Double Fourier Transformation Research Articles

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

The elastic half-space problem has been solved previously using Boussinesq, Papkovich, Love, and Green and Zerna, potential function methods. In this work, the Trefftz displacement potential function method is used to obtain the stress and displacement fields in an elastic half-space subjected to boundary loads. Point load and various distributed loads are considered. The problem is presented using displacement formulation as Navier–Lame equations. It is proved that the Trefftz functions are solutions of the Navier–Lame displacement equations. Strain fields are derived in terms of the Trefftz function using the strain-displacement relations. The stress fields are similarly derived. The Trefftz function for the case of a point load acting at the origin of the elastic half-space is derived using the double exponential Fourier transformation technique. Stress equilibrium boundary conditions are used to fully determine the Trefftz function. Stress and displacement fields for the point load are then determined. The solutions to stress and displacement fields for point load are then used as Green functions to obtain stress and displacement fields for uniformly distributed load over a finite line, circular area and rectangular area. It is found that the solutions obtained for the stress and displacement fields in the elastic half-space due to point and distributed loads are identical with previously obtained expressions, thus validating this work.

Analytical solution considering the tangential effect for an infinite beam on a viscoelastic Pasternak foundation

In this study, the dynamic response of an infinite beam resting on a Pasternak foundation subjected to inclined travelling loads was developed in the form of the analytical solution wherein the tangential effect between the beam and foundation and the damping were taken into consideration. Three parameters were used to model the mechanical resistance of the viscoelastic Pasternak foundation, one of them accounts for the compressive stress in the soil, the other accounts for the shearing effect of soils, and the last one accounts for the damping of the foundation. By contrast, the Pasternak model is more realistic than the Winkler model that just considers the compressive resistance of soil. In the paper, the tangential effect between the beam and foundation was simulated by a series of separate horizontal springs, the damping was also considered to obtain the dynamic response under forced vibration. The theory of elasticity and Newton's laws were used to derive the governing equation. To simplify the partial-differential equation to an algebraic equation, the double Fourier transformation was used wherein the analytical solution in the frequency domain for the dynamic response of the beam is obtained. And its inversion was adopted to convert the integral representation of the solution into the time domain. The degraded solution was then utilized to verify the validity of the proposed solution. Finally, the Maple mathematical software was used for further discussion. The solution proposed in this study can be a useful tool for practitioners.

Electron Irradiation Effect in Surface Modification of Hydroxyapatite Ceramics

The structure of ultrathin sections of dense submicrocrystalline hydroxyapatite (HA) (Ca10(PO4)6(OH)2) ceramics has been studied by X-ray diffraction and high-resolution transmission electron microscopy (TEM). The results demonstrate that high-energy electron irradiation can lead to surface modification with thin CaO, α-Ca3(PO4)2, and HA nanoparticles. Two-level interference contrast and double Fourier transformation of TEM images have been shown to be effective tools for revealing and identifying nanoparticles, respectively.

Development of the method of frequency­temporal representation of fluctuations of gaseous medium parameters at fire

The method of operative frequency-temporal representation of fluctuations of gaseous media parameters at an early stage of fire at premises was developed. The basic assumptions about the peculiarities of dynamics of hazardous factors of gaseous medium at early ignition at premises were stated. The authors created theoretical framework for development of the method, based on the fact that violation of equilibrium state of gaseous medium is translated by the medium to the zone of sensors’ localization and responds to emergence of an ignition in premises. The fire source in this case is considered a moving source of disturbances and parameters of the medium carry information about temporal and frequency shifts of disturbances. It was shown that these shifts of disturbances are characterized by the correspondent uncertainty function, which is an invariant with respect to the double Fourier transformation, determined by squared modulus of frequency-temporal energy density of the parameter. The proposed method is a further development of frequency-temporal representations of the Cohen class in case of fluctuations of gaseous medium parameters at early ignitions in premises. The main features of the method are its relative simplicity and the use of data in real time. The verification of the developed method was performed based on the experimental data of the main parameters of the gaseous medium at an early ignition of alcohol, paper, wood, and textiles in the simulation chamber.

