Fourier Integral Transformation Research Articles

Sliding frictional and adhesive coupling contact analysis of FGPM layered half-space under an insulating indenter

This paper proposes an analysis model for the sliding frictional and adhesive coupling contact problem in the plane strain state between the FGPM layered half-space and an insulating indenter. The electro-mechanical properties of FGPM layer vary exponentially along the thickness direction. By applying the Fourier integral transformation and superposition principle to the governing boundary value problem, the general solution for the sliding frictional and adhesive coupling contact problem can be derived by using the Maugis type of adhesion theory and extended Amonton’s law of friction. A Cauchy singular integral equation is further derived for present problem which is then numerically solved. The primary aim of this paper is to provide insight into the likely behavior for the effect of the gradient index, friction coefficient and adhesion parameters on the surface electro-mechanical response of FGPM layered half-space. For given values of parameters R, w, βh and μ, when value of σ0 decreases from infinity to zero there is a continuous transition from the JKR approximation to the DMT approximation. The research not only helps to further understand the frictional and adhesive damage mechanisms of MEMS devices composed of FGPM, but also provides reference basis for FGPM layer experimental analysis and intelligent structure design.

Analytical modeling and simulations of dynamic mode-III fracture in pre-stressed dry sandy elastic continuum

A mathematical model has been developed to investigate the SIF (stress intensity factor) and COD (crack opening displacement) for the propagation of mode-III fracture influenced by propagating SH-wave in a pre-stressed dry sandy elastic continuum. The Wiener-Hopf mathematical method along with the Fourier integral transform (FIT) technique have been successively implemented to obtain the solution of the boundary value problem associated with the mathematical model. Furthermore, the closed form solutions of SIF and COD for non-static and static conditions of mode-III fracture have been obtained, and are found to be in well agreement with the existing results present in the literature. The sound effects of various affecting parameters viz. horizontal and vertical (compressive/tensile) pre-stresses, crack depth, and crack velocity on SIF and COD have been explored. In order to delineate the influences of these aforementioned parameters on SIF and COD graphically, numerical simulation has been accomplished.

Harmonic Analysis of Quasi-2-Level Medium Voltage Super-Switch

This paper focuses on the harmonic analysis of the Quasi-2-Level Super-Switch (Q2L SS), considering different delay times and the number of steps. The harmonic performance is investigated in-depth using Double Fourier Integral Transformation analysis, simulations, and experiments. According to the analysis, increasing the number of steps facilitates a smoother transition between the negative and positive states of the output voltage. As a result, the harmonic performance of the Q2L SS is significantly improved. On the other hand, an increased number of steps sets the boundaries to the switching frequency and the maximum modulation index. Moreover, multi-step Q2L SS requires more capacitors, which can significantly increase the volume of the converter. The study establishes the ground basis for understanding the Q2L SS operation in the harmonic domain. It can be used for the future research in the domain of Q2L SS and approaches for designing EMI filters for Q2L SS-based converters.

Analytical determination of the dynamic response of layered saturated frozen foundation to moving loads under plane strain conditions

Based on the theory of solid porous media with pores, a calculation model of two-dimensional layered saturated frozen soil foundation is established. The study focuses on the dynamic response of layered saturated frozen soil medium placed on an impermeable rigid bedrock when subjected to a uniform surface moving strip load in motion. Firstly, the governing equation of saturated frozen soil is established. Then, the Helmholtz vector decomposition theorem, coordinate transformation, and Fourier integral transformation are used to decouple the governing equation of saturated frozen soil, and the general solution of the potential function is obtained, and the general solution of the stress and displacement of each layer of saturated frozen soil is derived. Using the transfer and reflection matrix (TRM) method, the semi-analytical solution of layered saturated frozen soil medium in the frequency domain is derived. Finally, the dynamic response of each phase of the saturated frozen soil in the time domain is obtained by the fast Fourier transform. After verifying the accuracy of the solution by comparing the model degradation with the existing literature, the effects of soil shear modulus, load moving speed, temperature, porosity, contact parameter, and frequency on the dynamic response of homogeneous foundation, soft interlayer foundation, and hard interlayer foundation are analyzed in detail. The results show that the stratification of the foundation is of great significance for accurately evaluating the dynamic response of saturated frozen soil.

