Two-dimensional Fourier Series Research Articles

Vibroacoustic analysis of submerged fluid-filled cylindrical shell

This study presents a comprehensive Chebyshev-Fourier spectral method for analyzing the free and forced vibroacoustic characteristics of submerged fluid-filled cylindrical shells in heavy fluids. The vibroacoustic interaction system of submerged cylindrical shells consists of three discrete regions: The Sanders shell theory is applied for the shell structure, whereas the wave equation is used to determine the fluid inside. The external fluid pressure on the shell is described using the Helmholtz boundary integral equation. The governing equations for the shell's motion, the formulation of the external fluid pressure on the fluid-structure-interaction (FSI) surface using first-class Chebyshev polynomial and Fourier series expansions, and the internal sound pressure employing two-dimensional Chebyshev polynomial and Fourier series expansions, are seamlessly integrated with Gauss-Chebyshev-Lobatto sampling techniques. Artificial springs are implemented to replicate the boundary conditions of the cylindrical shell. The precision of the proposed methodology is corroborated by comparisons with existing literature and finite element method (FEM) results. Furthermore, the research examines how the properties of the acoustical medium on the vibroacoustic responses of the cylindrical shell. Results indicate that the presence of air inside the shell slightly influence on its acoustic radiation. However, replacing the internal medium with water significantly alters the shell's original radiation characteristics.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

A Novel Analytical Method for Coupled In-Plane and Out-of-Plane Vibrations of Sandwich Plates with Arbitrary Boundary Conditions

Sandwich plates are popular in the research field of vibration damping and are widely used in numerous engineering domains. Sandwich plates have been studied extensively for their excellent performances by a large number of researchers. Due to the interaction between the out-of-plane and in-plane vibrations of the layers of sandwich plates, it is difficult to solve the displacements of each layer in all directions analytically. To deal with this problem conveniently and accurately, a novel analytical method is presented in this study for coupled out-of-plane and in-plane vibrations of sandwich plates, which can be applied to both free and forced vibrations of sandwich plates with arbitrary boundaries. In this method, the core plate is treated as a three-dimensional (3D) problem, and it is assumed that the displacement of the core plate varies linearly along the thickness. Based on the Kirchhoff hypothesis, the displacement solutions of the base and constrained plates of the sandwich plate in the [Formula: see text], [Formula: see text] and [Formula: see text] directions are expressed as a superposition of one-dimensional (1D) and two-dimensional (2D) Fourier series, respectively. By comparison to the published analytical solutions and numerical results of the finite element method, the proposed method achieves excellent accuracy and reliability. In addition, the influence of out-of-plane and in-plane vibrations of sandwich plates on each other is studied, and the effects of geometrical and material parameters on the dynamic behaviors of a sandwich plate are investigated. The result shows that the out-of-plane vibration affects the in-plane vibration significantly, which means that the coupling effect of the out-of-plane and in-plane vibrations must be taken into account when analyzing in-plane vibration.

New Natural Frequency Studies of Orthotropic Plates by Adopting a Two-Dimensional Modified Fourier Series Method

The free vibration behavior of orthotropic thin plates, which are clamped at three edges and free at one edge, is a matter of great concern in the engineering field. Various numerical/approximate approaches have been proposed for the present problem; however, lack precise analytic benchmark solutions are lacking in the literature. In the present study, we propose a modified two-dimensional Fourier series method to effectively handle free vibration problems of plates under various edge conditions. In the given solution, the adopted trial function automatically satisfies several boundary conditions. After imposing Stoke’s transformation in the trial function and letting it satisfy the remaining boundary conditions, we can change the present plate problem into calculating several systems of linear algebra equations which are easily handled. The present method can be regarded as an easily implemented, rational, and rigorous approach, as it can exactly satisfy both the governing equation and the associated edge conditions. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through the utilization of different types of Fourier series, and it can be extended for the further dynamic/static analysis of plates under different shear deformation theories. Finally, all the novel analytical solutions are confirmed to be sufficiently accurate since they match well with the FEM results. The new analytic solution obtained may serve as a benchmark for validating other numerical and approximate methods.

An analytical model for dynamic analysis of inhomogeneous sandwich plate

Sandwich plates are widely used in the engineering domain and are extensively studied by researchers for their special performances. Among them, the sandwich plate with coordinate-dependent material parameters has excellent vibration isolation performance. However, there are few effective and convenient analytical methods for solving the dynamic response of this type of sandwich plate. To solve this problem, an equivalent spring-based model for the dynamic analysis of the sandwich plate is proposed in this paper. Assuming that the sandwich plate is a linear elastic structure, and considering the core plate as equivalent springs, whose stiffness coefficients are derived based on the classical elasticity theory and Kirchhoff hypothesis. Besides, displacement solutions of base and constrained plates of the sandwich plate in the x, y and z directions are expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series, respectively. Then, according to the orthogonality of trigonometric functions and selectivity of Dirac functions, the displacements of the sandwich plate are solved accurately. By comparing with available analytical and numerical results, the proposed method is proven to have excellent performance in dealing with the sandwich plate with coordinate-dependent material parameters. Finally, the dynamic characteristics and vibration isolation performance of the inhomogeneous sandwich plate are discussed. The results show that the proposed method can be used to design the inhomogeneous sandwich plate and improve their vibration isolation performance significantly.