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution. The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction. Thus, it is more realistic than the Winkler model, which only represents compressive soil resistance. However, the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation; hence, the beam under inclined loads cannot be considered in the model. In this study, a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model, which tends to disregard the tangential interaction between the beam and the foundation. The horizontal spring reaction is assumed to be proportional to the relative tangential displacement. The governing equation can be deduced by theory of elasticity and Newton's laws, combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition. Double Fourier transformation is used to simplify the geometric equation into an algebraic equation, thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam. Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution. The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS. Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam. Results are realistic and can be used as reference for future engineering designs.

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

• Analytical solution for dynamic response of infinite beam on foundation is developed. • The elastic foundation considered the horizontal tangential effect. • Double Fourier transformation is used for deduction and the residue theorem is employed to obtain the solution. • The analytical solution for the beam under inclined dynamic load is presented. The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.

A novel explicit solution for twisting control of smart laminated cantilever composite plates/beams using inclined piezoelectric actuators

In the present work, a novel explicit analytical solution is proposed for obtaining twisting deformation and optimal shape control of smart laminated cantilever composite plates/beams using inclined piezoelectric actuators. The linear piezoelectricity and plate theories are adapted for the analysis. A novel double integral multivariable Fourier transformation method and discretised higher order partial differential unit step function equations are employed. For the first time, an exact solution is developed to analyse electro-mechanical twisting moments in smart composite structures. Since there are no published benchmark results for verification, a series of simple, accurate and robust finite element (FE) analysis models and realistic electro-mechanical coupled FE procedures are developed for the effective prediction of the structural behaviour of the smart laminated piezo-composite structures under arbitrary loads. In addition to the novelty of the explicit solution, more comprehensive FE simulations of smart structures and step-by-step guidelines are discussed. The effect of various parameters including electro-mechanical twisting coupling, layup thickness, actuators size, placement, and inclination angle, electrical voltage, stacking sequence, and geometrical dimension are taken into account. The comparison of results shows an excellent agreement. Unlike the earlier studies, the proposed method does not require the characteristic and trial deflection function to be predetermined.

New explicit solution for static shape control of smart laminated cantilever piezo-composite-hybrid plates/beams under thermo-electro-mechanical loads using piezoelectric actuators

In this paper, a new explicit exact analytical solution is proposed for obtaining static deformation and optimal shape control of smart laminated cantilever piezo composite hybrid plates and beams under thermo-electro-mechanical loads using piezoelectric actuators. The linear piezoelectricity and plates theories were adapted for the analysis. A novel double integral multivariable Fourier transformation method combined with discretised higher order partial differential unit step function equations were employed. The effect of various parameters including arbitrary loads such as non-uniform thermal stresses, electrical and mechanical loads, layup thickness, piezoelectric actuators size and placement, stacking sequence, and geometrical dimension were considered. The results were then compared with some published benchmark results and good agreement was observed. Unlike the earlier studies, the proposed method does not require the characteristic and trial deflection function to be predetermined. Both, the embedded and bounded actuators are considered. Until now, the shape control task of reducing mid-plane deformation at free end in smart laminated cantilever plates and beams was unsolvable and approximations were typically employed in numerical analysis Yu et al. (2009). This problem becomes more complicated for wider and longer plates but the method proposed herein successfully resolves this issue.

