CAUCHY PROBLEM FOR TECHNOLOGICAL CUMULATIVE CHARGE DESIGN. Christo Christov, Svetozar Botev

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1 SENS'6 Second Scientific Conference with International Particiation SPACE, ECOLOGY, NANOTECHNOLOGY, SAFETY 4 6 June 6, Varna, Bulgaria CAUCHY PROBLE FOR TECHNOLOGICAL CUULATIVE CHARGE ESIGN Christo Christov, Svetoar Botev 8 Evlogi Georgiev Blvd, G.S. Rakovski efence and Staff College 54 Sofia, tel , fax , e-ail: hhristov67@yahoo.co technological cuulative charge, design otiiation The cuulative liner velocity indeendence assution fro tie in its work under detonation roducts is ade on a foundation to the flat-radial Orlenko-Stanukovit s schea about cuulative charge oerating. The Cauchy roble for ordinary differential equation of first order is forulated relative to an unknown function of the technological cuulative charge geoetry. The equation solution allows constructing a technological rofile of a cuulative liner at a given rofile of the charge body. Introduction The availability of a velocity gradient lengthwise of cuulative jet is the factor of decrease of oeration erforance of a shaed charge on defocused distances. If the velocity gradient can be decreased or got to ero then the charge erforance of oerating will be increased in a large distances fro the target. This is advantage of a long-focus shaed charge. It is ossible to otiie charge oerating with intent by usage of different rofiles for art of the charge. The flat-radial Orlenko-Stanukovit s schea for siulation of a cuulative action allows contacting between geoetry of charge eleents and their echanical erforances. The solution of a roble for otiied eleent of a charge is reduced to deterination of a rofile of a surface of this eleent, at the given erforances and rofiles of other eleents. In result it also is the Cauchy roble for a differential equation. The utiliation of such roble takes lace at designing of new aunition at geoetrical liitations for its construction. The external liitations are a cylindrical rofile of the shell in artillery aunition for exale. The internal liitation is technological conical liner of the cuulative charge.. A Cauchy roble Let's analye a rocess of the cuulative liner reduction in following reorts of the writers [, - fig.. In the Cartesian Oy syste the equations of curves are given, they deterinate eleents surfaces of a shaed charge at rotation around an axis O. The equation y F is for an external surface of the shell. The equation y Φ is for an

2 internal surface of the shell. The equation y 3 ϕ is for an external surface of a cuulative liner. The equation y 4 f is for an internal surface of a cuulative liner. The equation y Φ siultaneously describes an external surface and equation y 3 ϕ describes an internal surface of an exlosive charge. The functions y, y, y 3 and y 4 are bounded they are continuous and have continuous firs derivatives and they execute following conditions: F Φ ϕ y f; F y h > ; [; h ; y. y Φ y 3 ϕ y 4 f O h Fig.. Cuulative charge schea and its ain identifications. It is acceted the exlosive charge is isotroic and hoogeneous we leave out technological defects. The front of a detonation wave oves on ass of exlosive charge fro left to right and it is a flat and erendicular to axis O. In tie t the detonation wave reaches to of a cuulative liner and at t > the wave are oving along a liner with detonation velocity. The schea of a cuulative liner reduction is utilied on fig. [3: y A B f α B L L fdf O A d B

3 Fig.. Reduction schea and cuulative liner collase. The eleentary art of a liner with coordinate and length d is selected. In tie t / the oint A of this art starts oving to a charge syetry axis with velocity W. uring tie dt d/ while the detonation wave will reach a oint B the oint A will be dislaced in a direction of axis О on distance dy which one deterinates itself under the forula [3: d dy W dt W and will be on distance fro axis О: d R f dy f W. We ake an assution. The velocity W does not deend on tie and it is a function only coordinate [,. Then oint A will ass a distance R for tie: T R d f W W W. 3 For the sae tie the oint B is dislaced in a direction of an axis O with velocity W dw and oint B is traveled a ath: d W dw L T[ W dw f W 4 W and B will be fro an axis O at distance L : L [ f df. 5 L The collase angle of a cuulative liner art on fig. is deterined under the forula: or d f df f W [ W dw L W tgα 6 d d df f dw W tgα. d W d The reduced velocity W of an eleentary art of liner with length d is connected to cuulative jet velocity W in considered cross-section by a kineatics roortion [,: Then α ki W [ W tg. 7 3

4 α α d tg W tg df f dw α tg tg W α. 8 d α d d W tg If we shall deand the condition W W const, we take following equation: In an equation 9 functions tg, sin, functions so: tg sin df dα W α tgα f tg. 9 d sinα d W tg α α tg arctg [ tg[ arctg W d α и are undefined. We show these d. ; α sin arctg [ [ arctg sin ; α tg tg arctg W [ tg [ arctg ; W d arctg [ α [ arctg d d where is soe interediate function d The first derivative fro is W d d, 3 d d [. 4 3 d { [ } d d. 5 d W Then the equation 9 accets a following view: 3 W { [ γ } ' f f '. 6 W ensities of aterials of the shell and the exlosive charge and the cuulative liner ρ K,,ρ ВВ,,ρ О are constant values. Then the eleentary asses of the shell, the exlosive charge and the liner н,, articiate in following exression so: d d d d 4

