AIRCRAFT EQUIVALENT VULNERABLE AREA CALCULATION METHODS

Transcription

1 4 TH ITERATIOAL COGRESS OF THE AEROAUTICAL SCIECES AIRCRAFT EQUIVALET VULERABLE AREA CALCULATIO METHODS PEI Yag*, SOG B-Feg*, QI Yg ** *College of Aeoautcs, othweste Polytechcal Uvesty, X a, Cha, ** Depatmet of Mathematcs, othweste Polytechcal Uvesty, X a, Cha Keywods: vuleablty, equvalet vuleable aea, edudacy Abstact Based o mathematcal expectato theoy, a method has bee deduced to calculate the equvalet sgly vuleable aea of acaft by oe theat ht. By smulatg the kll evet of multple vuleable compoets to Model of Fllg Boxes wth Balls, the expected umbe of hts equed to kll a acaft has bee gve though cluso-excluso pcple dscete mathematcs. The equvalet sgly vuleable aea thus ca be attaed. The cocept of equvalet vuleable aea solves the poblem of cosdeg the effect of vuleable compoets o the total acaft vuleable aea by oe theat ht ad may povde valuable advces o whethe the edudacy desg techque s adopted o how to deteme the umbe of edudat compoets the acaft coceptual desg. Itoducto Acaft combat suvvablty (ACS) [] s defed as the capablty of a acaft to avod o wthstad a ma-made hostle evomet. Suvvablty s composed of two focus aeas: ) Susceptblty ) Vuleablty. Fo theats those that must ht the acaft to kll t, the pobablty of kll of the acaft P K (the acaft s kllablty) s the poduct of the pobablty of ht (the acaft s susceptblty) P H ad the codtoal pobablty of kll gve a ht (the acaft s vuleablty) P K/H. Thus, P K =P H P K/H () The vuleablty of the acaft (fo a patcula theat aspect) s usually expessed as the pobablty the acaft s klled gve a adom (ufomly dstbuted) ht aywhee o the peseted aea of the acaft P K/H, o the sgle-ht vuleable aea of the acaft A V. Vuleable aeas povde a bass fo compag the cotbuto of dffeet compoets to acaft vuleablty ad ae theefoe useful acaft desg. Kowledge of the most vuleable compoets ca be assstace povdg the modfcato advces as the edudacy desg techque fo example. If the acaft s subected to a sgle adom ht, the the total vuleable aea ca be obtaed by smply summg the sgly compoet vuleable aeas gve by [,] m A V = Av () = whee: A v = vuleable aea of the th sgly vuleable ctcal compoet gve a ht o the compoet m = total umbe of sgly vuleable ctcal compoets, each capable of poducg a specfed kll level of acaft [] Cae must be execsed to detfy all ctcal compoets, ad whethe they ae multply o sgly vuleable. All ctcal compoets have some level of vuleablty, level of edudacy, f ay. If t s assumed a sgle ht ca damage, at the most, oe compoet, the the fst ht upo a multply vuleable compoet acaft ca ot kll the acaft by defeatg oe of the multply vuleable compoets sce the lost

2 PEI YAG, SOG BI-FEG, QI YIG fucto of the klled compoet ca be compesated by aothe compoet of the set of compoets. Hece, Eq.() caot clude the cotbuto of multple vuleable compoets to the total acaft vuleable aea. Thus, the fst ht s ot a elable cteo as to the vuleablty of the acaft. It s fo ths easo that a equvalet vuleable aea cocept [] based o the expected umbe of hts E(X) equed to kll a acaft has bee devsed fo cosdeg the effect multply vuleable compoets o the vuleablty of acaft. But, to ou best kowledge, thee s o publc lteatue publshed to gve the devato of the equvalet fomula of efeece []. Wthout clealy udestadg the mathematcal fomula devato ad all the assumptos volved, t s vey had to put t to easoable applcato. I ths pape, we has deduced aothe equvalet vuleable aea fomula based o Model of Fllg Boxes wth Balls, whch s compaable to the fomula efeece [] computato esults. The detaled devato gves a full udestadg of the kll evet of acaft wth multply vuleable compoets. I the est of the pape, we fst dscuss the equvalet vuleable aea method efeece []. The a detaled devato of ou poposed fomula s peseted. Followg s a example to demostate the compaso of the two equvalet vuleable aea fomulas. Coclusos ad ecommedatos ae gve the fal secto of ths pape. Acaft Equvalet Sgly Vuleable Aea The cocept of equvalet sgly vuleable aea s applcable oly to mpactg ouds, ad sequetal compoud damage s excluded. A lage umbe of hts ae assumed ad espectve locatve locatos of the vaous hts o the taget ae assumed to be take fom a ufom populato. The equvalet sgly vuleable aea A VE fo a acaft cosstg of oe o moe sgly vuleable compoets ad oe set of detcal multply vuleable compoets s gve by [] A VE =A V /E(X) (3) whee: A VE = equvalet sgly vuleable aea A V = A V +A V A V = summed sgly vuleable aea of the acaft gve by Eq.() = umbe of detcal compoets costtutg the set of multply vuleable compoets A V = vuleable aea of each multply vuleable compoet obtaed as though the tem wee a sgly vuleable compoet E(X) = expected umbe of hts o A V equed to kll the acaft fo the umbe of hts x E(X)= ( ) ( ) ( )( ) η η η ( )...( k + ) + ( )( )...( k + ) η η η.. (4) whee: k = umbe of tems the multply vuleable set whch must be defeated to esult the specfed level of acaft kll η = facto of the summed vuleable aea epeseted by the set of multply vuleable compoets η =( A V) /A V (5) The quatty η ca also be tepeted as the facto of the lethal hts o the summed vuleable aea A V that compses lethal hts o the set of multply vuleable compoets. 3 Devato of Aothe Equvalet Vuleable Aea Calculato Method Based o mathematcal expectato theoy, by smulatg the kll evet of detcal multply vuleable compoets to Model of Fllg Boxes wth Balls as s show Fg., the expected umbe of hts equed to kll a acaft has bee deduced though clusoexcluso pcple dscete mathematcs. The devato s o the assumptos that:

