Arbitrary Cross-sectional Shape Research Articles

Semi-analytical calculation of the force between two magnets with rhombus-shaped cross-sections

Analytical calculations of interaction forces between permanent magnets are essential for the fast and accurate modeling of magnetic springs and bearings. Whereas extensive research has been conducted on interaction forces between cuboidal magnets, there is a lack of research on force calculations between other prismatic magnets in existing literature. This paper newly develops a semi-analytical method for force calculation between two magnets with rhombus-shaped cross-sections. The method is based on interaction force between two rectangular magnetically charged surfaces which are inclined toward each other. Under a Cartesian coordinate system, two components of the force between inclined rectangular magnetically charged surfaces are presented with fully analytical expressions, whereas the other component is expressed with one numerical integral. By applying these expressions repeatedly, the force between two magnets with rhombus-shaped cross-sections is derived. Moreover, a test rig is constructed to validate this semi-analytical method. Furthermore, the method can be extended to force calculations between prismatic magnets with arbitrary-shaped cross-sections, which allows a simple and fast modeling of many magnetic applications.

Momentum transfer across an open-channel, turbulent flow

The distribution of stress generated by a turbulent flow matters for many natural phenomena, of which rivers are a prime example. Here, we use dimensional analysis to derive a linear, second-order ordinary differential equation for the distribution of stress across a straight, open channel, with an arbitrary cross-sectional shape. We show that this equation is a generic first-order correction to the shallow-water theory in a channel of large aspect ratio. It has two adjustable parameters – the dimensionless diffusion parameter, $\chi$ , and a local-shape parameter, $\alpha$ . By assuming that the momentum is carried across the stream primarily by eddies and recirculation cells with a size comparable to the flow depth, we estimate $\chi$ to be of the order of the inverse square root of the friction coefficient, $\chi \sim C_f^{-1/2}$ , and predict that $\alpha$ vanishes when the flow is highly turbulent. We examine the properties of this equation in detail and confirm its applicability by comparing it with flume experiments and field measurements from the literature. This theory can be a basis for finding the equilibrium shape of turbulent rivers that carry sediment.

Phenomenological comparison between the flexural performance of steel- and CFRP-reinforced concrete elements

In recent decades, FRP reinforcement has received increasing attention in engineering research of reinforced concrete composites due to its advantages over steel reinforcement, such as corrosion resistance and high strength. Due to brittle failure and high strength-to-stiffness ratio of the FRP reinforcement, the flexural behavior of FRP-reinforced beams exhibits a qualitative and quantitative difference compared to conventional steel-reinforced elements. In particular, understanding and improving material utilization is crucial to reduce material consumption and maximize design efficiency in view of the increasing environmental concerns. In this paper, a phenomenological comparison of the flexural behavior and material utilization for various design configurations for steel- and CFRP-reinforced elements is presented. The investigations include different concrete grades, FRP reinforcement properties, cross-section shapes, and hybrid steel–FRP reinforcement cross-sections. In order to deliver a comprehensive perspective to design efficiency in practical context, serviceability deflection limit has been derived for FRP reinforced elements and integrated into the evaluation scheme. Additionally, a general modeling framework for the investigation of the moment–curvature and load–deflection response of steel- and FRP-reinforced beams has been developed supporting the presented studies. The modeling approach is based on a numerical evaluation of the cross-section equilibrium and it supports different material laws as well as arbitrary cross-section shapes and reinforcement layouts. The obtained results show fundamental differences between the bending capacity and material utilization of steel- and FRP-reinforced elements, which can stimulate the development of resource efficient designs. A deeper understanding of the phenomenology based on the present studies can streamline the current research on innovative design concepts for FRP-reinforced concrete elements with optimal utilization of these high-performance materials.

Fully Developed Laminar Flow of Heat Transfer Non-Newtonian Flow in Ducts with Arbitrary Cross-Sectional Shape

The pressure drop and heat transfer properties of a laminar flow with circular, square, and triangular cross sections and a constant heat flux boundary condition are given by a numerical solution. A 3D Navier-Stokes incompressible flow and energy equation serves as the foundation for the numerical model. Results are tabulated and given visually for the velocity field, Poiseuille number, and Nusselt number product for various power-law indices (0.5 n 1.0). One of the analyzed ducts causes a greater heat transfer intensification along with a significant pressure loss. A different type of duct allows for a better balance between pressure loss and heat transfer. In order to verify the accuracy of the numerical codes created in the current study, critical comparisons with earlier findings in the literature are also carried out.

