Annotation: The scientific and methodological propositions for the designing single-stage guided missiles with the solid rocket motors for advanced multiple launch rocket systems are defined. The guided missiles of multiple launch rocket system are intended for delivering munitions to the given spatial point with required and specified kinematic motion parameters at the end of flight. The aim of the article is an analysis of the development trends of the guided missiles with the solid rocket motors for the multiple launch rocket systems, identifying the characteristics and requirements for the flight trajectories, design parameters, control programs, overall dimensions and mass characteristics, structural layout and aerodynamic schemes of missiles. The formalization of the complex task to optimize design parameters, trajectory parameters and motion control programs for the guided missiles capable of flying along the ballistic, aeroballistic or combined trajectories is given. The complex task belongs to a problem of the optimal control theory with limitations in form of equa lity, inequality and differential constraints. To simplify the problem, an approach to program forming is proposed for motion control in the form of polynomial that brings the problem of the optimal control theory to a simpler problem of nonlinear mathematical programming. When trajectory parameters were calculated the missile was regarded as a material point of variable mass and the combined equations for center-of-mass motion of the guided missile with projections on axes of the terrestrial reference system were used. The structure of the mathematical model was given along with the calculation sequence of the criterion function that was used for determination of the optimal parameters, programs and characteristics. The mathematical model of the guided missile provides adequate accuracy for design study to determine depending on the main design parameters: overall dimensions and mass characteristics of the guided missile in general and its structural comp onents and subsystems; power, thrust and consumption characteristics of the rocket motor; aerodynamic and ballistic characteristics of the guided missile. The developed methodology was tested by determining design and trajectory parameters, overall dimensions and mass characteristics, power and ballistic characteristics of two guided missiles with wings for advanced multiple launch rocket systems produced by the People’s Republic of China, using the limited amount of information available in the product catalog.

Annotation: The article is devoted to optimization of a trajectory of the antiaircraft guided missile performed in design phase. The review of existing solutions on this issue confirmed the topicality of the problem. The analytical solution cannot be obtained, therefore, according to modern tendencies, optimization by numerical method of original development was performed. The basis of the method is two-level optimization which is carried out, in turn, by two different numerical methods and for two different criteria functions. At the top level, by method of random search and as a variant, by method of coordinate descent, the search was carried out for a fixed set of intermediate for the specified flight range trajectory points which co-ordinates in aggregate provide the necessary optimum. At the bottom level, for each pair of consecutive intermediate points, the boundary problem of falling into distant point by one-dimensional optimization was solved. The coordinate descent method was used for search for the simplified flight program. As optimization criteria for top level, minimum flight time or maximum final speed, for bottom  terminal criterion were used. The control program selected the angle of attack  program. As a result, the optimum and suboptimum (additionally ensuring minimum calculation time) trajectories and flight programs to maximum range and different altitudes were obtained. The analysis of results showed practical proximity of trajectories of minimum flight time and maximum final speed.

Annotation: The mathematic modeling was performed of the flight of light-class three-stage launch vehicle injecting a payload into sun-synchronous orbit of 700 km altitude and a cluster of observed space debris objects in the conditions of dynamically changing cataloged space situation. It is shown that as the launch moment becomes closer, the cataloged space situation is ascertained, which leads to the constant change of the quantity of hazardous space debris objects observed in the vicinity of launch vehicle trajectory and to the change of the parameters of their approach to the launch vehicle: minimal relative distance, relative velocity, rendezvous angle and launch moment for which hazardous approach is revealed. The hazardous approaches for the launch vehicle trajectory under consideration are more often observed with the relative velocities of more than 8 km/s and rendezvous angles less than 90 deg and their variations within the launch window do not exceed 1.2 m/s and 0.035 deg respectively. In this case, the histograms of distribution of relative distance, relative velocity, and rendezvous angle from catalog to catalog vary insignificantly. The distribution of hazardous approaches in launch time within launch window is not uniform, the regions are observed with low quantity of hazardous approaches and with high quantity. The hazard of launch vehicle collision with observed space debris objects in a launch is confirmed. In all, in the launch day time window under consideration, more than ten hazardous approaches are revealed, for two of them the approach to minimal distance of less than 1 km is predicted. This testifies to the necessity of taking measures to increase safety of launch vehicle flight through observed space debris cluster. In order to increase Ukrainian launch vehicles miss ion safety in the conditions of near space pollution, it is proposed to create the system of pre -flight space analysis, whose tasks are periodic analysis of space situation not less than once in a day, revealing of hazardous approaches, determination of their parameters, and preparation of data to make decision on launch time.

