The article deals with the issue of computer research of the mathematical model of the artillery fire process. Russia’s war against Ukraine showed that the success of the Armed Forces of Ukraine (AFU) largely depends on artillery units. The effectiveness of the actions of the artillery units depends on the normal functioning of artillery weapons.  The phenomenon of the shot is characterized by short duration, high pressure and high temperature. The duration of the shot phenomenon is determined by tenths and even hundredths of a second. A pressure of up to 400 MPa and a temperature of more than 2000 С develop in the bore of the weapon. It should be noted that the processes of gunpowder combustion are very complex. They depend on the manufactures, namely, on the type of gunpowder, chemical composition, its type (geometry) and pressure. So, the process of gunpowder combustion is described by equations that were obtained experimentally. Such experiments were carried out a long time ago, so most of the constants currently used in the calculation of the combustion process are expressed in the old ICGSS system of units. Therefore, it should be taken into account in the calculations when determining, for example, the pressure in MPA. The mathematical model of the artillery shot process takes into account all the factors that affect the processes occurring in the barrel. The mathematical model of an artillery shot consists mainly of the gunpowder combustion equation, the energy conversion equation, the projectile motion equation, the initial and boundary conditions. The numerical solution of the given systems of equations was carried out by numerical methods and the differential equations were solved by the Runge-Kutta method. The obtained values of the maximum pressures of powder gases and their comparison with the experimental ones showed that the error does not exceed 3 % which confirms the adequacy of the mathematical model of the artillery shot. Thus, the proposed mathematical model of an artillery shot and the method of its solution can be used to study the influence of various parameters (gunpowder parameters, design parameter of the guns, mass of the projectile and others) on the processes occurring during a shot.

The article deals with the survivability of the artillery unit during the combat operation. The brutality of the artillery unit is regarded as a random process. In our view, this random process is a Markov process that depends on time. Mathematical model of survivability of artillery unit based on non-stationary Markov process described by Kolmogorov equation was developed. The mathematical model allows to change probability of impressions of the artillery unit, time of their presence in combat positions and time of determination of positions of the enemy, which allows to determine vitality of the artillery unit in time during combat operations. The article considers two cases of survivability of an artillery unit during hostilities. In the first case, the entire artillery unit begins to fight on the uncertain positions of the enemy's artillery unit. The latter for some time determines the exact position of the attackers and, despite the fact that they entered the battle later, inflicts significant losses on the former. In the second case, the artillery unit is divided into two parts. The first part also begins the battle on the uncertain positions of the enemy's artillery unit. The second part enters the battle after determining the positions of the enemy artillery unit. In this case, the overall survivability of the first unit will be greater than the enemy's artillery unit. Ways to increase the survivability of the artillery unit during combat operations were shown. In our view, this approach can be used to create an interactive simulation of the artillery unit’s command. Such a complex is applicable in the training of commanders of artillery units.

Polyethylene is increasingly used in various industries such as light, medical and others as a variety of packaging and structural materials. Rheological properties of polyethylene affect products quality. Therefore, knowledge of these properties affects the calculation methods of equipment that produces materials from polyethylene, as well as the control system of technological processes of such industries.When designing equipment for processing polyethylene, namely the forming channels of extruders, flat slotted heads, the question of accurate determination of its rheological properties depending on the temperature and shear rate arises. Depending on the specified rheological properties the geometrical sizes of the specified channels change. These rheological properties are usually determined on the basis of experiments conducted at certain shear rates and temperatures. When designing, there is a need to calculate the geometric dimensions of the forming channels at shear rates and temperatures other than experimental. Therefore, there is a problem of developing a method of calculating these dimensions on the basis of experimental data obtained at other shear rates and temperatures. On the basis of the conducted research, the equation of the dependence of the shear stress of polyethylene on its shear rate and temperature was obtained. In the equation, the shear stress is directly proportional to the conditional viscosity and the shear rate gradient in power, which is the flow index. It is shown that the conditional viscosity of polyethylene is well described by the Arrhenius equation, and the flow index has a quadratic dependence on temperature. The obtained generalized equation of shear stress of polyethylene from its shear rate and temperature will allow to more accurately calculate the geometric dimensions of the forming channels of extruders, slotted heads, as well as to improve control systems for these devices.