Cylindrical Coordinate System Research Articles

Self and mutual radiation impedances between translated spheroids. Application to parallel disks

This article derives the radiation impedance for two parallel pistons in two different configurations: baffled and unbaffled. For the former, the cylindrical system of coordinates is adapted to compute the self and mutual impedance of a piston between perfectly rigid infinite parallel planes. For the latter, spheroidal coordinates are used and declined for the case of two disks in free-space. The formulae for Green's function that respects Neumann boundary conditions are established in both configurations in order to compute the pressure field using the Huygens-Fresnel principle. Contrasting with the radiation impedance obtained for one single radiator, both systems exhibit resonant frequencies. The result of this study, validated in each case by a numerical method, can be used to predict the acoustic interaction of two vibrating systems for which their radiation surfaces are facing each other.

Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution

Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid “transfer” vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also “transfer” the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.

Dynamic analysis of rocket motor case

Rocket Motor Case (RMC) is a major component of Rockets and other Satellite launching vehicles. So, these structures should be of low weight, i.e. Low weight to high strength ratio materials, i.e., composite and alloys materials are used for main components. In the present analysis, 3-Dimensional finite element based static structural analysis and modal analysis are performed on the rocket motor case using aluminum alloy, titanium alloy, alloy steel, carbon epoxy and e glass epoxy. The problem is modeled in student version of ANSYS software. Stresses under radial, hoop, longitudinal directions are calculated with cylindrical co-ordinate system and list of natural frequencies and mode shapes are noted with respect to pre-stressed constrained and free- free boundary conditions. Later rectangular, semicircular, semi hexagonal and triangular types of stiffeners are introduced to the rocket motor case to perform static structural and modal analysis. The effect of stiffeners to the rocket motor case has studied for different materials under static and dynamic loading conditions. Best stiffener for rocket motor case is recommended based on the results, i.e., rectangular section having higher natural frequencies and semi hexagonal shape having less stresses in their respective directions.

Axisymmetric motion of a slip spherical particle in the presence of a Brinkman interface with stress jump

The axisymmetric motion of a spherical particle in the presence of a porous interface is considered in the limit of small Reynolds and capillary numbers where the interface is of negligible deformation. We consider the translation along and the rotation about an axis perpendicular to the interface. The flow through the porous medium is modeled by Brinkman equation with a tangential stress jump condition applied at the interface and a slip boundary condition is used at the surface of the particle. A semi-analytical approach based on a collocation technique is used. Due to the linearity of the present problem, the flow variables for the two flow regions are constructed by superposing basic solutions in both cylindrical and spherical coordinate systems. The collocation solutions for the normalized hydrodynamic drag force and torque acted on the particle are calculated with good convergence for various values of the separation parameter, the stress jump coefficient, viscosity ratio and the permeability parameter. The results for the normalized drag and torque coefficients are in good agreement with the available solutions in the literature for the limiting cases.

SAR Imaging Distortions Induced by Topography: A Compact Analytical Formulation for Radiometric Calibration

Modeling of synthetic aperture radar (SAR) imaging distortions induced by topography is addressed and a novel radiometric calibration method is proposed in this paper. An analytical formulation of the problem is primarily provided in purely geometrical terms, by adopting the theoretical notions of the differential geometry of surfaces. The novel and conceptually simple formulation relies on a cylindrical coordinate system, whose longitudinal axis corresponds to the sensor flight direction. A 3D representation of the terrain shape is then incorporated into the SAR imaging model by resorting to a suitable parametrization of the observed ground surface. Within this analytical framework, the area-stretching function quantitatively expresses in geometrical terms the inherent local radiometric distortions. This paper establishes its analytical expression in terms of the magnitude of the gradient of the look-angle function uniquely defined in the image domain, thus resulting in being mathematically concise and amenable to a straightforward implementation. The practical relevance of the formulation is also illustrated from a computational perspective, by elucidating its effective discrete implementation. In particular, an inverse cylindrical mapping approach is adopted, thus avoiding the drawback of pixel area fragmentation and integration required in forward-mapping-based approaches. The effectiveness of the proposed SAR radiometric calibration method is experimentally demonstrated by using COSMO-SkyMed SAR data acquired over a mountainous area in Italy.

