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 gives the analysis results for the servo drives dynamics, obtained from the theoretical calculations and during the development testing of the high power electric drives. Theoretical research was conducted, using the complete mathematical model of the servo drive, which included the equations of the control signal shaping path, electric motor, reducer and load. The equations of the control signal shaping network include only the characteristics of the compensating element in the assumption that all other delays in the transformation path are minimized. The electric motor equations are assumed in the classical form, taking into account the influence of the following main parameters on the motor dynamics: inductance and stator winding resistance, torque and armature reaction coefficients and rotor moment of inertia. Interaction of the motor with the multimass system of the reducer and load is presented in the form of force interaction of two masses – a reduced mass of the rotor and mass of the load through the certain equivalent rigidity of the kinematic chain. To describe the effect of gap in the kinematic connection the special computational trick, which considerably simplifies its mathematical description, is used. Efficiency of the reducer is presented in the form of the internal friction, proportional to the transmitted force. Calculation results with the application of the given mathematical model match well with the results of the full-scale testing of different specimens of servo drives, which makes it possible to use it for the development of new servomechanisms, as well as for the correct flight simulation when testing the aircraft control systems. In particular, based on the frequency response calculations of the closed circuit with the application of the given mathematical model, it is possible to define optimal parameters of the correcting circuit. Reaction on the step action with the various values of circular amplification coefficient in the circuit gives complete information on the stability regions of the closed circuit and influence of various drive parameters on these regions. Based on the conducted theoretical and experimental studies, the basic conclusions and recommendations were obtained and presented, accounting and implementation of which will provide high dynamic characteristics of the newly designed servo drives.

Annotation: The use of servo drives on flying vehicles determines the requirements to their dynamic characteristics. The problems of dynamics of drive with jet motor are not practically covered in publications. The task arises of selection of structure and parameters of devices consisting of several subsystems whose dynamic characteristics must be brought into agreement with each other in optimal way. The purpose of this work is to develop mathematical dependences for calculation of dynamic characteristics. The functional arrangement of the drive is considered consisting of jet motor based on Segner wheel with de Laval nozzle, mechanical transmission, pneumatic distributing device – jet pipe controlled by electromechanical converter. The layout is presented of mechanical segment of servo drive with jet motor with screw-nut transmission. The dynamic model is presented and the algebraic relations to determine natural frequencies of the drive are given. The motion equations of output rod at full composition of load are given. Using Lagrange transformation as applied to ball screw transmission, the expression for reduced mass of output element was derived. The reduced mass of load depends on the jet motor design and exerts basic influence on the drive’s natural frequencies. The evaluation is given of reduced mass change from the jet motor moment of inertia and reducer transmission coefficient. Based on the proposed algorithms, the dynamic characteristics of servo drive were constructed: transient process and amplitude-frequency characteristic. The drive has relatively low pass band, which is explained by the value of reduced mass of load.

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 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: Being a final executive element of rocket control systems, a hydraulic actuator is at the same time the main source of various non-linear dependencies in rocket dynamic design whose availability dramatically com plicates theoretical analysis of their dynamics and control systems synthesis. The required accuracy and complexity of mathematical models of hydraulic servo mechanisms are different for different design phases of guided rockets. The paper deals with the simplest models of hydraulic servo actuators intended to calculate rocket controllability and to define requirements to response and power characteristics of the actuators. To calculate the rocket stability regions and to evaluate own stability of servo actuators, a linearized mathematical model of hydraulic servo actuator is used that takes into account the most important parameters having impact on stability of the servo actuator itself and on that of the rocket: hardness of working fluid, stiffness of elastic suspension of the actuator and control element, slope of mechanical characteristic of the actuator in the area of small control signals, which, as full mathematical model analysis showed, is conditioned only by dimensions of initial axial clearances of slide’s throats. The full mathematical model constructed based on accurate calculations of the balance of fluid flow rate through the slide’s throats allows, as early as at designing phase, determining the values of most important static and dynamic characteristics of a future hydraulic actuator, selecting optimal characteristics of slides based on specified degree of stability and response of servo actuator and conducting final modeling of rocket flight on the integrated control system test benches without using real actuators and loading stands. It is correct and universal for all phases of rockets and their control systems designing and testing. Using this mathematical model, the powerful actuators of a line of intercontinental ballistic missiles with swinging reentry vehicle and the main engines actuators of Zenit launch vehicle first stage were developed. The results of their testing separately and in rockets practically fully comply with the data of theoretical calculations.

Annotation: The measurement errors upon conducting flight tests for launch vehicles are evaluated by considering the interferences and uncertainties in the measurement system procedure. Formal use of this approach can lead to unpredictable consequences. More reliable evaluation of errors upon conducted measurements can be achieved if the measurement process is regarded as a procedure of successive activities for designing, manufacturing, and testing the measurement system and the rocket including measurements and their processing during the after-flight analysis of the received data. The sampling rates of the main controlled parameters are three to ten times higher than the frequency range of their changing. Therefore, it is possible to determine the characteristics of the random error components directly on the basis of registered data. The unrevealed systematic components create the basic uncertainty in the evaluation of the examined parameter’s total measurement error. To evaluate the precision and measurement accuracy of a particular launch, the article suggests specifying the preliminary data on measurement error components determined during prelaunch processing and launch. Basic structures of algorithms for evaluation of precision and measurement accuracy for certain mathematical models that form the measured parameters were considered along with the practical case when static correlation existed among the measured parameters.

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 results of work are described to determine optimal materials for one of the elements of middle position lock to increase load bearing characteristics and contact resistance of the middle position lock. The results are presented of experimental check of impact of material of rod with hydraulic actuator piston on contact resistance and load capacity of the middle position lock of thrust vector control system two-channel hydraulic actuator. As replacer, the 18ХГТ steel was selected allowing (after carbonization and hardening) obtaining in surface layer of material the HRCэ 56-62 hardness with plastic core, instead of HRCэ 36-42 after hardening of applied 09Х16Н4Б steel. The comparative results were obtained in the tests of experimental sample of the lock completed with two rods with piston: the rod with piston manufactured according to DD and the experimental rod with piston that passed carbonization to the depth 0.9-1.3 mm and hardened to HRCэ 56-62. The rod’s ring groove – one of the elements of lock was subjected to carbonization and hardening. Both rods with piston were tested in the lock’s dummy in the load range: up to 1200 kgf –standard rod with piston and up to 3000 kgf – experimental rod with piston under static and cyclic loading. The test results are positive: the standard rod with piston confirmed its serviceability at the loads up to 1200 kgf inclusive; the experimental rod with piston withstood the loads up to 3000 kgf under static and cyclic loading. The evaluation of contact resistance was made by comparison of dimensions of traces left by the balls on the surface of rod’s grove under lock loading. The dimensions of traces on the experimental rod with piston under the load 3000 kgf inclusive did not exceed the dimensions of traces on the standard rod with piston, which testifies to the increase of contact resistance. We believe that the direction of search for steel brands in combination with advanced methods of thermal treatment is promising in increasing the lock’s load-bearing characteristics.