Double Fourier Series Method Research Articles

Optimization of time-dependent displacements of the axially graphene nanoplatelets reinforced micro-curved panels via swarm and genetic algorithm approaches

Micro-curved panels with moving micro/nanoparticles included are essential microstructures in the engineering field. So, improving its stability for use in various situations is an essential key for biomechanical engineers. There is potential for using graphene nanoplatelets (GPLs)-reinforced epoxy as the matrix phase in a fiber composite as it has been shown that GPLs successfully improve the properties of epoxy materials when compared to clean epoxy. So, in the current work, dynamic deflection of the axially reinforced thick micro-curved panel under a moving load is presented. Optimizing the geometry of composite materials is essential in structural design if you want to achieve the highest possible levels of strength or stability. In the present study, we use the genetic algorithm and particle swarm optimization to find the optimal radius ratio for lowering the displacement fields. For modeling the micro-size structure, modified couple stress theory with one length scale parameter is presented. The displacement fields are modeled using Taylor’s series higher-order terms considering strain in the thickness direction. Using forced vibration analysis, the space and time-dependent equations are solved via double Fourier series and state-space method, respectively. It is shown that reinforcing a microstructure with GPLs in the axial direction has lower deflection than in other directions. The results show that velocity and external loading of moving micro/nanoparticle, geometry properties, and length scale parameters have an important role in the time-dependent deflections of the axially reinforced thick micro-curved panel under moving micro/nanoparticles. The current outputs can be used in the biomechanical engineering field for modeling a micro-curved panel under moving micro/nanoparticles.

Generalized Accurate Harmonic Calculation Method Based on Discretized Double Fourier Series to Solve Double-Pulse Phenomenon

Harmonic issues have always been paid attention to in electrical systems, and accurate calculation of the amplitude and phase of each harmonic has become one of the research focuses of scholars. This article develops a generalized harmonic closed-form analytical solution for a single-phase voltage source inverter (VSI) under digital sinusoidal pulse width modulation (DSPWM) and dead-time effects. This article is the first to point out the double-pulse phenomenon that can occur when using a double Fourier series to solve the harmonic spectrum due to the discontinuity of the dead-time effects and half-cycle integrals. Analysis and data comparison shows that this phenomenon can affect the accuracy of harmonic calculations. An discretized double Fourier series method is proposed to solve this phenomenon. Moreover, the problems of missing pulse caused by the modulation under a specific current phase and current zero-crossing are also solved, improving the harmonic calculation's accuracy and generality. Simulation analysis and the results of inverse fast Fourier transform (IFFT) proved the correctness of the optimization method. And the experimental results are obtained on the DSP and SIC platforms, which are in good agreement with the expression calculation results

Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells

This study aimed to provide a static solution to the boundary value problem presented by symmetric (0°/90°/0°) and antisymmetric (0°/90°) cross-ply composite, moderately thick shallow shells and plates (a special case of the shells) subjected to mixed-type unsolved boundary conditions. The boundary-discontinuous double Fourier series (BDM) method, in which displacements are expressed in trigonometric functions, is employed in a well-established framework. The analytical solution obtained using the BDM is compared with the successful integration of the generalized differential quadrature (GDQ) method for the static analysis of composite shells with a roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. Comprehensive results are presented in order to show the effects of curvature on the deflections and stresses of moderately thick shallow shells made up of symmetric and antisymmetric cross-ply laminated composite materials. The validity of the proposed model is authenticated through the available HSDT-based literature review, and the convergence characteristics are demonstrated. The changing trends of displacements and stresses are explained in detail by investigating the effect of various parameters such as lamination, material properties, the effect of curvature, etc. Based on the results obtained using the proposed static solution, analytical BDM results were found to be in very close agreement with the numerical GDQ method, especially for symmetric lamination. However, the results obtained using the BDM and GDQ methods for antisymmetric lamination show differences, possibly due to the presence of a discontinuity in the derivatives originating from the bending–stretching matrix in antisymmetric lamination. Important numerical results presented include the sensitivity of the predicted response quantities of interest to material properties, lamination, and thickness effects, as well as their interactions. The results presented here may also serve as benchmark comparison points with numerical solutions such as finite elements, boundary elements, etc.

