Polar Coordinate System Research Articles

Variable stroke fatigue test for aircraft landing gears

The fixed stroke fatigue test cannot simulate the real loading conditions of the landing gear, and the variable stroke fatigue test has increasingly become mainstream. A modular test device for fatigue tests for aircraft landing gears was designed, and the automatic control of the buffer stroke was realized with ectopic displacement control technology effectively. The horizontal loading cylinder could follow the wheel axis actively with the displacement active control technology. The follow-up loading along the vertical direction was realized by constructing a polar coordinate system and making the loading line automatic alignment. A split-type dummy wheel was designed to improve the loading accuracy, and a booster cylinder was designed to realize the control loading of high pressure combined with the hydraulic control system. The test results show that the technical scheme meets the design requirements, and the fatigue test was completed successfully. The device for landing gear variable stroke fatigue testing is appropriate for other landing gear static/fatigue tests.

Modeling of the Nanofiltration Process Based on Convective Diffusion Theory

The article formulates the state of the problem of improving the theoretical calculation of the nanofiltration kinetic characteristics in the time cycle of separation of industrial solutions containing copper(II), iron(III), trisodium phosphate and OP-10 (a wetting agent used in electroplating, a product of treating a mixture of mono- and dialkylphenols with ethylene oxide) using the equations of convective diffusion, hydrodynamics and mass transfer. To calculate the kinetic characteristics of the nanofiltration process, the mathematical model was improved by numerically solving the equations of convective diffusion, the Navier–Stokes equation and the flow continuity equation in a polar coordinate system with initial and boundary conditions. The theoretical results obtained in the process of an analytical solution of the system of equations allow calculating changes in concentrations in the permeate and retentate tracts and the permeate volume during nanofiltration separation. The acceptability of the developed nanofiltration method for separating solutions is assessed by comparing the calculated data according to the mathematical model with the experimental data obtained on the nanofiltration unit during separation of solutions containing copper(II), iron(III), trisodium phosphate and OP-10.

A Fast Obstacle Detection Algorithm Based on 3D LiDAR and Multiple Depth Cameras for Unmanned Ground Vehicles

With the advancement of technology, unmanned ground vehicles (UGVs) have shown increasing application value in various tasks, such as food delivery and cleaning. A key capability of UGVs is obstacle detection, which is essential for avoiding collisions during movement. Current mainstream methods use point cloud information from onboard sensors, such as light detection and ranging (LiDAR) and depth cameras, for obstacle perception. However, the substantial volume of point clouds generated by these sensors, coupled with the presence of noise, poses significant challenges for efficient obstacle detection. Therefore, this paper presents a fast obstacle detection algorithm designed to ensure the safe operation of UGVs. Building on multi-sensor point cloud fusion, an efficient ground segmentation algorithm based on multi-plane fitting and plane combination is proposed in order to prevent them from being considered as obstacles. Additionally, instead of point cloud clustering, a vertical projection method is used to count the distribution of the potential obstacle points through converting the point cloud to a 2D polar coordinate system. Points in the fan-shaped area with a density lower than a certain threshold will be considered as noise. To verify the effectiveness of the proposed algorithm, a cleaning UGV equipped with one LiDAR sensor and four depth cameras is used to test the performance of obstacle detection in various environments. Several experiments have demonstrated the effectiveness and real-time capability of the proposed algorithm. The experimental results show that the proposed algorithm achieves an over 90% detection rate within a 20 m sensing area and has an average processing time of just 14.1 ms per frame.

A prediction model for surface settlement during the construction of variable cross-section tunnels under existing structures based on stochastic medium theory

When predicting ground surface settlement caused by constructing a new tunnel beneath existing structures, traditional stochastic medium theory inadequately accounts for the effects of abrupt changes in the cross-sectional areas of variable-section tunnels and the presence of existing structures, potentially leading to significant errors. In this paper, an enhanced ground surface settlement prediction model, based on stochastic medium theory, is proposed. The model equates the settlement of the existing structure’s base slab to that of the overlying soil and divides the horseshoe-shaped tunnel cross-section into eight arc segments, which are calculated using a polar coordinate system. Furthermore, the model treats soil loss at the junctions of variable-section tunnels as a linear transition, introduces the concept of a linear transition segment for variable sections, and accounts for the superimposed effects of closely spaced twin-tunnel excavations. Based on this model, a general-purpose calculation program has been developed that enables the rapid prediction of ground surface deformation caused by a variable-section tunnel passing beneath existing structures, simply by inputting engineering parameters. Finally, the accuracy of the prediction model was validated through comparisons with field measurement data, finite element analysis results, and calculations based on traditional stochastic medium theory. The results indicate that the proposed prediction model demonstrates high consistency with field data and finite element analysis results, whereas traditional stochastic medium theory results exhibit significant errors. This model is scientifically valid and provides a reliable reference for predicting ground surface settlement in comparable variable-section metro tunnel construction projects.