Anisotropy and magnetoelasticity effects on the propagation of the SH-wave-induced semi-infinite crack in a magnetoelastic orthotropic medium

The crux of the present study is to investigate semi-infinite crack propagation in magnetoelastic strip influenced by propagating SH-wave under orthotropic geometrical configurations. The Wiener-Hopf (WH) technique and two-sided Fourier integral transforms (FIT) are applied in the proposed analytical model. A closed-form expression of the stress intensity factor (SIF) has been established for the constant concentrated force. The SIF in a magnetoelastic orthotropic strip is shown to be significantly affected by the crack length and crack speed, medium anisotropy, and magnetoelastic parameter. The proposed model is compared to the case of a magnetoelastic isotropic strip in order to demonstrate the effect of orthotropy as well as other anomalies. Among other observations, it is noted that regardless of whether orthotropy exists in the strip or not, the value of the SIF rises along with the crack speed until it reaches its maximum, at which point it rapidly declines.

Analytical solutions for the plane thermoelastic problem of a nano-open crack in one-dimensional hexagonal quasicrystal non-periodic plane

As a stable material at high temperatures, the presence of cracks and other defects greatly reduces the performance of quasicrystals (QCs). The plane thermoelastic problem of one-dimensional hexagonal quasicrystals (1DHQ) non-periodic plane containing a nano-open crack is analyzed using surface elasticity theory. The closed form solutions are obtained to describe the thermoelastic field corresponding to the nano-crack by the Fourier integral transformation technique. The analytical solutions of the thermal stress intensity factors (TSIFs) and the strain energy density factor (SEDF) at the crack tip are derived explicitly. The influence of the applied loads, the surface effects and the multi-field coupling effects on the TSIFs and SEDF are discussed. The numerical results reveal that the surface effects have an obvious impact on TSIFs with small applied loads, and the surface effects will inhibit the propagation of nano-open cracks. The research results of this study provide some reliable theoretical references for the development of nano-QCs.

Analytical study on the propagation of semi-infinite crack due to SH-wave in pre-stressed magnetoelastic orthotropic strip

This study deals with the propagation of semi-infinite crack impacted by SH-wave propagation in a pre-stressed magnetoelastic orthotropic strip (PMOS). The proposed analytical model makes use of the Wiener-Hopf (WH) approach and two-sided Fourier integral transforms (FIT). The exact formulation of the stress intensity factor (SIF) for constant concentrated force (CCF) has been established in closed form. The effect of influencing factors, including the crack length, crack speed, magnetoelastic parameter, horizontal compressive pre-stress (HCP)/horizontal tensile pre-stress (HTP), vertical compressive pre-stress (VCP)/vertical tensile pre-stress (VTP), and anisotropy parameter on the SIF in the considered strip has been unraveled. The proposed model is compared with the case of a pre-stressed magnetoelastic isotropic strip (PMIS) and some peculiarities, along with the influence of orthotropy have been reported. The significant results reported in the present manuscript reveal that magnetoelasticity and pre-stress is a prominent factor impacting crack propagation in elastic material with orthotropy configuration properties and has a significant impact on the SIF, which has not been taken into account in the existing literature on the subject of fracture mechanics. The present investigation helps with structural and reliability analysis, fracture propagation, material durability and failure of engineering structures.

Parameter identification method of nonuniform and under-sampled blade tip timing based on extended DFT and compressed sensing

Vibration amplitude and frequency are the two most important indicators that characterize the health status of high-speed rotating blades, but the signal obtained by blade tip timing (BTT) technology, one of the best rotating blade vibration monitoring methods, is seriously nonuniform and under-sampled, which makes these two indicators difficult to identify. In view of this problem, the paper proposes a parameter identification method for the nonuniform and under-sampled BTT signal based on extended Discrete Fourier transform and compressed sensing (CS), with the Fourier integral transformation as the goal. It realizes the frequency analysis of nonuniform under-sampled signals by constructing and optimizing the transformation basis function instead of the exponential basis in the traditional FFT transformation in the extended frequency range, and then constructs a CS model through the obtained blade vibration frequency. The complete waveform of the blade vibration is restored by using a small number of under-sampled signals, thus obtaining the blade vibration amplitude and vibration frequency. On the one hand, the method proposed in this paper breaks through the limitation of Nyquist’s sampling theorem, and the number of analytical spectral lines is no longer limited to the number of sampling points, which improves the frequency resolution. On the other hand, only a small number of measurement signals can be reconstructed to achieve a complete vibration signal. The feasibility and reliability of the proposed method are verified by mathematical modeling, simulation analysis, and experimental testing. The results indicate that when the number of sensors is greater than or equal to four, the time domain and frequency domain signals of blade vibration can be accurately analyzed based on the proposed method, the vibration amplitude error is less than 0.01 mm, the frequency error is less than 0.1 Hz, and it has good anti-interference performance.