5 5 dfi Fi df F df f dfi fi dfi fi dfi Fi, 7 [ [ A ; 8 B [ ;9 C [ ; F a F k π ; b Оπϕ ϕ ; c BB Φ Φ π ; 3 d k Φ Φ π ; 4 f e f Оπ ; 5 g BBπϕ ϕ. 6 Then we can receive a following equation: [ Φ d d c C d A E d df a A E d df [ W d df e B E d d g C b B E ϕ, 7 where [ { } 3 W f E i. 8 The equation 7 is a differential equation of first order for unknown one of functions F, Ф, ϕ or f at given other three and initial conditions for an unknown function. This equation is assued for forulation of a Cauchy roble about deterination of the geoetrical erforance of a cuulative charge, which one ensures no-gradient foring of a cuulative jet.

6 This way allows sulying no-gradient foring of a cuulative jet by a solution of a Cauchy roble. The following two cases are baseline for such otiiation. 3. Exales 3.. Cuulative Liner Profile esign for No-Gradient Cuulative Jet Let's study one of ossible version of realiation of a roble 7. At designing of cargo aunition cuulative charge, it is necessary to link of geoetry of the subunition shell with a rofile of the caera of the rojectile. This is a internal liitation. It is introduced a new designation for an equation of thickness of a cuulative liner on a noral to an axis О by the function δ ϕ -f δ const is a secial case. The roble is decided at the indicated assutions and liitations above. By analogy 7, it is ossible to show { [ } 3 df f. X d W or df d X A a [ A d C c dδ d i df d dφ d W i [ B. e C. g. [ B. b C. g. df Wi df dφ E [ A a A d C c d d d where { E [ B b C g } dδ d [ B e C g Q 9, 3 Q 3 and the values b, e, g are deterined siilarly - 6 with allowance for links δ. In result, the Cauchy roble is forulated for a differential equation of first order concerning an unknown function f. The nuerical solution of a roble allows constructing a rofile of a cuulative liner for no-gradient foring of a cuulative jet. For its solution it is necessary to adit the hyothesis: f - is sall on value. Then the first value of an unknown function f can be received with the hel of the forula of the Euler: f f tgα. 3 Fig.3 deonstrates a solution of a roble 3. for no-gradient cuulative jet with vary of a liner thickness δ δ const: δ ; δ ; δ 3 3 ; δ 4 4, for a cuulative charge by constant thickness of the cylindrical shell. Fig.4 deonstrates a solution of a roble 3. for no-gradient cuulative jet with vary of velocity of a cuulative jet W const: W, 5 /s; W, 7 /s; W,3 9 /s; W,4 /s; W,5 3 /s for a cuulative charge by constant thickness of the cylindrical shell. 6

7 But the decided cuulative liner is not technological. In this case the new cuulative liner for could ake a roble in serial anufacture. That is why we have to deterinate a artial solution for this roble where the liitation for cuulative liner ust be a conical for and no-gradient jet. 3.. Conical Cuulative Liner Profile esign For No-Gradient Cuulative Jet Let's study another version of a roble realiation 7. In this case the we are going required a condition for technological and conical tye of liner and a cylindrical tye of the shell. At designing of cargo aunition cuulative charge, it is necessary to link of Fig.3. Fig.3 7

8 Fig.4. geoetry of the subunition shell with a rofile of the caera of the rojectile. This is an internal liitation. It is introduced a new designation for an equation of thickness of a cuulative liner on a noral to an axis О by the function δ ϕ -f. In this case δ const and ϕ const and f const and F x Ф x. The roble is decided at the indicated assutions and liitations above and by analogy 7 we can write or E dϕ d df d [ B b C g [ E B e W dδ. ε W. ε df df [ B b C g. Q dx ε d d,33 34 where { E [ B b C g } Q 35 and the values b, e, g are deterined siilarly - 6 with allowance for links δ. In result, the Cauchy roble is forulated for a differential equation of first order concerning an unknown function δ. The nuerical solution of a roble allows constructing a conical rofile of a cuulative liner for no-gradient foring of a cuulative jet in case of cylindrical shell. Fig.5 deonstrates a solution of a roble Fig. 5. Fig. 6 shows the decision of a straight line roble for cuulative construction which we designed. 8

9 4. Conclusion The obtained results are tested by exerient and the criterion for no-gradient cuulative jet was checked u through the deth of aror enetration in a hoogeneous steel aror. At variation of distances between a cuulative charge and barrier the deth iercing was saved aroxiately constant, which one is the evidence of obtained theoretical results fig 7 Fig. 6. Fig. 7. References. Orlenko L.P.: Gas dynaics robles. Part. oskow, "Bauan" TU, 977, 48.. Orlenko L.P., Babkin A.V., Kolakov V.I.: Probles of alied gas-dynaics. oskow, "Bauan" TU,

10 3. Baranov V.L., Nikolov G.N.: Cuulative charge constructive design otiiation. Tula State University News, TSU, 99.