Mathematical model of the dynamics of a hydrofoil vessel as a complex technical system

Определена структура единой среды моделирования, состоящая из трёх блоков: блок, где задаются или формируются значения исследуемых параметров, влияющие на выходные показатели судна, как объекта моделирования, блок представляющий собой ядро единой среды моделирования и блок, где формируется совокупность тех или иных показателей, подлежащих анализу. Определена математическая модель динамики возмущенного движения СПК, при этом использованы следующие системы координат: земная прямоугольная горизонтальная правая, связанная с судном прямоугольная правая и вспомогательная нецентральная прямоугольная правая. Определены основные допущения математической модели. Представлены уравнения динамики судна на подводных крыльях в общем виде и определены силы и моменты, действующие на судно на подводных крыльях в крыльевом режиме движения. Гидродинамические силы и моменты, возникающие на каждом из крыльевых устройств, определены расчетным путем. Работа движителей моделируется заданием среднего упора, направленного по оси вала движителя и параллельного диаметральной плоскости судна. В модели динамики предусмотрена возможность задания аэродинамических сил и моментов, действующие на СПК в крыльевом режиме. Разработана математическая модель электрогидравлического привода, состоящая из суммирующего устройства, электрогидроусилителя и силового интегрирующего привода, охваченных общей обратной связью по положению и скорости перемещения, а также модель системы управления движением, которая является одной из важнейших подсистем СПК, формирующей алгоритмы управления, поступающие на входы ЭГП соответствующих ИО, расположенных на несущих поверхностях КУ. При решении некоторых задач, связанных с проектированием СПК и его технических систем, особенно для получения оценочных значений фазовых координат судна на начальных этапах проектирования или решения специальных задач, разработана линеаризованная система дифференциальных уравнений объекта. The structure of a unified modeling environment has been determined, which consists of three blocks: a block where the values of the studied parameters are set or formed, which affect the output indicators of the vessel as an object of modeling, a block that is the core of a unified modeling environment and a block where a set of certain indicators is formed. analysis. A mathematical model of the dynamics of the disturbed motion of the SPK was determined, with the following coordinate systems used: earth rectangular horizontal right, rectangular right connected to the ship and auxiliary off-center rectangular right. The basic assumptions of the mathematical model are determined. The equations of the dynamics of a hydrofoil ship in general form are presented and the forces and moments acting on a hydrofoil ship in the wing mode of motion are determined. The hydrodynamic forces and moments arising on each of the wing devices are determined by calculation. The operation of the propellers is modeled by setting the middle stop directed along the axis of the propeller shaft and parallel to the diametral plane of the vessel. The dynamics model provides for the possibility of setting aerodynamic forces and moments acting on the HFV in the wing mode. A mathematical model of an electrohydraulic drive has been developed, consisting of a summing device, an electrohydraulic amplifier and a power integrating drive, covered by a general feedback on the JJposition and speed of movement, as well as a model of a motion control system, which is one of the most important subsystems of the SPC that forms control algorithms entering the EGP inputs of the corresponding EUT located on the bearing surfaces of the KU. When solving some problems related to the design of the HFV and its technical systems, especially for obtaining the estimated values of the phase coordinates of the vessel at the initial stages of design or solving special problems, a linearized system of differential equations of the object was developed.

CONTROL WITH REDUCING OF DISTURBING FACTORS

The structural and dynamic features of the space (moving outside the dense layers of the atmosphere) stages of rockets - carriers of spacecraft as control objects are analyzed. The reasons are investigated - disturbing factors that generate external forces and moments that determine the disturbed motion of space rocket stages. For space rocket stages, disturbing factors are: mass asymmetry of the stage relative to its longitudinal axis and angle of mismatch of the line of action of the thrust vector of the propulsion system of the stage with the longitudinal axis of the stage. It is shown that when using the stage control deviating in the hinge of the marching engine as the executive organs of the control system, the effect of auto-reduction of the mentioned disturbing factors arises. The consequence of the autocompensation of disturbing factors is the reduction of disturbing forces and moments that violate the programmed motion of the step in the pitch and yaw planes. Mass asymmetry and the angle of mismatch of the line of action of the thrust vector of its engine and the longitudinal axis of magnitude are constant. Therefore, a decrease in perturbing forces and moments is accompanied by a decrease in the amount of energy (fuel) spent on processing (zeroing) perturbations of the parameters of the perturbed motion of the stage. It is shown that if the thrust of a space-stage engine is 8000 kgf, the engine operating time (flight time of the stage) is 500 sec, the specific engine thrust is 330 sec, the mass asymmetry is 0.05 m, the angle of mismatch is 0.25 degrees, then fuel economy can reach 200 kgf. The studies were performed using mathematical modeling methods.

