DU Zheng,CHEN Wan-chun

(School of Astronautics,Beijing University of Aeronautics and Astronautics,Beijing 100191,China)

0 Introduction

With the continuous development of science and technology,multi-purpose guided missile becomes an important development direction of the next generation missile. There are two main methods to get kill performance of multi-purpose guided missile against gunship currently. One method focuses on computing kill probability of missile against the cabin of the gunship by using kinematics and ignoring rotors when the kill probability models of missile against gunship are established[1,2]; the other method focuses on the mechanical property changes and damage assessments of the cabin and the rotor blades by combining the finite element method with materials and dynamics to establish a collision model of gunship in the study of kinetic energy rod or other shape fragments collision with gunship[3-5].

This paper combines the advantages of the two methods,estimates and verifies armor thickness of gunship. We introduce the important coefficient of vulnerability blade unit and try out new ideas to establish laser fuze startup model of rotors of gunship and kill probability model of warhead against rotor blades. The kill probability model of multi-purpose guided missile against gunship is systematic and compact under the conditions to ensure a high analytical precision. Analysis of sensitivity and optimal designs of the main parameters that affect kill probability of multi-purpose guided missile against gunship are conducted with the model,which can provide basis for the design of multi-purpose guided missile.

1 Coordinate system

1.1 Cabin coordinates

We set AH-64D gunship as the example to establish the cabin coordinates of gunshipozxzyzzz,as shown in Fig.1. Whereozis the origin of coordinates,which locates in the gunship centroid;xzaxis points to the nose of gunship;yzaxis is perpendicular to thexzaxis and vertical upwards;zzaxis satisfies the right-hand rule.

According to the geometry distribution of AH-64D,the cabins of AH-64D are simplified as cuboid structure. Main parameters of cabins of AH-64D are shown in Table 1.

Fig.1 Diagram of cabin coordinates

Table 1 Equivalent data of key parts of AH-64D gunship

1.2 Rotor coordinates

It is assumed that four blades are in the same plane during the flight,and the missile velocity vector is consistent with the missile body axis. The rotor coordinatesorxryrare defined in Fig.2.

Fig.2 Rotor coordinates

It is assumed that the angle between No.1 rotor blade andx-axis of the rotor coordinates isξ,when the furthest trigger point from the rotor shaft touches trigger side. Given that the randomness of the blade position and the consistency of the four blades properties,the angleξis subject to the uniform distribution [0°,90°]. The parameters of AH-64D rotor geometry are shown in Table 2.

Table 2 Rotor geometry parameters of AH-64D

2 Laser fuze triggering process

2.1 Calculation of cabin laser fuze triggering process

With applications of the “trigger line” method[6],each cabin of the gunship is meshed. If intersections of the grids are taken as trigger points and grids are treated as vulnerable units,size of the grid is 0.01 m×0.01 m in this paper.

The trajectory of trigger points in the body coordinates is

(x-xbi)/ν1=(y-ybi)/ν2=

(z-zbi)/ν3,

(1)

whereν1,ν2,ν3are the target relative speed components of the missile in the missile body coordinates;xbi,ybi,zbiare the coordinates component of trigger points in the missile body coordinates which take the center of fuze as the origin point.

The equations of trigger side in the missile body coordinates which take the center of fuze as the origin point are

(2)

The corresponding timetithat is intersection (xci,yci,zci) of the trigger points trajectory with trigger side is calculated. The judgment of whether the point (xci,yci,zci) is in the trigger side is that the intersections are not only within the fuze field of vision,but also within the fuze detection distance.

2.2 Calculation of rotor blade laser fuze triggering process[7]

The trajectory of rotor shaft endpoint in the missile body coordinates is

(x-xa)/ν1=(y-ya)/ν2=

(z-za)/ν3=T,

(3)

wherexa,ya,zaare coordinate components of rotor shaft endpoint in the missile body coordinates;ν1,ν2,ν3are the target relative speed components of missile in the missile body coordinates;Tis the time when endpoint of rotor shaft touches the fuze trigger side.

