficient Cmis a function of angle of attack α，time rate of change of the angle of attack˙α and pitching velocity q.

The shape of decoy is bilaterally and vertically symmetrical,as shown in Fig.11.Consequently,the longitudinal and lateral aerodynamic characteristics of decoy will be consequently the same.Aerodynamic characteristics of decoy obtained by the CFD method are illustrated in Fig.12.

5.2.Influence of segment number

Simulations are performed to study the influence of segment number under real flight conditions.Assume that during t=0–2 s,the aircraft flies at 9 km altitude and a speed of Mach number 0.6.Then at t=2 s,the aircraft begins to accelerate uniformly at an acceleration rate of 10 m/s2and eventually speeds up to Mach number 0.8.The acceleration process lasts for 6 s.After that,it keeps uniform motion at a speed of Mach number 0.8.

Fig.9 Tension of cable and pitching angle of decoy during acceleration movement.

Table 4 Parameters of towed cable in numerical simulations.

Here we pay attention to two physical parameters in this process:maximum tension at connection point and oscillation amplitude of pitching angle which is defined as the difference value between the maximum and minimum values.Results indicate that the number of segments in thin rod model has great effect on the accuracy of simulation during nonstationary motion,as illustrated in Figs.13 and 14.But when segment number exceeds over 300,simulation results remain almost unchanged.Hence,in subsequent calculations,the segment number of cable is chosen to be 300 so as to yield accurate results.

5.3.Results and discussion

5.3.1.Steady flight

(1)Influence offlight conditions:

Assuming that the aircraft flies a steady-level straight path at 9 km altitude,the shapes of the towed cable at different speeds are illustrated in Fig.15.It indicates that the cablenearby the decoy exhibits a large curvature,while the cable close to the aircraft approximates a straight line.Tension gradually increases along the length of cable from the decoy to the aircraft(see Fig.16).As the flight speed increases,the pitching angle of decoy decreases but aerodynamic force on the decoy barely changes,as shown in Fig.17.Therefore,the tension of cable at the towing point almost remains unchanged with speed(see Fig.18).However,since the aerodynamic force on the cable is proportional to the square of speed,the maximum tension of cable non-linearly increases.The vertical offset of decoy decreases with speed as the aerodynamic force on cable is mainly in normal direction.

Table 5 Parameters of towed decoy in numerical simulations.

Fig.10 Schematic diagram of aerodynamic force on cable segment.

Fig.11 Shape of decoy.

Fig.12 Aerodynamic force coefficients of decoy vs angle of attack.

Assuming that the aircraft flies a steady-level straight path at Mach number 0.7,as shown in Fig.19,the angle between cable and horizontal line increases with flight altitude.Fig.20 illustrates the distribution of tension along the cable at different flight altitudes.The density of air decreases with flight altitude,but the pitching angle of decoy increases(shown in Fig.21).As a result,the aerodynamic force on the decoy barely changes,so does the tension of cable at towing point.Figs.21 and 22 show that the vertical offset of decoy and the maximum tension of the cable non-linearly vary with altitude.

(2)Influence of towing point and decoy’s center of gravity:

For conventional air vehicle,the aerodynamic center locates behind the center of gravity to ensure the longitudinal static stability.The aerodynamic center of decoy hardly changes as the flight Mach number increases within subsonic range.In this instance,the aerodynamic center of decoy locates far behind the decoy’s center of gravity due to the position of wing.Consequently,the aerodynamic center barely affects the stability of towed decoy system.Therefore,in this section,studies mainly focus on the influence of towing point and the decoy’s center of gravity.Simulation results show that xtand xgwill affect whether the system can reach an equilibrium state,where the position of decoy and shape of cable keep stable.

Forexample,atH=9 km,Ma=0.7,whenfixing xg=0.27 m,as Fig.23 suggests,the pitching angle of decoy decreases with xt.The maximum tension of cable almost keeps constant.If xt＞0.26 m,the decoy will not achieve a stable equilibrium position.Similarly,when fixing xt=0.12 m,as shown in Fig.24,the pitching angle of decoy increases with xg.The maximum tension of cable almost keeps constant.If xg＜0.13 m,the decoy will not converge.

Fig.13 Maximum tension at connection point vs segment number.

Fig.14 Oscillation amplitude of pitching angle vs segment number.

Fig.15 Shape of cable at different flight speeds(H=9 km).

Fig.16 Distribution of tension along cable at different flight speeds(H=9 km).

Results show that the towing point has to locate before the center of gravity.For the stability of system,the stability margin is suggested to be 5%of the length of decoy’s body.Results also indicate that the convergence time increases with the distance between the towing point and the decoy’s center of gravity.It is because that the coupling effects of cable’s tension and the oscillation amplitude of decoy’s position are proportional to|xg-xt|.Therefore,the value of|xg-xt|is recommended to be less than 20%of the length of decoy’s body.However,this threshold value varies with the physical and aerodynamic characteristics of cable and decoy.

Fig.17 Pitching angle and vertical offset of decoy vs flight Mach number(H=9 km).

Fig.18 Maximum tension of cable and tension at towing point vs flight Mach number(H=9 km).

Fig.19 Shape of cable at different flight altitudes(Ma=0.7).

5.3.2.Accelerated motion

Assume that during t=0–2 s,the aircraft flies at 9 km altitude and a speed of Mach number 0.6.Then at t=2 s,the aircraft begins to accelerate uniformly at the acceleration rate of 10 m/s2and eventually speeds up to Mach number 0.8.The acceleration processlastsfor6 s.Afterthat,itkeepsuniformmotionataspeed of Mach number 0.8.

Fig.20 Distribution of tension along cable at different flight altitudes(Ma=0.7).

