Normal Cauchy Form Research Articles

Load of the wagon-platform for transportation of bulk cargoes

Purpose. To substantiate the improvement of the load-bearing element of the wagon-platform for the possibility of transporting bulk cargoes. Methodology. In order to be able to transport bulk cargo on the wagon-platform, it is proposed to install a composite boiler module on it. In order to determine the dynamic load of the improved load-bearing structure of the wagon-platform, mathematical modeling was performed. The mathematical model formed by professor Bohomaz H.I. was used. However, within the framework of the research this model was refined to determine the load of the wagon-platform of the proposed design. The solution of the system of differential equations is carried out in the MathCad software package. To do this, the mathematical model was reduced to the normal Cauchy form, and then integrated by the Runge-Kutta method. The obtained acceleration is taken into account when calculating the strength of the advanced load-bearing structure of the wagon-platform. The calculation is performed in the SolidWorks Simulation software package, which implements the finite element method. Also, within the research the modal analysis of a load-bearing structure of the wagon-platform is carried out. Findings. Based on the calculations, it is established that the acceleration acting on the load-bearing structure of the wagon-platform car is 0.38 g, i.e. it is within acceptable limits. The results of the calculation of the strength of the improved design of the wagon-platform showed that the maximum equivalent stresses occur in the area of interaction of the spine beam frame with the pivot and is about 340 MPa, the maximum displacement made 8.6 mm. That is, the obtained stresses do not exceed the yield strength of the structural material. The results of the modal analysis showed that the values of the natural frequencies of oscillations are within acceptable limits, because the first natural frequency has a value greater than 8 Hz. Originality. The scientific substantiation of improvement of a load-bearing structure of a universal wagon-platform to transportations of bulk cargoes is carried out. Practical value. The conducted research will promote increase in efficiency of operation of railway transport and creation of developments concerning planning of innovative designs of a rolling stock.

Voltage Stabilisation of a Drive System Including a Power Transformer and Asynchronous and Synchronous Motors of Susceptible Motion Transmission

A mathematical model is developed of the master circuit of an electric driver system including a power transformer and susceptible motion transmission of asynchronous and synchronous drives. Electric motors drive water pumps by means of motion transmission that comprises two elastic couplings of lumped mechanical parameters and a long shaft of distributed mechanical parameters. Differential equations for oscillatory processes for the long shaft and the elastic couplings are different. The shaft is described with partial derivative Euler–Poisson equations, which, combined with the boundary conditions, form mixed problems from the mathematical point of view. The elastic couplings, on the other hand, are described with the ordinary second type Lagrange equations. Based on the theory of electromagnetic field, the partial differential equations describe the skin effects across the rotor age bars. Vertical pumps are presented by means of a loading torque waveform as a function of the input shaft’s angular velocity. The complex mathematical model serves to analyse electromechanical transient processes across the integrated drive system. Starting from there, conditions of stabilisation of the drive system voltage are determined. Electromechanical state equations are presented in the normal Cauchy form and integrated using the fourth-order Runge–Kutta method. Results of computer simulations are shown with graphics, which are interpreted and described.

Численный расчет неавтономных динамических моделей с нелинейностями в виде нестепенных элементарных функций

The numerical calculation method of dynamic models with nonlinearities described by elementary non-power functions is represented. The computational difficulties in applying power series are analyzed. To overcome computational difficulties, a constructive procedure based on the analytical-numerical method of analysis developed by the authors is proposed. The numerical procedure is applicable to the models described by a system of nonlinear differential equations in the normal form of Cauchy. The proposed procedure, while preserving all the advantages of the power-series apparatus, eliminates the necessity for constructing the power-series compositions and special estimates, as well as raises the formalization of dynamic model calculations with non-power nonlinearities. The proposed numerical procedure uses the evaluations of the desired solutions and manages the boundaries of one-dimensional domains of exact solutions in order to bring them to unknown exact solutions. The advantages of the procedure are highlighted during the calculation of nonlinear autonomous dynamic model of the type «damped pendulum».

Mathematical Modeling of Transient Processes in Magnetic Suspension of Maglev Trains

On the basis of a generalized interdisciplinary method that consists of a modification of Hamilton–Ostrogradski principle by expanding the Lagrange function with two components that address the functions of dissipation energy and the energy of external conservative forces, a mathematical model is presented of an electromechanical system that consists of the force section of a magneto-levitation non-contact maglev suspension in a prototype traction vehicle. The assumption that magnetic potential hole, generated naturally by means of cryogenic equipment, is present in the levitation suspension, serving to develop the model system. Contrary to other types of magnetic cushion train suspensions, for instance, maglev–Shanghai or Japan–maglev, this suspension does not need a complicated control system, and levitation is possible starting from zero train velocity. As high-temperature superconductivity can be generated, the analysis of levitation systems, including the effect of magnetic potential holes, has become topical. On the basis of the model of a prototype maglev train, dynamic processes are analyzed in the levitation system, including the effect of the magnetic potential hole. A system of ordinary differential equations of the dynamic state is presented in the normal Cauchy form, which allows for their direct integration by both explicit and implicit numerical methods. Here, the results of the computer simulations are shown as figures, which are analyzed.