Capacitor current analysis of a three-level neutral point clamped converter under unbalanced loading conditions

A neutral point clamped (NPC) converter is considered a forefront in industrial applications. Supplying a typically balanced stand-alone load is one of those applications. However, the loading may become unbalanced which can impact the capacitors’ current and voltage ripple. In this work, an approach is proposed that analyze the capacitor current under unbalanced loading. The proposed method is based on a combination of two-dimensional Fourier series and symmetrical components. Since, two-dimensional Fourier series sectorize the spectrum into harmonics that are either defined by multiples of, fundamental, carrier or both frequencies, the method derives the Fourier coefficients for each sector, corresponding to a sequence current. Therefore, based on the presented approach, each harmonic amplitude in the spectrum sector is defined by three Fourier coefficients reflecting zero, positive and negative sequence current. The capacitor current spectrum is obtained by vector summing sequence coefficients. The method is tested on unbalanced load conditions with (out) a ground path. Results verify feasibility of the proposed method in deriving Fourier coefficients of capacitor currents that accurately reflects the loading status of the NPC. This is vital for converter design in terms of proper sizing of DC capacitor and can help in avoiding components failure.

Raster image processing using 2D Padé-type approximations

We have developed a method called the two-dimensional Padé-type approximants method, which can be used to reduce the Gibbs phenomenon in the harmonic two-dimensional Fourier series. This method can be applied to both monochrome and color raster images. To do this, we implement the generalized two-dimensional Padé approximation proposed by Chisholm. In this approach, we select the range of frequency values on the integer grid according to the Vavilov method. We propose a definition of a Padé-type functional and provide examples of its application to simple discontinuous templates represented as raster images. Through this study, we are able to draw conclusions about the practical usage and advantages of the Padé-type approximation. We demonstrate that the Padé-type approximant effectively eliminates distortions associated with the Gibbs phenomenon, and it is visually more appropriate than the Fourier approximant. Additionally, the application of the Padé-type approximation reduces the number of parameters without sacrificing precision.

Tangential Force Noise Reduction of Vehicular PMSM Based on Reference Harmonic Current Analytical Computation

The harmonic current injection can remarkably reduce the noise of the vehicular electric drive system. However, the optimal reference harmonic current is difficult to obtain because of the vehicular motor’s non-linear electromagnetic characteristics. The existing methods are always time-consuming and cannot satisfy the real-time requirement of the vehicular application. This article proposed an analytical computation method of the reference harmonic current for tangential force noise reduction. Specifically, by combining two-dimensional Fourier series decomposition and polynomial fitting, a data-based non-linear model is constructed to accurately describe the tangential force harmonic under arbitrary current excitation. Then, considering the dominant effect of fundamental current, a simplified tangential force model is obtained by local linearization at each fundamental current working point. Meanwhile, by representing the stator current amplitude with a Fourier series combination, the suppression of tangential force harmonics is transferred to solving linear equations concerning each harmonic current vector. Finally, simulation and bench test experiments have been done to verify the effectiveness of the proposed method. The results demonstrate that the reference harmonic current at different speeds/torque can be computed in real-time, and the corresponding tangential noise can be reduced remarkably.

Fourier acoustics: Uncovering the origins of sound

No doubt, Jean-Baptiste Joseph Fourier has had a great impact in science. At the root of Fourier Acoustics, the Fourier series, when employed as a starting point and manipulated, allows us to uncover the origins of sound in a simple and elegant way, not only in acoustics but also in electromagnetics. A well-known stepping stone on the path is nearfield acoustical holography (NAH) and its theory can be recast almost completely, and simply, in terms of two-dimensional (2D) Fourier series and applied to any geometry. For separable coordinate systems, each of the components of the series can be propagated to a surface conformal to the measurement surface using an analytically defined propagator. But even when the surfaces are not part of a separable system, the spirit of Fourier’s expansions is mirrored with the singular value decomposition, applied to a numerically derived propagator constructed using the equivalent source method. A comprehensive demonstration of Fourier acoustics applied to aeroacoustics was in the cabin of a Boeing aircraft, in flight, using a two-dimensional spherical array. Reconstructions of vector intensity in a volume from correlations of the measured pressure to panel accelerometers showed the direction and location of the dominant fuselage sources of noise—uncovering the origin of sound. [Work supported by the Office of Naval Research.]

