Orthogonal Coordinate System Research Articles

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

For the sake of minimizing fuel consumption, a rocket typically keeps flying in or near a plane specified by the Earth's center, launch pad, and orbit-insertion point, where the plane is called the Reference Flight Plane (RFP) here. To facilitate trajectory analysis, a series of generalized position and velocity coordinates are defined based on the RFP. To develop the Ordinary Differential Equations (ODEs) of the generalized coordinates, a novel dynamics modeling method based on tensor theory is proposed. As the first step of the new method, the rotating Earth is taken as the reference frame and a flight dynamics model described by orthogonal coordinates is developed. Then, according to the definitions of the generalized coordinates, two special curvilinear coordinate systems are put forward, and the relationships between the curvilinear and orthogonal coordinate systems are found. Finally, skillfully using the relationships as well as tensor theory, the dynamics model for the orthogonal coordinates are transformed into that for the generalized coordinates. The benefit of the new method is that it can derive the ODEs of multiple coordinates simultaneously because it is entirely based on matrix operations. By contrast, the traditional modeling methods from Newtonian mechanics and analytical mechanics need to use the infinitesimal approach to develop the equation of each generalized coordinate one by one, which may result in a large amount of repeated derivation. Therefore, the new method has higher efficiency in developing flight dynamics model described by generalized coordinates, and helps to prevent mistakes in derivation.

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

Orthogonal coordinate systems represent an attractive alternative for the formulation of two-dimensional composition space equations for partially premixed combustion: They avoid the need of closure models for a cross scalar dissipation rate and allow for a direct recovery of the corresponding flamelet equations in the asymptotic limits of non-premixed and premixed combustion. Despite these remarkable features, this kind of coordinate system still presents some important unsolved issues, which have limited their application so far. These difficulties are mainly associated with i) the lack of an appropriate formalism for the definition of two variables having orthogonal gradients in the entire flame domain and ii) the absence of corresponding closure models for the gradients of these variables in two-dimensional composition space. In the present work, it is shown how a Lagrangian interpretation of the flamelet derivative allows solving both problems. More specifically, after the mixture fraction, Z, is adopted as first coordinate, the proposed approach allows to derive i) two-dimensional composition space equations for all reactive scalars, ii) a transport equation for a modified reaction progress variable, φ, satisfying the desired orthogonality condition, ∇Z·∇φ=0, and iii) two-dimensional composition space equations for gZ=|∇Z| and gφ=|∇φ|. The obtained set of two-dimensional equations in orthogonal composition space is general and it can describe different flames of interest. In order to illustrate the capabilities of the formulation, the equations are then specialized for planar flames with unity Lewis number, obtaining in this way a solvable set of composition space equations in orthogonal coordinates. After an appropriate numerical approach is introduced, the resulting 2D equations are solved to analyze the interaction between premixed flamelets with a strain rate prescribed along the Z-dimension controlling the interaction. Both flame structures and budgets of the scalar gradient equations are studied and the results provide new insights into the physics of scalar gradients in two-dimensional composition space. Finally, conceivable coupling strategies of the present formulation with CFD codes for the simulation of turbulent flames are discussed.

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

This paper presents a novel analytical solution for the magnetic fields of large permanent magnet (PM) wind generator with bread loaf PMs under normal and eccentricity conditions. To simplify the calculation, the generator is first approximately modeled in the Cartesian coordinate, and then the bread loaf PMs are equivalently considered by pairs of coils so that their effects can be expressed by mathematic equations. By applying the Fourier separation method, with the boundary and interface conditions, the magnetic fields in the calculation regions can be obtained by solving system equations. Afterward, the rotor eccentricity is investigated by transforming the corresponding model to a new orthogonal coordinate system by using a conformal mapping technique. Numerical examples validate the proposed method and the calculated results have a satisfactory agreement with the finite element analysis. The established model in this paper has a significant value for modeling large PM wind power generators, and the calculation time is deduced remarkably.

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

The vertical axis of a multi-story building, at which the rigidity centers of the floors are aligned, is called real torsion axis. In particular, the real torsion axis is characterized as bending; twist; and shear axis, as well. If the local axes of the vertical structural elements are not aligned with the axes X and Y of the global orthogonal coordinate system (O,X,Y,Z), the principal horizontal axes, I and II of the coordinate system defined at the real torsion axis, present an angle, ao, which is usually determined according to axis X. The principal vertical axis III coincides with Z axis. The present paper proposes a new method for the determination of the orientation angle of the real torsion axis. The method refers to single-story and special classes of multi-story buildings, which possess a real torsion axis, in contrast with the majority of the multi-story buildings, at which an optimum torsion axis is usually sought. Firstly, the stiffness matrix of the building is determined at the location of the real torsion axis. Secondly, the rotation matrix around the vertical axis is determined and the rotated stiffness matrix is determined, as well. Three new criterions are proposed in order to define the critical rotation angle, at which the rotated stiffness matrix receives a diagonal form, and therefore, the critical angle defines the orientation of the two principal bending planes I,III and II,III. The optimization matrix method can be easily implemented in any programming language. Numerical examples are presented. Their results satisfy the three characteristic axis properties of the real torsion axis and are in complete agreement with the results of other researchers.