Stress-strain state of thick-walled anisotropic cylindrical shells under thermal power load, protected by the functionally graded material

In the article the stress-strain state of thick-walled structurally anisotropic composite cylindrical shells under thermal power load, which are protected by a functionally graded material, are analysed. Based on the interrelations of the spatial theory of elasticity, a system of inhomogeneous differential equations in three-dimensional formulation, which describes the stress-strain state of thick-walled anisotropic cylindrical shells, was obtained. To reduce the dimensionality of this system, the Bubnov-Galerkin analytical method was used. Thus, the obtained one-dimensional system of twelve equations of normal Cauchy form was implemented using the numerical method of discrete orthogonalization. To represent the possibilities of the proposed approach, there were used stress-strain states of two, four and five-layered anisotropic cylindrical shells of fibrous composites, protected from temperature by a layer of transversely isotropic functionally graded material.

Thermal-Magnetic-Elastic Effect Analysis of a Thin Current-Carrying Conical Frustum Shell*

A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using Newmark’s stable finite equidifferent formulas and the quasi-linearization method, the nonlinear partial differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The temperature field in a thin conical frustum shell and the integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic field and the thermal equilibrium equation. The change of stresses, displacements, and temperatures in the thin current-carrying conical frustum shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example. These results are expected to be a theoretical reference for further analysis of this case.

Modified Method of State Variables

Formal mathematical model development of an object with lumped parameters is an integral part of CAE systems. The computer-aided design (CAD) system functionality largely depends on the formation method used. The article proposes to modify a method of state variables, which is a pioneer in formal mathematical models development of technical objects whose behaviour is described by a system of ordinary differential equations.The classical method of state variables uses a graph-based representation of the object structure and allows development of its mathematical model in the normal Cauchy form without incorrect locations. The incorrect locations mean situations when capacitance-type branches fall in the graph chords or inductance-type branches fall in the graph tree branches. If there are incorrect locations in the object’s scheme then, to have a model of correct development, additional elements are included to eliminate them. Such an approach is possible if the object description is performed at the level of the basic two-poles, but it is assumed that in all modern CAD systems there is a pre-processor in which it is possible (and as a rule) to use multi-pole components. In this case, it is challenging for an unsophisticated user to make scheme correction.The modified method of state variables proposed in the article is free from this drawback and allows us to obtain a mathematical model in the normal Cauchy form and with incorrect locations available. This will allow us to use both explicit and implicit integration methods, conduct modal analysis, and simply have a model version in the FMI standard.

Analysis of periodic processes of electrical energy devices on the basis of matrix of transition of the states

The mathematical model of the synchronous machine, the differential equations of which are presented in the normal form of Cauchy, an accelerated method of obtaining periodic processes on the basis of the transition matrix of states is proposed, and the algorithm for calculating periodic processes is given. Thanks to the calculation of the state transition matrix, the method of finding the initial conditions that lead to a periodic process is expanded to analyze the modes of power devices with saturated magneto resources. In addition, the efficiency of the method is described by the calculation of the electromechanical process, which is the two-phase short circuit of the synchronous generator.

DETERMINATION OF THE TANK CONTAINER STABILITY RELATIVE TO THE FRAME OF A FLAT CAR DURING TRAIN FERRY TRANSPORTATION

Purpose. The research is aimed at revealing the peculiarities of stability determination for a tank container relative to the frame of a flat car during transportation by train ferry. Methodology. In order to reach the purpose, the mathematical modelling of dynamic loading of a tank container located on a flat car during the train ferry transportation was conducted. The model takes into account the tank container displacements relative to the flat car frame and liquid cargo in the tank, the displacements of which are limited by the tank’s walls. It was assumed that the flat car, with tank container on it, was rigidly fixed relative to the deck of the train ferry and moved together with it. A 1CC type tank container located on a 13-4012 flat car was chosen as a prototype. While designing the model, the following aspects were considered: the trochoidal law of motion of the disturbing action (sea waves) on the train ferry with containers, the dissipative component occurring due to the train ferry oscillations under conditions of sea rolling, the relative bearings of sea waves to the train ferry body, and the wind force to the above-water projection of the train ferry, flat car located on the upper deck and tank containers. The differential equations of motion were solved in the Mathсad with their reduction to the normal Cauchy form with subsequent integration by the Runge–Kutta method. The acceleration values obtained, as the components of the dynamic loading, were considered in the stability research for the tank container relative to the flat car frame. Findings. The author obtained the clarified values of dynamic loadings, acting on the tank container located on a flat car during the train ferry transportation. Originality. A mathematic model of displacements of a tank container located on a flat car during the train ferry transportation was developed. Practical value. The findings of the research can be used at designing tank containers of a new generation with improved technical and economical, as well as ecological indices.

