Orthogonal Coordinate System Research Articles

Presentation of solutions of three-dimensional dynamic problems of the theory of elasticity in a curvilinear orthogonal coordinate system

To describe the processes of distribution of elastic waves, a model of a three -dimensional isotropic body is used under the action of dynamic loads. A well known presentation of the general solution of equations had been considered in a vector form, which contains four functions. It is established that the function that describes the expansion waves is uniquely determined by the volume deformation. It is shown that the dynamic tense-deformed state of the body with zero volumetric extension can be expressed through two independent functions that satisfy the equation that describes the waves of shift. It is proved that the overall solution of equations can be expressed through three four dimensional displacement functions, which are defined as the solutions of wave equations of the second order. This solution was used and an analytical expression of the general solution of the equations of dynamic theory of elasticity in the curvilinear orthogonal coordinate system was found. This submission has been used and a clear expression of elastic displacements in the cylindrical coordinate system was recorded. However, there are multipliers near one displacement function, which depend on the angular variable, which complicates its practical use. The general solution is regulated in the cylindrical coordinate system in such a way that the coefficients of the expansion in the Fourier rows do not depend on the angle. This made it possible to significantly simplify the expression of solution. The components of a stress-deformed state in the cylindrical coordinate system are recorded.

Improvement of methods of interception of enemy UAVs

The main result of this work is the creation of a methodology for determining the coordinates, velocities and trajectories of spatial movements of moving objects by means of kinematic design and the development of principle schemes for the optimal placement of radar search equipment for the successful detection of enemy UAVs. Its purpose is to increase the defense capability of military units, populated areas of Ukraine and their infrastructure by resisting attacks by attack and sabotage reconnaissance unmanned aerial vehicles (UAVs). The protection scheme is based on the determination of coordinates, movement speeds and trajectories of spatial movements of enemy aircraft by means of kinematic design. The established coordinates of enemy UAVs are reported to command posts equipped with liquidator drones (kamikaze) and small arms or to control points of anti-aircraft missile systems (SAMS), which eliminate these aircraft. The proposed improvement of the widespread method of radar search for moving objects is based on the introduction of additional search radar equipment and mathematical apparatus for precise calculations of coordinates and parameters of spatial movements of moving objects by means of kinematic design. Two paired radar search systems, equipped with modern high-speed computing equipment with appropriate software, are able to "screen out" false return electromagnetic signals. This makes it possible to clearly single out the wanted moving object and subsequently track its movement in space. Corresponding mathematical dependencies are proposed for calculating the distance of a moving object to radar stations, its coordinates in the introduced orthogonal coordinate system, speed and acceleration of movement, that is, instantaneous coordinates and parameters of spatial movements of moving objects. This makes it possible to increase the effectiveness of detecting enemy attack and sabotage-reconnaissance unmanned aerial vehicles, including Iranian-made drones of the Sahed-136 model.

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal signed-distance coordinates and vector calculus near evolving curves and surfaces

We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and collate useful vector calculus identities for these coordinates. These results and provided code enable consistent accounting of geometric effects to arbitrary order for boundary layer asymptotics in a wide range of physical systems.

Fractal Analysis and FEM Assessment of Soft Tissue Affected by Fibrosis

This research shows an image processing method to determine the liver tissue’s mechanical behavior under physiological damage caused by fibrosis pathology. The proposed method consists of using a liver tissue CAD/CAE model obtained from a tomography of the human abdomen, where the diaphragmatic surface of this tissue is compressed by a moving flat surface. For this work, two tools were created—the first to analyze the deformations and the second to analyze the displacements of the liver tissue. Gibbon and MATLAB® were used for numerical analysis with the FEBio computer program. Although deformation in the scenario can be treated as an orthogonal coordinate system, the relationship between the total change in height (measured) and the deformation was obtained. The outcomes show liver tissue behavior as a hyperelastic model; the Mooney–Rivlin mathematical characterization model was proposed in this case. Another method to determine the level of physiological damage caused by fibrosis is fractal analysis. This work used the Hausdorff fractal dimension (HFD) method to calculate and analyze the 2D topological surface.