Bending deformation during rotation of rotor blade is ignored. TimeTcan be obtained by calculating intersection (xb,yb,zb) of the trajectory of rotor shaft endpoint when endpoint of rotor shaft touches the fuze trigger side,as shown in Fig.3.

Fig.3 Fuze start point of rotor blades geometric relationship

The relative motion of multi-purpose guided missile and the rotor blades is shown in Fig.3.OBCis the fuze trigger side;δis the angle between gunship velocity and the rotor plane;νris the target relative velocity of the missile;νtis the gunship velocity;νmis the missile velocity respectively;γis the intersection angle between missile and target;φsis the fuze inclination angle,OC≤Rs;tis the time interval that the furthest trigger point of the adjacent rotor blade from the rotor blade shaft touches the trigger side;ωis rotor angular velocity;AC=DHis the length of blade.

And there are

Since the time interval that rotor blade trigger point touches trigger side twice is short,so there is approximately relationship ofGH≈FG. To solve Eq.(4) can obtain the time intervaltthat the furthest trigger point of the adjacent rotor blade from the rotor blade shaft touches the trigger side. So the timet′ that laser fuze begins to accumulate signal is

t′=T-AB/νr+t·κ,

(5)

whereκis a uniformly distributed random number between [0,1].

It is assumed that the laser fuze starts working when it is away from the targetL(L>0) along thex-axis of relative velocity coordinates. The location of the start point of the laser fuze in the target relative coordinates is

(6)

whereνris the relative velocity of missile and target;ρ,ηθare miss distance and miss orientation respectively;τis the average fuze delay time;tn=min(t′,ti).

The angle between No.1 rotor blade andx-axis of the rotor coordinates changes to (ξ+ω(t·κ+τ)) when the laser fuze starts working.

3 Establishment of kill probability models

3.1 Modeling of kill probability of cabin

3.1.1 Equivalent armor thickness estimate of gunship

Armor thickness estimate requires thickness ratio of the composite armor,so the data acquisition is very difficult to obtained because of various reasons. Therefore,formulas and data are needed to be reverse derived and optimized to obtain armor thickness and thickness ratio which shoule be validated by experiments. According to U.S. military specifications for the bulletproof level requirements for the fifth class of lightweight ceramic/composites armor used for rotor aircraft:V50is 1 600 ft/s,and incident angle is 0°[8].

Taking B4C/Kevlar armor of AH-64D gunship as example,assuming that it is attacked by 12.7 mm flat tungsten alloy rod. The diameter of flat tungsten alloy rod is 12.7 mm; the length is 24 mm; penetration velocity is 487 m/s. The density of B4C is 2 510 kg/m3; the density of Kevlar is 1 650 kg/m3; the ultimate tensile strength of Kevlar is 9.664×108Pa; the maximum failure strain of Kevlar is 0.019.

Taking the areal density of B4C/Kevlar as the objective function by the improved Ben-Dor formula[9],that is

whereV50is the ballistic limit of B4C/Kevlar；mis the quality of projectile;Ris the radius of the tungsten alloy;h1,h2are the thicknesses of panel and backplane;σis ultimate tensile strength of Kevlar;εis the maximum failure strain of Kevlar;ρis the density; subscripts 1,2 are panel and backplane; generallya=1.

Using simulated annealing genetic algorithm (SAGA) to optimize the objective function,the optimization process is shown in Fig.4.

The optimum areal density is 63.95 kg/m2; the optimum thicknesses of panel is 19.4 mm; the optimum thickness of backplane is 9.3 mm.

Equivalent thickness of duraluminhdcan be obtained after getting armor thickness ratio according to the Eq.(8)[10]

(8)

whereσb1,σb2are ultimate strength of panel and backplane;σbcis ultimate strength of standard duralumin;η1,η2are equivalent correction factors of panel and backplane;h1,h2are actual thicknesses of panel and backplane.