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.

Planar Conformal Deformation of Robotic S-Isothermic Surface

Robotic surfaces can design the space they cover by deforming and can be used in laparoscopic surgery to secure a surgical field space. In medical applications, a coordinate transformation is crucial to cooperate with other robots. However, allowing shear in the robot surfaces' deformations makes accurate coordinate transformations difficult. Robotic S-isothermic surfaces, given their mathematics, are only slightly deform in shear. In a system that combines this robot with rigid-body robots, each coordinate system can be entirely unified in the orthogonal coordinate system, simplifying the control. However, it has not been thoroughly investigated whether the deformed robot's local coordinate system is Cartesian or not. In this study, we use a rubber-like resin robotic S-isothermic surface and demonstrate that its deformation in a plane is conformal. We also show that when the deformation to the desired shape is small, there are conformal parameters close to the shape even during the deformation.

Parametric study of the Giesekus fluid flow in a curved duct with square cross section

In this paper, the log-conformation representation method (LCR) is applied in an orthogonal curvilinear coordinate system to study the Giesekus fluid flow in a curved duct. Derivations for evolution equations of LCR in this curvilinear coordinate system are presented. Secondary flow patterns and oscillation solutions are computed by using the collocation spectral method. The influence of a wide range of Dean number, Weissenberg number, and dimensionless mobility parameter α on fluid behaviors is studied. A six-cell secondary flow pattern is found under very low Dean number and relatively high Weissenberg number and α. Moreover, both Weissenberg number and α are able to facilitate the development of the secondary flow. In addition, simulations under critical Reynolds number for oscillation imply that Giesekus fluid flow with We≥0.1 is not able to retain a four-cell secondary flow pattern in a steady state, which is different from Newtonian fluids.

A review on design of scaffold for osteoinduction: Toward the unification of independent design variables.

Biophysical stimulus quantifies the osteoinductivity of the scaffold concerning the mechanoregulatory mathematical models of scaffold-assisted cellular differentiation. Consider a set of independent structural variables ($) that comprises bulk porosity levels ([Formula: see text]) and a set of morphological features of the micro-structure ([Formula: see text]) associated with scaffolds, i.e., [Formula: see text]. The literature suggests that biophysical stimulus ([Formula: see text]) is a function of independent structural variables ($). Limited understanding of the functional correlation between biophysical stimulus and structural features results in the lack of the desired osteoinductivity in a scaffold. Consequently, it limits their broad applicability to assist bone tissue regeneration for treating critical-sized bone fractures. The literature indicates the existence of multi-dimensional independent design variable space as a probable reason for the general lack of osteoinductivity in scaffolds. For instance, known morphological features are the size, shape, orientation, continuity, and connectivity of the porous regions in the scaffold. It implies that the number of independent variables ([Formula: see text]) is more than two, i.e., [Formula: see text], which interact and influence the magnitude of [Formula: see text] in a unified manner. The efficiency of standard engineering design procedures to analyze the correlation between dependent variable ([Formula: see text]) and independent variables ($) in 3D mutually orthogonal Cartesian coordinate system diminishes proportionally with the increase in the number of independent variables ([Formula: see text]) (Deb in Optimization for engineering design-algorithms and examples, PHI Learning Private Limited, New Delhi, 2012). Therefore, there is an immediate need to devise a framework that has the potential to quantify the micro-structural's morphological features in a unified manner to increase the prospects of scaffold-assisted bone tissue regeneration.

Dual-Time-Scale Sliding Mode Control for Surface-Mounted Permanent Magnet Synchronous Motors

The permanent magnet synchronous motors (PMSMs) as the completely symmetrical three-phase machines, which are usually driven by symmetrical voltage signals. Unfortunately, a PMSM system usually suffers from the different lumped disturbances, such as internal parametric perturbations and external load torques, the speed regulation problem should be addressed within the different operation situations. Characterizing by the current variation speed of the motor winding is much faster than that of the mechanical dynamic velocity, a dual-time-scale sliding mode control (SMC) method for the surface-mounted PMSMs is proposed in this paper. Firstly, the mathematical model of PMSMs is established in the two-phase synchronous rotating orthogonal reference coordinate system, and the slow and fast variation subsystems are obtained based on the quasi-steady-state theory. Secondly, a tracking differentiator (TD)-based and exponential reaching law-based sliding mode controllers are individually designed within dual-time-scale, respectively. As a result, the eventual SMC strategy is presented, and the stability of control system is analyzed by applying the Lyapunov stability theory. The main contribution of this study is to present an alternative control framework for the PMSMs servo system, where the dual-time-scale characteristic is involved, and thus a non-cascade control structure that different from the traditional drive strategy is proposed in the motor community. Finally, the model of whole system is built and carried out on the simulation platform. Research results demonstrate that the presented servo control system can accurately track the reference angle velocity signal, while the strong robustness and fast response performance are guaranteed in the presence of external disturbances. In addition, the three-phase current transient response values are completely symmetrical with the rapid adjustment characteristic.