3 AIRCRAFT EQUIVALET VULERABLE AREA CALCULATIO METHODS (ⅰ) Ay oe theat ht s take fom a ufom dstbuto. (ⅱ)The compoet whe ht has oly two states, amely kll o o kll. (ⅲ) The edudacy acaft has oly oe set of multple vuleable compoets ad the edudacy s acheved though the use of smla compoets whch each pefoms detcal fuctos ad each has the same vuleable aea. (ⅳ) Equal o moe tha k (k>=) boxes havg ball (balls) the boxes wll esult the kll of acaft. A V s < < ()... A A... A =, =, =,... = () I the above equatos, A deotes the umbe of combatos of the elemets of A, amely, cadalty of A. Let = Ω () W = = A = C( = W ) (3) A A = C ( (4) W = ) < BOX BOX BOX3 BOX W A <... < = A... A = C ( ), =,, (5) A V Let P be the popety that the th box s empty whe balls ae put to the boxes adomly. Let A be the subset cotag the elemets that have popety P the uvesal set Ω, wtte as Thus, Fg.. Model of fllg boxes wth balls A ={W Ω :W has popety } (6) = (, =,,3, (7) A ) A A = ( ), < A A... A P (8) = ( ), < <... <, =,, (9) A A... A = [ ( )] =, whee, Let W A A... A ( m ) = = (6) C! = ( )!! = (7) be the umbe of elemets that has at least m popetes of the popetes,,, uvesal set Ω ad be the P P umbe of elemets that has ethe moe o less tha m popetes of the popetes,,, uvesal set Ω. Fo example, P P P P [ m ] ( ) = A A... A (8) [ ] = A A A (9)... Based o cluso-excluso pcple [3,4] dscete mathematcs, we have ( ) = ) = ( W () 3

4 PEI YAG, SOG BI-FEG, QI YIG [ ] = ) W = m [ m] = ) = ( () ( C W () m m+ m+ m m ( m) = ) Cm + Wm+ = ( (3) Accodg to the assumpto (ⅳ)ad the kll pocess of acaft wth multply compoets that acaft wll be klled whe oe of the sgly-ht compoets s klled o equal k o moe tha k compoets ae klled the set of multply compoets, we have f x k, the f x k, the P(X=x) = x P(X=x) = η ( η) (4) x ( x ) k x ( ( )) x [ ( k )] + ) ( η) + η x x ( k η η (5) whee P(X=x) deotes the kll pobablty fo the adom vaable X= x, x=,,+. The expected value E(X), also kow as the mea ad the expectato, whch s a weghted aveage of the adom vaable x, ad the weghts ae the pobabltes, s gve such that E(X)= P(x=)+ P(x=)+ 3 P(x=3)+ + x= = xp( X = x) (6) Substtutg Eqs.(4) ad (5) to Eq.(6) gves k x= + x k + xp( X = x) x k x= k E ( X ) = xp( X = x) (7) Accodg to Eqs. () though (3), smplfyg expesso (7) gves whee, E( X ) = + η + η η + k = ( ) C k k + k C + ( k) η k + + kη k k ( k ) η η k (8) k η = η (9) As a specal case, the equvalet vuleable aea educes to the sum of the compoet vuleable aeas fo a acaft cosstg oly of sgly vuleable compoets. Thus, fo the case whee = ad k= efeg to sgly vuleable compoet acaft, E(X)= (3) Summazg, fomulas (8) though (3) ae ou poposed fomulas to calculate the expected umbe of hts. Substtutg them to Eq. (3) ca gve the equvalet sgly vuleable aea of acaft wth oe set of multply edudat compoets. 4 Example A sample calculato of A VE s llustated as follows []. Gve a sgle-place tw-ege fghte whch the eges ae cosdeed to be the oly oe set of multply edudat compoets ad both eges must be klled to esult a kll of the acaft. A V = ft =sgly vuleable aea of ethe ege = = umbe of edudat compoets k= = umbe of edudat compoets whch must be klled to esult a kll of the acaft A V = 4ft =total vuleable aea fo a sgly-vuleable compoet Eqs. (4) ad (8) all gve E(X)=.4=expected umbe of hts o A V equed to kll the acaft ad A VE =4.85=Equvalet sgly vuleable aea Thus, the cotbutg of multply compoets to the total vuleable aea of acaft s 4