Fig.4 Optimization process of surface density by SAGA

3.1.2 Kill probability of fragments

Residual velocity and residual mass of fragments estimation can be obtained by THOR equation based on the experimental data fitting[11]. THOR equation can apply to the steel fragments that velocity is less than 2 500 m/s,and the aspect ratio is less than 3 penetrate metallic and non-metallic target plate,it is

(9)

whereνris average residual velocity of the fragments after penetrating the target;mris average residual mass;νsis average dynamic striking velocity;hdis equivalent thickness of duralumin;Asis the contact area when penetrating the target;msis initial mass of the fragments;θis the average bank angle when striking the target;c11-c25are coefficients of THOR equation of different materials.

Kill Probability of fragmentsPac1,ican be calculated in accordance with the ratio dynamic energy formula in the case of lacking relevant experimental data

Pac1,i=F(Ei)·Ki,

(10)

whereF(Ei) is the empirical formula;Kiis correction factor associated with vulnerability of the cabin.

Generally,cumulative kill probability is used to calculat kill probability of fragmentsPac2,iof vulnerable units. The vulnerable unit is not damaged when the number of hit fragments ofi-th vulnerable unitKiis less thanni,but it is inevitable damaged when the numberkiis more thanmi.

wheremiandniare related with position of vulnerable unit on the rotor blade and the mass and velocity of the fragments.

Kill probability of fragments of the cabinPacis

(12)

whereNc1,Nc2are the numbers of vulnerable units mentioned above.

3.1.3 Kill probability of shockwave

Overpressure and specific impulse of shockwave can be obtained in Ref.[12]. Kill probability of shockwave of vulnerable units is

(13)

where ΔPi,Iiare overpressure and specific impulse of shockwave on thei-th vulnerable unit; ΔP*,I*are critical overpressure and critical specific impulse of wavefront when the target is destroyed;Di=(ΔPi-ΔP*)(Ii-I*),K=α(ΔP*/2+I*/2)β,α,βare related with materials of blade and the damage level,and they can be obtained experimentally.

Kill probability of shockwavePbyis

(14)

wherePby,iis kill probability of shockwave of vulnerable units.

3.2 Modeling of kill probability of rotor blades

3.2.1 Important coefficient of vulnerability blade unit

This paper introduces the important coefficient of vulnerability blade unitλ. The coefficient is used to determine whether the blade is incapable when a vulnerable unit along the rotor blade is damaged. The coefficient is related with rotor structure,materials and spanwise location of the vulnerable units along the rotor blades,and can be obtained experimentally.

For the 60 vulnerable units along the blade spanwise,50 rotor instability experiments after destruction are conducted respectively,and average results are shown in Table 3.

Table3Partofinstabilityexperimentaldataofrotorvulnerableunitsafterthedestruction

xi(m)λijxi(m)λijxi(m)λij1.30.9771.50.9961.71.0003.70.9923.91.0004.10.9416.10.6206.30.5696.50.456

The least-squares curve is used to fit the experimental data,and the function prototype is

λij(xi)=1-1/(1+e-(axi+b)),

(15)

wherea,bare fitting constants;xiis the distance of thei-th vulnerable unit of each rotor blade from the shaft;i=1,2,…,Nis the serial number of vulnerable units,Nis the total number of vulnerable units of each rotor blade;j=1,2,3,4 is the serial number of four rotor blades.

Fitting curve of the important coefficientλijof vulnerability blade unit of AH-64D is shown in Fig.5.

Fig.5 Fitting curve of important coefficient λij of vulnerability blade units

Fitting parameters are obtained asa=1.796，b=-11.55,so

λij=λij(xi)=1-1/(1+e-(1.796xi-11.55)).

(16)

From Fig.5,the important coefficient of vulnerability blade unitλdecreases with an increasing distance along the blade spanwise,then it remains at about 1 m before 4.5 m and decreases obviously after 4.5 m,which means the closer the destroyed vulnerability unit from the shaft is,the greater the kill probability of the rotor blade is. And the best detonation point of warhead should be near the rotor shaft.

