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Simulation and Software Development for Solving Internal Ballistics Problems

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Simulation and Software Development for Solving Internal Ballistics Problems INTERNATIONAL SCIENTIFIC JOURNAL "SCIENCE. BUSINESS. SOCIETY" WEB ISSN 2534-8485; PRINT ISSN 2367-8380 SIMULATION AND SOFTWARE DEVELOPMENT FOR SOLVING INTERNAL BALLISTICS PROBLEMS МОДЕЛИРОВАНИЕ И ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ ЗАДАЧ ВНУТРЕННЕЙ БАЛЛИСТИКИ PhD. Stelia O., PhD. Potapenko L., eng. Sirenko I., Faculty of Computer Sciences and Cybernetics, Taras Shevchenko National University of Kyiv, Ukraine. [email protected], [email protected], [email protected] Abstract: The problem of internal ballistics is considered, including basic pyrodynamic equation, law charge combustion, gasification law and equations of the projectile motion. A numerical method is used to solve a system of three algebraic and three ordinary differential equations. The software was developed and a number of computational experiments were made to analyze the influence of the charging parameter on the processes occurring during the projectile movement in the barrel channel. The problem of gunpowder burning and a finite-difference method to solve it are considered. The results of this work can be applied to the design of new types of trunks and charges. KEYWORDS: INTERNAL BALLISTICS, MODEL, NUMERICAL SIMULATION, PYRODYNAMIC CHARACTERISTICS. 1. Introduction 2. Mathematical model of internal ballistics Modern approaches based on ballistic computer models should Mathematical model of internal ballistics consists of six be used to improve the efficiency of artillery defeat means. These equations: three differential and three algebraic ones. According to developments are also required to create new artillery systems, [3], the system can be written as shells and charges. Fundamental research in the field of ballistic theory, mathematical and cybernetic methods for solving problems (1) λ+χ=ψ zz )1( , of ballistics, and software development are required for this. This paper is a continuation of works [1, 2] devoted to the θϕqv2 −ψ development of mathematical methods, algorithms and software for 2 fw solving ballistics problems. (2) p = fw , Internal ballistics investigates the phenomena occurring in the ψ + lls )( barrel weapons under powder gases and other processes at a shot in dv the barrel [3–6]. Solving the problem of internal ballistics allows (3) q =ϕ sp , determining the dependence of the projectile velocity in the barrel dt on the time and the path which projectile passing in channel under dl different conditions of charging and calculating the parameters of (4) = v , new artillery systems. dt The papers [7– 10] are dedicated to using of mathematical dz p models of internal ballistics problems. The work [7] describes a (5) = , mathematical model of shot for low pressure in space behind the dt Ik projectile. The paper [8] considers a mathematical model of internal ∆ 1 ballistics based on the physical laws of powder combustion, and (6) ψ ll 0 −= (1 −α− ) ψ∆ . experimental results obtained by burning powder in the closed δ δ vessel. In [9] the mathematical model for the weapon system with additional propelling charges along the barrel to increase muzzle Equation (1) describes the formation of powder gases. Here velocity were developed. ψ is the relative mass of burned gunpowder, z is the relative Paper [11] analyzes the changing characteristics of ballistic powder charges artillery ammunition of naval artillery systems for thickness of the burned powder, and ,λχ are the powder grain small caliber at post-warranty stages of storage on the basis of form factors. These values are connected by the ratio internal ballistics model. Papers [12–14] describe sophisticated /)1( χχ−=λ . models of artillery shot, using one- and two-dimensional gas Equation (2) is the pyrodynamic basic equation that describes dynamic equations of transient multiphase environments. Computer the process of expanding powder gases. model of internal ballistics is widely used not only for military Here is the pressure of powder gases, is the force of applications but also for civilians. For example, in [15] internal p f ballistics model was used to calculate the damper in the safety powder, w is the mass of powder charge, θ is the expanding grooves of the car. powder gases parameter, ϕ is the fictitious factor, q is the mass of The aim of this work is to develop a numerical algorithm for the projectile, v is the projectile velocity, s is the chamber solving the basic problems of internal ballistics, to develop the software for pyrodynamic processes simulation depending on the sectional area, lψ is the effective length of a chamber free volume, charging parameters. The mathematical model of gunpowder which given by (6), l is the effective length of the chamber, l is combustion is considered too. To implement this