DYNAMIC MODEL OF THERMAL PROCESSES IN A LIQUID FLOWING THROUGH A CHANNEL

The article presents a dynamic thermal process model for an incompressible fluid moving through a channel of arbitrary cross-sectional shape. The study focuses on expressing the non-steady law of conservation of internal energy of the fluid by considering the dependence of parameters on both spatial coordinates and time. The one-dimensional spatial formulation of the problem is simplified by disregarding any changes in flow parameters in the direction perpendicular to the channel axis. The resulting equation represents a non-stationary, non-homogeneous, quasi-linear hyperbolic partial differential equation, which can be solved with the method of characteristics in the case of constant coefficients. The solution consists of two distinct solutions that are glued along the line x/t=v, with isolines shown in the figure provided. The research is relevant for the design of heat exchangers, rocket and aircraft engines, and other heat engineering devices.

Geometrically Exact Beam-based Aeroelastic Modeling and Solution Of Composite Rotor Blades in Forward Flight

In this paper, the geometrically exact beam model and aeroelastic solution methods for composite rotor blades in forward flight by the latest variational asymptotic beam sectional analysis (VABS) have been employed. The geometrically exact beam equations of motion in the mixed variational form and the latest VABS are used to deal with one-dimensional blade analysis and the structural property of blade cross section, respectively. The methods can be used for the aeroelastic solution of composite rotor blades with arbitrary cross-sectional shape and material distribution, large deflections and significant nonclassical effects such as cross-sectional warping, transverse shear deformation, and elastic couplings caused by anisotropic material properties. The Peters–He finite state dynamic inflow model and the Peters finite state airloads theory are used to calculate the induced velocity and blade airloads, respectively. An auto-pilot trim scheme is used for calculating the blade pitch controls to meet the trim requirements. The convergence issue encountered when solving the geometrically exact, mixed variational aeroelastic equations in time domain has been successfully addressed. The values of the empirical parameters in the auto-pilot trim scheme for the presented aeroelastic model have been properly selected. The accuracy of the presented aeroelastic modeling and solution methods has been verified against the SA349/2 flight-test data. The influence of transverse shear deformation on the aeroelastic response of composite rotor blades was also investigated, indicating that this effect has a nonnegligible influence on the aeroelastic response of the five different kinds of elastically coupled hingeless composite rotors investigated in this paper.

Torsional vibration of Timoshenko-Gere non-circular nano-bars

The displacement field and governing equation completely depend on the cross-section shape in the torsion of nano-bars. Due to the torsion of structure and warping of its cross-section, axial strain is created. A review of the literature demonstrates that the effect of the normal strain in the torsional analysis of nanostructures with non-circular cross-sections is ignored for the sake of simplicity and causes a computational error. While in the torsion of short nano-bars due to the great warping of the cross-section the normal strain appears and it shouldn’t be ignored. Therefore, for the first time, the effect of normal strain based on the torsion of Timoshenko-Gere theory is considered in this research. In this theory, the twist rate of non-circular sections in the axial direction is not considered constant, unlike studies in the literature. The governing equation of torsional vibration is extracted using the Hamilton principle, and the nonlocal strain gradient theory is used to show the size-dependent effects. The fundamental frequency is calculated for the arbitrary cross-section shape by using Galerkin’s method. The effect of parameters such as size effects, the thickness of nano-bar, mode number, and dimensions changes of cross-section versus natural frequency of nano-bar for both the Timoshenko-Gere and typical theories are investigated. Results indicate that neglecting normal strain due to warping the cross-section causes a significant error in the short nano-bars, especially at higher mode numbers. Also, a comparison is made between the obtained natural frequencies and those of the results reported in the literature.

