Fourier Series Research Articles

Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices.

Thermoelastic response of multilayer cylinders: The direct integration and single-solid approach

An approach for solving the plane nonaxisymmetric thermoelasticity problems in multilayer cylinders is suggested from the single-solid standpoint, which implies considering a multilayer structure as a single unity with a step-wise variation of material properties. The direct integration method was used to the advantage of our approach by applying toward the compatibility equation in terms of strains instead of the one in terms of stresses. As a result, the governing equation is derived without involving generalized derivatives by the radial coordinate and then reduced to an integral equation of the second kind. Using the resolvent-kernel for solving the latter equation, the thermal stresses are computed as the Fourier series whose constituents are explicitly determined by the corresponding constituents of the thermal loading.

Determining the optimal sampling interval for 3D morphology measurements of different-sized natural rock joints

Efficiently studying the scale effect of surface roughness and peak shear strength of natural rock joints, especially unfilled and well-matched ones, requires an optimal sampling interval to realize cost-effective three-dimensional morphology measurements for different-sized natural rock joints with advanced non-contact optical devices. However, rationally determining such an optimal sampling interval remains controversial. To this end, inspired by digital signal processing, the optimal sampling interval is creatively determined by Shannon's sampling theorem using the Fourier series' optimum order. An empirical formula relating the joint size to the optimum order is correspondingly developed. The proposed empirical formula was found to work well under conditions of high normal stress, where the relative errors for the surface roughness and peak shear strength of series-sized natural rock joints were mostly less than 10% and 5%, respectively, but should be used cautiously under conditions of low normal stress. Furthermore, different from previous perceptions, a higher joint roughness level was discovered to not necessarily imply the need for a smaller optimal sampling interval, and conversely. The proposed empirical formula can facilitate the determination of the optimal sampling interval for measuring different-sized natural rock joints' surface morphologies in engineering practice.

RPAS operations in urban airflow: Efficient modelling of representative wind speed variations along a flight path through a flow field with changing turbulence characteristics

For remotely piloted aircraft systems (RPAS) operating in the urban environment, the flight path will most likely pass through the turbulent airflow of building wakes. To attenuate or reject instabilities caused by the turbulent airflow, control systems strategies are being developed in preparation for increased demand on the smaller urban air mobility vehicles. In this paper, results from urban airflow measurements of four 1:300 scaled Canadian city model flow fields are examined to provide detailed understanding of flow characteristics that can be found in typical Canadian cities. The results show common patterns for the range of inner city wind speed and turbulence intensity but that within the urban boundary layer (UBL) the variation of flow characteristics does not have a gradient with respect to altitude and that flow characteristics are highly variable along representative flight paths. For the purpose of simulation of RPAS response to the complex urban airflow, methods for using real urban flows data as inputs to simulations is presented. Demonstration of an Advancing Fourier Series (AFS) method, using the measured city data, shows successful preservation of the spatial- and temporal-correlations for reconstruction of a flow time series offering a novel, efficient and flexible approach for implementation of a time-based simulation of RPAS response in complex flow environments such as the urban environment.

Comparison of approximation methods for bending of finite and infinite elastic plates: The traction boundary value problem

A generalized Fourier series method is designed to approximate the solution of the Neumann problem in a finite domain and in an infinite domain with a hole, for the mathematical model of bending of elastic plates with transverse shear deformation. The results in the two cases are compared and contrasted, and the analysis is illustrated by examples with specific boundary conditions.

On the discrepancies between POD and Fourier modes on aperiodic domains

The application of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. In this approach to turbulence decomposition, which has recently been termed Spectral POD (SPOD), Fourier modes are considered as solutions to the corresponding Fredholm integral equation of the second kind along homogeneous-periodic or homogeneous coordinates. In the present work, the notion that the POD modes formally converge to Fourier modes for increasing domain length is challenged. A family of analytical correlation functions parametrized by the Taylor macro/micro scale ratio (MMSR) is investigated numerically. The results of the analysis indicate that the discrepancy between POD and Fourier modes along locally translationally invariant coordinates is coupled to the MMSR of the flow. Increasing discrepancies are observed for smaller MMSRs, which are characteristic of low Reynolds number flows. The asymptotic convergence rate of the eigenspectrum matches the corresponding convergence rate of the exact analytical Fourier spectrum of the kernel in question, even for extremely small domains and small MMSRs where the corresponding DFT spectra suffer heavily from windowing effects. The Taylor micro scales are consistently underestimated when reconstructed using Fourier modes—failing to converge to the correct value even if all Fourier modes are used—while these are accurately reconstructed using POD modes.