Fig.21 Pitching angle and vertical offset of decoy vs flight altitude(Ma=0.7).

Fig.22 Maximum tension of cable and tension at towing point vs flight altitude(Ma=0.7).

As illustrated in Fig.25,the tension of the cable increases gradually during the acceleration motion.The maximum tension reaches 325 N at t=7.8 s.After that,it fluctuates within the range of 260–320 N and eventually converges to a constant value until t=16 s.Fig.26 presents that the pitching angle of decoy starts increasing at t=2 s,then decreases with fluctuation.After the acceleration motion,it fluctuates within a narrow range and eventually converges to a constant value until t=20 s.The shape of cable during this process is shown in Fig.27.

Fig.23 Pitching angle of decoy and maximum tension of cable vs xt.

Fig.24 Pitching angle of decoy and maximum tension of cable vs xg.

Fig.25 Maximum tension of cable during aircraft constant acceleration movement.

There are high-frequency oscillations in the simulation results.It is because once the flight speed changes,the aerodynamic force on cable and decoy changes immediately,which will result in the simultaneous altitude change of decoy.Due to the dynamic stability,the additional aerodynamic force drives towed decoy to previous state again.But this period is very short(about 0.1 s).That’s how the high-frequency oscillations occur.Finally,the oscillations disappear on account of aerodynamic damping.

Fig.26 Pitching angle of decoy during aircraft constant acceleration movement.

Fig.27 Shape of cable during aircraft constant acceleration movement.

Fig.28 Maximum tension of cable during aircraft up-risen maneuver.

This phenomenon is also found in wind tunnel tests,as shown in Fig.9.But different from simulation results of physical model,the oscillations in wind tunnel tests are smaller because of the structural damping of cable,which is not taken into consideration in the physical model.

5.3.3.Up-risen maneuver

Fig.29 Shape of cable during aircraft up-risen maneuver(ω=0.20 rad/s).

Fig.30 Shape of cable during aircraft turning movement(ω=0.20 rad/s).

Fig.31 Maximum tension of cable during aircraft turning movement.

Suppose that during t=0–2 s,the aircraft flies at 9 km altitude and a speed of Mach number 0.6.Then at t=2 s,the aircraft begins to rise up at the angular velocity ω=0.20 rad/s.As the pitching angle of aircraft increases,the angle between gravity and aerodynamic force reduces.Accordingly,the maximum tension of cable increases with fluctuation,which is shown in Fig.28.As a result,the cable is straightened as illustrated in Fig.29.

Researches indicate that the cable and flight path becomes closer as the angular velocity of aircraft gets larger.Note that in this process,the cable may enter the hot exhaust jet of aircraft,risking of being blown.So the value of angular velocity is suggested to be less than 0.10 rad/s.However,this restriction also depends on the area of hot exhaust jet,which varies with the engine state.

5.3.4.Turning movement

Assume that the aircraft begins to turn right at the angular velocity of 0.20 rad/s at t=0 s.Fig.30 demonstrates the flight path and shape of cable.The horizontal projection of cable is located outside of the circular path.During the turning movement of aircraft,the cable’s extent of stretching outwards is proportional to the aircraft’s angular velocity,which is benef icial to avoiding the aircraft exhaust jet.Meanwhile,the decoy,aircraft and missile form a triangle,which promotes the decoy’s performance.There is an extra centrifugal force during this process,so that the tension of cable is greater than that during the straight flight at the same speed magnitude(see Fig.31).The average differential value is about 20 N.It will increase with the angular velocity of the aircraft.

6.Conclusions

(1)The computation complexity of tension recurrence algorithm for calculating the towed decoy system is directly proportional to the number of rigid segments,which is lower than the usual numerical solution methods.The computing speed is significantly improved when segment number is up to 100.The results of simulation based on tension recurrence algorithm are basically consistent with wind tunnel tests when the segment number of cable amounts to 300.So the demand of rapid analysis for engineering is easily satisfied.

(2)The tension gradually increases along the length of cable from the decoy to aircraft.The maximum tension increases with flight speed and decreases with altitude while tension at the towing point almost remains unchanged at different flight speeds and altitudes.The vertical offset and pitching angle of decoy decrease with flight speed and increase with altitude.

(3)The relative position of the towing point and the decoy’s center of gravity barely affect the cable’s shape and tension,but will greatly affect whether the system can reach equilibrium.The towing point is suggested to locate before the center of gravity for the stability of system.Distance between the towing point and the decoy’s center of gravity is recommended to be 5%-20%of the length of decoy body.However,this threshold value varies with the physical and aerodynamic characteristics of cable and decoy.

(4)In up-risen maneuver process,the cable and flight path become closer as the angular velocity of aircraft gets larger.The cable may enter the hot exhaust jet of aircraft in this process,risking of being blown.The value of angular velocity is suggested to be less than 0.10 rad/s.However,this restriction depends on the area of hot exhaust jet which varies with the engine state.

(5)In turning movement process,the tension of cable is greater than that during the straight flight at the same speed magnitude.The cable’s extent of stretching outwards is proportional to the aircraft’s angular velocity.This is bene ficial to avoiding the aircraft exhaust jet.Meanwhile,the decoy,the aircraft and the missile form a triangle,which promotes the decoy’s performance.

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25 September 2015;revised 30 November 2015;accepted 10 February 2016

Available online 20 October 2016

Numerical simulation;

Spinor;

Tension recurrence algorithm;

Towed cable;

Towed decoy

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses:madongli@buaa.edu.cn(D.Ma),wangshaoqi@buaa.edu.cn(S.Wang),yangmuqing@buaa.edu.cn(M.Yang).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.09.003

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).