Влияние ширины канала на вязкоупругие колебания ледового покрова под действием движущейся нагрузки

In this paper, the channel width influence on hydroelastic waves in an ice-covered channel is investigated. Waves are generated by a load moving along the ice sheet. External load is modeled by a localized smooth pressure distribution. The differential equation of oscillations of viscoelastic ice plate and the Laplace equation for the velocity potential of fluid flow under the ice cover are used as a mathematical model. These equations are supplemented with impermeability boundary conditions on channel walls and bottom, clamped conditions for ice on the channel walls, and the kinematic and the dynamic condition on the ice – liquid interface. The time-independent travelling wave solution in a coordinate system moving with the external load is studied. Applying the Fourier transformation allows the initial problem to be reduced to the two-dimensional problem of wave profile across the channel, which is solved by the normal mode method for a fixed beam. The problem of deflections in the infinite ice sheet is solved by double Fourier transformation. Numerical results of the channel width influence are discussed. The ice defection and strains in the ice sheet are calculated and compared with the results of problem solution for infinite ice plate.DOI 10.14258/izvasu(2016)1-35

Full-field analysis of three dimensional piezoelectric layered half-space with point loading

Abstract In this study, a three dimensional transversely isotropic piezoelectric layered half-space subjected to concentrated loading and electric charge was analyzed. This paper provides an effective analytical methodology to obtain the explicit full-field solutions for this problem. Due to the dissimilar structures of the solutions, the applied loadings are separated into axis-symmetric loadings and non-symmetric loadings. By using the double Fourier transformation technique in conjunction with series expansion, the explicit closed-from solutions for displacements, stresses, and the electric field in the layered half-space are obtained. The analytical solutions are valid for loadings applied either in the interior or at the surface of the structure. It can be seen that the complete solutions obtained by the method used in this study comprise only the fundamental solutions of an infinite space with a concentrated loading. One of the fundamental solutions is the Green's function of an infinite space and the others are solutions induced to satisfy the boundary and interface conditions. The mathematical meaning of the method used in the study is in fact the image method or Mindlin's superposition method. Instead of finding image singularities manually, the intensity and locations of the image singularities are determined automatically from the mathematical technique used in this paper. Taking PZT materials as an example, the numerical computation for the full-field distributions of the displacement and electric field in the layered half-space medium subjected to a point force and a point charge was discussed.

Dual-modulator broadband infrared Mueller matrix ellipsometry

A broadband mid-infrared Mueller matrix ellipsometer is described based on two photoelastic modulators and a step-scan interferometer. The data are analyzed using a combination hardware-software double Fourier transformation. Obtaining spectra of the Mueller matrix elements requires that the infrared wavelength-dependent retardation amplitude of the modulators be known through calibration and subsequently incorporated into the data processing. The spectroscopic capability of the instrument is demonstrated in transmission and reflection geometries by the measured Mueller matrices of air, an anisotropic quartz crystal, and the ZnSe-water interface, each from 2500-4000 cm(-1).

Solution of a quasistatic problem of thermoelasticity for a half space with moving mechanical and thermal load locally distributed on the surface

Using the double integral Fourier transformation, we construct solutions of space problems of the theory of elasticity and thermoelasticity for a half space with a locally distributed moving mechanical and thermal load on the surface. The obtained formulas allow one to find displacements and stresses in a half space for a velocity of the load smaller than the velocity of the Rayleigh wave. In the limiting case of the action of a fixed load, the obtained solutions coincide with known ones.

The action of a strip punch on a transversely isotropic half-space

Three-dimensional contact problems on the action of an absolutely rigid punch, in the form of a strip in plan, on a transversely isotropic elastic half-space, when the isotropy planes are perpendicular to the boundary of the half-space are investigated. In view of the fact that the elastic rigidity of the boundary of the half-space, characterizing the normal displacement under the action of a given concentrated force, depends considerably on the chosen direction on this boundary, two cases of the arrangement of the punch are considered: along the first or second axis of a Cartesian system of coordinates on the body boundary (Problems A and B). The normal displacement of the body boundary under the action of a specified concentrated normal force, after using a double Fourier transformation, is obtained in a form that is free of quadratures, which enables the stiffness of the boundary in different directions to be determined without difficulty, as well as the direction of the extremal stiffness. Assuming that the function, which describes the form of the base of the punch, can be represented by a Fourier series, one-dimensional integral equations of contact Problems A and B, the symbols of the kernels of which are independent of the number of the term of the Fourier series, are obtained. For a special approximation of the symbol of the kernel, a closed solution of the contact problem is derived in terms of Mathieu functions using Rvachev's method, which is a closed solution of the analogous contact problem of the action of a strip punch on an isotropic elastic half-space. To solve the integral equations of contact problems, regular and singular asymptotic methods are used with the introduction of a dimensionless geometrical parameter λ, which represents the ratio of the value of the period of the wavy base of the punch to the thickness of the contact strip. Using the method of orthogonal functions, the integral equations are also reduced to an infinite system of linear algebraic equations, to solve which the reduction method is employed for any values of λ.