Non-linear radiative bioconvection flow of cross nano-material with gyrotatic microorganisms and activation energy

Bioconvection with gyrotactic microorganisms has a significant role in biotechnology and biosensors. This paper communicates a theoretical numerical investigation regarding the transient bioconvection flow of chemically reactive cross nanofluid over a permeable cylinder under the impact of nonlinear radiation and binary chemical reactions. Additionally, Arrhenius activation energy influences are taken into account. Further, revise model of nanoparticles is considered in present investigation. Boundary layer theory is invoked to develop the basic PDEs of microorganisms field, nanoparticles concentration, energy, momentum and mass for bioconvection flow of cross nanofluid in cylindrical coordinate system and then altered into nonlinear ODEs by utilizing the transforming variables. Numerical solutions of obtained highly nonlinear ODEs system are achieved through shooting Fehlberg approach for both the cases of stretching cylinder aw well as shrinking cylinder. Numerical computations for surface drag force, mass transfer rate, density number of microorganisms and heat transfer rate are executed. It is interesting to found that the density number of microorganisms is depreciated for higher estimation of Peclet number, microorganisms difference parameter and bioconvection Rayleigh number in both cases of stretching cylinder as well as shrinking cylinder. Furthermore, nanoparticles concentration is accelerated for the augmented values of activation energy parameter in shrinking as well as stretching cylinder. For the authenticity of considered study, we have also made a comparative analysis with existing study and noticed to be in excellent agreement.

Solving anisotropic subdiffusion problems in annuli and shells

This paper presents a family of numerical solvers for anisotropic subdiffusion problems in annuli and also cylindrical and spherical shells. The fractional order Caputo temporal derivative is discretized based on linear interpolation. The spatial Laplacian is discretized by utilizing Chebyshev and Fourier spectral collocation. Detailed discussion and useful formulas are presented for polar, cylindrical, and spherical coordinate systems. Numerical experiments along with a brief analysis are presented to demonstrate the accuracy and efficiency of these solvers. These solvers represent a continuation of our work in Tan and Liu (2020) and shall be useful for numerical simulations of subdiffusion problems in cellular cytoplasm and other similar settings.

A 3D complexity-adaptive approach to explore sparsity in elastic wave propagation

We have developed an adaptive approach for seismic modeling by which the computational cost of a 3D simulation can be reduced while retaining the resolution and accuracy. This azimuthal complexity adaptation (ACA) approach relies upon the inherent smoothness of wavefields around the azimuth of a source-centered cylindrical coordinate system. Azimuthal oversampling is thereby detected and eliminated. The ACA method has recently been introduced as part of AxiSEM3D, an open-source solver for global seismology. We use a generalization of this solver that can handle local-scale Cartesian models and that features a combination of an absorbing boundary condition and a sponge boundary with automated parameter tuning. The ACA method is benchmarked against an established 3D method using a model featuring bathymetry and a salt body. We obtain a close fit when the models are implemented equally in both solvers and an expectedly poor fit otherwise, with the ACA method running an order of magnitude faster than the classic 3D method. Furthermore, we evaluate maps of maximum azimuthal wavenumbers that are created to facilitate ACA. We determine how these maps can be interpreted in terms of the 3D complexity of the wavefield and in terms of seismic resolution. The expected performance limits of the ACA method for complex 3D structures are tested on the SEG/EAGE salt model. In this case, ACA still reduces the overall degrees of freedom by 92% compared with a complexity-blind AxiSEM3D simulation. In comparison to the reference 3D method, we again find a close fit and a speedup of a factor of 7. We explore how the performance of ACA is affected by model smoothness by subjecting the SEG/EAGE salt model to Gaussian smoothing. This results in a doubling of the speedup. Thus, ACA represents a convergent, versatile, and efficient method for a variety of complex settings and scales.