On Stability of a Class of Linear Systems with Distributed and Lumped Parameters

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

МЕТОДИКА МОДЕЛИРОВАНИЯ РЕГИОНАЛЬНО ОБОСОБЛЕННОГО ЭЛЕКТРОТЕХНИЧЕСКОГО КОМПЛЕКСА

Актуальность исследования обусловлена необходимостью устранения разрыва между существующими подходами к моделированию изолированных систем электроснабжения и необходимостью учета динамических процессов в их физически разнородных подсистемах, образующих совокупность, при наличии слабых связей и слабых взаимодействий между единичными системами. Цель исследования: сконструировать проблемно-ориентированный численный метод, непосредственно применимый к математическим моделям совокупности изолированных систем электроснабжения и существенно превосходящий по критерию «время счета – точность счета» универсальные численные методы. Объект: электротехнический комплекс, состоящий из совокупности изолированных систем электроснабжения, объединенных слабыми связями и слабыми взаимодействиями. Методы. При анализе формы и структуры исходной математической модели совокупности электротехнических комплексов, состоящих из изолированных систем электроснабжения, объединенных слабыми связями и слабыми взаимодействиями, и при конструировании проблемно-ориентированного численного метода использованы: энергетический подход к построению математических моделей электротехнических комплексов и систем, теория численных методов решения жестких систем дифференциальных уравнений, корректные допущения при составлении математических моделей и компьютерное моделирование. Результаты. Введено в рассмотрение понятие «Регионально обособленный электротехнический комплекс», и выявлены его характерные признаки. Показана возможность создания новых алгоритмов управления «Регионально обособленным электротехническим комплексом», в целях снижения возможного экологического ущерба и средней расчётной себестоимости производства электроэнергии электростанциями, входящими в «Регионально обособленный электротехнический комплекс». Выявлена исходная структура математической модели «Регионально обособленного электротехнического комплекса». Показано принципиальное отличие структуры данной модели от модели в нормальной форме Коши. Обосновано принципиальное наличие свойств жесткости и сверхжесткости у анализируемого класса моделей. Сконструирован и реализован в программе BKM-3 проблемно ориентированный трехшаговый численный метод, применимый к исходной математической модели «Регионально обособленного электротехнического комплекса». Проведен численный эксперимент, показавший, на идеализированных математических моделях «Регионально обособленного электротехнического комплекса», существенное преимущество программы BKM-3, по критерию «время счета – точность счета» в области применимости.

SYNTHESIS OF CONTROLLERS FOR NONLINEAR CONTROL OBJECTS ON BASIS OF NUMERICAL METHODS FOR SOLVING DIFFERENTIAL EQUATIONS

The algorithm of parametric synthesis of regulators for nonlinear control objects is presented in the article, in which knowledge of the real trajectory of the system motion is not required in an explicit form. The essence of the algorithm is as follows: the dynamic properties of the control system are always determined by the right-hand side of the system of differential equations written in the normal form of Cauchy. Depending on the values of the regulator parameters that are also included in the right-hand side, the system of equations may have one or another solution. If we substitute the reference trajectory into the scheme of numerical integration of differential equations describing the dynamics of the system, then at each step of integration it can be considered as a system of algebraic equations for the desired parameters of the regulator. For each discrete value of the reference signal, there is a set of desired regulator parameters from which the desired values are determined, for example, as weighted averages over the entire study interval, or, as some limiting values. In the proposed algorithm, in an implicit form, the criterion of optimality is the norm in the space of convergent numerical sequences.