Magnetic Shunt Design and Their Effects on Transformer Winding Electromagnetic Forces

The major design purpose of the magnetic flux shunts is to reduce stray losses in the power transformer. Their influence on the surrounding leakage field, and thus the winding electromagnetic (EM) forces are also serious concerns, which are relevant to transformer operation safety. In this paper, the EM forces of transformer windings in the occurrence of magnetic flux shunts were studied based on the finite element method, which was validated by a double Fourier series method. The sensitivities of transformer EM forces to the shunt size and position were also investigated through parametric studies. The discussion on factors that influence the EM force should be of interest to practicing engineers in the transformer industry in winding strength calculation and magnetic shunt design.

Theoretical harmonic analysis of cascaded H‐bridge inverter under hybrid pulse width multilevel modulation

This study investigates the analytical approach for determining the spectral components of a hybrid pulse width modulation (HPWM) strategy for cascaded H-bridge (CHB) multilevel inverters. By decomposing the HPWM voltage into a staircase waveform and a non-sinusoidal pulse width modulation waveform and applying the one-dimensional Fourier series as well as double Fourier series methods, the closed-form solution for arbitrary level HPWM voltage is derived. Based on the derived solution, analysis of HPWM output with a changed modulation index is also performed, with good stability and control dynamic demonstrated. The spectral analytical expression is verified extensively by comparison of theoretical calculation results with those of the simulation and laboratory experiment, which coincide with each other very well. Furthermore, a brief switch function rotation balancing method for HPWM-based CHB inverter is utilised, and is also validated by experimental results.

Static analysis of laminated and sandwich composite doubly-curved shallow shells

A new analytical solution based on a third order shear deformation theory for the problem of static analysis of cross-ply doubly-curved shells is presented. The boundary-discontinuous generalized double Fourier series method is used to solve highly coupled linear partial differential equations with the mixed type simply supported boundary conditions prescribed on the edges. The complementary boundary constraints are introduced through boundary discontinuities generated by the selected boundary conditions for the derivation of the complementary solution. The numerical accuracy of the solution is compared by studying the comparisons of deflections, stresses and moments of symmetric and anti-symmetric laminated shells with finite element results using commercially available software under uniformly distributed load. Results are in good agreement with finite element counterparts. Additional results of the symmetric and anti-symmetric laminated and sandwich shells under single point load at the center and pressure load, are presented to provide data for the unsolved boundary conditions, benchmark comparisons and verifications.

Lebesgue points of double Fourier series and strong summability

A general summability method of double Fourier series is given with the help of a double sequence θ(k,n). Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f∈L1(T2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so-called modified Lebesgue points of f∈Lp(T2), whenever 1<p<∞. The sufficient conditions of θ are proved for the Fejér, Abel and Cesàro summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong θ-summability. Some other special cases of the θ-summation are considered as well, such as the Weierstrass, Picard, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.

Thermal wrinkling of thin membranes using a Fourier-related double scale approach

The thermal wrinkling behavior of thin membranes is investigated in this paper. Wrinkles often occur at multiple length scales where induced compressive stresses are located during thermal loading. In the present study, the method of double scale Fourier series is used to deduce the macroscopic membrane wrinkling equations. The obtained equations account for the global and local wrinkling modes. Numerical examples are conducted to assess the validity of the approach developed. The present membrane׳s model needs only few degrees of freedom to recover more accurately the post-buckling equilibrium curves and the wrinkling shapes. Different parameters such as membrane׳s aspect ratio, wave number, and pre-stressed membranes are discussed from a numerical point of view and the properties of the wrinkles (critical load, wavelength, size, and location) are presented.