Analysis of Free-Interaction Theory in Swept Shock Wave/Turbulent Boundary Layer Interaction

The flow structures of swept shock wave/turbulent boundary layer interactions are parametrically studied over the sharp fin and swept compression ramp, by means of the Reynolds-averaged Navier–Stokes equation at the same moderate Reynolds number. The general flow characteristics and quasi-conical symmetry are presented in the context of a spherical coordinate system for clarity. The commonality of these two flow patterns is performed at the same incoming conditions of Mach number M∞=2.0. The quasi-conical symmetry of wall-surface pressure is discussed. Besides, a mathematical formula of wall pressure makes an acceptable agreement with the numerical results in the spherical coordinate system, providing a potential mathematical method for rapid prediction of wall pressure. Finally, the derivation of free-interaction theory is extended on the polar coordinate system to describe the swept interaction, and its independence from radial position is also verified. All numerical data collapse well with theoretical line for the pressure jump across the separation. Hence, the extension of free-interaction theory is also applicable to the swept shock wave/turbulent boundary layer interaction.

Методика перетворення кодових послідовностей у відповідності до системи координат модуляційного диска

An analysis of current approaches used in the design of optical recording systems for modulation disks has been conducted. By adapting mathematical models, software algo-rithms were developed to transform code sequences represented in polar, homogeneous, and parametric coordinate systems, and the unique characteristics of each approach in de-veloping a universal method for ensuring accurate and efficient modulation pattern trans-formation were established. It was noted that the polar coordinate system is the most suit-able for practical applications in code sequence transformations. Its ability to effectively represent circular and symmetrical structures, as well as the convenient representation of objects with radial symmetry, provides advantages in the creation of modulation disks. As a result of the study, effective software solutions for automating data processing in the formation of modulation patterns were identified. The implementation of the developed software algorithms for the presented approach involves adapting the code sequence to the current metric within the design of an optical system based on modulation disks. The pre-sented methodology ensures the ability to transform code sequences regardless of the cho-sen coordinate system, which significantly enhances its flexibility and versatility for prac-tical applications.

Inverse problem for 2d Laplace equation in polar coordinate system

The article presents the results of study on finding the thermophysical characteristics of a heterogeneous material based on the heat conductivity equation in polar coordinates, carried out within the framework of the grant project of the Ministry of Science and Higher Education of the Republic of Kazakhstan AP19677594 «Development of methods for finding all nonlinear moisture-heat-conducting characteristics of nonhomogeneous medium, realization of methods by creating machine learning program». The temperature state of a body of a multilayer system of bodies can be characterized using a temperature field, which is understood as a set of instantaneous temperature values at all points of the studied space. It occurs when two bodies with different temperatures come into contact or within one body with areas with different temperatures.

A novel directional simulation method for estimating failure possibility

Failure possibility plays a crucial role in assessing the safety level of structures under fuzzy uncertainty. However, the traditional fuzzy simulation method suffers from computational inefficiency as it requires a large number of samples for accurate estimation. To address this issue, a directional simulation method is proposed to improve the efficiency of estimating failure possibility. The directional simulation method reformulates the failure possibility estimation into two key steps: the generation of direction samples and the estimation of conditional failure possibility under each direction sample in the polar coordinate system of the standard fuzzy space. To ensure direction uniformity, these direction samples are generated by adopting a good lattice point set based on stratified sampling on the unit hypercube. The conditional failure possibility under each direction sample is estimated by solving the minimum root of a nonlinear equation. The proposed method effectively reduces the dimensionality of the fuzzy input and greatly improves the computational efficiency. To further enhance efficiency, an adaptive Kriging model is embedded into the directional simulation method to reduce the number of performance function evaluations. Four examples are performed to illustrate the accuracy and efficiency of the proposed method. The results highlight the superiority of the directional simulation method over the traditional fuzzy simulation method, offering substantial improvements in computational efficiency while maintaining high estimation accuracy.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

