Application Of Fourier Series Research Articles

Application of Fourier Series for Evaluation of Roundness Profiles in Metrology

Application of Fourier Series for Evaluation of Roundness Profiles in Metrology

Applications of Fourier Series and Fourier Transformation

This paper investigates into the application of fourier transformation and series, which converts time domain signal to frequency domain signals, at which signals can be analyzed. Unlike Laplace transform, Fourier Transforms does not have full S plane, it just have the frequency j2πf plane. Fourier Transforms helps to analyze spectrum of the signals, helps in find the response of the LTI systems. (Continuous Time Fourier Transforms is for Analog signals and Discrete time Fourier Transforms is for discrete signals). Discrete Fourier Transforms are helpful in Digital signal processing for making convolution and many other signal manipulations. Overall, the paper will conclude the impact of Fourier Transforms in life.

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Conservation of time scale for one-dimensional pulsating flow

In the work by analysis of one-dimensional unsteady flows, based on the fundamental law of conservation with application of Fourier series is shown that in the presence of periodic, steady pulsations along the flow, the main frequency as well as all higher frequencies remain constant and only the amplitude of oscillations is changed that is in full agreement with the results of analysis of more complex three-dimensional flows. Thus, is confirmed the validity of the principle of conservation of frequencies or time scale along the flow. So, is obtained very interesting result for turbulence problem solution.

Application of Fourier series in managing the seasonality of convective and monsoon rainfall

ABSTRACTThis study assesses the performance of Fourier series in representing seasonal variations of the tropical rainfall process in Malaysia. Fourier series are incorporated into a spatial-temporal stochastic model in an attempt to make the model parsimonious and, at the same time, capture the annual variation of rainfall distribution. In view of Malaysia’s main rainfall regime, the model is individually fitted for two regions with distinctive rainfall profiles: one being an urban area receiving rainfall from convective activities whilst the other receives rainfall from monsoonal activities. Since both regions are susceptible to floods, the study focuses on the rainfall process at fine resolution. Fourier series equations are developed to represent the model’s parameters to describe their annual periodicity. The number of significant harmonics for each parameter is determined by inspecting the cumulative fraction of total variance explained by the significant harmonics. Results reveal that the number of significant harmonics assigned for the parameters is slightly higher in the region with monsoonal rains. The overall simulation results show that the proposed model is capable of generating tropical rainfall series from convective and monsoonal activities.Editor D. Koutsoyiannis Associate editor K. Hamed

Применение рядов Фурье для технического состояния здания

The article discusses the possible application of Fourier series to predict the technical condition of buildings. It is noted that one of the problems of the present time in Russia – a condition of the housing stock, the deterioration of which about 80 %. For the Northern territories of the country this issue is even more acute. The inspection of buildings is a significant component of the work package to predict and assess the state of buildings when making repairs, modernization and reconstruction. It is noted that an adequate model for prediction of technical state of knowledge in General or of individual structural members must consider many factors, such as the presence of periodic oscillations in the dynamics of a number of indicators caused by annual temperature fluctuations. The authors of this paper note that the use of Fourier series in solving practical problems and in forecasting various phenomena have found wide application in science and technology. It is shown that one of the possible solutions to the problem of predicting the technical condition of the building on the survey results is forecasting using Fourier series for different numbers of harmonics. It is concluded that with increasing number of harmonics, the forecast becomes the most accurate.

A study on the application of Fourier series in IMRT treatment planning

In intensity-modulated radiotherapy, a set of x-ray fluence profiles is iteratively adjusted until a desired absorbed dose distribution is obtained. The purpose of this article is to present a method that allows the optimization of fluence profiles based on the Fourier series decomposition of an initial approximation to the profile. The method has the advantage that a new fluence profile can be obtained in a precise and controlled way with the tuning of only two parameters, namely the phase of the sine and cosine terms of one of the Fourier components, in contrast to the point-by-point tuning of the profile. Also, because the method uses analytical functions, the resultant profiles do not exhibit numerical artifacts. A test case consisting of a mathematical phantom with a target wrapped around a critical structure is discussed to illustrate the algorithm. It is shown that the degree of conformality of the absorbed dose distribution can be tailored by varying the number of Fourier terms made available to the optimization algorithm. For the test case discussed here, it is shown that the number of Fourier terms to be modified depends on the number of radiation beams incident on the target but it is in general in the order of 10 terms.

Fourier Series and the cartographic line

The application of Fourier Series to cartography and GIS, in particular to shape analysis, line simplification and enhancement, and the measurement of cartographic scale, is explored. Vector representations of the Australian coastline are manipulated using Fourier techniques and the output appraised in terms of its ability to encapsulate the shape of the original coastline, or as a simplified or enhanced version of the original. Using Fourier techniques on vector representations of parts of the Australian coastline at six different cartographic scales, a relationship between cartographic scale and line complexity is found to exist, and an empirical expression of that relationship developed. The intent of developing an empirical relationship is to bring greater consistency to the process of assigning cartographic scale to data, and thereby to the process of map production, data simplification and GIS analysis. Fourier techniques are found to be a poor encapsulator of cartographic shape, a good method of line simplification and enhancement, and a promising method of measuring cartographic scale. As background, some revision of the history and mathematics of Fourier Series, and previous research into the application of Fourier Series to cartography, is given.