One-Dimensional Spectral Analysis of Complex PWM Waveforms Using Superposition

This paper takes a new look at the mechanisms underlying the double-edge pulse-width modulation (PWM) process. It presents a novel way of deriving equations for the spectrum of double-edge PWM using basic mathematical techniques. In the process the underlying nonlinearities that generate the PWM sidebands are identified. Unlike the classical double Fourier series approach, the proposed method of deriving the PWM spectrum does not require the construction of the so-called unit cell. The interaction between this new model of the pulse-width modulator and the regular sampling process is studied, and generalized equations for the Fourier transforms of regularly sampled PWM waveforms are derived. A general solution to the important question of what happens to the PWM spectrum when the PWM reference consists of a summation of signals is presented. It is shown that the addition of reference signals in the time domain results in a double convolution of the PWM sidebands in the frequency domain. As an application of this result, it is shown how new analytic equations for the harmonics of third-harmonic injection PWM and space vector modulation can easily be derived. Finally, the new theoretical results are benchmarked against results from the well-established double Fourier series method.

Analytical Modeling of the Effect of Nonlinear Switching Transition Curves on Harmonic Distortion in Class D Audio Amplifiers

Practical switching transition times typically occupy less than one percent of the switching period in class D audio amplifiers. These transitions, however, are a very nonlinear function of current, which give rise to significant levels of distortion. This paper presents a method for determining the effect of nonlinear switching transition curves on harmonic distortion within half-bridge and full-bridge topologies, taking into account the interaction between the various dominant pulse-timing errors. An accurate analytical model, based on the well-established double Fourier series method of analysis, for determining the effect on harmonic distortion is introduced. The analytical model is accompanied by an equivalent simulation model using an existing strategy, and the results are verified experimentally.

Frequency-Domain Modeling of Multipulse Converters by Double-Fourier Series Method

Operation of multipulse converters can be conveniently described in the frequency domain by using their voltage and current mapping functions. Applications of such mapping functions include input impedance modeling of multipulse rectifiers, analysis of harmonic transfer and impedance coupling through high-voltage dc transmission lines, and input harmonic current distortion of multipulse rectifiers under unbalanced and/or distorted source conditions. The existing approach to spectral modeling of the mapping functions requires small-signal assumption and commensurability between the ac source fundamental frequency and the harmonic frequency. A new spectral modeling method based on double-Fourier series analysis is presented. The proposed method overcomes the limitations of conventional Fourier analysis and provides a more rigorous basis for frequency-domain analysis of multipulse converters.

Magneto-convection in an anisotropic porous layer with Soret effect

Thermal instability in an electrically conducting two component Boussinesq fluid-saturated-porous medium has been investigated, in the presence of Soret coefficient. The porous medium is confined between two horizontal surfaces, and subjected to a constant vertical magnetic field. Flow in the porous medium is characterized by generalized Darcy model, which includes the time derivative term. Performing linear and non-linear stability analysis, the effect of magnetic field on the stability of flow through porous medium has been investigated. The normal mode method is used in linear stability analysis, while a weak non-linear analysis based on a minimal representation of double Fourier series method is used in non-linear analysis. The critical Rayleigh number, wave number for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. Effects of various parameters on stationary, oscillatory and finite amplitude convection, rate of heat and mass transfer have been obtained analytically and presented graphically.

A radial basis function method for the shallow water equations on a sphere

The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

Solution to Boundary-value Problems in Fabrication of High-precision Reflector Panels

The fabrication of high-precision panels for the compact antenna test range (CATR) with a sandwich construction of two aluminum skin-plates and one aluminum middle plate, which are bonded to two aluminum honeycomb core-layers poses a lot of tricky problems. Of them, the force analysis of individual skin-layers and the springback calculation of sandwich are of utmost importance. Under reasonable assumptions, by using Fourier expansion of stress function and power series expansion of deflection function, two boundary-value problems of skin-layer and sandwich, i.e. force analysis and springback calculation, are solved. An analytical solution of stress function in series form and an approximate solution of deflection function are obtained. Compared with finite element method (FEM), the generalized double Fourier series method used in individual skin force analysis is superior in accuracy. Compared with experimental results, the weighted residual method used in springback calculation of sandwich-structured panel is acceptable from view of precision. Based on calculation results, processing parameters to panel fabrication are determined more efficient and accurate.