Research on the high-performance computing method for the neutron diffusion spatiotemporal kinetics equation based on the convolutional neural network

Due to the uncertainty of computational results and the lack of interpretability of models in solving physical field equations in current deep learning, this paper designs a convolutional neural network that can be used to solve the neutron diffusion spatiotemporal kinetics equation in polar and cylindrical coordinate systems. This algorithm directly utilizes the macroscopic cross-section of the material without using the lattice homogenization method, replaces the finite volume method with the extended matrices, and solves the extended matrices using the convolutional kernels instead of the iterative algorithms. Taking the simplified Tsinghua High Flux Reactor (THFR) as an example, the feasibility of the algorithm is verified on the PyTorch platform and compared with the calculation results of the source iteration method running on the GPU. The calculation results show that when the number of grids in the radial and axial sections of the simplified THFR model is 804,600 and 3,576,000, respectively, and the algorithm is iterated 3000 times, the normalized power of the convolutional neural network and the source iteration method converges to 10−10, and the maximum point by point error of the neutron flux density of the above two algorithms converges to 10−5. The computational time consumed by the convolutional neural network is approximately 880.64 s and 3729.62 s, which reduces the computational time by 4.66% and 5.05% compared to the GPU parallel accelerated source iteration method, and the former consumes 43.75% less memory compared to the latter. The convolutional neural network is mainly used as the virtual physics engine for the THFR digital twin system, in addition to solving the neutron diffusion spatiotemporal kinetics equation and further improving computational speed. The algorithm directly utilizes the neutron macroscopic cross-section of the material to calculate the neutron flux density distribution without using the lattice homogenization, providing theoretical guidance and algorithm support for developing the high-precision multi-physical field coupling model.

Corrective and new research frontier of dual-phase -lag non-fourier heat conduction in functionally graded cylindrical materials with bi-directional property variations

This study presents a novel and corrective analysis of Dual-Phase-Lag (DPL) non-fourier heat conduction in functionally graded cylindrical materials with bi-directional property variations. For general purposes, the 2-D DPL model was employed and solved in a polar coordinate system for an FGM cylinder whose material properties vary exponentially in the axial and radial directions. The proposed model's analytical and numerical solutions were obtained through the SOV method and implicit FDM with a non-uniform grid. Moreover, the effect of the inhomogeneity parameter in the Fourier, Cattaneo–Vernotte (C–V), and DPL models has been analyzed. The results indicate that the DPL model achieves temperature stability in less time when contrasted with the C–V model. In addition, the reduced inhomogeneity parameters result in quicker attainment of a steady temperature and the induction of higher temperatures. The thermal wave propagation in the DPL model is consistently greater than that in the C–V model. In addition, an increase in time lag for heat flux enhances thermal wave properties (amplitude, wavelength, propagation speed etc.); conversely, an increase in time lag for temperature gradient counteracts wave properties and augments heat release. Nevertheless, the present outcomes offer a straightforward multivariate analytical and numerical solution for a finite cylinder's non-Fourier heat conduction equation under diverse boundary conditions.

An overset-grid finite-difference algorithm for seismic wavefield propagations modelling in the polar coordinate system with a complex free-surface topography

Summary Nowadays, finite-difference method has been widely applied to simulate seismic wavefield propagation in a local seismic model with a complex topography by utilizing curvilinear grids and traction image method in Cartesian coordinates system. For a global seismic model in polar coordinate system, it is still a challenge for the conventional finite-difference method to deal with the grid singularity at the center and the topographic free surface at the top. In this study, we develop a finite-difference method in polar coordinates for seismic wavefield propagation in the 2D global model. In the proposed finite-difference method, the overset-grid algorithm is used to handle the grid singularity at the center, and the curvilinear grid technique as well as the traction image method are applied to implement free-surface boundary condition on the complex topography. The proposed finite-difference method is validated in flat and topographic free-surface models by comparing synthetic waveforms with reference solutions. The seismic wavefield propagation in a realistic Mars profile is computed by the proposed finite-difference method. The proposed finite-difference method is an efficient and accurate method for seismic wave modeling in the polar coordinate system with a complex free-surface topography.