Applications of Fourier Series in Classical Guitar Technique

In elementary differential equations courses, the model of a (transversely) vibrating string is frequently used to motivate the one-dimensional wave equation and Fourier series. For musically-inclined students, the motivation can be strengthened by applying this model to techniques used by classical guitarists to vaiy tone quality. This note describes two such applications. The first application (harmonics) is well-known, eliciting nods of familiarity from guitar-playing students. The second application (sol tasto and sol ponticello) is less well-known outside the specific context of trained classical guitarists, but it is equally interesting, suggesting a surprisingly simple mathematical explanation for the technique where it is not immediately obvious that such an explanation exists. The physical model of a string in the xy-plane, with ends fixed at the points (0,0) and (tt,6), displaced to the shape of a curve y=f(x) at time ?=0 and then released to move in the j;-direction only (see Figure l), gives rise to the boundaiyvalue problem

Applications of Fourier Series in Classical Guitar Technique

Applications of Fourier Series in Classical Guitar Technique

Fourier analysis and its applications

This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.

Application of the mathieu series to the boundary integral method

An application of Fourier series as interpolation functions in the boundary integral method of structural analysis is presented. The stress field and the boundary displacements are independently interpolated. Equilibrium is locally satisfied and the kinematic boundary conditions are locally enforced. The compatibility and static boundary conditions are enforced through weighted residual equations. The interpolation criteria and the identification of auxiliary variables are so selected as to preserve symmetry in the governing system. The biharmonic stress function of Mathieu is used to interpolate the stress field thus ensuring that all intervening structural operators have boundary integral expressions. For regular domains, numerical integration can be avoided by providing general, closed form solutions for the boundary expressions of the structural operators. The structure of the governing system is well suited to the use of efficient algorithms to store and solve sparse systems of linear equations. Numerical testing and validation of the formulation is based on two-dimensional elastostatic problems. Macro elements are adopted to exploit the hierarchical nature of finite Fourier series.

Magnetic field of a direct current in a cylindrical-shaped conductor imbedded in a resistive half-space beneath a conductive surface layer

New applications of Fourier series and integral equation methods have yielded results of significance for electrical prospecting. This paper presents the first model calculation in three dimensions of the magnetic field due to a direct current flow in a conductor imbedded in a half-space beneath a conductive surface layer. The conductor is in the shape of a cylinder of semi-infinite length, and the current flow is from a point source in the surface layer. The vertical component of the magnetic field is given here by a Fourier series integral over a function which satisfies a Cauchy singular integral equation. For the two special cases, namely, an outcropping cylinder with no surface layer and a nearly invisible (in the sense of negligible conductivity contrast with its surrounding) cylinder, this integral equation can be solved exactly to yield closed-form expressions for the vertical component of the magnetic field. The exact solution in the first special case is based on a new generalization of the Hankel integral transform (published separately), and it extends the class of exactly solvable singular integral equations. This exact solution has also yielded as useful byproducts two new standard integrals involving Bessel functions. The integral equation approach used here, in contrast to the well-established surface integral equation approach, is suitable for numerical calculations, and typical magnetic field profiles are presented. The magnetic field results given here would be valuable for interpretation in electrical prospecting (based on the measurement of magnetic fields of direct currents), and the mathematical techniques used here would be useful for future calculations in electromagnetism.

Application of fourier series in the calculations of the nonlinear deformation of a circular ring of variable rigidity

Application of fourier series in the calculations of the nonlinear deformation of a circular ring of variable rigidity

Solution of a magnetoviscous channel flow problem using integral transforms

A previous paper establishes a general treatment of integral transforms on the basis of Sturm‐Liouville theory and associated eigenfunction expansions [1]. In this paper the general theory is applied to the problem of flow under uniform pressure gradient of an electrically conducting incompressible viscous fluid through a uniform channel across which an initially uniform transverse magnetic field acts. The treatment is an alternative to the application of Fourier series which is often made.

An Approximate Solution to the Shuttle Heat-Transfer Losses in a Reciprocating Machine

The axial heat transfer due to the oscillatory motion of a long displacer in a cylinder with an axial temperature gradient is treated. The effect of the resistance to heat transfer due to the radial clearance between the displacer and the cylinder is considered, as well as the effect of a limited thermal diffusivity in the walls. A method for the estimation of the effect of the gas flow in the clearance volume is presented. A simple approximate solution is obtained by linearization and application of Fourier series. The effect of radial clearance, material properties, length, and wall thickness on the net heat transfer rate is discussed.

Application of Fourier Series to Summation of Series, I

Application of Fourier Series to Summation of Series, I

Application of Fourier Series to Summation of Series, I

Application of Fourier Series to Summation of Series, I

Space Requirements for Equipment Items Subjected to Random Excitation

If an equipment item is supported on a low-frequency mounting, there must be sufficient space allowance around the item to permit unrestricted vibratory motion. Also, the motion of the equipment item must be within the allocated travel of the mount. If the system is linear, the required space can be computed for random motion by the application of Fourier series. The solution includes motion of the adjacent structure and phase angles. After a review of basic random theory showing the application of random theory to design, an expression is developed for calculating the space required for an equipment item to vibrate. A numerical example is included to illustrate a method for estimating structural response and structural phase.

Some Applications of Fourier Series in the Numerical Treatment of Linear Behaviour

Earlier work is here continued. Certain functional relations common to various phenomenological theories, like the theory of linear visco-elasticity and the theory of linear electrical networks, may be written as integral equations of the convolution type. As the circular functions are eigenfunctions of the convolution type kernels, such equations can be solved by Fourier expansion. A number of possible applications of such Fourier expansions are indicated and the relevant details are worked out. The method centres on the evaluation of the eigenvalues for each kernel. The examples discussed refer to the theory of visco-elasticity and include the determination of the relaxation spectrum either from the dynamic elasticity as a function of frequency or from the stress relaxation as a function of time.