Discretely Distributed Scheduled Jumps and Interest Rate Derivatives: Pricing in the Context of Central Bank Actions

Interest rate dynamics are influenced by various economic factors, and central bank meetings play a crucial role concerning this subject matter. This study introduces a novel approach to modeling interest rates, focusing on the impact of central banks’ scheduled interventions and their implications for pricing bonds and path-dependent derivatives. We utilize a modified Skellam probability distribution to address the discrete nature of scheduled interest rate jumps and combine them with affine jump-diffusions (AJDs) in order to realistically represent interest rates. We name this class the AJD–Skellam models. Within this class, we provide closed-form formulas for the characteristic functions of a still broad class of interest rate models. The AJD–Skellam models are well-suited for using the interest rate version of the Fourier-cosine series (COS) method for fast and efficient interest rate derivative pricing. Our methodology incorporates this method. The results obtained in the paper demonstrate enhanced accuracy in capturing market behaviors and in pricing interest rate products compared to traditional diffusion models with random jumps. Furthermore, we highlight the applicability of the model to risk management and its potential for broader financial analysis.

Model Parameters Identification and Backstepping Control of Lower Limb Exoskeleton Based on Enhanced Whale Algorithm

Exoskeletons generally require accurate dynamic models to design the model-based controller conveniently under the human-robot interaction condition. However, due to unknown model parameters such as the mass, moment of inertia and mechanical size, the dynamic model of exoskeletons is difficult to construct. Hence, an enhanced whale optimization algorithm (EWOA) is proposed to identify the exoskeleton model parameters. Meanwhile, the periodic excitation trajectories are designed by finite Fourier series to input the desired position demand of exoskeletons with mechanical physical constraints. Then a backstepping controller based on the identified model is adopted to improve the human-robot wearable comfortable performance under cooperative motion. Finally, the proposed Model parameters identification and control are verified by a two-DOF exoskeletons platform. The knee joint motion achieves a steady-state response after 0.5 s. Meanwhile, the position error of hip joint response is less than 0.03 rad after 0.9 s. In addition, the steady-state human-robot interaction torque of the two joints is constrained within 15 N·m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{N}\\cdot\ ext{m}}$$\\end{document}. This research proposes a whale optimization algorithm to optimize the excitation trajectory and identify model parameters. Furthermore, an enhanced mutation strategy is adopted to avoid whale evolution's unsatisfactory local optimal value.

Boolean Function Analysis on High-Dimensional Expanders

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only |X(k-1)|=O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X(k-1)|=O(n)$$\\end{document} points in contrast to nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) $$\\end{document} points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Shape-based method for low-thrust transfers between periodic orbits in cislunar space

Low-thrust trajectory design in the context of the three-body problem, particularly in the vicinity of liberation points, is addressed in this paper with a focus on efficiency and accuracy. The Fourier series, previously successful in approximating near-coplanar low-thrust trajectories within two-body dynamics, is applied to the challenging landscape of the three-body problem, marked by instability and nonlinearity. This paper introduces a novel approach for constructing an initial guess trajectory, that combines coast arcs from periodic orbits and intermediate trajectory arcs from other natural dynamical structures. The resulting initial guess trajectory is instrumental in generating efficient initial estimates for the coefficients of the Fourier series. The study encompasses the analysis of homoclinic and heteroclinic transfers in both two-dimensional and three-dimensional scenarios. The approximated trajectories derived from this methodology serve as invaluable starting points for high-fidelity optimization solvers, offering promise for enhancing the precision and efficiency of low-thrust transfers in cislunar space.