Tensorial-formulation-based method for modeling of flight dynamics with generalized coordinates

Tensorial-formulation-based method for modeling of flight dynamics with generalized coordinates

A self-consistent extension of flamelet theory for partially premixed combustion

A self-consistent extension of flamelet theory for partially premixed combustion

Modeling of Non-Isothermal Polymer Melt Flow in a Conical Annular Channel of a Disk Extruder

Purpose. The homogenization zone consists of various channels with different configurations, for each of which it is necessary to determine the passage of the melt flow process, and on its basis - the velocity fields, which determine the quality of mixing and distribution of components in the melt. To ensure a flexible and controllable homogenization process with the possibility of improving the quality of the melt, it is necessary to study the flow processes in the channels of a disk extruder. Therefore, the main objective of this work is to perform hydrodynamic modeling of processes during melt flow in a conical channel. Methodology. To achieve this goal, we propose a methodology for determining the flow processes in a conical channel, and find out which zones are convenient to consider in a special conical orthogonal coordinate system. For this purpose, the change in the radial coordinate , which has the same meaning in both the straight and the conical gap, was described - it is a coordinate along the width of the gap. This makes it possible to further apply this coordinate for the width of the disk gap - between the moving and stationary disks. Findings. A method has been proposed that describes the flow processes in the conical channel of the homogenization zone of a disk extruder. The calculation procedure is presented in an analytical form, and graphical dependences of the distribution of tangential and longitudinal velocities and shear velocities of the melt flow along the width of the annular channel at the nominal and maximum disk speeds and at the nominal and maximum disk gap are also given. Originality. For the first time, a methodology for calculating the conical channel of the homogenization zone of a disk extruder is presented, which describes the flow processes in a conical orthogonal coordinate system, which allows taking into account the common coordinate for the entire homogenization zone. The general procedure for calculating the channels of the homogenization zone has been supplemented. Practical value. The procedure for calculating the channels of the homogenization zone, which was started earlier, was extended and applied to the flow in a conical annular channel. This coordinate allows us to describe the flow processes along the width of the channel for all channels of the homogenization zone of a disk extruder, which greatly simplifies the calculations. Important results of hydrodynamic and thermal processes were obtained for the annular channel, which makes it possible to design disk extruders with greater accuracy and calculate their optimal operating modes.

ГЕОМЕТРИЯ ПРЯМЫХ КОНОИДОВ С ОРТОГОНАЛЬНОЙ СИСТЕМОЙ КООРДИНАТ.

The geometry and methods of forming right conoids are considered in the article. Right conoids are formed by the movement of a rectilinear generatrix perpendicular to a fixed straight line – the axis of the conoid, along which the rectilinear generatrix moves. When moving, the rectilinear generatrix rotates according to a given law around the directrix curve. Usually, the generatrix line is connected to some reference curve that the generatrix line touches. The coordinate grid along the generatrix lines is also connected with the reference curve. In this case, a non-orthogonal coordinate system of a right conoid is formed. Coefficients of quadratic forms of surfaces with a non-orthogonal coordinate system, usually are complex. Methods for analyzing shells with a non-orthogonal coordinate system of the middle surface of the shell are becoming more complicated. The article considers the possibility of forming right conoids with an orthogonal coordinate system, including with directrix reference curves. A vector equation is given for specifying right conoids with an orthogonal nonconjugate coordinate system. Formulas of coefficients of quadratic forms characterizing the internal geometry and curvature of right conoidal surfaces in space with the help of this vector equation are obtained. Based on the vector equations of the surface assignment, drawings of right conoids with various geometric parameters are constructed in the MathCAD program. The difference in the construction of right conoids with orthogonal and non-orthogonal curved coordinate systems is clearly demonstrated, so with different methods of specifying the angles of inclination of rectilinear generatrix of surfaces.