Bending of an elastoplastic circular sandwich plate on an elastic foundation in a temperature field

Today, the development of the general theory of quasi-static deformation of three-layer structural elements, including plates, is not yet complete and is being intensively studied. Mathematical models of deformation under complex thermo-force and thermo-irradiation loads are created. The problems of strength, stability, and dynamic behaviour are considered. In strength calculations of three-layer structural elements, it is necessary to take kinematic hypotheses for each layer separately, which complicates the mathematical side of the problem but leads to significant refinement of the stress-strain state. The reaction of an elastic foundation is described by the Winkler model. The use of variational methods allows one to obtain a refined system of three differential equations of equilibrium in internal forces. The thermo-force bending of an elastoplastic circular sandwich plate with a light core connected to an elastic foundation is considered. The polyline normal hypotheses are used to describe the kinematics of a plate package that is not symmetric in thickness. In thin base layers, the Kirchhoff-Love hypotheses are accepted. In a light relatively thick core, the Timoshenko hypothesis is true, while the normal remains rectilinear, but rotates at some additional angle, the radial displacements change linearly in thickness. The differential equations of equilibrium are obtained using the Lagrange variation method. The statement of the boundary value problem in displacements is given in a cylindrical coordinate system. Numerical results for circular metal-polymer sandwich plates are presented.

A numerical study of the expansion of a gas-particles mixture with axial symmetry

The article deals with the study of new phenomena that accompany the free or wall-bounded expansion of a nonequilibrium in terms of velocities and temperatures mixture of gas and particles of various sizes with axial symmetry. The dynamics of the gas suspension is considered in the multifluid model of a calorically perfect inviscid gas and incompressible monodisperse spherical particles. The Eulerian approach is used to describe the motion of each phase of the mixture. For numerical simulation, a hybrid large-particle method of the second order of approximation in space and time with nonlinear correction of artificial viscosity, an additive combination of fluxes, and a semi-implicit scheme for calculating interfacial friction and heat transfer are implemented. The efficiency and accuracy of the method for a two-dimensional problem with axial symmetry are confirmed by comparing the solutions obtained in a one-dimensional formulation in a cylindrical coordinate system. In the case of small particles (with a diameter of d = 0.1 µm), the relaxation time of the phases is much less than the characteristic time of the problem and the gas and particles mixture behaves as a homogeneous medium similar to the gas flow. For sufficiently large particles (d = 20 µm), the effects of the difference in the inertia of the phases and nonequilibrium, associated with the mismatch of the velocities and temperatures of the gas and particles, are manifested. These effects cause the splitting of the initial interface of the media into a contact discontinuity in the gaseous phase and the surface between the suspension and the pure gas (a jump in porosity). At subsequent moments, the flow pattern changes to the opposite, which is explained by the deceleration of the gas, the appearance of a reverse flow to the center of expansion due to rarefaction in the vicinity of the axis of symmetry, and the formation of a secondary shock wave. Then there are fluctuations with a change in the relative position of the phase boundaries. In this case, fractures of the gas contact trajectories (the break of the first derivative) are observed, which are associated with the passage of the shock wave reflected from the wall and the plane of symmetry. Over time, due to baroclinic instability (the mismatch of density and pressure gradients), vortex structures begin to appear at the interface boundaries. In addition, in the case of expansion of the gas suspension in a closed volume, a complex shock-wave structure is formed, due to multiple reflections of shock waves from the walls and their interaction with the contact surfaces. The practical significance of the results obtained is to identify the fundamental physical effects that should be taken into account when setting and solving problems in chemical technologies, pneumatic transport, and other areas. In addition, the numerical solutions can be useful in testing the resolution of other difference schemes for reproducing the vortex instability of contact boundaries and shock wave structures in the flows of relaxing gas suspensions.