Discover an accurate approximation of dynamical system without prior information and customized design

Modeling dynamical systems is fraught with challenges when data can be collected but thorough analysis of the mechanism is absent. We design a method to discover unknown dynamical systems from data. The method discovers an accurate approximation of the model without the prior information and the customized design for each problem. The identification steps are straightforward as bringing in the data and then obtaining the model. The method begins with the simple idea that the equations of motion of many practical problems are Riemann integrable functions. For this reason, the Fourier series can decompose the equations of motion. In order to improve the accuracy, we design an extension that helps us to approximate unknown functions by the Fourier series with a high rate of convergence. The idea converts the difficulty of modeling the dynamical system into finding its Fourier series approximation. Convenient procedures enable the modeling of different problems. Numerical examples show that the new method discovers linear and nonlinear dynamical systems in the same steps and without the prior information.

Heart failure prognostic algorithm using Spectral Analysis and MATLAB software.

Fourier analysis for biological signals is based on the use of the infinite sum of sines and cosines that allows modeling: the periodic functioning of the heart, its amplitude, frequency, and phase period, transforming these signals into images called ECG, based on studies and programs that model the ideal functioning of the heart. In this work, a mathematical algorithm has been designed to predict the cardiac pathology called bradycardia, which relates the prolongation of the monthly QT interval in the order of 10-4 seconds/month in a time of 10 years with the ventricular alteration. MATLAB software and spectral analysis are used to contrast the spectrum without pathology, which contains harmonics of greater amplitude, with a spectrum already with pathology that reaches heart failure, where the most significant number of harmonics are grouped in the first values, and then the model with an exponential function the delay of the QT interval in the ECG, concluding that up to 40 months after the onset of the pathology, the patient can counteract the disease. In comparison, by 80 months, difficulties arise, even the disease becomes irreversible in the last months, and the blood-propelling organ ceases to function. Keywords: Heart Failure Prognostic, Bradycardia, Fourier Series, Spectral Analysis.

Heart failure prognostic algorithm using Spectral Analysis and MATLAB software.

Fourier analysis for biological signals is based on the use of the infinite sum of sines and cosines that allows modeling: the periodic functioning of the heart, its amplitude, frequency, and phase period, transforming these signals into images called ECG, based on studies and programs that model the ideal functioning of the heart. In this work, a mathematical algorithm has been designed to predict the cardiac pathology called bradycardia, which relates the prolongation of the monthly QT interval in the order of 10-4 seconds/month in a time of 10 years with the ventricular alteration. MATLAB software and spectral analysis are used to contrast the spectrum without pathology, which contains harmonics of greater amplitude, with a spectrum already with pathology that reaches heart failure, where the most significant number of harmonics are grouped in the first values, and then the model with an exponential function the delay of the QT interval in the ECG, concluding that up to 40 months after the onset of the pathology, the patient can counteract the disease. In comparison, by 80 months, difficulties arise, even the disease becomes irreversible in the last months, and the blood-propelling organ ceases to function. Keywords: Heart Failure Prognostic, Bradycardia, Fourier Series, Spectral Analysis.

An Overview of Electricity Consumption in Europe: Models for Prediction of the Electricity Usage for Heating and Cooling

Although aggregate electricity consumption provides valuable information for market analysis, demand composition, including industrial, residential, illumination, and other uses, and special days, such as national or religious holidays and annual industrial shutdowns, differ for each country. This paper analyzes the hourly electricity consumption of European countries in the European Transmission System Operation for Electricity (ENTSO-E) grid from 2006 to 2018. We propose an outlier detection method to identify special days and a modulated Fourier Series Expansion model to determine the breakdown of industrial versus household consumption and heating versus cooling consumption. The proposed outlier detection method uses the time series for each hour and checks whether a day has more than a threshold number of hours with exceptional electricity consumption levels. The proposed demand prediction model has a 3% average error when electricity usage for heating is not dominant. It also allows country classification based on consumption patterns to efficiently manage regional or country-based electricity markets.

