This article reports on the results of a study of the optimum form of three-dimensional bodies for the penetration of dense media in cases in which, given certain assumptions, the interaction of the medium and the body can be examined within the framework of the law of locality [1]. The method of local variations [2] was used to develop a numerical algorithm to search for forms of the body that would maximize the depth of penetration of the medium.Examples are presented of the solution of a variational problem with different isoperimetric conditions on the geometry of the body. The examples show that, in terms of depth of penetration, three-dimensional bodies whose form has been optimized may have a significant advantage over equivalent traditional solids of revolution.1. Classes of Bodies. We will examine the motion of a body whose form is described by the following equation in a cylindrical coordinate system (r. 0, x), with its origin at the tip of the body and the x-axis directed oppositely to the direction of motionwhere ~(x), R(0) are functions determining the longitudinal and transverse contours of the body, respectively. Here, ~o(0) = 0, ~(L) = 1 (where L is the specified characteristic length of the head of the body).We will henceforth assume that the longitudinal contour of the head is known and is given by the equationIf the head, the area of the center section of which is S m, is a cone of length L, then R(0) = q~m/a" = Lr/2 (r is the relative thickness of the head) and ~o(x) ---1 at x > L.If we want to shape the head so as to maximize depth of penetration for a given impact velocity v 0 and cross-sectional area Sm, we need to consider that different isoperimetric conditions may be imposed on the body, depending on the practical requirements. The conditions most often encountered are specification of the mass (volume) of the body and limitation of its transverse dimensions.There are two basic approaches to resolution of the problem. In the first approach, depth of penetration is increased by the "deformation" of a certain frontal region of the body (Fig. 1). This region has a circular midsection and a head with a specified relative thickness r (dashed lines). In this case,:the surface of the new, conical head will contact with the circular cylinder along a space curveGiven a known function g(0) ~ r/2, the head will be conical for ~ = ~1 = z/(2gl) (see Fig. 1) and will consist of conical and cylindrical surfaces with ~I < ~ < ~2 = r/(2g0), where gl and go are respectively the largest and smallest values of g(0) on the segment [0, 2~r].Moscow.
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