Ballistic Problem Research Articles

Account of interior ballistic parameters of the launch by slurry/liquid propellant with ETC-ignition

A mathematical model and algorithms of the basic problem of interior ballistics of the electrothermal-chemical (ETC) launcher on slurry/liquid propellant (SP) is presented. Simplifying assumptions permit the equations to be reduced to the numerical integration of a quasilinear equations system in partial derivatives of a hyperbolic type for functions of two arguments, with appropriate boundary conditions. Calculated and experimental results are compared.

Fluid–object interactions in interior ballistics

A fluid–object interaction model for an interior ballistics problem is presented. The fluid is a compressible gas and is modeled using the Deformable-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation. The objects can move axially within the model domain, and their motion is determined by the fluid pressure forces and collisions with other objects and rigid boundaries. The model is implemented assuming axisymmetry of the geometry and the flow field. The fluid mesh is composed of structured regions of quadrilateral elements and unstructured regions of triangular elements. The structured elements are used near the surface of the objects in order to better resolve the boundary layer, while the unstructured elements are used elsewhere in the domain. As the objects move, the mesh deformation needed to accommodate these motions takes place only in the unstructured parts of the mesh. Application to an interior ballistics problem is presented and discussed.

CROSS-VALIDATION OF NUMERICAL SCHEMES FOR EXTENDED HYDRODYNAMICAL MODELS OF SEMICONDUCTORS

The numerical integration of the hydrodynamical model of semiconductors based on extended thermodynamics has been tackled. On account of the mathematical complexity of the system, no theoretical conditions of convergence are available for the existing schemes. Therefore in order to obtain numerical solution it was almost mandatory to resort to a cross-validation comparing the results given by two different methods. The kinetic scheme and the finite difference method represented by a suitable modification of the Nessyahu–Tadmor scheme furnish numerical results for the ballistic diode problem in good agreement even for non-smooth solutions.

Analytical solution of the inverse problem of the interior ballistics of a controlled solid rocket motor

The inverse problem concerning the interior ballistics of a solid rocket motor is considered for the case where the pressure in the motor is a given time function. Given the pressure, the problem reduces to calculation of the temperature and cross-sectional area of the nozzle. An analytical solution of the inverse problem is obtained. It is shown that as the pressure changes, the combustion temperature also changes. It is found that the increase in the combustion temperature with an intense increase in pressure is greater than that under constant-volume conditions.

The Great Explanandum

ALFRED W. CROSBY'S The Measure of Reality: Quantification and Western Society, 1250-1600 is a broad survey of measurement animated by captivating examples drawn from four centuries of mathematics, astronomy, music, painting, and bookkeeping. The thesis he develops from this survey is that the unique success of European imperialism can be explained by a shift in Western mentalite in the late thirteenth century from a qualitative to a quantitative perception of reality. The larger significance of this book is that it is one of the more careful attempts to provide rigorous argument and historical documentation for a set of themes common in the historical literature of the twentieth century-radical breaks in Western thought and the uniqueness of the West. And as such, the shortcomings of Crosby's book suggest some of the larger problems in an entire genre of world history. The more focused historical problem that Crosby addresses-setting aside for a moment larger claims about the West and its uniqueness-is an exceedingly complex one: the explication of a series of historical contingencies that led to the superiority of specific European empires (and not others) as of the nineteenth century in weaponry, navigation, and bureaucratic administration. Tracing any facet of these developments in Europe is a monumental task. Weaponry alone, for example, requires analyses of the developments in chemistry, metallurgy, and technology, to name a few, that transformed cannons from ineffective to deadly weapons.' Another aspect of the explanation would be the enormous manpower invested in the study of cannons by Italian mathematicians,2 and the fortuitous fact that the problems of ballistics turned out to have mathematical solutions. But a complete explanation would also require comparative analysis, examining why other nations invested less in developing these particular technologies. The most obvious approach to explaining superiority in weapons, then, might be a comparative history of weapons. However, victors of military conquest often prefer loftier explanations for their superiority; and Europeans are hardly unique in their attempts to attribute their success to more noble cultural traits claimed to be uniquely their own. Crosby

The Great Explanandum

ALFRED W. CROSBY'S The Measure of Reality: Quantification and Western Society, 1250-1600 is a broad survey of measurement animated by captivating examples drawn from four centuries of mathematics, astronomy, music, painting, and bookkeeping. The thesis he develops from this survey is that the unique success of European imperialism can be explained by a shift in Western mentalite in the late thirteenth century from a qualitative to a quantitative perception of reality. The larger significance of this book is that it is one of the more careful attempts to provide rigorous argument and historical documentation for a set of themes common in the historical literature of the twentieth century-radical breaks in Western thought and the uniqueness of the West. And as such, the shortcomings of Crosby's book suggest some of the larger problems in an entire genre of world history. The more focused historical problem that Crosby addresses-setting aside for a moment larger claims about the West and its uniqueness-is an exceedingly complex one: the explication of a series of historical contingencies that led to the superiority of specific European empires (and not others) as of the nineteenth century in weaponry, navigation, and bureaucratic administration. Tracing any facet of these developments in Europe is a monumental task. Weaponry alone, for example, requires analyses of the developments in chemistry, metallurgy, and technology, to name a few, that transformed cannons from ineffective to deadly weapons.' Another aspect of the explanation would be the enormous manpower invested in the study of cannons by Italian mathematicians,2 and the fortuitous fact that the problems of ballistics turned out to have mathematical solutions. But a complete explanation would also require comparative analysis, examining why other nations invested less in developing these particular technologies. The most obvious approach to explaining superiority in weapons, then, might be a comparative history of weapons. However, victors of military conquest often prefer loftier explanations for their superiority; and Europeans are hardly unique in their attempts to attribute their success to more noble cultural traits claimed to be uniquely their own. Crosby

