Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

Analytical Expressions for the Relaxation Moduli of Linear Viscoelastic Composites With Periodic Microstructure

Abstract: Many micromechanical models have been used to estimate the overall stiffness of heterogeneous- materials and a large number of results and experimental data have been obtained. However, few theoretical and experimental results are available in the field of viscoelastic behavior of heterogeneous media. In this paper the viscoelastostatic problem of composite materials with periodic microstructure is studied. The matrix is assumed linear viscoelastic and the fibers elastic. The correspondence principle in viscoelasticity is applied and the problem in the Laplace domain is solved by using the Fourier series technique and assuming the Laplace transform of the homogenization eigenstrain piecewise constant in the space. Formulas for the Laplace transform of the relaxation functions of the composite are obtained in terms of the properties of the matrix and the fibers and in function of nine triple series which take in account the geometry of the inclusions. The inversion to the time domain of the relaxation and the creep functions of composites reinforced by long fibers is carried out analytically when the four parameters model is used to represent the viscoelastic behavior of the matrix. Finally, comparisons with experimental results are presented.

Approximation Theory: From Taylor Polynomials to Wavelets

Abstract: The book is a concisely written introduction to approximation theory. The presentation of results demonstrates the dynamic nature of mathematics and is enriched by illustrative examples, which help to develop intuition. The book consists of five chapters, each followed by exercises. The first is devoted to function approximation with polynomials. It begins with a general introduction to the idea of approximation, then deals with the Weierstrass and Taylor theorems. In the second chapter the authors focus on the approxima- tion properties related to infinite series of functions. The basic idea of signal transmission is also presented. The next chapter indicates how Fourier series and Fourier transform can be used as tools to represent functions and how pro- perties of a function are reflected in the expansion coefficients. The purpose of the last two chapters is to introduce wavelets as natural continuation of the previous material. They give the reader an understanding of the fundamental concepts, sketch the history of wavelets and present their application in signal processing. The explanation for how wavelet expansions reflect the local beha- viour of a function and how wavelets can be used to detect jumps in a signal is also provided. The book ends with an outline for the frames and Gabor systems, which are frequently used as alternatives to wavelet systems. The authors focus on ideas rather than on technical details, so that proofs are generally omitted, and just a few are included in the appendices. Minimal prerequisites (elementary calculus) make the book comprehensible to undergra- duate students of mathematics, mathematical physics and engineering. Readers are, at the same time, led towards the advanced literature in approximation theory.

Dynamic modeling of nonlinear SRM drive with Pspice

Abstract: In this paper, a comprehensive nonlinear dynamic model for a switched reluctance motor (SRM) drive using Pspice is presented. The SRM is represented by its nonlinear dynamic equations and the magnetic model of SRM includes the representation of inductance-current-position characteristics which closely match those obtained experimentally. The variation of the phase inductance with rotor position is expressed by a limited number of Fourier series terms. The coefficients of the Fourier series are determined by the values of the inductance at the aligned position, unaligned position and a position midway between the two. The nonlinear relationship between the phase inductance and the current is represented by polynomial functions whose coefficients are derived by static characteristics obtained from finite element analysis or experimental results. Since Pspice is a circuit oriented package, any type of converter and control scheme can be modeled with ease and hence the component ratings can be selected based on simulation results. Further, any type of converter/motor faults as well as phase asymmetries can also be simulated. It is also possible to simulate various control strategies for low and higher speed operations and obtain optimum control angles. The details of the developed model and the simulation results obtained for a 300 W, 12 V, 8/6 SRM drive in various operating regions are presented in this paper. To validate the model, the simulation results are compared with experimental and finite element analysis results.

Reconstruction of crack shape by optimization using eddy current field measurement

Abstract: The eddy current field perturbation due to a thin crack may be described as the field generated by a current dipole layer located on the surface of the crack. In this paper the Fourier-transform of the eddy current field by a known dipole layer is evaluated analytically if the dipole layer density function is given as, for example, a Taylor's or Fourier series. This result is used for the calculation of the impedance change of the exciting coil due to a crack by solving an integral equation. In the case of an unknown crack the measured impedance is used for reconstruction. By zeroth order optimization the shape of the crack is varied to fit the calculated impedance data to the measured ones. Several local minima of the objective function are found and statistically processed to give reliable approximation of the crack shape even in the case of sparse and noisy data. >