Analysis of a column with variable, arbitrary shaped cross section, including initial imperfections, geometric and material nonlinearity

In this paper, we propose a general method for solving structural mechanics problems by developing a “functional model”. It is implemented by defining the required geometrical and physical relationships in their general form via analytical and numerical functions. For that purpose, a special programming language Calcpad is used. The obtained data structure is processed by a core engine, which generates the computer code, required for the solution. This model can be used for calculation of different partial cases, after specifying the specific input functions and parameters. In this way, the need for developing a different procedure for each separate case is eliminated. The proposed method is applied for the analysis of a column with variable, arbitrary shaped cross section, accounting for initial imperfections, geometric and material nonlinearity.

Одномерная математическая модель колебаний ствола с поперечным сечением произвольной формы

The problem of longitudinal and transverse vibrations of a barrel with arbitrary cross-sectional shapes is considered and solved in the framework of a one-dimensional model. The study shows that the amplitude of transverse vibrations in the vertical plane significantly exceeds that in the horizontal plane. This paper proposes to reduce the amplitude of vibrations by changing the shape of the barrel cross-section, namely by adding stiffeners. The numerical algorithm for solving the problem is developed on the basis of the integro-interpolation method. The verification of the numerical integration method is carried out, and the grid convergence is verified by means of the modeling of barrel vibrations for a 30 mm automatic cannon. The study of the impact of the barrel cross-section shape shows that the use of stiffeners can reduce the initial deflection and the amplitude of muzzle vibrations when firing in bursts. The obtained results demonstrate a narrow spread of projectile departure angles, and, consequently, the improved shooting accuracy of the automatic cannon.

Deformation approach to the calculation of compression resistance of reinforced stone elements

Introduction. When calculating stone and reinforced stone elements, only the limiting stage before destruction is considered, the nonlinearity of deformation of materials and, accordingly, the redistribution of forces between masonry and longitudinal reinforcement in cross sections are not taken into account. A deformation approach to the calculation of compression resistance of reinforced stone elements is considered. It allows us to take into account the physical nonlinearity of deformation of materials of reinforced stone elements.
 
 Materials and methods. Accounting for physical nonlinearity is based on using deformation diagrams under “compression – tension”. Approximations of diagrams are given. A sample of experimental studies of the most characteristic armoured stone elements has been compiled according to the data of studies published in open sources.
 
 Results. The prerequisites for using a deformation approach to the calculation of reinforced stone elements are formulated. As a criterion for calculating the compression resistance at the ultimate strength stage, it is proposed to take the maximum force calculated from the condition of compatibility of deformations of masonry and longitudinal steel reinforcement. Algorithms for calculating the compression resistance and parameters of the stress-strain state of reinforced stone elements at any level of loading are compiled.
 
 Conclusions. The proposed calculation method makes it possible to obtain the parameters of the stress-strain state of elements of arbitrary cross-section shape with any reinforcement parameters and various eccentricities of longitudinal force application at all loading levels. A calculation criterion is proposed to take into account the redistribution of forces in the cross-section of armoured stone elements. Verification of the calculation method was performed on a sample of experimental data, which showed good convergence of theoretical and experimental values of load-bearing capacity and deformability.

A novel model for calculating the impedance of arbitrarily-shaped coils with perpendicular to the planar medium

Some studies have shown that coils placed perpendicular to conductors have excellent detection effect for specific defects in eddy-current testing (ECT), where the impedance is an important parameter. The impedance calculation for perpendicular coil is more complex than that for parallel coil, and currently the main focus is on the impedance analysis of circular and rectangular coils. In this paper, a closed formula for the impedance of a coil with arbitrary-shape perpendicular to the conductor is given based on the second order vector position method. In the proposed model, a novel shape and position function is defined to extend the coil application to arbitrary cross-sectional shapes. Meanwhile, experimental comparisons are conducted to evaluate the impedance of various coils on an aluminum plate. The results illustrated that the experimental values are in perfect agreement with the calculated ones. Therefore, the proposed model can provide guidance for the impedance calculation of arbitrarily-shaped coils in eddy current testing.