ВЫПИСЫВАНИЕ ФОРМУЛ ВЫЧИСЛЕНИЯ СИЛ В СОЧЛЕНЕНИЯХ МАНИПУЛЯТОРОВ В СТАТИКЕ

The problem of bulkiness of mathematical models of manipulative systems of industrial robots is solved. Here we consider formulas for calculating static reactions in joints and formulas for active forces that balance the forces of gravity acting on the manipulator's bodies in its stationary state. The manipulator can be in such a state when it is before capturing the object of manipulation and releasing it, or when it is performing some assembly operations, or it is during spot welding and in slow (quasi-static) arc-welding and painting processes. Aim. The aim is to derive general recur-rence and finite formulas for calculating the reaction forces in joints and their projections to the ax-es of the coordinate system rigidly connected with the selected body. Express the formulas of force projections in terms of guiding cosines and justify their optimality in terms of the minimum of arithmetic operations. Derive general inverse recurrence formulas for writing out the guide cosines of the axes associated with the moving bodies of the coordinate system with respect to the stationary coordinate system. Research methods. The methods of research relate to vector mechanics and sys-tems analysis, and the algorithmization of calculations by reducing them to the use of recurrent formulas. Results. A systematic analysis of general formulas, in which all possible regular expres-sions are highlighted which are corresponding unambiguously to the kinematic parameters of ma-nipulators, is performed. These regular expressions are used in software for analytical modeling of manipulator, in particular, for the analytical solution of problems of statics of a manipulator. The method of analytical verification of the prescribed formulas is described. The tasks of writing out optimal formulas for calculating the projections of static reaction forces in joints have been solved. And the tasks of writing out optimal formulas for calculating active forces in progressive joints of universal manipulators with six degrees of freedom, operating in Cartesian, cylindrical, spherical and angular coordinate systems, have been solved also. Analytical verification of the derived equations of stat-ics is performed. Examples of the reuse of the derived formulas for manipulators with the same kin-ematic schemes of their subsystems. Conclusion. Expressions of the equations of statics of manipu-lators through the guide cosines of the axes of the associated coordinate systems of their bodies al-low us to write these equations through the known parameters of body orientation. The recurrent formulas for calculating directional cosines allows to use recursive functions in their software im-plementation, i.e. to increase the computational efficiency of the software.

THE STATIC RESPONSE OF AN AXISYMMETRIC FUNCTIONLLY GRADED PIEZOELECTRIC/PIEZOMAGNETIC CYLINDER

For infinite-length functionally graded piezoelectric/piezomagnetic (FGPEPM) hollow cylinders, the statics of the electro-magnetic-elastic coupling for an axisymmetric hollow cylinder is studied. In the cylindrical coordinate system, assuming that the material parameters are distributed along the radial direction as a power function, analytical solutions of the physical fields such as displacement, stress, electric and magnetic potentials in the hollow cylinder under external excitation are derived. In the numerical discussions, for the BaTiO3-CoFeO4 composite material hollow cylinder, the distribution of stress, electric and magnetic potentials for the hollow cylinder with different gradient parameters and different thicknesses are obtained under different boundary conditions. The results show that the material failure is mainly caused by the circumferential stress for the FGPEPM hollow cylinder, and that the cylindrical thickness has a significant influence on the circumferential stress. In addition, the gradient parameters strongly affect the electrical/magnetic output and the mechanical properties for FGPEPM sensors/actuators. The results have a certain guiding effect on the design and analysis for FGPEPM cylindrical sensors/actuators.

Study on arc and flow characteristics of a non-transferred DC steam torch

ABSTRACT A turbulent magnetohydrodynamic model is proposed to predict the steam plasma flow inside a direct-current and non-transferred torch. The continuity, momentum, energy and current continuity equations as well as the turbulence model are expressed in the cylindrical coordinate system and numerically solved with a finite volume method. The operation conditions of mass flow rate ranging from 3 g/s to 8 g/s and current varying between 125 A and 200 A are systematically analyzed via a proposed numerical approach. An explicit determination of the arc root location and the maximum current density of the cathode spot via the experimental measurements is demonstrated. The maximum current density at cathode is found to be on the order of 107 A/m2. A least square fitting approach is then employed to analyze the experimental and numerical results where the quantitative formulation of the arc and flow characteristics in terms of the arc current and mass flowrate are presented. The main arc length falls in the range between 220 mm and 330 mm. At the torch exit, the mean gas temperature is predicted to be between 7.5 kK and 10 kK, the mean axial velocity varies from 750 m/s to 2 km/s and the mean azimuthal velocity is found between 160 m/s and 360 m/s.