An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq fluid, which is heated and salted from below in the presence of Soret coefficient is studied analytically using both linear and nonlinear stability analyses. The normal mode technique is used in the linear stability analysis while a weak nonlinear analysis based on a minimal representation of double Fourier series method is used in the nonlinear analysis. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameters, solute Rayleigh number, Soret parameter and Lewis number on the stationary, oscillatory, finite amplitude convection and heat and mass transfer are shown graphically.

The frequency spectrum of pulse width modulated signals

The determination of the frequency spectrum of a pulse width modulated (PWM) signal with general band-limited input x( t) has been an open problem for many years. We describe a new approach that gives exact analytical expressions for the spectra of uniform-sampling PWM signals and natural-sampling PWM signals with single-edge as well as with double-edge modulation. For the special case of single tone modulating signals, our results reduce to those obtained previously using a double Fourier series method. We also show that if the maximum magnitude of the derivative of x( t) is smaller than twice the carrier frequency, then a PWM signal consists of a baseband signal y( t) together with y( t) phase-modulated onto each carrier harmonic, where, for uniform-sampling PWM, y( t) is a nonlinear function of the modulating signal x( t), while for natural-sampling PWM, y( t) is just x( t) itself, that is, there is no distortion in the baseband when natural sampling is used.

A biperiodic interfacial pattern of misfit dislocation interacting with both free surfaces of a thin bicrystalline sandwich

The elastic field, of a thin bicrystalline sandwich with parallel faces when the interface between the two anisotropic crystal components is parallel to the faces and is composed of a biperiodic pattern of defects is numerically computed, using a double Fourier series method, an explicit formulation of each harmonic term of series is derived that depends explicitly on the anisotropic elasticity coefficients and thickness of each crystal. Application is made to the particular case of intrinsic dislocation meshes using a InAs/( 1 1 1 )GaAs related system by a regular array of MDS at the interface, cf. J.G. Belk et al., Phys. Rev. Lett. 77 (1997) 475 and H. Yamaguchi et al., Phys. Rev. B 55 (1997) 1337, to determine free surface deformations through different thickness of the bicrystal. Graphic

Influence of applied electric field on the energy release ratefor cracked PZT/elastic laminates

Since the delamination induced failure of a laminated smartstructure always occurs under the action of mechanical and electricalfields, a generalized two-dimensional model for a piezoelectric/elasticlaminate is established in order to analyze the effect of an applied electricfield on its delamination. First, based on the double Fourier series method, ageneral analytical solution of the mid-plane displacement is derived for anyboundary condition. Furthermore, the energy release rate is employed to studythe influence of the applied electric field and mechanical loading on thefracture characteristics of the piezoelectric/elastic laminate. The energyrelease rates for modes I and II fracture behavior of the PZT/elasticlaminates are calculated in detail. Such calculations indicate that it isfeasible to choose not only suitable material properties of the piezoelectricand elastic layers, but also their thicknesses, to improve the smartstructure's fracture strength within a specified range.

Analytical solution to harmonic characteristics of traction PWM converters

The paper presents an analytical technique for calculating the AC input current harmonic spectrum of single-phase PWM converter drives used in railway traction. The technique is based on a 3-D modulation model and the double Fourier series (DFS) method. Formulas in the form of Bessel functions representing the harmonic spectra for several combinations of interlaced PWM converters are derived. These give a useful insight into the harmonic generation and cancellation process which occurs with interlaced configurations. They are also used in a theoretical explanation of the interlacing angle, system parameter symmetry and load balance between interlaced converters. The accuracy of this analytical technique is confirmed by comparing the results with those obtained from a frequency domain model (FDM) of the PWM converter drive.

The debonding effect in saddles: comparison of the performance of Flügge's and Sanders' shell theories

The aim of the present paper is the calculation of the unilateral contact stresses between the cylindrical vessels and the saddles by formulating the linear complementarity problem (LCP). Both the theories of Flügge and Sanders are applied for the consideration of shell behaviour. The LCP problems are solved by using the Mangasarian algorithm, the flexibility coefficients being obtained for both shell theories by using the method of double Fourier series. Numerical applications are given for different sizes of saddles and a comparison between the results of the two shell theories are shown and discussed.