Interaction among interfacial offset cracks in composite materials under the anti‐plane shear loading

AbstractThe present article considers an anti‐plane stress problem of three cracks at different orthotropic materials' interfaces. According to the geometry of the problem, the governing equations and mixed boundary conditions have been formulated. Fourier integral transformation is used to convert the mixed boundary value problem into dual integral equations, which gives two equations containing infinite series. The investigation of the problem concerning anti‐plane cracks subjected to static loadings is done with the help of the Schmidt method to satisfy the given boundary conditions. The difference in displacements is expanded to proceed further in the problem, which becomes zero outside the cracks. Numerical computations are carried out for the graphical representation of stress intensity factors (SIFs) at all cracks' tips. The Interaction among the cracks as those are in close proximity to each other or move away are represented pictorially. Detailed numerical results and discussion are done for the considered materials, which include aluminium, epoxy and graphite epoxy. The novelty of the present article is the numerical analysis and pictorial presentation of SIFs at the tips of interfacial offset parallel cracks for various crack lengths and normalised heights for different combinations of materials. The authors have obtained variations in SIFs for the cracks at the interfaces of dissimilar composite materials.

A method for optimizing the precision of weighted fourier integral transformation algorithm

RCS refers to the radar cross section, which is an important index to reflect the electromagnetic scattering characteristics of the object. RCS measurement is the only method to obtain the real RCS of objects. Since test range often cannot reach the far-field condition for large size object, the near-field-to-far-field transformation(NFFFT) technology is significant in the RCS measurement. The core of NFFFT are different algorithms. Among them, the weighted fourier integral transformation is a universal algorithm, but the data in the aspect angle is segmented in the process, which leads to the discontinuity of the transformation results at the splicing point. This paper introduces a method to optimize the precision of weighted fourier integral transformation algorithm, which can fully suppress the discontinuity of transform results and improve the precision of weighted fourier integral transformation algorithm under the paraxial constraint conditions. Finally, the result of simulation demonstrates the precision and efficiency of this method.

Couple Stress Sodium Alginate-Based Casson Nanofluid Analysis through Fick’s and Fourier’s Laws with Inclined Microchannel

Casson nanofluid plays a vital role in food industries with sodium alginate nanoparticles. That is why many researchers used Casson nanofluid in their study. Due to this, the main objective of this study is to investigate the inclined microchannel flow of a Casson nanofluid based on sodium alginate (SA) under a few stresses. Because the plate at y = d is stationary and the plate at y = 0 is in motion, the fluid flows. Physically existent things utilize partial differential equations as a method of derivation. By using dimensionless variables, the underlying PDEs are dimensionless. Applying Fourier’s and Fick’s laws to the time-fractional model makes the classical model dimensionally stable by generalization. A generalized fractional model is solved using the Laplace and Fourier integral transformations. In addition, the parametric influence of other physical elements, such as the Casson parameter, coupling velocity, temperature, and stress parameters, is considered (Grashof, Schmidt, and Prandtl numbers). Concentration distributions are shown using graphs and discussed with accompanying text. We compute and describe the Sherwood number, rate of heat transfer, and skin friction. It is concluded that skin friction and Nusselt number can be enhanced by adding nanoparticle. Also, the fractional derivative makes the study more realistic by incorporating Fick’s and Fourier’s laws as compared to the classical one.

Transient Deformation of Anisotropic Timoshenko’s Plate

In this paper, we will present an approach to constructing of dynamical spatial Green’s function (elementary solutions, dominant function) for a thin infinite elastic plate of constant thickness. The plate material is anisotropic with a single plane of symmetry, geometrically coinciding with plate’s middle plane. The Timoshenko theory was used for describing the plate movement. Transient spatial Green’s functions for normal displacements and angles of orthogonal alteration to middle surface before deformation of material fiber are built in the Cartesian coordinate system. To construct Green’s function, direct and inverse Laplace and Fourier integral transformations are applied. The originals of Laplace Green’s functions were analytically found with the theorem of residues. To construct Fourier originals, a specific method was used based on Fourier series transformation inversion integral connection with Fourier series on a variable interval. Green’s function found for normal displacement made it possible to represent the normal transient function as three-fold convolution of Green function with distant load function. The functions of normal distant displacements were constructed in case of the impact of transient total loads concentrated and distributed across rectangular courts. The numerical method of rectangles was used to calculate the convolution integrals. The influence of the concentrated load speed on transient normal displacements of the anisotropic plate was analyzed. As a verification of constructed transient spatial Green’s functions, the results of numerical solutions were compared with the results found using known transient Green’s functions for isotropic thin elastic rectangular simply supported Timoshenko’s plate which solutions are constructed using Laplace integral transformation in time and its decomposition into Fourier series on coordinates. Besides, its confidence was proved analyzing the nature of waves in anisotropic, orthotropic and isotropic plate, found in the process of numerical calculations. The results are represented as diagrams. Examples of calculations are given.