Analytical modeling and experimental verification for linearly gradient thickness disk springs

Even though disk springs with non-uniform thickness have the advantages of improving the load capacity, optimizing the stress distribution, and adjusting the stiffness characteristics, their exploration remains sparse. Prior analytical models for non-uniform disk springs were based on the Almen–Laszlo theory, and the model verifications lacked the support of experiments. A mechanical model of linearly gradient thickness (LGT) disk springs is established in this study in the cylindrical coordinate system based on the theory of plates and shells. A new analytical equation as a function of material and geometric parameters is developed to describe the load–deflection characteristics of LGT disk springs. The load–deflection characteristics of disk springs with different thickness variation parameters are compared through cases with a constant mid-radius thickness and constant flattened load, respectively. Moreover, the stiffness, deformation energy, and stress distribution of the disk springs are also systematically analyzed. To verify the proposed model and characterize the nonlinear load–deflection properties of LGT disk springs, specimens were fabricated, and experimental tests were conducted. Comparisons between the analytical results and experimental data demonstrate the superior effectiveness and accuracy of the developed analytical model in predicting the load–deflection characteristics of disk springs with uniform thickness or LGT. The results from the analytical and finite element methods indicate that the proposed model can describe the load–deflection characteristics of disk springs with different cone angles.

Theoretical fractional formulation of a three-dimensional radio frequency ion trap (Paul-trap) for optimum mass separation

We investigate the dynamics of an ion confined in a Paul–trap supplied by a fractional periodic impulsional potential. The Cantor–type cylindrical coordinate method is a powerful tool to convert differential equations on Cantor sets from cantorian–coordinate systems to Cantor–type cylindrical coordinate systems. By applying this method to the classical Laplace equation, a fractional Laplace equation in the Cantor–type cylindrical coordinate is obtained. The fractional Laplace equation is solved in the Cantor–type cylindrical coordinate, then the ions is modelled and studied for confined ions inside a Paul–trap characterized by a fractional potential. In addition, the effect of the fractional parameter on the stability regions, ion trajectories, phase space, maximum trapping voltage, spacing between two signals and fractional resolution is investigated and discussed.

Building an analytical model of the gravitational grain movement in an open screw channel with variable inclination angles

Existing technical means for loading silos with grain material do not fully meet the needs of production. The issue related to grain injury remains urgent, which predetermines the need to design a gravitational loader of another principle of operation with the appropriate theoretical justification for the movement of grain material in it. This paper has presented and substantiated the model of the gravitational movement of grain in the peripheral open screw channel with two variable angles of inclination. The model is based on the system of forces in the cylindrical coordinate system, acting on the volume of grain flow in the peripheral screw channel. The grain speed at the end of the braking section of the channel should be as low as possible but not less than the initial flow rate at the beginning of the acceleration section. The model takes into consideration this condition and ensures the optimal passage of grain along any part of the channel. The reported model makes it possible to obtain the speed of grain movement at any time, takes into consideration the height of the bunker hole and the dependence between the angles of inclination of the spirals of acceleration and brake sections. A mathematical dependence is given for these angles that ensures the passage of grain without its discharge and, at the same time, prevents injury to the grain mass due to a controlled decrease in the resulting speed. A separate dependence is provided to find the time at which the grain increases its speed on the acceleration section, reaching the maximum value. Based on the model, a peripheral open screw channel with two angles of inclination of spirals α and β has been proposed. For this channel, the relationships between its key parameters have been established, in particular, values have been substantiated for the recommended angles of 41°…45° for the acceleration section and 39°…35° for the brake section, respectively, as well as the hо/r ratio not less than 0.6...0.7.