Branching of solutions for the problem of mean-square approximation of a real finite function of two variables by the modulus of double Fourier transformation

We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya, 51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental equations for finding the points of their branching. We also present, in the first approximation, the analytical representations of complex solutions branched-off from the real solution for the two-dimensional case of branching.

Statistical characteristics of AGW wave packet propagation in the lower atmosphere observed by the MU radar

Abstract. We study the horizontal structure of the atmospheric gravity waves (AGW) in the height ranges between 6 and 22 km observed using the MU radar at Shigaraki in Japan, during a 3 day period in January and a 4 day period in August 1988. The data were divided by double Fourier transformation into a data set of upward moving waves and a data set of downward moving waves for independent analysis. The phase and group velocity tracing technique was applied to measure the vertical group and phase velocity as well as the characteristic period of the gravity wave packet. Then the dispersion equation of the linear theory of AGW was solved to obtain its intrinsic wave period – horizontal wavelength and horizontal group velocity – and the vertical flux of horizontal momentum associated with each wave packet was estimated to help determine the direction of the characteristic horizontal wave vector. The results showed that the waves with periods in the range of 30 min~6 h had horizontal scales ranging from 20 km to 1500 km, vertical scales from 4 km to 15 km, and horizontal phase velocities from 15 m/s to 60 m/s. The upward moving wave packets of wave period of 2 h~6 h had horizontal group velocities mainly toward east-south-east and northeast in winter, and mainly in the section between the directions of west-north-west and north in summer.

Magnetic Field Analysis of an Arbitrary Shaped Coil Using Shape Functions

In eddy-current sensors and eddy-current testing, it is important to analyze the magnetic field of the coil. Toward this end, we analyzed magnetic fields in various coils, such as circular and rectangular coils. We used the double Fourier transformation and applied it to the eddy-current problem of coils facing a sheet conductor. However, only a limited number of coil shapes could be analyzed by this method. We therefore devised a method of magnetic field analysis that could be applied to a variety of coils by double Fourier transformation. We focused attention on the generated magnetic field of the coil, which is a basic characteristic. Our analysis method is made applicable to a much larger variety of coil shapes by defining a completely new type of function, called shape functions. We verified the validity of the analytical results experimentally. Although the Biot-Savart law may be used to analyze magnetic fields of coils, the law is not applicable when the coil faces a sheet conductor.

Studies of vertical fluxes of horizontal momentum in the lower atmosphere using the MU-radar

Abstract. We study the momentum flux of the atmospheric motions in the height ranges between 6 and 22 km observed using the MU radar at Shigaraki in Japan during a 3 day period in January 1988. The data were divided by double Fourier transformation into data set of waves with downward- phase- velocity and data set of waves with upward-phase-velocity for independent momentum flux calculation. The result showed that both the 72 h averaged upward flux and downward flux of zonal momentum were negative at nearly each height, meaning that the upward flux was dominated by westward propagating waves while the downward flux was dominated by eastward propagating waves. The magnitude of the downward flux was approximately a factor of 1.5 larger than the upward flux for waves in the 2~7 h and 7~24 h period bands, and about equal to the upward flux in the 10–30 min and 30 min–2 h period bands. It is also observed that the vertical flux of zonal momentum tended to be small in each frequency band at the altitudes below the jet maximum (10~12 km), and the flux increased toward more negative values to reach a negative maximum at some altitude well above the jet maximum. Daily averaged flux showed tremendous variation: The 1st 24 h (quiet day) was relatively quiet, and the fluxes of the 2nd and 3rd 24 h (active days) increased sharply. Moreover, the upward fluxes of zonal momentum below 17 km in the quiet day for each period band (10~30 min, 30 min~2 h, 2~7 h, and 7~24 h) were dominantly positive, while the corresponding downward fluxes were dominantly negative, meaning that the zonal momentum below 17 km in each period band under study were dominantly eastward (propagating along the mean wind). In the active days, both the upward fluxes and downward fluxes in each frequency band were dominantly negative throughout the whole altitude range 6.1–18.95 km.

Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load

A two-dimensional elasticity analysis for steady-state axisymmetric dynamic response of an arbitrarily thick elastic homogeneous hollow cylinder of infinite length, which is imperfectly bonded to the surrounding fluid-saturated permeable formation, subject to an axially moving ring load, is presented. The problem solution is derived by using Biot’s dynamic theory of poroelasticity in conjunction with double Fourier transformation with respect to time (frequency) and axial coordinate (axial wave number). The analytical results are illustrated with numerical examples in which a concrete tunnel lining of uniform wall thickness is imperfectly bonded to a surrounding water-saturated poroelastic formation of soft/stiff frame characteristic. Numerical solutions for the radial shell mid-plane and formation displacements are calculated by analytical (numerical) inversion of the Fourier transformation with respect to the frequency (axial wave number). Primary attention is focused on the influence of bonding condition at the liner/soil interface, formation material type, and load velocity on the system’s dynamic response. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.

Combined Analysis of C-18 Unsaturated Fatty Acids Using Natural Abundance Deuterium 2D NMR Spectroscopy in Chiral Oriented Solvents

The quantitative determination of isotopic (2H/1H)i ratios at natural abundance using the SNIF-NMR protocol is a well-known method for understanding the enzymatic biosynthesis of metabolites. However, this approach is not always successful for analyzing large solutes and, specifically, is inadequate for prochiral molecules such as complete essential unsaturated fatty acids. To overcome these analytical limitations, we use the natural abundance deuterium 2D NMR (NAD 2D NMR) spectroscopy on solutes embedded in polypeptide chiral liquid crystals. This approach, recently explored for measuring (2H/1H)i ratios of small analytes (Lesot, P.; Aroulanda, C.; Billault, I. Anal. Chem. 2004, 76, 2827-2835), is a powerful way to separate the 2H signals of all nonequivalent enantioisotopomers on the basis both of the 2H quadrupolar interactions and of the 2H chemical shift. Two significant advances over our previous work are presented here and allow the complete isotopic analysis of four mono- and polyunsaturated fatty acid methyl esters: methyl oleate (1), methyl linoleate (2), methyl linolenate (3), and methyl vernoleate (4). The first consists of using NMR spectrometers operating at higher magnetic field strength (14.1 T) and equipped with a selective cryoprobe optimized for deuterium nuclei. The second is the development of Q-COSY Fz 2D NMR experiments able to produce phased 2H 2D maps after a double Fourier transformation. This combination of modern hardware and efficient NMR sequences provides a unique tool to analyze the (2H/1H)i ratios of large prochiral molecules (C-18) dissolved in organic solutions of poly(gamma-benzyl-L-glutamate) and requires smaller amounts of solute than previous study on fatty acids. For each compound (1-4), all 2H quadrupolar doublets visible in the 2D spectra have been assigned on the basis of 2H chemical shifts, isotopic data obtained from isotropic quantitative NAD NMR, and by an interspectral comparison of the anisotropic NAD spectra of four fatty acids. The NMR results are discussed in terms of (2H/1H)i isotopic distribution and molecular orientation in the mesophase. For the first time, we show that the investigation of natural isotopic fractionation of complete fatty acids is possible without the need of chemical modifications, hence providing an alternative method to probe the mechanisms of enzymes implied in the biosynthetic pathway of unsaturated fatty acids.