Disturbed motion of the hypersonic first stage of an aerospace system in climb

Disturbed motion of the hypersonic first stage of an aerospace system in climb is analyzed. Deviations of atmospheric density from standard values and deviations of aerodynamic force coefficients from reference values are taken as disturbances. Disturbance motion of the hypersonic first stage of a hypersonic vehicle with the optimal angle-of-attack schedule obtained for reference atmosphere and nominal aerodynamic characteristics is modeled. Deviations of terminal conditions of disturbed motion from the target values of velocity, altitude and flight path inclination are determined. The problem of minimum propellant mass consumed in the climb with acceleration to hypersonic velocity is solved for disturbed motion by the method of Pontryagin’s maximum principle. Optimal angle-of-attack schedules, optimal flight paths and finite values of the mass of the hypersonic first stage are determined. Comparative analysis of optimal control programs and flight paths for disturbed and undisturbed motion is made.

Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians

Non-standard Lagrangians play an important role in the systems of non-conservative dynamics or nonlinear differential equations, quantum field theories, etc. This paper deals with quasi-fractional dynamical systems from exponential non-standard Lagrangians and power-law non-standard Lagrangians. Firstly, the definition, criterion, and corresponding new conserved quantity of Mei symmetry in this system are presented and studied. Secondly, considering that a small disturbance is applied on the system, the differential equations of the disturbed motion are established, the definition of Mei symmetry and corresponding criterion are given, and the new adiabatic invariants led by Mei symmetry are proposed and proved. Examples also show the validity of the results.

Disturbed motion of a hypersonic vehicle in climb

Disturbed motion of a hypersonic vehicle in climb is analyzed. Deviations of atmospheric density from standard values and deviations of aerodynamic force coefficients from nominal values are taken as disturbances. Disturbed motion of a hypersonic vehicle with the optimum angle-of-attack schedule and nominal flight characteristics is modeled. Deviations of terminal conditions of disturbed motion from the target values of velocity, altitude and path inclination are determined. Using the method of Pontryagin’s maximum principle the problem of fuel mass minimum consumed in hypersonic acceleration climb is solved for disturbed motion. Optimal angle-of-attack schedules, optimal flight paths and finite values of the hypersonic vehicle’s mass are determined. Comparative analysis of optimal control programs and flight paths obtained for disturbed and undisturbed motion is carried out.

About the decision regularized equations of perturbed motion body

The article deals with determination of the second- and higher-order perturbations in Cartesian coordinates and body motion velocity constituents. A special perturbed motion differential equations system is constructed. The right-hand sides of this system are finite polynomials relative to an independent regularizing variable. This allows constructing a single algorithm to determine the second and higher order perturbations in the form of finite polynomials relative to some regularizing variables that are chosen at each approximation step. Following the calculations results with the use of the developed method, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the results of numerical integration of the equations of disturbed motion shows close agreement of the results. The developed methods make it possible to calculate, by the approximating polynomials, any intermediate point of the motion trajectory of the body.

Design procedure of first-order perturbations for collisions trajectories with a perturbing body

The article is devoted to the determination of firstand secondorder perturbations in rectangular coordinates and velocity components of body motion. Special differential equation system of perturbed motion is constructed. The right-hand sides of this system are finitesimal polynomials in powers of an independent regularizing variate. This allows constructing a single algorithm to determine firstand second-order perturbations in the form of finitesimal polynomials in powers of regularizing variates that are chosen at each approximation step. Following the calculations results with the developed method use, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the numerical results of the disturbed motion equations shows their close agreement. The developed method make it possible to calculate any visa point of body motion by the approximating polynomials.