Application of the inverse Laplace transform techniques to solve the generalized Bagley–Torvik equation including Caputo’s fractional derivative

This article employs the Laplace transform approach to solve the Bagley–Torvik equation including Caputo’s fractional derivative. Laplace transform is a powerful method for enabling solving integer and non-integer order ODEs and PDEs in engineering and science. Using the Laplace transform for solving integer and non-integer order ODEs and PDEs, however, sometimes leads to solutions in the Laplace domain that are difficult to convert back into the time domain using analytical techniques. Therefore, the best option to convert the Laplace domain solution back into real domain is to use numerical inversion techniques. But it is important to remember that the inverse Laplace transform is not a well-posed problem. There is not a single, general approach that can address every challenge in this matter. Numerous algorithms are devised for numerically inverting the Laplace transform. In this paper, we have evaluated three numerical inverse Laplace transform techniques. These methods include the Fourier series method, the Gauss–Hermite quadrature method, and Zakian’s method. Different test problems are used to evaluate these inversion methods. The effectiveness and efficiency of these methods are demonstrated through graphical representations and tabular data. The comparison of the obtained results with the results of other methods available in literature led us to conclude that the proposed methods exhibit rapid convergence with optimal accuracy without any time instability.

Long-term Load Forecasting for Optimal Power System Planning and Decision-Making

Forecasting the future load growth of an area based on its load demand is often a proactive measure to ensure a steady electricity power supply to that area. The study focused on long-term load forecasting for power system planning, specifically examining the electric load demand from consumers on distribution transformers within Port Harcourt City, located in Rivers State, Nigeria. The study encompassed a comprehensive review of both statistical and artificial intelligence-based approaches. Historical load data for distribution transformer readings spanning 2008 to 2017 were acquired from the Port Harcourt Electricity Distribution Company (PHEDC) and subjected to analysis using the curve-fitting technique. For the period between 2015 and 2030, a yearly load forecast simulation was conducted using the Fourier Series model, implemented with MATLAB software. This simulation aimed to provide insights into future load demand, facilitating careful and informed decision-making in the investment, operation, and maintenance of power system equipment. The effectiveness of the forecasting investigation was assessed using the Root Mean Square Error (RMSE), confirming the efficiency, reliability, and validity of the employed model. The study's forecasted results are presented as a valuable guide and practical tool for policymakers and the utility company (PHEDC) to enhance proper planning and decision-making processes. Considering the observed trend in the results, it is suggested that installing additional transformer units in the region would be necessary to alleviate the loads on existing overloaded transformer units within the power system network.

Free Vıbratıon of a Carbon Nanotube-Reınforced Nanowıre/Nanobeam wıth Movable Ends

ObjectiveThe purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter.MethodsIn this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends.ConclusionsThe eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

The conditional Lyapunov exponents and synchronisation of rotating turbulent flows

The synchronisation between rotating turbulent flows in periodic boxes is investigated numerically. The flows are coupled via a master–slave coupling, taking the Fourier modes with wavenumber below a given value $k_m$ as the master modes. It is found that synchronisation happens when $k_m$ exceeds a threshold value $k_c$ , and $k_c$ depends strongly on the forcing scheme. In rotating Kolmogorov flows, $k_c\eta$ does not change with rotation in the range of rotation rates considered, $\eta$ being the Kolmogorov length scale. Even though the energy spectrum has a steeper slope, the value of $k_c\eta$ is the same as that found in isotropic turbulence. In flows driven by a forcing term maintaining constant energy injection rate, synchronisation becomes easier when rotation is stronger. Here, $k_c\eta$ decreases with rotation, and it is reduced significantly for strong rotations when the slope of the energy spectrum approaches $-3$ . It is shown that the conditional Lyapunov exponent for a given $k_m$ is reduced by rotation in the flows driven by the second type of forcing, but it increases mildly with rotation for the Kolmogorov flows. The local conditional Lyapunov exponents fluctuate more strongly as rotation is increased, although synchronisation occurs as long as the average conditional Lyapunov exponents are negative. We also look for the relationship between $k_c$ and the energy spectra of the Lyapunov vectors. We find that the spectra always seem to peak at approximately $k_c$ , and synchronisation fails when the energy spectra of the conditional Lyapunov vectors have a local maximum in the slaved modes.