System Of Three-dimensional Equations Research Articles

Model of transverse-transverse type piezoelectric transformer

The mathematical model of a piezoelectric transformer of the transverse-transverse type and describes the method of its construction has been presented. Although mathematical modeling programs for piezoelectric devices can achieve any predetermined modeling accuracy, the simulation results cannot be directly used in the development of electronic equipment, because the programs are not integrated with CADs, for this reason most often in calculations and in modeling circuits based on piezotransformers, the simplest equivalent circuit is used. But its adequately reflects currents and voltages in the piezotransformer circuit only in the vicinity of the operating resonant frequency. The proposed model is based on a one-dimensional approximation of the equations of state and dynamics of the piezoelectric medium for flat plates of constant thickness and width, which is obtained from a three-dimensional system of equations by averaging the width and thickness. While the usual approximate model often allows to model a piezotransformer with two pairs of electrodes and only in the vicinity of one resonant frequency, the model constructed in the article allows to take into account the presence of several electrodes on piezotransformer surfaces and their different relative positions on the upper and lower surfaces. 'esoplastin. Compared with the usual, the proposed model is more convenient for modeling by means of circuit modeling systems. In the developed model, the piezotransformer is represented as a set of interconnected sections that carry one pair or several pairs of electrodes on the surfaces. Also, in contrast to the usual, the proposed model allows to take into account the presence of several resonant frequencies of the piezotransformer, which allows more adequate modeling of electronic equipment that uses in its structure a piezoelectric transformer of the transverse type. On the basis of the mathematical model the scheme of substitution of separate sections of the piezoelectric transformer is constructed and formulas for calculation of parameters of elements of the scheme are given. In in the article as example the implementation of the developed model in the computer-aided design system MicroCAP has been showed.

Uniqueness of a three-dimensional stochastic differential equation

In order to extend the study of the uniqueness property of multidimensional systems of stochastic differential equations, we look at the following three-dimensional system of equations, of which the two-dimensional case has been well studied: dXt=Ytdt, dYt=Ztdt, dZt=|Xt|αdBt. We prove that if (X0,Y0,Z0)≠(0,0,0) and 34<α<1, then the system of equations has a unique solution in the strong sense.

A Numerical Algorithm for Time Integration of the Problems of Ideal Magnetohydrodynamics Based on Analyticity of Their Solutions

Abstract—We show that a solution of the system of three-dimensional equations of ideal magnetohydrodynamics is analytic in the spatial and temporal variables on a certain time interval with a strictly positive length provided that the initial flow velocity and magnetic field are analytic functions of spatial variables. Utilizing the property of frozenness of the magnetic field, we construct time Taylor expansions of the solution in the Eulerian and Lagrangian coordinates. We derive recurrence relations for the coefficients of these expansions and employ them for developing the Eulerian and Lagrangian algorithms for numerical time integration of the equations of ideal magnetohydrodynamics. The Largangian algorithm has been tested in computations; the formation of the structures of a smaller dimension is observed in the solution.

Numerical Simulation of Curved Flames in Cylindrical Tubes

Dynamics of a curved flame front in cylindrical tubes is studied by means of numerical simulation of the complete system of hydrodynamic equations including thermal conduction, fuel diffusion, viscosity and chemical kinetics. The complete three-dimensional system of equations is reduced to a two-dimensional system with the account of cylindrical symmetry of the problem. The dependence of the flame velocity on the tube diameter and the expansion coefficient of the fuel is investigated. It is obtained that the velocity increase due to the curved shape of the flame front is considerably larger for the case of cylindrical tubes compared to the two-dimensional curved flames. The regime of strong initiation of the flame instability is obtained for narrow tubes, when all perturbation modes of small amplitude are stable, but a curved stationary flame is still possible. The simulation results indicate that there is no maximal velocity for curved flames in cylindrical tubes unlike two-dimensional flames. Instead the flame velocity increases with no limit with the increase of the tube diameter. The case of a flame propagating upwards in a gravitational field is also investigated.

Method of determining the aerodynamic coefficients of asymmetric bodies with allowance for nonlinear body shape influence factors

The numerical method proposed makes it possible to determine the aerodynamic coefficients of asymmetric bodies of fairly arbitrary shape (including those with discontinuities of the transverse contour) at small solid angles of attack. The method allows an aerodynamically sound transition from the three-dimensional system of equations of gas dynamics to a two-dimensional system, which considerably simplifies the problem and reduces by an order the machine time required. The method takes into account the nonlinear body shape influence factors, which substantially improves the accuracy of the calculations. The efficiency and accuracy of the method are demonstrated by comparing the results of the calculations with the results of a numerical solution of the three-dimensional problem.

A method for evaluating the energy characteristics of the air in an open pit mine

The physical analysis of the processes and means of ventilation in studies of the air in open pit mines involves evaluation and comparison of the energy characteristics of the natural and artificial ventilation and also the estimates of energy required for disrupting intrapit air inversions, artificial air flows, etc. In this study, the authors proceed from results of a numerical modeling of ventilation conditions to compare the energy of natural and artificial processes. The study is based on a mathematical model previously described which represents interactions in the system and also a model evaluating the sensitivity of the air in the pit to the various artificial and natural factors when taking the various energy indicators as the functionals. A system of three-dimensional equations of air dynamics is presented. An evaluation of the thermal energy of incoming radiation used to disrupt the inversion in the pit atmosphere is presented. It is shown that the atmosphere is more sensitive to natural effects acting on the thermal regime of the surface rather than to artificial ventilation by local pulse sources.