Analytical modeling for rotor eccentricity solution in wind power generator with bread loaf PMs

Analytical modeling for rotor eccentricity solution in wind power generator with bread loaf PMs

Principal Stress Trajectories in Plasticity under Plane Strain and Axial Symmetry

The two families of principal stress trajectories can be regarded as an orthogonal curvilinear coordinate system under plane strain and axial symmetry. Under plane strain, the equilibrium equations in conjunction with a yield criterion comprise a statically determinate system. Under axial symmetry, a statically determinate system results from the above equations supplemented with the hypothesis of Haar von Karman. In both cases, the compatibility equations for mapping the principal line coordinate system to a given coordinate system show that the scale factors of the former satisfy a simple algebraic or transcendental equation for many yield criteria. Using this equation, one can develop a method for reducing boundary value problems in plasticity to purely geometric problems. The method is independent of any flow rule that can be chosen to calculate displacement or velocity fields, as well as independent whether elastic strains are included. The present paper summarizes available results related to using principal stress trajectories in plasticity and emphasizes the advantages of the method above.

Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR

Geosynchronous (GEO) spaceborne–airborne very high-frequency ultra-wideband bistatic synthetic aperture radar (VHF UWB BiSAR) can conduct high-resolution and wide-swath imaging for ocean scenes. However, GEO spaceborne–airborne VHF UWB BiSAR imaging faces some challenges such as the geometric configuration, huge amount of echo data, serious range–azimuth coupling, large spatial variance, and complex motion error, which increases the difficulty of the high-efficiency and high-precision imaging. In this paper, we present an improved bistatic fast factorization backprojection (FFBP) algorithm for ocean scene imaging using the GEO satellite-unmanned aerial vehicle (GEO-UAV) VHF UWB BiSAR, which can solve the above issues with high efficiency and high precision. This method reconstructs the subimages in the orthogonal elliptical polar (OEP) coordinate system based on the GEO satellite and UAV trajectories as well as the location of the imaged scene, which can further reduce the computational burden. First, the imaging geometry and signal model of the GEO-UAV VHF UWB BiSAR are established, and the construction of the OEP coordinate system and the subaperture imaging method are proposed. Moreover, the Nyquist sampling requirements for the subimages in the OEP coordinate system are derived from the range error perspective, which can offer a near-optimum tradeoff between precision and efficiency. In addition, the superiority of the OEP coordinate system is analyzed, which demonstrates that the angular dimensional sampling rate of the subimages is significantly reduced. Finally, the implementation processes and computational burden of the proposed algorithm are provided, and the speed-up factor of the proposed FFBP algorithm compared with the BP algorithm is derived and discussed. Experimental results of ideal point targets and natural ocean scenes demonstrate the correctness and effectiveness of the proposed algorithm, which can achieve near-optimal imaging performance with a low computational burden.

Optimization matrix method for the determination of the real torsion axis orientation angle

Optimization matrix method for the determination of the real torsion axis orientation angle

INFINITESIMAL AND FINITE DEFORMATIONS IN THE POLAR COORDINATE SYSTEM

The deformation problem of elasticity theory with regard to nonlinear deformations is examined. The expressions of deformations through displacements in the orthogonal curvilinear coordinate system are recorded. The relations for finite deformations in cylindrical and polar coordinate systems are derived. Physical relations for finite deformations and corresponding generalized stresses are recorded. 

Robotic Arm Position Computing Method in the 2D and 3D Spaces

This paper presents a method on how to compute the position of a robotic arm in the 2D and 3D spaces. This method is slightly different from the well-known methods, such as forward or inverse kinematics. The method presented in this paper is an optical method, which uses two video cameras in stereo vision configuration to locate and compute the next move of a robotic arm in space. The method recognizes the coordinates of the markers placed at the joints of the robotic arm using the two video cameras. The coordinate points of these markers are connected with straight lines. Around certain points, circles are drawn. From the tangent to the circles, a non-Cartesian (orthogonal) coordinate system is drawn, which is enough to compute the target position of the robotic arm. All of these drawings are overlaid on the live video feed. This paper also presents another method for calculating the stereo distance using the triangulation method. An alternative method is also presented when a non-Cartesian (orthogonal) 3D coordinate system is created, which is used to compute the target position of the robotic arm in the 3D space. Because the system is in a loop, it can make micro-adjustments of the robotic arm, in order to be exactly in the desired position. In this way, there is no need to make calibrations for the robotic arm. In an industrial system, there is no need to stop the production line, which can be a really big cost saver.