3.2.2 Kill probability of fragments

Kill probability of fragmentsParcan be obtained by the related theories after introducing the important coefficient of vulnerability blade unitλ. That is

(17)

wherePa,i1,Pa,i2,Pa,i3,Pa,i4are kill probabilities of fragments of vulnerability units of the four blades;λi1,λi2,λi3,λi4are the important coefficients of vulnerability blade units.

3.2.3 Kill probability of shockwave

The destruction effect of rotor blade shockwave is divided into “overpressure-specific impulse” destruction and “bending-shearing” destruction. Kill probability ofi-th blade vulnerability unit is

Kill probabilitiy of blade destruction is

wherePby,i1,Pby,i2,Pby,i3,Pby,i4are the kill probabilities of “overpressure-specific impulse” destruction of four blades;Pbs,i1,Pbs,i2,Pbs,i3,Pbs,i4are the kill probabilities of “bending-shearing” destruction of four blades.

Kill probability of shockwavePbris

Pbr=1-(1-Pby)(1-Pbs).

(20)

3.2.4 Collision analysis of rotor blades against missile

Collision injury of missile against the rotor blades is mainly caused by relative speed. Fracture of the blades caused by the collision is considered only. Impact damage of material is related with fracture toughnessKIC,cross section of the blades and impact valueaK. In the district ofV-gap platform[13]

whereqsis yield strength of the material;CVNis charpy all-sample impacting energy of the material;Sis cross section of the charpy sample;S′ is cross section of the rotor blades;Evis energy consumption when the rotor blades are broken.

So kill probabilities of the rotor blades collided with missilePdris

(22)

whereEis kinetic energy of rotor blades collided with the missile.

It is supposed that the warhead detonates when the missile collides with the rotor blades.

4 Calculation process of kill probability model

Based on the analysis mentioned before,condition kill probabilityPK,iof thei-th Monte Carlo sampling is

PK,i=1-(1-Par,i)(1-Pbr,i)(1-

Pdr,i)(1-Pac,i)(1-Pbc,i),

(23)

where the calculation ofPar,i,Pbr,i,Pdr,i,Pac,i,Pbc,ican be obtained by Eqs.(11)-(22); If the missile hits the target directly,soPK,i=1.

Kill probability of the multi-purpose guided missile against gunship can be obtained by the models deduced before,that is

(24)

whereNis the total number of Monte Carlo sampling.

Kill probability calculation process of multi-purpose guided missile against gunship is shown in Fig.6 with specific steps as follows:

1) Inputting main parameters of fuze and warhead of multi-purpose guided missile: fuze inclination angleφs,average fuze delay timeτ,filling coefficient of warheadKa,number of fragmentsN,mass of single fragmentm,static scattering angle of fragmentsφαand static scattering direction angle of fragmentsφβ.

2) Inputting guidance precision of multi-purpose guided missile,and conducting guidance precision sampling and position of rotor blades sampling.

3) Calculating laser fuze start timet″.

4) Calculating detonation point of warhead (x,y,z).

5) Estimating armor thicknesses of gunshiph1,h2; Estimating armor equivalent thickness of duraluminhd; Calculating the important coefficients of vulnerability blade unitsλ1,λ2,λ3,λ4.

6) Calculating kill probabilities of fragments of the cabin and rotor bladesPac,i,Par,i.

7) Calculating the parameters of shockwave of warhead ΔP,I,ΔP*,I*; Calculating shockwave kill probabilities of the cabin and rotor bladesPbc,i,Pbr,i.

8) Calculating the kinetic energy of rotor blades collided with missileE; calculating the energy consumptionEvwhen the rotor blades are broken; calculating the kill probability of rotor blades collided with missilePdr,i.

9) Calculating the condition kill probability of thei-th Monte Carlo samplingPK,i.

10) Judging whether the sampling frequency reached the maximum limit,if yes it is transferred to 11),otherwise transferred to 2).

11) Clculating kill probability of the multi-purpose guided missile against gunshipPK.

