Annotation: The purpose of this study is the enhancement of Euler equation possibilities in order to solve the brachistochrone problem that is the determination of a curve of fastest descent. There are two circumstances: 1) the first integral of an Euler equation does not contain a partial derivative of integrand with respect to y in an explicit form; 2) when the classical Euler equation is derived, only the second term of integrand is integrated by parts. This allowed formulating a problem of determination of new conditions of functional extremality. It is assumed that the integrand of the first variation of a functional is equal to zero. Taking into account this pro vision and some other assumptions, the procedures have been determined for simultaneous application of the Euler equation and its analogue being non-invariant in relation to the coordinate system. The brachistochrone problem was solved using these equations: the curves that satisfy the conditions of weak minimum optimality were plotted. The time of a material point’s descent along the suggested curves and the classic extremals was numerically compared. It is shown that the application of suggested curves ensures short descent time as compared to the classic extremals.

Annotation: Launch vehicle lift-off is one of the most critical phases of the whole mission requiring special technical solutions to ensure trouble-free and reliable launch. A source of increased risk is the intense thermal and pressure impact of rocket propulsion jet on launch complex elements and on rocket itself. The most accurate parameters of this impact can be obtained during bench tests, which are necessary to confirm the operability of the structure, as well as to clarify the parameters and configuration of the equipment and systems of complex. However, full-scale testing is expensive and significantly increases the development time of the complex. Therefore, a numerical simulation of processes is quite helpful in the design of launch complexes. The presented work contains simulation of liquid rocket engine combustion products jet flowing into the gas duct at the rocket lift-off, taking into account the following input data: the parameters of propulsion system, geometric parameters of launch complex elements, propulsion systems nozzles and gas duct. A three-dimensional geometric model of the launch complex, including rocket and gasduct, was constructed. The thermodynamic parameters of gas in the engine nozzle were verified using NASA CEA code and ANSYS Fluent. When simulating a multicomponent jet, the equations of conservation of mass, energy, and motion were solved taking into account chemical kinetics. The three-dimensional problem was solved in ANSYS Fluent in steady-state approach, using Pressure-based solver and RANS k-omega SST turbulence model. The calculation results are the gas-dynamic and thermodynamic parameters of jets, as well as distribution of gas-dynamic parameters at nozzle exit, in flow and in boundary layer at gas duct surface. The methodology applied in this work makes it possible to qualitatively evaluate the gas-dynamic effect of combustion products jets on gas duct for subsequent optimization of its design.

Annotation: The strength of real solids depends essentially on the defect of the structure. In real materials, there is always a large number of various micro defects, the development of which under the influence of loading leads to the appearance of cracks and their growth in the form of local or complete destruction. In this paper, based on the method of singular integral equations, we present a unified approach to the solution of thermal elasticity problems for bodies weakened by inhomogeneities. The purpose of the work is to take into account the heterogeneities in the materials of the elements of the rocket structures on their functionally-gradient properties, including strength. The choice of the method of investigation of strength and destruction of structural elements depends on the size of the object under study. Micro-research is related to the heterogeneities that are formed in the surface layer at the stage of preparation, the technology of manufacturing structural elements. Defectiveness allows you to adequately consider the mechanism of destruction of objects as a process of development of cracks. In studying the limit state of real elements, weakened by defects and constructing on this basis the theory of their strength and destruction in addition to the deterministic one must consider the probabilistic – statistical approach. In the case of thermal action on structural elements in which there are uniformly scattered, non-interacting randomly distributed defects of the type of cracks, the laws of joint distribution of the length and angle of orientation of which are known, the limiting value of the heat flux for the balanced state of the crack having the length of the “weakest link” is determined. The influence of heterogeneities of technological origin (from the workpiece to the finished product) that occur in the surface layer in the technology of manufacturing structural elements on its destruction is taken into account by the developed model. The strength of real solids depends essentially on the defect of the structure. In real materials, there are always many various micro defects, the development of which under the influence of loading leads to the appearance of cracks and their growth in the form of local or complete destruction. In this paper, based on the method of singular integral equations, we present a unified approach to the solution of thermal elasticity problems for bodies weakened by inhomogeneities. The purpose of the work is to take into account the heterogeneities in the materials of the elements of the rocket structures on their functionally gradient properties, including strength. The choice of the method of investigation of strength and destruction of structural elements depends on the size of the object under study. Micro-research is related to the heterogeneities that are formed in the surface layer at the stage of preparation, the technology of manufacturing structural elements. Defectiveness allows you to adequately consider the mechanism of destruction of objects as a process of development of cracks. In studying the limit state of real elements, weakened by defects and constructing on this basis the theory of their strength and destruction besides the deterministic one must consider the probabilistic – statistical approach. With thermal action on structural elements in which there are uniformly scattered, non-interacting randomly distributed defects of the cracks, the laws of joint distribution of the length and angle of orientation of which are known, the limiting value of the heat flux for the balanced state of the crack having the length of the “weakest link” is determined. The influence of heterogeneities of technological origin (from the workpiece to the finished product) that occur in the surface layer in the technology of manufacturing structural elements on its destruction is taken into account by the developed model. The solution of the singular integral equation with the Cauchy kernel allows one to determine the intensity of stresses around the vertexes of defects of the cracks, and by comparing it with the criterion of fracture toughness for the material of a structural element, one can determine its state. If this criterion is violated, the weak link defect develops into a trunk crack. Also, a criterion correlation of the condition of the equilibrium defect condition with a length of 2l was got, depending on the magnitude of the contact temperature. When the weld is cooled, it develops “hot cracks” that lead to a lack of welding elements of the structures. The results of the simulation using singular integral equations open the possibility to evaluate the influence of thirdparty fillers on the loss of functional properties of inhomogeneous systems. The exact determination of the order and nature of the singularity near the vertices of the acute-angled imperfection in the inhomogeneous medium, presented in the analytical form, is necessary to plan and record the corresponding criterion relations to determine the functional properties of inhomogeneous systems.