Annotation: Basic dynamic properties of the reentry part of the aircraft-type first stage were examined when turbojet engine is used in the recovery phase. Such configuration can be of interest because turbojets have considerably smaller rate of flow in comparison to rocket engines. Moreover, they are launched in the lower stratosphere or in the troposphere so that there is no need to place oxidizer supply on board. This recovery plan differs from an alternative rocket recovery system and, from our point of view, provides more efficient usage of the fuel stores because it doesn’t require the main propulsion to be started in the recovery phase. Besides the analysis of qualitative characteristics of the descend phase for this stage, the efficiency of a wing with moderate values of maximum aerodynamic characteristics and a turbojet was studied. In this case three ways for stage recovery were investigated. The first one implied unguided descend with zero angle of attack assuming that the stage is statically stable. This descend trajectory was considered as standard and was used to evaluate the efficiency of the wing and turbojet with relatively small propulsion. The second and the third design cases offered the gliding guided descend with turbojet being started only in the lower stratosphere. The last two cases used the same program for the angle of attack. The possibility to ensure permissible overload values at the critical points of the descend trajectory and acceptable values of kinematic characteristics at the earth surface tangency point are also of great interest. Thereby the program for the angle of attack was developed in a way that allowed kinematic characteristics on touchdown be as close as possible to the corresponding values, shown by civil and/or military-transport heavy aircraft. Simulation was conducted on Microsoft Visual Studio 2010.

Annotation: The purpose of the article is development of a modified variational method to determine extremals in the tasks of search for optimal solutions. The method has been developed using the results of investigations of the first variation of functional with autonomous subintegral function for the problem with fixed ends. The assumption of non-zero values of variation of function at boundary points has been introduced. It is shown that when using this assumption and introducing some other assumptions and limitations, it is possible to expand the class of permissible functions, among which the extremal curves should be sought for. With this expansion, to construct one extremal it is necessary to use two conditions of extremeness, one of which is Euler equation. To fulfill them, it is necessary to realize the constancy of partial derivative from subintegral function of desired variable at each point of interval considered. The new condition of extremeness unlike Euler equation is noninvariant relative to coordinate system. The use of this property allows, at presentation of the second variation of functional in parametrical form, constructing the solutions that satisfy the necessary and sufficient conditions of local minimum (maximum). It is noted that the proposed method is the first step in the development of a new approach to solution of multidimensional variational problems. The use of the latter will allow obtaining new solutions of various problems of technical mechanics, such as the task of determining optimal trajectory parameters of launch vehicles in the phase of designing and development of technical proposals, selection of optimal flight modes et al. The efficiency of the proposed method is demonstrated by example of solving the known problem about brachistichrone – determination of the curve of quickest descent. Using the method, two curves have been constructed that satisfy the necessary and sufficient conditions of optimality. The results are presented of comparison of time of material point descent along the proposed curves and descent along classical extremals. It is shown that the time of descent along the proposed curves is shorter than that at descent along classical exteremals.

Annotation: The main scientific and methodological propositions for designing single-stage guided missiles with main solid rocket motors that are intended for delivering payload to the given spatial point with required and specified kinematic motion parameters are defined. The aim of the article is to develop methodology for the early design phase to improve the basic characteristics of guided missiles, including formalization of complex problem to optimize design parameters, trajectory parameters and motion control programs for guided missiles capable of flying along the ballistic, aeroballistic or combined trajectories. The task is defined as a problem of the optimal control theory with limitations in form of equality, inequality and differential constraints. An approach to program forming is proposed for motion control in the form of polynomial that brings the problem of the optimal control theory to a simpler problem of nonlinear mathematical programming. When trajectory parameters were calculated the missile was regarded as material point of variable mass and the combined equations for center-of-mass motion of the guided missile with projections on axes of the terrestrial reference system were used. The structure of the mathematical model was given along with the calculation sequence of criterion functional that was used for optimization of design parameters, control programs and basic characteristics of the guided missile. The mathematical model of the guided missile provides adequate accuracy for design study to determine: overall dimensions and mass characteristics of the guided missile in general and its structural components and subsystems; power, thrust and consumption characteristics of the main engine; aerodynamic and ballistic characteristics of the guided missile. The developed methodology was tested by solving design problems. Applications of the developed program were studied to present the research results in a user-friendly form.

Annotation: The tubes of rectangular section made of composite material find ever- growing use. Winding of tubes without using dowels for small winding angles has a number of disadvantages. The technology of winding through dowels does not have these disadvantages. The paper presents the authors-developed calculation technique for programs of winding the tubes of rectangular section made of composite materials using dowels. To ensure winding continuity, the dowels at base must have equal distances between them. The external diameter of comb is selected with margin taking into account band stranding on the dowels. The scheme of winding on entire layer is determined that represents a table which indicates band position between the dowels of front and rear comb during winding a layer for each coil. Then the coordinates of machine tool are determined. For this purpose, we determine the angles of coil winding, corresponding angles of mandrel turn in determined coordinates X, the coordinates of band escape point, angle of inclination of free section of band, then from geometrical considerations we obtain the components of unit vector of tangent to reinforcing trajectory and directly the coordinated of machine tool actuators. As a result, we obtain the table of coordinates of machine toll actuators during winding of the first coil (at forward and back travel). Using the presented logics, similarly it is easy to obtain for all coils the coordinates of machine tool actuators. According to this technique, the Vitok auto-programming system was developed in MathCad environment. The Vitok system was used during development of winding program for tube 28x28x2 mm under Sich-2M program. The programs operated successfully. The test samples of tubes were wound. The authors performed winding of the tubes of rectangular section made of composite material to obtain a structure with zero linear thermal expansion coefficient.