A General Approach for the Analysis of Transverse Vibration Characteristics of a Rectangular Plate with Arbitrary-shaped Openings

Based on the Rayleigh-Ritz method, the transverse vibration characteristics of a rectangular plate with an arbitrarily shaped inner opening are investigated. A modified two-dimensional Fourier series was chosen as the trial function and springs of different stiffnesses were used to equate the arbitrary boundary conditions of the rectangular plate. The Lagrange generalization of the transverse vibration of the open rectangular plate is obtained by equating the open area to a special film with physical properties of zero according to the energy superposition principle. The variational method is used to derive the characteristic equations of the thin plate and to find the intrinsic frequencies and corresponding vibration modes. The convergence of this theoretical method is analyzed by means of an example, and the accuracy of the method is verified by comparison with the results of finite element software calculations, which provides a reference for practical engineering problems.

APPLICATION OF TWO-DIMENSIONAL PADÉ-TYPE APPROXIMATIONS FOR IMAGE PROCESSING

Context. The Gibbs phenomenon introduces significant distortions for most popular 2D graphics standards because they use a finite sum of harmonics when image processing by expansion of the signal into a two-dimensional Fourier series is used in order to reduce the size of the graphical file. Thus, the reduction of this phenomenon is a very important problem.
 Objective. The aim of the current work is the application of two-dimensional Padé-type approximations with the aim of elimination of the Gibbs phenomenon in image processing and reduction of the size of the resulting image file.
 Method. We use the two-dimensional Padé-type approximants method which we have developed earlier to reduce the Gibbs phenomenon for the harmonic two-dimensional Fourier series. A definition of a Padé-type functional is proposed. For this purpose, we use the generalized two-dimensional Padé approximation proposed by Chisholm when the range of the frequency values on the integer grid is selected according to the Vavilov method. The proposed scheme makes it possible to determine a set of series coefficients necessary and sufficient for construction of a Padé-type approximation with a given structure of the numerator and denominator. We consider some examples of Padé approximants application to simple discontinuous template functions for both formulaic and discrete representation.
 Results. The study gives us an opportunity to make some conclusions about practical usage of the Padé-type approximation and about its advantages. They demonstrate effective elimination of distortions inherent to Gibbs phenomena for the Padé-type approximant. It is well seen that Padé-type approximant is significantly more visually appropriate than Fourier one. Application of the Padétype approximation also leads to sufficient decrease of approximants’ parameter number without the loss of precision.
 Conclusions. The applicability of the technique and the possibility of its application to improve the accuracy of calculations are demonstrated. The study gives us an opportunity to make conclusions about the advantages of the Padé-type approximation practical usage.

New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

This article aims at analytically solving the free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. By utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation processes, new, precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.

Finite-Region Approximation of EM Fields in Layered Biaxial Anisotropic Media

A new algorithm is developed to accurately compute the electromagnetic (EM) fields in the layered biaxial anisotropic media. We enclose the computational region in an infinitely long rectangular region by four vertical truncation planes and establish the corresponding algorithm to approximate the EM fields in the entire space. The EM fields in this region are expanded as a two-dimensional (2-D) Fourier series of the transverse variables. By using the spectral state variable method, the generalized reflection coefficient matrices and transmission matrices are then derived to determine the Fourier coefficients per layer. Therefore, we can obtain the spatial-domain EM fields by summing the 2-D Fourier series. To enhance the accuracy and efficiency of this algorithm, we apply the method of images to estimate the influence of the artificial boundaries on the EM fields at the observer. We then further develop a quantitative principle to choose the proper size of the region according to the desired error tolerance. With the proper choice, the summation of the series can achieve satisfactory accuracy. This algorithm is finally applied to simulate the responses of the triaxial logging tool in transversely isotropic and biaxial anisotropic media and is verified through comparisons to the other method.

Frozen spatial coherence.

Inspired by the concept of coherent frozen waves, this paper introduces one possible theoretical framework of its partially coherent version, a frozen spatial coherence, in which a desired two-point correlation structure of an optical field is created on the propagation axis by superposing partially coherent zero-order Bessel beams. It is shown that the cross-spectral density can be given a description in terms of a two-dimensional Fourier series, analogous to the one-dimensional approach of coherent frozen waves. The formalism is applied to the design of a partially coherent field which is highly coherent only if the pair of points in the propagation axis belong to a predetermined and finite range and highly incoherent outside that range.