Core depth determination of a slightly buried Si-wire waveguide with a reduced bend loss

The effects of adjusting the distance between a core and an air-cladding interface on the propagation properties are investigated in a bent Si-wire waveguide. The eigenmode analysis is carried out using the imaginary-distance beam-propagation method based on Yee’s mesh in the cylindrical co-ordinate system. The pure bend loss (PBL) is evaluated as a function of core depth for several core aspect ratios. Since a waveguide bend causes polarization coupling, the polarization crosstalk (PCT) is also evaluated by the propagation analysis with the aid of the FDTD method. It is found that a core depth with a minimized PBL exists for the TM mode when the air-cladding interface is placed at the position where the field amplitude decays to approximately 5% of its peak, with a PCT being less than −30 dB.

Analysis of Seismic Wavefield Characteristics in 3D Tunnel Models Based on the 3D Staggered-Grid Finite-Difference Scheme in the Cylindrical Coordinate System

The accurate prediction of the geological conditions ahead of a tunnel plays an important role in tunnel construction. Among all forward geological prospecting methods, the seismic detection method is widely applied. However, due to the characteristics of the tunnel and the complexity of the geological conditions, the seismic wavefield is complicated. Carrying out a more realistic forward modeling method is vital for fully understanding the law of seismic wave propagation and the characteristics of seismic wavefield in the tunnel. In this paper, the 3D staggered-grid finite-difference scheme in the cylindrical coordinate system based on the decoupled nonconversion elastic wave equation is used to carry out the numerical simulation. This method can avoid the diffraction interferences produced at the edges of the tunnel face in the Cartesian coordinate system. Based on this forward modeling method, the characteristics of wavefield and propagation laws of seismic waves under three kinds of common typical unfavorable geological models were explored, which can provide theoretical guidance to seismic data interpretation of tunnel seismic forward prospecting in practice.

Motion of a slip spherical particle near a planar micropolar-viscous interface

An analytical–numerical method based on collocation techniques is presented to investigate the problem of determining Stokes flow caused by a spherical particle moving perpendicularly to a plane interface separating two semi-infinite immiscible fluid phases. Attention is focused on the case when one of the two fluid phases is of a microstructure nature (micropolar fluid). A linear slip boundary conditions, of Basset type, is used on the surface of the particle. The motion is considered in the limit of small Reynolds and capillary numbers where in this case the interface is of insignificant deformation. To solve the axisymmetric creeping motion of the micropolar fluid for velocity and microrotation components, a general solution is constructed from the superposition of some basic solutions in both cylindrical and spherical coordinate systems. Boundary conditions are satisfied first at the fluid–fluid interface using the Fourier Bessel transforms and then on the surface of the particle by the collocation scheme. Numerical results of the normalized drag force acting on the particle are obtained with good convergence for various values of the relevant parameters and presented both in graphical and tabular. Our results of the normalized drag force are compared with the available data in the literature for the limiting cases.

Efficient Computation of Heat Distribution of Processed Materials under Laser Irradiation

In this paper, a solution is provided to solve the heat conduction equation in the three-dimensional cylinder region, where the laser intensity of the material irradiation surface is expressed as a Gaussian distribution. Based on the symmetry of heat distribution, firstly, the form of the heat equation in the common rectangular coordinate system is changed to another form in the two-dimensional cylindrical coordinate system. Secondly, the ADI with the backward Euler method and with Crank–Nicolson method are established to discretize the model in the cylindrical coordinate system, after which the simulation results are obtained, where the first kind of boundary value condition is used to verify the accuracy of these two algorithms. Then, the above two methods are used to solve the model with the third kind of boundary value condition. Finally, the comparison is performed with the results obtained by the MATLAB’s PDETOOL, which shows that the solution is more feasible and efficient.