Annotation: The use of Langrangian multipliers at solution of optimal control problems in linear statement with qua dratic quality criterion leads to the necessity of solving boundary value problem with conditions for multipliers at the right end of control interval. Solution of the obtained equations for the purpose of regulation synthesis in forward time in this case does not produce stabilizing effect, as a rule. For regulation synthesis, the met hod is widely used of analytical construction of optimal regulator based on stabilizing matrix, which is obtained by solution of algebraic Riccati equation. However, in this case, there are some difficulties ‒ the necessity of calculating the stabilizing matrix, impossibility of calculating this matrix in non-stationary problem. The article proposes the regulation synthesis method by way of solving boundary value problem on regulation cycle i nterval. For this purpose, the differential equations for state parameters and Langrangian multipliers are expressed in the form of finite-difference linear relations. Taking into account that the state parameters and Langrangian multipliers are equal to zero at the end of cycle, the Langrangian multipliers at the beginning of cycle are determined by known values of state parameters for the same moment through solving the above linear system. The obtained values form the regulation law. In consequence of small duration of regulation cycle, an amplifying coefficient is introduced in the regulation law. Its value is determined based on results of preliminary modeling. Efficiency of the proposed method was verified by the example of adopted dynamic system, including non-stationary. The amplifying coefficient is fairly simply selected by the type of stabilization process. The proposed method may be used in the control systems of rockets of various purpose for motion parameters regulation.

Annotation: The shell structures widely used in space rocket hardware feature, along with decided advantage in the form of optimal combination of mass and strength, inhomogeneities of different nature: structural (different thicknesses, availability of reinforcements, cuts-holes et al.) and technological (presence of defects arising in manufacturing process or during storage, transportation and unforseen thermomechanical effects). The above factors are concentrators of stress and strain state and can lead to early destruction of structural elements. Their different parts are deformed according to their program and are characterized by different levels of stress and strain state. Taking into consideration plasticity and creeping of material, to determine stress and strain state, the approach is effective where the calculation is divided into phases; in each phase the parameters are entered that characterize the deformations of plasticity and creeping: additional loads in the equations of equilibrium or in boundary conditions, additional deformations or variable parameters of elasticity (elasticity modulus and Poisson ratio). Then the schemes of successive approximations are constructed: in each phase, the problem of elasticity theory is solved with entering of the above parameters. The problems of determining the lifetime of space launch vehicles and launching facilities should be noted separately, as it is connected with damages that arise at alternating-sign thermomechanical loads of high intensity. The main approach in lifetime determination is one that is based on the theory of low-cycle and high-cycle fatigue. Plasticity and creeping of material are the fundamental factors in lifetime substantiation. The article deals with various aspects of solving the problem of strength and stability of space rocket objects with consideration for the impact of plasticity and creeping deformations.

Annotation: This article contains information on the calculation of the thermodynamic and translational properties of the gaseous xenon in the amount sufficient for the most engineering applications. The equation of state of xenon was obtained in dimensionless form, enabling calculations of the thermodynamic values using already known methods, developed for the air and other extensively used gases. An example is made on the implementation of the method based on this equation for calculation of the density, enthalpy and entropy of the gaseous xenon. The accuracy of calculation of these properties in the temperature range from 300 to 3000 K from 0,1 up to 120 MPa pressure was from 0,2 to 1,6%. Sufficiently accurate and simple dependencies were obtained for calculation of the enthalpy and heat of vaporization at the saturation line. Accuracy of the enthalpy calculation of the liquid xenon at the saturation line is not below 0,2%, the accuracy of the calculation of the heat of vaporization is not below 0,5%. New, simpler method, as compared to standard reference data, to calculate the translational properties (thermal conductivity, viscosity) of xenon at atmospheric pressure has been proposed. It is shown that thermal conductivity and viscosity can be calculated from the expression of the same type with different coefficients. Accuracy of the calculation of these properties using the proposed method is not below 2,2%. Considering the unsatisfactory test results of the well-known methods of calculation of the translational properties at high pressures, the effective method for approximating the table values of these properties has been proposed. In this case, at first, the temperature data at fixed pressures are approximated, then using these approximations, the values of the properties are calculated at the given temperature and various pressure values. After this, the value of the property is interpolated at the given high pressure. As an example of the implementation of this method, the Mathcad software for calculations of the thermal conductivity of gaseous xenon at high pressure is given. The materials of the article are intended for the specialists dealing with heat exchange processes.