Estimating Radiation Scattering Around Plasmonic Nanowires Using Engineered Geometric Features

Abstract This study offers an analytical estimation model for radiative scattering at nanoscale. The study focuses on isolated nanowires of arbitrary shape cross sections and uses predictive geometric features and statistical regression to model the wavelength-dependent light-particle interaction. This work proposes to estimate the radiative properties of nanowires based on engineered geometric features, potentially leading to new understandings of how the geometric attributes impact light scattering at nanoscale. A predictive model is designed and tested for estimating radiative scattering around nanowires. Random polygon-shaped cross sections with high degrees-of-freedom are chosen to train and test the models. The derived model can successfully explain scattering across out-sample synthetic plasmonic objects with a 90% R-squared metric.

Finite-element dynamic-matrix approach for propagating spin waves: Extension to mono- and multi-layers of arbitrary spacing and thickness

In our recent work [Körber et al., AIP Adv. 11, 095006 (2021)], we presented an efficient numerical method to compute dispersions and mode profiles of spin waves in waveguides with translationally invariant equilibrium magnetization. A finite-element method (FEM) allowed to model two-dimensional waveguide cross sections of arbitrary shape but only finite size. Here, we extend our FEM propagating-wave dynamic-matrix approach from finite waveguides to the important cases of infinitely extended mono- and multi-layers of arbitrary spacing and thickness. To obtain the mode profiles and frequencies, the linearized equation of the motion of magnetization is solved as an eigenvalue problem on a one-dimensional line-trace mesh, defined along the normal direction of the layers. Being an important contribution to multi-layer systems, we introduce interlayer exchange into our FEM approach. With the calculation of dipolar fields being the main focus, we also extend the previously presented plane-wave Fredkin–Koehler method to calculate the dipolar potential of spin waves in infinite layers. The major benefit of this method is that it avoids the discretization of any non-magnetic material such as non-magnetic spacers in multilayers. Therefore, the computational effort becomes independent of the spacer thicknesses. Furthermore, it keeps the resulting eigenvalue problem sparse, which, therefore, inherits a comparably low arithmetic complexity. As a validation of our method (implemented into the open-source finite-element micromagnetic package TETRAX), we present results for various systems and compare them with theoretical predictions and with established finite-difference methods. We believe this method offers an efficient and versatile tool to calculate spin-wave dispersions in layered magnetic systems.

Design of an Intermediate Die for the Multi-Pass Shape Drawing Process.

The multi-pass shape drawing process is mainly used in metal forming processes to manufacture long components with constant arbitrary cross-sectional shapes along their lengths. The cross-roller guide is a typical component that is manufactured by a multi-pass shape drawing process. The cross-roller guide is mostly used in optical measurement equipment where high-precision movement is required. Therefore, the dimensional accuracy of the cross-roller guide is very important since it can influence precision linear motion. However, the unfilled defects can occur in a case where the product has a complex cross-sectional shape. In this study, a new design method for an intermediate die is suggested by using an equal-radial-velocity variation method in order to reduce the unfilled defects. The proposed design method can reduce the unfilled defects by minimizing the radial velocity variation in the deformation zone of the drawing die. The intermediate die was designed by geometrical information of the final product without prior finite element (FE) analysis. The suggested method was applied to design the multi-pass shape drawing process for manufacturing the cross-roller guide. FE analysis was performed to validate the effectiveness of the proposed method in comparison to the conventional design method that uses equipotential lines in the multi-pass shape drawing process. Finally, a shape drawing experiment was performed to compare the target shape and the FE analysis with the experimental data.

Unsteady convective–diffusive transport in semicircular microchannels with irreversible wall reaction

The unsteady dispersion of a solute band by a steady pressure-driven flow in a semicircular microchannel is theoretically studied via the generalized dispersion model. Considering an irreversible first-order reaction at the curved wall while assuming a no-flux boundary condition at the flat wall, analytical solutions are obtained for the exchange, convection and dispersion coefficients as well as the dimensionless forms of solute concentration and mean solute concentration. The solutions are obtained assuming an initial solute band of arbitrary cross-sectional shape and axial distribution and the results are presented for both circular and semicircular shapes with the uniform distribution being a special case of the latter. Besides the general solutions, special solutions are also derived for uniform velocity and no-reaction cases. In the following, the influences of the initial concentration distribution and the Damköhler number, a measure of the reaction rate, on the transport coefficients and the concentration distribution are investigated in depth. It is demonstrated that the combination of the initial concentration distribution and the Damköhler number specifies the variations of the transport coefficients in the short term but the Damköhler number is the only parameter dominating the long-term values: the exchange and convection coefficients are increasing functions of the Damköhler number whereas the opposite is true for the dispersion coefficient. Moreover, we show that, provided the solute injection is appropriately positioned and shaped, liquid-phase transportation with little dispersion is possible in typical microchannels utilizing semicircular geometry.