Analytical prediction of the open‐circuit magnetic field for the V‐shaped interior PM machine by the improved subdomain method

AbstractThe open‐circuit magnetic field of the V‐shaped interior PM (IPM) machine is predicted by the analytical method. The method is implemented following the basic principle of the subdomain method. The machine model is divided into five different regions, including rotor slot, magnetic barrier, air gap, stator slot, and stator slot opening. To perform the analytical prediction process in the polar coordinate system, these regions should be transformed into the appropriate subdomains. The last four regions can be transformed into the subdomains with the required shape based on the existing method. The rotor slot cannot be treated as a single subdomain, for the shape of which is irregular. Herein, the PM and vacuum region in the rotor slot is divided into several fan‐shaped subdomains. The division method is described clearly. Then, the boundary conditions alongside the interface between adjacent subdomains are derived based on function transformation. On this basis, the open‐circuit magnetic field of the IPM machine can be predicted. The magnetic field of the 6p/36 s V‐shaped IPM machine is obtained by the proposed method, finite element method simulation, and experiment, respectively. Additionally, the comparison results verify the accuracy and efficiency of the proposed method.

Circular Clustering With Polar Coordinate Reconstruction.

There is a growing interest in characterizing circular data found in biological systems. Such data are wide-ranging and varied, from the signal phase in neural recordings to nucleotide sequences in round genomes. Traditional clustering algorithms are often inadequate due to their limited ability to distinguish differences in the periodic component θ. Current clustering schemes for polar coordinate systems have limitations, such as being only angle-focused or lacking generality. To overcome these limitations, we propose a new analysis framework that utilizes projections onto a cylindrical coordinate system to represent objects in a polar coordinate system optimally. Using the mathematical properties of circular data, we show that our approach always finds the correct clustering result within the reconstructed dataset, given sufficient periodic repetitions of the data. This framework is generally applicable and adaptable to most state-of-the-art clustering algorithms. We demonstrate on synthetic and real data that our method generates more appropriate and consistent clustering results than standard methods. In summary, our proposed analysis framework overcomes the limitations of existing polar coordinate-based clustering methods and provides an accurate and efficient way to cluster circular data.

Tomographic task solution using a dichotomous discretization scheme in polar coordinates and partial system matrices invariant to rotations

Objectives. The purpose of this work was to create an effective iterative algorithm for the tomographic reconstruction of objects with large volumes of initial data. Unlike the convolutional projection algorithm, widely used in commercial industrial and medical tomographic devices, algebraic iterative reconstruction methods use significant amounts of memory and typically involve long reconstruction times. At the same time, iterative methods enable a wider range of diagnostic tasks to be resolved where greater accuracy of reconstruction is required, as well as in cases where a limited amount of data is used for sparse-view angle shooting or shooting with a limited angular range.Methods. A feature of the algorithm thus created is the use of a polar coordinate system in which the projection system matrices are invariant with respect to the rotation of the object. This enables a signification reduction of the amount of memory required for system matrices storage and the use of graphics processors for reconstruction. Unlike the simple polar coordinate system used earlier, we used a coordinate system with a dichotomous division of the reconstruction field enabling us to ensure invariance to rotations and at the same time a fairly uniform distribution of spatial resolution over the reconstruction field.Results. A reconstruction algorithm was developed on the basis of the use of partial system matrices corresponding to the dichotomous division of the image field into partial annular reconstruction regions. A 2D and 3D digital phantom was used to show the features of the proposed reconstruction algorithm and its applicability to solving tomographic problems.Conclusions. The proposed algorithm allows algebraic image reconstruction to be implemented using standard libraries for working with sparse matrices based on desktop computers with graphics processors.

Dynamic Analysis and PD Control in a 12-Pole Active Magnetic Bearing System

This paper conducts an in-depth study on the dynamic stability and complex vibration behavior of a 12-pole active magnetic bearing (AMB) system considering gravitational effects under a PD controller. Firstly, based on electromagnetic theory and Newton’s second law, a two-degree-of-freedom control equation of the system, including PD control terms and gravitational effects, is constructed. This equation involves not only parametric excitation, quadratic nonlinearity, and cubic nonlinearity but also a more pronounced coupling effect between the magnetic poles due to the presence of gravity. Secondly, using the multi-scale method, a four-dimensional averaged equation of the system in Cartesian and polar coordinates is derived. Finally, through numerical analysis, the system’s amplitude–frequency response, motion trajectory, the relationship between energy and amplitude, and global dynamic behaviors such as bifurcation and chaos are discussed in detail. The results show that the PD controller significantly affects the system’s spring hardening/softening characteristics, excitation, amplitude, energy, and stability. Specifically, increasing the proportional gain can quickly suppress the rotor’s motion, but it also increases the system’s instability. Adjusting the differential gain can transition the system from a chaotic state to a stable periodic motion.