Features of ultrafiltration process engineering calculation in concentration and purification of pectin polysaccharides

In this paper, we propose a mathematical model of the process of changing the productivity of a membrane, both through the formation of a continuous gel layer, and by blocking individual pores. Also, the results of modeling the process of dia- ultrafiltration (DUF) purification of pectin solutions and their concentration by ultrafiltration (UF) using hollow fiber membranes are presented. The effect of periodic washing of the membrane and the process of gelation along the membrane on the productivity of the plant was studied. Thus, the process is characterized by two parameters, and A is the main parameter of the problem, which is the same for all stages, and the second parameter, the ratio (k =F0/F'), characterizes the degree of pore purification with a general cleaning of the membrane surface and is thus one from the characteristics of a single cycle. The valuesof a number of parameters are determined experimentally. It is shown that when the system of differential equations, represented in the normal Cauchy form describing the given technological process, is jointly solved, the productivity of the DUF system can be increased by the frequency of flushing. The conducted experiments demonstrated the advantage of the use of the DUF process in the production of pectin and the features of purification and concentration of apple and sunflower pectins in this process. It is established that the use of DUF causes the introduction of an energy-saving technological process in the production of pectin to produce the target product that meets the required quality and environmental safety of production. The introduction of the proposed methods of DUF and UF into the production of pectin leads to an improvement in the quality and cost of the product.

FORMATION OF CONTROL STYLE BY THE OBJECT IN THE TRAINING COMPLEXES: ENGINEERING AND PSYCHOLOGICAL BASES

Ponte Academic JournalJan 2017, Volume 73, Issue 1 FORMATION OF CONTROL STYLE BY THE OBJECT IN THE TRAINING COMPLEXES: ENGINEERING AND PSYCHOLOGICAL BASESAuthor(s): Irina Garkina ,Alexander DanilovJ. Ponte - Jan 2017 - Volume 73 - Issue 1 doi: 10.21506/j.ponte.2017.1.12 Abstract:We consider the engineering-psychological aspects of formation at the training complex dynamic management style ergatic system operator. We give a formal description of the skills and control style. It is shown that the parameters of the simulation model must be selected taking into account the specific problems to be solved at the training complex. Simulation characteristics are determined by comparing the functional values for ergatic systems: �operator-object�, �operator-object simulator�. The methods of structural and parametric identification of the object are proposed. Is given an example of assessing the quality of the simulation control objects described by a system of ordinary differential equations in normal form of Cauchy. Download full text:Check if you have access through your login credentials or your institution Username Password

Identification Method for Nonlinear Dynamic Models of Gas Turbine Engines on Acceleration Mode

The form and identification method of nonlinear dynamic gas turbine models on acceleration mode on the base of experimental transition processes is proposed. The nonlinear dynamic model of a gas turbine engine is represented as a system of nonlinear differential equations as in the normal form of Cauchy. The parameters of the system of differential equations are presented as a function of the acceleration parameter α, and the identification method is based on the determination of parameter α using numerical optimization method, allowing to simulate an acceleration mode with high accuracy. The experimental data are approximated by cubic splines for reduction of identification errors. The developed fast-calculating real-time nonlinear dynamic model is approved to implement in the software for hardware-in-the-loop test-beds and in the on-board software for automatic control systems of gas turbines.

Analysis of Thermal-Magnetic-Elastic Stresses and Strains in a Thin Current-Carrying Cylindrical Shell

A thermal-magnetic-elastic problem for a thin current-carrying cylindrical shell in a magnetic field is studied. The normal Cauchy form nonlinear differential equations, which include eight basic unknown functions in all, are obtained by the variable replacement method. Using the difference and quasi-linearization methods, the nonlinear differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The numerical solutions are obtained. The change of stresses, temperatures, and deformations in the thin current-carrying cylindrical shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example

Analysis of Nonlinear Stresses and Strains in a Thin Current-Carrying Elastic Plate

Two-dimensional magnetoelastic problems of a thin current-carrying plate under the interaction of an unsteady electromagnetic field and a mechanical field are studied. The nonlinear magnetoelastic kinetic equations, geometric equations, physical equations, electrodynamics equations, and expressions for the Lorentz force of a thin current-carrying plate under the action of a coupled field are given. The normal Cauchy form nonlinear differential equations, which include ten basic unknown functions in all, are obtained by the variable replacement method. Using the difference and quasi-linearization methods, the nonlinear magnetoelastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by the discrete-orthogonalization method. Numerical solutions for the magnetoelastic stresses and strains in a thin current-carrying elastic plate are obtained by considering a specific example. The results that the stresses and strains in the thin current-carrying elastic plate change with variation of the electromagnetic parameters are discussed. The results show that the stress–strain state in thin plates can be controlled by changing the electromagnetic parameters. This provides a method of theoretical analysis and numerical calculation for changing the service conditions and intensity research of thin plates of engineering structures in the electromagnetic field