Фрикційний розігрів системи штамп-пружна півплощина при ковзанні вздовж твірної

Friction heating of system punch-elastic half plane when sliding along creative line is considered. Model of so-called “third body”, i.e., thin near-surface and intermediate layers, the physical and mechanical properties of which differ from those of the interacting bodies, and by the microgeometry of their surfaces in the contact zone, used for mathematical description of contact. The method of determination of thermal contact conductance in mathematical modelling of contact interaction with considering friction and hear generation by “third body” is presented. Using of modified conditions of heat contact in mathematical model of contact thermoelasticity, taking into account of friction and heat generation is proposed. The solution of the problem of thermoelasticity for a half-plane is obtained by means of the Fourier integral transformation. Heat conductivity problem for the punch is solved by method of straight lines. The system obtained of dual integral equations is reduced to the system of linear algebraic equations by means of points collocation method. Formulas for thermal fields, heat fluxes and contact stresses are proposed. In order to obtain the unknown contact area, the iterative scheme based on a control of a sign of normal stresses in the immediate contact interaction zones is used. Method of moving line of separation of boundary conditions is proposed.

The dynamical problem on acting distributed load on the elastic layer

The wave field of an elastic half-layer is constructed, when a dynamic normal load distributed over a rectangular area acts on upper face at the initial moment of time. The lower face of the half-layer is rigidly fixed to the foundation, and the side border is in the conditions of a smooth contact. The method of decomposing the system of motion equations into a system of equations and an independently solvable equation is used, this approach was proposed by Popov~G.~Ya. Laplace and Fourier integral transformations are applied directly to the motion equations and boundary conditions, which reduces the problem to a vector one-dimensional boundary value problem, which is solved by the matrix differential calculus method. The output displacements are obtained using inverse integral transformations. The case of steady oscillations was considered and the amplitude of vertical displacement occurring in the layer was analyzed depending on the shape of the distributed load section, the material of the layer medium and the values of the natural frequency of the layer oscillations.

Scattering of SH-wave by an impedance strip on the rigid wall of the plane elastic waveguide

The problem of SH-wave scattering from an impedance strip on the rigid wall of an elastic waveguide is considered. The opposite waveguide surface is free from tensions. This structure is illuminated by one of the normal SH-waves that propagate along the waveguide without attenuation. The displacement of the particles in this wave is perpendicular to the direction of wave propagation and has the harmonic dependence on time. The problem is two-dimensional and is reduced to the mixed boundary value problem for Helmholtz equation with respect to the unknown diffracted displacement field. The separation of variables and the Fourier integral trans-formation techniques are applied for the solution. Using these techniques, the problem is reduced to the functional equation with respect to the Fourier transforms of the unknown tensions and displacement at the particular intervals of integration. It is shown that this equation is valid in the strip of the complex plane which encompasses the real axis. It is proved that this equation is the equation of the Wiener–Hopf type. Using the factorization and decomposition techniques this equation is reduced to solving an infinite system of linear algebraic equations of the second kind. Its solution is applied to elucidate the behavior of the displacement field on the tensions-free surface of the elastic layer.

Efficient estimation of propagator anisotropy and non-Gaussianity in multishell diffusion MRI with micro-structure adaptive convolution kernels and dual Fourier integral transforms.