Explicitly correlated study of the torsional vibrations of HSOSH molecule. Comparison with MP2/CBS(T,Q) level of theory

Geometric parameters of equilibrium configurations of trans- and cis-conformers of HSOSH molecule, their dipole moments, rotational constants in the ground electronic and vibrational states, and kinetic parameters corresponding to two torsional coordinates and were calculated at CCSD(T)-F12/cc-pVDZ-F12 level of theory. In the latter case, at each node of the two-dimensional grid and all other geometric parameters of the molecule were optimized. The obtained configurations were used for calculation of 2D potential energy surface at CCSD(T)-F12/cc-pVTZ-F12 level without additional optimization. The energies of a number of excited torsional states were determined by numerically solving the vibrational Schrödinger equation of restricted dimensionality using two-dimensional complex Fourier series. The obtained data on the frequencies of torsional vibrations of S-H groups in the two conformers and the frequencies of tunneling in the ground and excited torsional states are compared with the results obtained earlier at MP2/CBS(T,Q) level of theory.

Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier series

Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier series

Vibroacoustic analysis of a thin laminated composite plate with surface-boned piezoelectric patches and subjected to general boundary conditions

In this paper, a semi-analytical model is proposed to deal with the vibroacoustic problems of laminated composite plates with surfaced-boned piezoelectric patches and subjected to general boundary condition using the modified Fourier series method. Based on Kirchhoff plate theory, the dynamic equation of the laminated composite plate is derived using Hamilton's principle. In order to satisfy general boundary conditions, the displacement solution of the plate is expressed in the form of two-dimensional Fourier series and serval auxiliary functions. The acoustic response of the laminated composite plate due to a harmonic concentrated force is obtained with the Rayleigh integral. Besides, the mass and stiffness contribution of the piezoelectric patch are taken into consideration in the present study. Through enough convergent studies and comparative studies, the convergence, accuracy and universality of the proposed method are validated. The developed semi-analytical model can be used for efficient and accurate analysis and design of laminated composite plates equipped with shunted piezoelectric patches. Finally, the effects of the resistor and inductor shunt damping circuits on the vibration and acoustic response is discussed.

In-Plane Vibration Analysis of Square Plate with Multiple Cutouts

A theoretical solution method for the in-plane vibration characteristics of the plate with cutouts is proposed in this paper. The energy principle in conjunction with the Rayleigh–Ritz solution technique is adopted for the theoretical modeling of the in-plane vibration of the structure. The energy functions for the plate with cutouts are established by subtracting the energies of the cutout domains from the total energies of the whole plate. To ensure continuity over the entire solution domain, the in-plane displacements are composed of two-dimensional standard Fourier series and supplementary functions. The in-plane eigenmodes of the plates with different square cutouts are compared with the results obtained from the finite element method (FEM), with good agreements. The influences of the cutouts on the in-plane vibration characteristics of the plates with cutouts are investigated by varying the number, size, and position of the cutouts.

Widening wave band gaps of periodic plates via shape optimization using spatial Fourier coefficients

Periodic media have been shown to exhibit wave attenuation in frequency ranges called band gaps. The challenge is designing feasible periodic structures that present total band gaps in the low-frequency range. Plate structures are two-dimensional media that can benefit from the application of these concepts. They can be modeled using either Kirchhoff’s or Mindlin’s plate theories. Since the periodic properties of plates (geometry and material properties) can be described by a two-dimensional spatial Fourier series, it should be possible to optimize its configuration using the series coefficients. The Fourier series representation is commonly used to compute the dispersion diagrams via the plane wave expansion (PWE) method. In this work, the Fourier series coefficients that describe the spatial distribution of the plate properties are used as optimization variables to obtain solutions that maximize an objective function capable of yielding low-frequency band gaps. In particular, the spatial distribution of the plate thickness is described by a two-dimensional Fourier series. Its coefficients are optimized with constraints on the minimum and maximum values to achieve the widening of low-frequency band gaps. Results show feasible solutions for several values of minimum and maximum thicknesses using Kirchhoff’s and Mindlin’s plate formulations.

Evolution of the Spatial Distribution of Hydrogen Atoms Channeling along Non-Chiral Carbon Nanotubes

The exact potentials of the interaction of channeled hydrogen atoms with non-chiral carbon nanotubes of $$\left( {n,0} \right)$$ and $$\left( {n,n} \right)$$ types are calculated using the Lennard-Jones potential. After expansion in a two-dimensional Fourier series of one-dimensional reciprocal lattice vectors and azimuthal harmonics, these potentials are used to analyze the motion of atoms in the channel. Numerical solution of the Schrodinger equation with the averaged electrostatic interaction potential for a $$\left( {10,10} \right)$$ nanotube ensures discrete transverse energy levels and corresponding wave functions. The evolution of the spatial distribution of channeled hydrogen atoms moving in the longitudinal direction under the influence of the periodic perturbing interaction potential as a function of the longitudinal velocity is studied by means of fundamental solution of the kinetic equation. The ability to induce the focusing effect at the center of a nanotube for moving hydrogen atoms is shown as well.