Analysis of hemodynamic parameters on two-layered blood flow in a curved artery

Arterial stenosis is the thickening of the arterial wall due to the growth of aberrant tissues that prevent adequate blood flow in the human circulatory system and induces cardiovascular diseases. Mild stenosis, may lead to serious or permanent damage if remains uncured. There are differences in the curvature response and material composition between the outer layers and the core. There are several locations in blood vessels where they are curved, which affects the blood flow and shear stress. The Navier-Stokes equation in the cylindrical polar coordinate system has been extended by incorporating curvature term in two-layered blood flow along the axial direction with appropriate boundary conditions. Mathematical expressions for hemodynamic parameters such as velocity profile, volumetric flow rate, pressure drop, and shear stress have been calculated analytically in the case of a curve artery with stenosis. Moreover, we have analyzed the effect of stenosis on different hemodynamic parameters with the variation of core and peripheral-layer viscosity and curvature. Flow quantities are affected by the habitancy of stenosis and stipulate different blood flow behavior in both layers in the case of curved artery. This modeling technique may help researchers in medicine, mathematical biology, and bio-engineering.

MODEL OF RANDOM-LIKE PLANAR TRAJECTORIES WITH INTERSECTIONS

Background. Recently the task of detecting and identifying trajectories of objects whose genuine purposes are uncertain or strike threatening has become extremely important. The known approaches produce insufficiently smooth trajectories. Objective. The purpose of the paper is to build a model of generating random-like planar trajectories, which would have sufficiently smooth curves. A trajectory may have self-intersections and may intersect other trajectories. Methods. Preliminarily two starting points on a plane are generated. The distance and angle between these points are calculated, which then are successively updated to calculate new trajectory points using the polar coordinate system. A trajectory of points is generated using values of normally distributed random variables with zero mean and unit variance and four values of -uniformly distributed random variables. Results. The random-like trajectory generator has the same time complexity as its predecessors, including the direction randomization generator and its modifications. Exemplary trajectories appear very realistic. Self-intersections are important to manoeuvre and confuse the opponent side. The trajectory has four parameters to adjust its heading, scattering of points, and intensity of turns and twists. These parameters serve as magnitudes to amplify the respective properties. The highest influence has the angle-scattering parameter. Four simple conditions can be embedded to fit the trajectory within a rectangular domain. Conclusions. The suggested model should serve either for generating trajectory datasets to train manoeuvring-object detectors on them or for masking reconnaissance. The model allows balancing the trajectory smoothness and randomness.

Subharmonic functions with separated variables and their connection with generalized convex functions

In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables, representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, i.e., convex with respect to two functions.

Fine-Grained Entity-Type Completion Based on Neighborhood-Attention and Cartesian–Polar Coordinates Mapping

Entities refer to things that exist objectively, and entity types are concepts abstracted from entities that have the same features or properties. However, the entity types in the knowledge graph are always incomplete. Currently, the main approach for predicting missing entity types is to learn structured representations of entities and types separately, which ignores neighborhood semantic knowledge of the entity. Therefore, this paper proposes the aggregation neighborhood semantics model for type completion (ANSTC), which extracts neighborhood triple features of target entities with two attentional mechanisms. Meanwhile, the spatial mapping module in ANSTC maps entities from Cartesian coordinate to Polar coordinate system, which can map similar vectors onto a concentric circle and then rotate the angle according to the fine-grained difference to achieve entity-to-type transformation. Moreover, we add semantic features from text to the entity representations to enrich semantics. Through experimental comparison on the FB15K and YAGO43K dataset, we get similar results to the baseline. We also construct person dataset in computer domain, and the values of MRR, Hit@1, Hit@3 and Hit@10 are improved compared with the ConnectE model. The experimental results demonstrate that our model can effectively predict the fine-grained entity types in the domain dataset, and achieve state-of-the-art performance.