The Analysis of Electric Machines Electro-Mechanic Processes

The rapid development of modeling of various electromechanic energy transformers (power transformers, electric machines) over the past years created the necessity to develop the systematic method of quality and quantity model analysis. Typical analysis of model includes the following stages: the calculation of transient processes, the calculation of established processes, the evaluation of static stability, and the parametric sensitiveness defining. Nowadays, only the analysis of transient processes is performed continuously, i.e. in the state cognate with the physical nature of a process. Other stages are studied in discrete time. The variety of mathematical apparatus used on each stage complicates the analysis and requires significant expenses. In this article, the author describes the method of solving all four tasks on the basis of a unified mathematic apparatus - general theory of differential equations based on a unified algorithm. The completion of first three stages is illustrated on example of the induction machine, the mathematical model of which is arranged to the normal Cauchy form in order to avoid inversion of coefficient matrix on each integration step. By using the affine coordinate system the physical meaning of stator variables has been preserved and the rank of model equation system has been reduced. Stationary electromechanic process is defined by solv ing the twopoint Tperiodic boundary problem leaded do wn on period to Cauchy problem with initial conditions clarification by Newton iterations. The response matrix for the initial conditions, that is the basic element of the Newton formula, is defined from the analytical solving of homogeneous system of linear variation equations. The transient electro� mechanic process will result from the integration of state equations defining on time longer than period. The static stability of the process is defined by the multipliers of response matrix to the initial conditions. The matrix should be calculated for variables values of the last step in the integration process on period. The suggested method can be freely generalized for the analysis of complicated electromechanic schemes. Keywords: energy transformer, power transformer, electric machine, model analysis, unified mathematic apparatus, induction machine, electromagnetic proc ess, Tperiodic boundary problem, Cauchy problem.

Synthesis of optimal digital controller of flocculant dosing

Purpose. The task of automatic process control of the slime water thickening and flotation tailings clarification is the stabilization of thicken product density within the given range and keeping up the solids content in the overflow not above the permissible level with minimum use of the flocculants. In existing systems for automatic control the flocculant dosing is carried out according to the solids content in the device input (the principle of open-loop control). This leads to the excess consumption of the flocculants and increase the dispersion density of the overflow. To perform the synthesis of the optimal digital controller in order to minimize the deviations from the master control and ensure the specified quality of the transition process. Over controlling value should not exceed 5 %. To perform the system operation modeling in order to determine the quality of transient processes. Methodology. Synthesis of the optimal digital controller is based on the method of dynamic programming. Findings. A mathematical model of the object control is represented in the normal form of Cauchy and further in the form of differential equations. The optimum period of quantization as the function from specified error of control and the output coordinate change is calculated. The differential equation of Bellman is obtained and the condition for minimization of the quality functional. Bellman function is represented as a quadratic form from the variables of the system condition. In order to limit possible control, the weight coefficients of the functional are calculated based on maximum permitted values of the system condition variables and the control actions during the transient process. Practical value. Using the modeling of ACS of the flocculant dosing it was established that the over controlling amount is 3.5%, the transient process life 5.6 sec, the transient process is aperiodical, non-static control, which meets the requirements imposed on the ACS.

СИНТЕЗ ОПТИМАЛЬНОГО ЦИФРОВОГО РЕГУЛЯТОРА ДОЗИРОВАНИЯ ФЛОКУЛЯНТА

Purpose. The task of automatic process control of the slime water thickening and flotation tailings clarification is the stabilization of thicken product density within the given range and keeping up the solids content in the overflow not above the permissible level with minimum use of the flocculants. In existing systems for automatic control the flocculant dosing is carried out according to the solids content in the device input (the principle of open-loop control). This leads to the excess consumption of the flocculants and increase the dispersion density of the overflow. To perform the synthesis of the optimal digital controller in order to minimize the deviations from the master control and ensure the specified quality of the transition process. Over controlling value should not exceed 5 %. To perform the system operation modeling in order to determine the quality of transient processes. Methodology. Synthesis of the optimal digital controller is based on the method of dynamic programming. Findings. A mathematical model of the object control is represented in the normal form of Cauchy and further in the form of differential equations. The optimum period of quantization as the function from specified error of control and the output coordinate change is calculated. The differential equation of Bellman is obtained and the condition for minimization of the quality functional. Bellman function is represented as a quadratic form from the variables of the system condition. In order to limit possible control, the weight coefficients of the functional are calculated based on maximum permitted values of the system condition variables and the control actions during the transient process. Practical value. Using the modeling of ACS of the flocculant dosing it was established that the over controlling amount is 3.5 %, the transient process life 5.6 sec, the transient process is aperiodical, non-static control, which meets the requirements imposed on the ACS.