Fig.6 Kill probability calculation process of multi-purpose guided missile against gunship

5 Numerical examples

It is supposed that directions of the missile velocity and missile body axis are the same,miss distance of missileρ～N(8.0 m,1.0 m),miss orientation of missileηθ～N(74.5°,33°); missile velocityνmis 450 m/s,and gunship velocityvtis 103 m/s within the horizontal plane. The selected sensitivity analysis and optimization design variables are： fuze inclination angleφs,average fuze delay timeτ,filling coefficient of warheadKa,number of fragmentsN,mass of single fragmentm,static scattering angle of fragmentsφαand static scattering direction angle of fragmentsφβ.

5.1 Analysis of sensitivity

This paper selects range analysis method of orthogonal experiment to determine sensitivity of each factorSiwith the kill probability model of multi-purpose guided missile against gunship. The factors should be normalized and dimensionless,so

(25)

wherea,bare the range of influence factors,and they are shown in Table 4;r,r′ are factors pre- and post- normalized and dimensionless.

From Table 4,descending order of factors affecting kill probability is:Ka>m>τ>φs>φα>φβ>N. Coefficients associated with the warhead are predominant,so filling coefficient of warheadKamainly influences destruction degree caused by fragments and shockwave of warhead,while mass of single fragment mainly influences penetration effect of fragments; the second important aspect is fuze inclination angleφs,average fuze delay timeτ,static scattering angle of fragmentsφαand static scattering direction angle of fragmentsφβ,which mainly influence warhead detonation time to make warhead detonated nearby the best position.

Table 4 Range of influence factors a,b

Table 5 Results of orthogonal experiment

5.2 Parameters optimization of fuze and warhead

Taking kill probability of multi-purpose guided missilePKand mass of warheadMas the optimization objectives with intelligent single particle optimizer(ISPO)[14],the objective functionFfitis

min(Ffit)=A×(1-PK)+B×M,

(26)

whereAandBare weight coefficients ofPKandM,hereA=10，B=0.1. Range of design variables are shown in Table 4.

Parameters comparison of fuze and warhead between U.S. CM-501G multi-purpose guided missile and values of corresponding design variables are shown in Table 6,of which kill probability is 0.862 1 and mass of warhead is 12.50 kg. Optimizing CM-501G multi-purpose guided missile with ISPO to redesign the fuze and warhead,the optimization process of objective functionFfitis shown in Fig.7. Optimization process ofPKandMare shown in Fig.8 and Fig.9.PKafter optimization is 0.975 7,andMis 11.95 kg.

Fig.7 Optimization process of objective function Ffit

Fig.8 Optimization results of kill probability PK

From Table 6 we can know that the mass of warhead decreases 4.4% after optimization while kill probability increases 13.17%,and the optimization effect is obvious; the filling coefficientKais changed from 0.52 to 0.68,which means the rotor blades are more sensitive than blast warhead; the total number of fragmentsNdecreases from 2 000 to 450; the mass of single fragmentmincreases from 3.0 g to 8.5 g; the static scattering angle of fragmentsφαdecreases from 40° to 22.5°. To some extent,the decrease of static scattering angle of fragmentsφαoffsets the density reduction of fragments which is caused by decrease of the total number of fragmentsN,but the kinetic energy of single fragment increases significantly,which improves the penetration ability of single fragment directly.

Fig.9 Optimization results of warhead mass M

Table 6 Preliminary design parameters of CM-501G and optimal design parameters

6 Conclusion

A complete mathematical model of kill probability of multi-purpose guided missile against gunship is established; laser fuze actuation model and warhead condition kill probability model of rotor blades are structured with new ideas and the important coefficientλof vulnerability blade unit is introduced well to calculate kill probability of warhead against rotor blades,which makes the results more consistent with the theoretical analysis.

Sensitivity analysis is conducted for the main factors influenced on kill probability of multi-purpose guided missile against gunship to find out the most influential factors. After the U.S. CM-501G multi-purpose guided missile is optimized,the results show that the killing performance of the missile increases 13.17% and mass of warhead decreases 4.4%. When the filling coefficient,the total number of fragments and the mass of single fragment change significantly are optimized,we can know that the killing effect of a blast warhead with greater mass and smaller scattering angle fragments to attack gunship is better than fragmentation warhead with similar mass.

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