Numerical investigation of a gas dynamic model for charge transport in semiconductors

A mathematical gas dynamic model for semiconductor devices is numerically analysed. The well‐known ballistic diode problem is taken as an example.

QUALITY IMPROVEMENT FOR PROCESSES WITH CIRCULAR AND SPHERICAL SPECIFICATION REGIONS

This article deals with specification limits involving circular and spherical regions as part of geometric tolerancing problems. Many of these types of issues appear related to machine boring, ballistic, and weapon targeting problems. Instead of traditi..

Flamespreading during Base Ignition of Propellant Charge: Theoretical and experimental studies

AbstractIn the paper theoretical modeling of flamespreading through granular propellant charge during base ignition is given. The theoretical model includes the balance equations for the gas and solid phase, as well as necessary constitutive laws. The stable convergent numerical procedure for solution of the system of equations is developed. This procedure is included in computational program FSPC (Flame‐spreading through Propellant Charge). The program FSPC enables investigations of influence of ignition material, physical and chemical characteristics of propellant and propellant charge loading density on the flamespreading process. The experimental investigations by an especially designed apparatus are performed. The verification of the theoretical‐numerical access through the comparison with the experimental data (pressure vs. time and locations, flamespreading velocity, propellant grains displacements) is carried out. The entire access makes possible more successful solutions of many interior ballistics problems.

The main problem of external ballistics

This paper is devoted to the main problem of external ballistics, i.e., the problem of finding a trajectory of a material point, which starts in the given initial set, reaches the given target set at time to be determined, and satisfies the dynamics law as well as some optimality constraint, while the absolute value of its initial velocity is given.

A PHENOMENOLOGICAL THEORY OF PROPELLANTS' IGNITION AND SUBSEQUENT UNSTEADY COMBUSTION IN HIGH PRESSURE CHAMBERS

The goal of this paper is to show significant influence of ignition conditions on subsequent combustion of a condensed propellant and on the pressure diagram of combustion chamber. Emphasis is placed on the effect of accelerated combustion (AC-effect) after ignition in a high pressure closed bomb and in a chamber with moving piston. An analysis of such typical problems is carried out using an original phenomenological theory. Intermediate asymptotic solutions for the temperature and composition of combustion products as well as for the transitional combustion rate are presented. These asymptotic solutions show how the ignition conditions influence the subsequent unsteady combustion and pressure diagram in a high pressure chamber. The AC-effect of propellant surface layers and its dependence on ignition characteristics is demonstrated. A modification of Drozdov’s solution of the basic problem of gun interior ballistics has been obtained taking into account the AC-effect. It is shown that AC-effect can be responsible for the anomalous increasing of pressure-peak with decreasing of initial temperature of propellant charge in a gun-chamber.

The motion of a body through a large-scale inhomogeneity in a stratified atmosphere

The free motion of a blunted body of revolution of considerable length through a large-scale cloud of heated gas (a thermal) floating in a stratified atmosphere is investigated. An effective numerical method of determining the trajectory of the body is proposed, based on the simultaneous solution of flow and ballistics problems. The flow problem is solved by a numerical method that is very economical in computer time, based on an expansion in a small parameter [1, 2], which is the angle of attack, in conjunction with the global-iteration method [2, 3]. The effect of the floating large-scale cloud of heated gas on trajectory, spatial orientation and stability of the body is determined. It is shown that under certain flight conditions the presence of a floating cloud of reduced density along the path of the body may lead to flipping of the body.

Mathematical simulation of the lagrange ballistic problem for high-velocity systems

The problem of a piston being pushed by expanding gases released during rapid burnup of a charge is simulated mathematically. The model includes the laws of conservation of the dynamics of a two-phase medium, supplemented by the equation of kinetics with a linear dependence of the combustion rate on the pressure. The results of calculations for three types of ballistic systems showed good agreement with experimental data. Analysis of the calculations indicated that the complete gasdynamic picture must be taken into account when the projection velocity is increased and that the simplified models used earlier are inapplicable.