model a finite- 0 difference scheme was developed using parabolic spline. the current position of the projectile in the barrel ( ≤≤ Ll ,0 The paper presents the results of calculations for influence of where L is the total path of the projectile in the barrel), δ is the the powder thickness tape on the pyrodynamic characteristics. powder density, α is the powder gases covolume, ∆ is the charge Unlike analytical methods for solving basic problems of internal ballistics [3], the computer implementation is advantageous because density, Ik is the final pressure impulse powder gases. it allows setting parameters of the model not only as a constant, but The values I , ∆ , and l are calculated by formulas: in the form of functional relationships and considering the k 0 increasing number of factors affecting the process shot. 111 YEAR II, ISSUE 3, P.P. 111-114 (2017) INTERNATIONAL SCIENTIFIC JOURNAL "SCIENCE. BUSINESS. SOCIETY" WEB ISSN 2534-8485; PRINT ISSN 2367-8380 Then the finire-difference scheme is written as e1 w W0 Ik =,,, ∆ =l0 = D V 2D V 2µ u1 W0 s (11) φτ ()( + µ − φτ −1( − )) + i−1 h2h 2 i h2 h h where e is the thickness of the powder burning layer, u is the 1 1 D V µ powder burning speed ratio, W is the chamber volume. + φτ ( − (1 − )) = −(fx + f x ) / 2, i= 1, N − 2. 0 i+1 h2 h h i+1 i This system is closed with initial conditions and used to where φ is the lattice function that approximates the solution of determine the pyrodynamic parameters at any time moment t . τi The initial condition for equation (5) is calculated from equation (11) at the grid. equations (1) and (2) at the end of pyrostatic period when some The finite-difference scheme is monotonous and has a unique powder burns. This amount of powder is determined by the values solution. This follows from the following theorem [19]. ψ and z . We have set forcing pressure p for calculate these Theorem. Let the following conditions be satisfied for the splitting 0 0 0 (9)–(10): values. So, with (2) we have τi+1 = τ i +h, i = 0, N − 2, h > 0, N = L / h + 1, f∆ψ0 p = . x1= x 0 + h − µ, xi+1 = x i + h, i = 1, N − 2, 0 ∆ 1 1− − ( α − )∆ψ xNN= x −1 + µ, 0 < µ < h . δ δ 0 D D Solving this equation with regard to ψ , we get Then, if µ >h − and V > 0 or if µ < and V > 0 the 0 V V 1 1 − difference maximum principle for (11) is valid. ψ = ∆ δ Since the right-hand side of Equation (7) contains the desired 0 f 1 solution, the system of the finite-difference equations is nonlinear. + α − p δ The combined Newton's method and successive upper relaxation 0 method is used for solution of the system [20]. The Kutta-Merson Knowing ψ0 , we found z0 from (1): method is used for solving the Cauchy problem for the equation (8). ψ = χz(1 + λ z ) 0 0 0 4. The results of computational experiments For solving the ordinary differential equations (3)–(5) we use 4λψ −1 + 1 + 0 the Kutta-Merson method [21] . The advantage of this method over χ the "standard" Runge-Kutta 4th order method is an opportunity to z0 = evaluate error at each step of calculations without additional 2λ computational cost. This allows choosing a necessary step for the The projectile motion starts at zero time. The system of series of sample calculations. This method worked well for solving equations (1)–(6) is solved with such initial conditions the problem of external ballistics [1]. The software was developed v=0, l = 0, z = z0 when t = .0 according to the constructed algorithm. To conduct the computational experiments we should set a significant amount of numerical parameters which are divided into 3. Mathematical model of stationary gunpowder groups such as gunpowder parameters, gun and projectile combustion characteristics, charging conditions. Stationary combustion of gunpowder is described of equations For these calculations we used data for 100 mm guns from [22]. [16, 17] Computational experiments were conducted for strip of d2 T dT nitroglycerine gunpowder with different thickness of strip (Table 1). (7) D −V = − f , x∈(0, L ) , We used the following gunpowder parameters: δ = 1600 dx2 dx kg/m3, α = 0,001 m3/kg, f = 950 000 J/kg, dy (8) = Φ()T , = ⋅ −9 3 θ = dx u1 0,6 10 m /(N·s), 2,0 . Charging conditions was here T is the temperature, D is the coefficient of thermal w = 7,15 kg. Gun and projectile characteristics were q =16 kg, conductivity, V= u ⋅ c ⋅ρ , u is the speed of the front of 2 3 s = ,0 0082 m , W0 = 0,011 m , L = 5,53 m, ϕ =1,15 , combustion, c is the heat capacity, ρ is the density, 7 l0 =1,3415 , p0 =3 ⋅ 10 Pa. f= Q ⋅Φ() T , Q is the heat of reaction, Examples of calculating the pyrodynamic characteristics such as E dependence the pressure of powder gases in the space behind the Φ(T ) = k0 exp( − ) , k0 is pre-exponential factor, E is the projectile and speed of the projectile on the time are shown at RT Figures 1 and 2 (numbers are corresponding to different rows of activation energy, R is the universal gas constant, and y is the Table 1).
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