Predicting the Thermal Behavior in Functional Textile Fibers Having Embedded Electronics

In this paper, both steady-state and transient thermal simulations were performed on functional fibers having an embedded electronic chip acting as a heat source. Simulations were conducted for a range of different fiber materials and arbitrary fiber cross-sectional shapes. We show that under steady-state heating conditions, the thermal response for any arbitrary fiber shape and fiber material system was convection dominated regardless of the effective thermal conductivity of the fiber, and that the corresponding temperature rise within the fiber can be predicted analytically allowing for the maximum temperature to be estimated for any known heat load and fiber geometry. In the case of transient heating, we show that for pulsed power operation of the embedded electronic device, the maximum temperature reached in the fiber is always greater than the maximum temperature of the equivalent steady-state average power. However, high peak powers can be safely achieved if the power-on pulse time and duty cycle are selected to limit the maximum temperature reached in the fiber. Based on the results from the transient simulations, a set of criteria was developed to determine whether the operating conditions would be: (1) allowable for the fiber system, thus requiring no transient simulations, (2) requiring a transient simulation to verify that the maximum temperature is acceptable, and (3) the operating conditions are too severe and device operation at these conditions are not practical.Graphical

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, offer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to interactions between stiffness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efficient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress field recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of effects where stiffness and external load distributions differ. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to 15^{circ }. The model is validated by comparison with finite element analysis and relevant analytical formulations.

A new distortion evaluation method based on geometric association for arbitrary cross-sectional shape of inlet

The distortion flowfields generated by convoluted inlets are major factors that compromise the stability and performance of the downstream engine. For better understanding the distortion evolution, the total pressure distortion from the arbitrary cross-sectional shape of inlets was efficiently calculated based on geometric association. First, the Poisson equation was solved to generate a grid on the original, arbitrarily shaped, non-circular plane; then, a previously simulated dataset was used as a sample to interpolate the grid, whose nodes were mapped into a circular plane according to the principle of equal areas of rings and blocks. Finally, the traditional distortion descriptors were used to evaluate the distortion of the newly generated plane. The proposed method guarantees that both the total pressure recovery coefficient of the transformed crossflow plane and the contribution of this coefficient from each grid unit remain constant. This method was also applied for the flowfield analysis of serpentine and submerged inlets; the various forms of distortion along the ducts were studied. The total pressure distribution coefficient DC(60) distributions along the internal flow direction differed completely under different exit Mach numbers and their drastic changes were located at the severe bends or expansions of the inlet profile. Such innovation in distortion evaluation can provide insights into flow distortion in various inlets and enable better integration between inlets and engines.

Analytical Solution for Seismic Response of Deep Tunnels with Arbitrary Cross-Section Shape in Saturated Orthotropic Rock

Analytical Solution for Seismic Response of Deep Tunnels with Arbitrary Cross-Section Shape in Saturated Orthotropic Rock

Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

We show that the one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the model. This result extends upon earlier work on the shallow water equations (SWE), with the important difference that we include non-conservative pressure terms in the analysis, and that we propose a formulation that holds for ducts with an arbitrary cross-sectional shape, with the 2D channel and circular pipe geometries as special cases.The second novel result of this work is the formulation of a finite volume scheme for the TFM that satisfies a discrete form of the continuous energy equation. This discretization is derived in a manner that runs parallel to the continuous analysis. Due to the non-conservative pressure terms it is essential to employ a staggered grid, which requires careful consideration in defining the discrete energy and energy fluxes, and the relations between them and the discrete model. Numerical simulations confirm that the discrete energy is conserved.