We seek to reformulate the so-called Propagator Anisotropy (PA) and Non-Gaussianity (NG), originally conceived for the Mean Apparent Propagator diffusion MRI (MAP-MRI), to the Micro-Structure adaptive convolution kernels and dual Fourier Integral Transforms (MiSFIT). These measures describe relevant normalized features of the Ensemble Average Propagator (EAP). First, the indices, which are defined as the EAP's dissimilarity from an isotropic (PA) or a Gaussian (NG) one, are analytically reformulated within the MiSFIT framework. Then a comparison between the resulting maps is drawn by means of a visual analysis, a quantitative assessment via numerical simulations, a test-retest study across the MICRA dataset (6 subjects scanned five times) and, finally, a computational time evaluation. Findings illustrate the visual similarity between the indices computed with either technique. Evaluation against synthetic ground truth data, however, demonstrates MiSFIT's improved accuracy. In addition, the test-retest study reveals MiSFIT's higher degree of reliability in most of white matter regions. Finally, the computational time evaluation shows MiSFIT's time reduction up to two orders of magnitude. Despite being a direct development on the MAP-MRI representation, the PA and the NG can be reliably and efficiently computed within MiSFIT's framework. This, together with the previous findings in the original MiSFIT's article, could mean the difference that definitely qualifies diffusion MRI to be incorporated into regular clinical settings.

Time fractional analysis of channel flow of couple stress Casson fluid using Fick\u2019s and Fourier\u2019s Laws

This study aim to examine the channel flow of a couple stress Casson fluid. The flow is generated due to the motion of the plate at y=0, while the plate at y=d is at rest. This physical phenomenon is derived in terms of partial differential equations. The subjected governing PDE’s are non-dimensionalized with the help of dimensionless variables. The dimensionless classical model is generalized by transforming it to the time fractional model using Fick’s and Fourier’s Laws. The general fractional model is solved by applying the Laplace and Fourier integral transformation. Furthermore, the parametric influence of various physical parameters like Casson parameter, couple stress parameter, Grashof number, Schmidt number and Prandtl number on velocity, temperature, and concentration distributions is shown graphically and discussed. The heat transfer rate, skin friction, and Sherwood number are calculated and presented in tabular form. It is worth noting that the increasing values of the couple stress parameter lambda deaccelerate the velocity of Couple stress Casson fluid.

NUMERICAL SIMULATION OF WAVE PROCESSES IN PARTIALLY SATURATED POROELASTIC MEDIA (REVIEW)

The review of a number of scientific works in the field of modeling wave processes in partially saturated porous media is proposed. Review consists of three parts. In the first part of overview, classification of porous media according to the degree of saturation with a wetting filler is stated. The fundamental works of the modern theory of porous continuums, which resulted in the Biot's theory and the mixture theory, are considered. The advantages, disadvantages, as well as the conditions for the correspondence between the two approaches are indicated. The second part is devoted to scientific publications of an applied and academic nature, which are mostly devoted to the study of the features of wave processes in poroelastic media by analytical and semi-analytical methods. Special attention is paid to the works devoted to the modeling of slow longitudinal waves as a distinctive feature of wave processes in partially saturated poroelastic media. The third part gives an overview of the application of boundary element and finite element methods to study the stress-strain state of saturated and partially saturated porous media, the behavior of waves during their propagation in the media under consideration, and the analysis of local dynamic effects. The two methods are compared. Experimental works confirming theoretical results obtained within the Biot's theory are indicated. A brief analysis of modern achievements in the problems of numerical simulation of dynamic effects in semi-infinite porous media with a layered structure such as the seabed is carried out. Works on the derivation of fundamental solutions for systems of poroelasticity equations and the construction of the corresponding integral equations in time and space of the Fourier and Laplace integral transformations are considered. Works are given that consider various aspects of boundary element modeling of unsaturated porous media and the application of boundary and finite element methods on specific examples.

Frictionless contact mechanics of an orthotropic coating/isotropic substrate system

This study has been performed to investigate the receding contact problem of a homogeneous orthotropic coating that is not bonded to a homogeneous isotropic substrate without any interfacial defects. The isotropic substrate is supported on a Winkler foundation. The problem is solved assuming that the contact between the rigid punch and orthotropic coating, and that between the orthotropic coating and isotropic substrate, are frictionless. Additionally, the effect of the body forces is neglected, and only compressive normal tractions can be transmitted through the interfaces. The contact analysis of the orthotropic coating, which is subjected to a contact load using a rigid cylindrical punch, is performed under plane strain conditions. The governing equations are analytically found using the theory of elasticity and Fourier integral transformation techniques. Subsequently, the governing equations are reduced to a system of two singular equations, wherein the unknowns are the contact stresses and contact widths. To numerically solve the resulting singular integral equations, Gauss-Chebyshev integration formulas are employed. It is analyzed the influence of the following parameters on the contact stresses and contact widths: orthotropicmaterial properties, punch radius, load ratio, Winkler foundation stiffness.

Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.