The internal lagrange ballistic problem for the controlled rate of layered combustion of powder

We have studied the flow of a gas—powder mixture in a discharge, using a moderated propellant with a controlled combustion rate. New gasdynamic effects are detected: formation of combustion waves from a moderated layer of powder grain, formation of a reflected compression wave at the bottom of the piston, etc. The calculated and experimental data are compared for a small-caliber ballistic apparatus and the need to take heat transfer in the barrel wall into account is noted.

Evangelista Torricelli

This article provides a brief introduction to Evangelista Torricelli and to one aspect his work. It is divided into three parts. Part 1 gives some biographical details and a short survey of his work and achievements. Part 2 is concerned with a geometrical approach towards the study of projectiles as seen in Torricelli's early work De motu. Part 3 shows how Torricelli applied his projectile studies to practical ballistic problems.

Boundary conditions for a moving interface in solving internal ballistics problems (phenomenological approach)

Boundary conditions are examined for two- and three-dimensional calculations of combustion product parameters in ignition channels. These problems are solved to determine the parameters in solid-fuel rocket engines, various types of gas generators, solid-fuel MHD generators, etc., and also in artillery systems, if special calculations are done on the distribution of parameters along powder elements or along channels within powder charges. When analogous boundary conditions must be used in nonigniting channels (for example for a period of gradual ignition), the following problem classes can be designated: pre-ignition, post-burnout, and interignition. It very more important that definite physical and chemical processes, which occur in the combustion layer or on the fuel surface, precede the combustion itself.

Super‐computer simulation of a multichambered muffler

AbstractThe problem of attenuating muzzle blast noise produced by firing a gun weapon is studied. A multichambered muffler is attached to a M242 25mm cannon. For this weapons configuration, the transitional ballistics problem is solved numerically using the CRAY XMP computer. The second‐order accurate total variation diminishing (TVD) shock capturing scheme of Harten is used to solve the Euler equations of compressible flow. The simulation yields a detailed picture of the flow field, as displayed by pressure and Mach contours. Pressure time histories at selected field locations are recorded for future comparison with experiment. Energy efflux time history, from which predictions of noise attenuation can be made, is also recorded. Noise reduction levels so obtained are seen to compare well with experimental data. From this evidence it appears that the simulation method can become an effective aid in muffler design.

On a moving boundary problem of transitional ballistics

AbstractA hybrid algorithm for calculating the flow around a flat‐nosed projectile moving through a fluid is established. At cells of fixed volume away from the projectile, Harten's total variation diminishing (TVD), second‐order accurate, shock capturing scheme is utilized. Due to projectile motion, cells adjacent to the projectile can be treated as compressing or expanding in volume. For such cells, a local finite volume approach has been employed to derive a cell update algorithm.Proof of concept for the expanding cell scheme is established through calculating the one‐dimensional flow behind a moving piston, whose theoretical solution is well‐known. The resulting hybrid scheme is applied to the problem of blast wave simulation in axisymmetric geometries. Flow around a vertical muzzle brake is calculated for the case of an embedded moving projectile.

Least squares model fitting to fuzzy vector data

We consider model fitting problems in which the mathematical model of an observed event is a system of equations involving model parameters and observables, and in which the data are fuzzy vectors. Such problems naturally arise in applications when data are scarce and information is vague about distributions of variances that are contained in the observations. Then it may only be possible to obtain estimates of membership functions of the data. A model can be fitted to such data by maximizing the membership values of the adjusted observations. We achieve this by minimizing the sum of squares of the deviations of the membership values from one. The optimization problem simplifies significantly if the membership functions of the fuzzy vectors belong to a class of conical functions, which are defined in terms of an elliptic norm. Properties of these functions are discussed and a membership propagation formula derived which component-wise obeys the extension principle. The least squares model fitting to the described fuzzy vector data deviates in important aspects from ordinary least squares. One obtains as a solution a fuzzy model parameter vector and a number of indicators which describe the quality of the fit, viz., the grade of compatibility between data and model, their discord and dilators of model and data membership functions. These new concepts are illustrated and discussed in the context of an example from a terminal ballistics problem.

External ballistic of volcanic explosions

In order to determine the kinetic energy of an explosion it is necessary to know the initial velocities of ejected fragments. Calculations of initial velocities made earlier with few exceptions did not take into account the resistance of the air and therefore, greatly underestimated the initial velocities, and consequently the energy of the explosions. A solution of the inverse problem of ballistics which takes into account air resistance makes it possible to find precise values of initial velocities of ejection, determined from the distance of ejected fragments and their ballistic coefficients. The method makes it possible to determine the kinetic energy of explosions for eruptions which were not directly observed. It is shown that the velocity field in volcanic explosions is not spherical,i.e. the velocities of ejection of fragments differ markedly in different directions. Accordingly, the maximum distance of ejection of fragments depends not only on the initial angle of ejection, but also on the character of distribution of initial velocities and the ballistic coefficients of the fragments. For fragments with diameters of 40 cm to 2 m the maximum distance of ejection is realized in the case of angles of ejection of 30–36°. For smaller fragments the optimal angles of ejection are still smaller. The ballistics of explosive eruptions of the volcanoes Asama, Arenal, and Shiveluch are considered.