New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton-Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.

Динамическое имитационное моделирование системы возбуждения синхронных генераторов стационарных дизель-генераторных установок аварийного электроснабжения атомной станции

In the article, using MatLab dynamic simulation modeling, a study was made of the excitation systems of powerful synchronous generators of stationary diesel generator sets, which are the main sources of emergency power supply for nuclear power plants. The optimal structural complexity mathematical model of a synchronous machine in relative units and orthogonal synchronous coordinate system is used. A comprehensive simulation of diesel generator sets was carried out with the reproduction of both the dynamics of the automatic control system for excitation of a synchronous generator and the diesel engine control system. The simulation takes into account the features of starting a diesel generator to accelerate a synchronous machine, its initial excitation from a battery. Particular emphasis is placed on the study of self-excitation modes through a transformer connected to the stator circuit of the generator and a thyristor rectifier with an excitation winding as a load, as well as parallel operation with the power system. As a result, the processes of starting a diesel generator set in idle mode, effective self-excitation, autonomous operation of the generator at idle, and applying a load to the generator up to the values of permissible overload were simulated. The work of all channels of the control system is shown, including the signals of the regulators of the automatic control system and mechanical variables that are inaccessible in practice. The adequacy of the developed model is proved by comparison with a real physical experiment when testing a diesel generator at a nuclear power plant. The possibility of using the model developed in MatLab as a virtual test site for testing a diesel generator set and a computer simulator for specialized engineering personnel of a nuclear power plant is demonstrated.

Lagrangian 3-form structure for the Darboux system and the KP hierarchy

A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the entire Kadomtsev–Petviashvili (KP) hierarchy in terms so-called Miwa variables. Thus, in providing a Lagrangian description of this multidimensionally consistent system amounts to a new Lagrangian 3-form structure for the continuous KP system. A generalisation to the matrix (also known as non-Abelian) KP system is discussed.

Parametric electro‐mechanical dispersive behaviors of guided waves in the functionally graded piezoelectric annular plate

AbstractThis paper investigates the propagation characteristics of guided waves in functionally graded annular plates considering the electro‐mechanical coupling. In the orthogonal circular curvilinear coordinate system, the elastodynamic wave equation of guided waves is derived for the functional graded piezoelectric annular plate. The Chebyshev spectral elements are utilized to model the annular plate, and the governing wave equation is discretized. The convergence and accuracy of the proposed method are evaluated by comparing with the results in previous publications. The parametric study on dispersive characteristics of the functionally graded piezoelectric annular plate is conducted systematically for the influences of various volume fraction index, curvature, and geometry size. Some exciting results are discussed including mode separation, mode conversion, and cutoff frequency. This research can benefit the design and development of advanced smart materials and structures.

Analytical solution of orthogonal similar oblate spheroidal coordinate system

Satisfactory description of gravitational and gravity potentials is needed for a proper modelling of a wide spectrum of physical problems on various size scales, ranging from atmosphere dynamics up to the movements of stars in a galaxy. In certain cases, Similar Oblate Spheroidal (SOS) coordinate system can be of advantage for such modelling tasks, mainly inside or in the vicinity of oblate spheroidal objects (planets, stars, galaxies). Although the solution of the relevant expressions for the SOS system cannot be written in a closed form, it can be derived as analytical expressions -- convergent infinite power series. Explicit analytical expressions for the Cartesian coordinates in terms of the curvilinear Similar Oblate Spheroidal coordinates are derived in the form of infinite power series with generalized binomial coefficients. The corresponding partial derivatives are found in a suitable form, further enabling derivation of the metric scale factors necessary for differential operations. The terms containing derivatives of the metric scale factors in the velocity advection term of the momentum equation in SOS coordinate system are expressed. The Jacobian determinant is derived as well. The presented analytical solution of SOS coordinate system solution is a tool applicable for a broad variety of objects exhibiting density, gravity or gravitation levels resembling similar oblate spheroids. Such objects range from the bodies with small oblateness (the Earth itself on the first place), through elliptical galaxies up to significantly flattened objects like disk galaxies.