5 AIRCRAFT EQUIVALET VULERABLE AREA CALCULATIO METHODS [( )/4.85]*%=6.65%. Refeece [] has poted out that the equvalet sgly vuleable aea A VE dffes oly slghtly fom the sum of the sgly vuleable compoet vuleable aea A V whe the multply vuleable compoets ae small. Hece, pactce ad depedg o the obectves of aalyss, t s fequetly possble to goe all o all but the most sgfcat multply vuleable compoets the acaft vuleablty assessmet. If the vuleablty of the set of multply vuleable compoets s goed, the expected umbe of hts equed to scoe a kll o the taget s E(X)=/(-η ) (3) Usg Eqs. (4) ad (8), the qualty /(-η ) ad E(X) ae plotted fo vaous combatos of ad k, as fuctos of η. We foud that the two equatos ca gve the same cuves. Fo coveece, the ecpocals of E(X) ad /(-η ) s plotted agast η Fg.. It ca be see that the dffeece the values vaes. A bass fo decdg whethe o ot to clude the cotbuto of a set of multply vuleable compoets to total acaft vuleable aea s accodg to the appoachg extet of ts / E(X)cuve to the cuve 6. I those cases whee the dffeece s acceptably small, the cotbuto of the multply vuleable set to total acaft vuleablty may be goed ad thus the umbe of the set of edudacy compoets ca also be detemed Fg.. 5 Coclusos ad Recommedatos Eqs. (8) though (3) ae ou poposed fomulas to calculate the expected umbe of hts equed to kll the acaft. The cocept of equvalet sgly vuleable aea should be used cautously, especally whe the multply vuleable compoets do ot have the same vuleable aea. I ths case, the Eqs (4), (8) though (3) ae.8.6 /E(X).4. k 3 4 -η (Facto of multply vuleable aea) η Fg.. Relatoshp betwee E(X) ad η fo vaous values of ad k 5

6 PEI YAG, SOG BI-FEG, QI YIG ot goously vald. Whe multply vuleable compoets do ot have the same vuleable aea, the cocept of equvalet sgly vuleable aea s also useful, ad ew equato ca be deduced though the abovemetoed Model of Fllg Boxes wth Balls by cosdeg the dffeet aeas of each multply compoet. The cocept of a equvalet sgly vuleable (A VE ) ca be geealzed to apply to acaft havg moe tha oe set of multply vuleable compoets []. A umbe of challeges exst whch could allow model cludg the effect of sequetal compoud damage compoet. Aothe topc of teest the equvalet vuleable aea calculato methods s the cosdeato of the case whee ovelap aea amog compoets exsts a gve theat aspect. Joual of othweste Polytechcal Uvesty, Vol., o. 6, pp 73-76, 3. (I Chese) 6 Ackowledge Ths eseach was suppoted pat by the atoal atual Scece Foudato of Cha (gat awad umbe: ad 378) ad the Doctoate Foudato of othweste Polytechcal Uvesty (gat awad umbe: CX-3). Refeeces [] Robet E.Ball. The fudametals of acaft combat suvvablty aalyss ad desg. Secod edto, Ameca Isttute of Aeoautcs ad Astoautcs, Ic., 3. [] Suvvablty acaft ouclea geeal ctea, Vol.,MIL-HDBK-336-, 988. [3] Rchad A.Buald. Itoductoy combatocs. Thd edto, Cha Mache Pess Petce Hall,. [4] Kebbeth H.Rose. Dscete mathematcs ad ts applcatos. Fouth edto, Cha Mache Pess McGaw-Hll, 999. [5] Pe Yag, Sog Bfeg ad L Zhake. O mpovg detemato of pobablty of blast kll of acaft. 6