Fourier Series Research Articles

A semi-analytical spectral element model for guided wave propagation in composite laminated conical shells

The conical shell is a typical non-uniform waveguide with its circumferential radius varies linearly. This work aims to provide physical insights for the composite laminated conical shell in terms of waves. A semi-analytical spectral element model is presented, which adopts the Fourier series in the circumferential direction and the higher-order shape functions in the axial direction. The first-order shear deformation theory and Hamilton principle are employed to derive the energy functions of the conical shell, and they are discretized using the spectral elements to achieve a weak-form variational problem. The dispersion equation and the group velocity expression are derived via the piecewise approximation concept and Bloch's periodic theorem. The accuracy of the developed spectral element is verified by comparison with published results. The deformation evolution mechanisms of the composite conical shell are revealed, and the wave propagation characteristics of uniform and ply drop-off composite conical shells are investigated. The results demonstrated that wavenumber distribution at 0–5 kHz is spatially dependent, and wavenumber becomes cut off and bifurcate near the cone vertex. When the frequency exceeds 2 kHz, the deformation of conical shell changes from rigid to flexible, and the structure tends to be a uniform waveguide. Compared to the geometric curvature effect, the asymmetric lay-up leads to more significant mode coupling.

Approximation of function in the Generalized Hölder Class using (TC^δ) means for Fourier Series and Conjugate Fourier Series

Approximation of function in the Generalized Hölder Class using (TC^δ) means for Fourier Series and Conjugate Fourier Series

Effect of spatio-temporal internal heat source on the thermal convection in a porous layer using a thermal non-equilibrium model

Convection induced by internal heating and adverse temperature gradient in a porous medium has various real-world applications, such as solar energy collectors, solar-based drying and cooking. Thus, it is crucial to analyze such flows to understand the physics behind the flow and analyze heat transfer through the system. We assume that volumetric heating occurs within the fluid phase due to radiation exposure from an overhead source or a stratified arrangement of heat-generating materials. For modeling convenience, we further assume that internal heating diminishes exponentially with the depth of the porous layer and fluctuates sinusoidally over time, affecting the mean value. Solid and fluid phases of the porous layer are considered out of thermal equilibrium. Floquet theory is employed to determine the linear instability bound of the system. By analyzing the eigenvalue spectrum, we identify the critical Rayleigh number that marks the onset of instability. Subsequently, a weakly nonlinear analysis is carried out using a truncated Fourier series expansion of the physical quantities to analyze the heat transfer within the system. The effect of governing nondimensional parameters on the system’s stability and heat transport is graphically illustrated and thoroughly discussed. The effect of modulation amplitude and volumetric heat is to enhance heat transfer, whereas depth coefficient and modulation frequency suppress heat transfer.

Pulse wave signal-driven machine learning for identifying left ventricular enlargement in heart failure patients

BackgroundLeft ventricular enlargement (LVE) is a common manifestation of cardiac remodeling that is closely associated with cardiac dysfunction, heart failure (HF), and arrhythmias. This study aimed to propose a machine learning (ML)-based strategy to identify LVE in HF patients by means of pulse wave signals.MethodWe constructed two high-quality pulse wave datasets comprising a non-LVE group and an LVE group based on the 264 HF patients. Fourier series calculations were employed to determine if significant frequency differences existed between the two datasets, thereby ensuring their validity. Then, the ML-based identification was undertaken by means of classification and regression models: a weighted random forest model was employed for binary classification of the datasets, and a densely connected convolutional network was utilized to directly estimate the left ventricular diastolic diameter index (LVDdI) through regression. Finally, the accuracy of the two models was validated by comparing their results with clinical measurements, using accuracy and the area under the receiver operating characteristic curve (AUC-ROC) to assess their capability for identifying LVE patients.ResultsThe classification model exhibited superior performance with an accuracy of 0.91 and an AUC-ROC of 0.93. The regression model achieved an accuracy of 0.88 and an AUC-ROC of 0.89, indicating that both models can quickly and accurately identify LVE in HF patients.ConclusionThe proposed ML methods are verified to achieve effective classification and regression with good performance for identifying LVE in HF patients based on pulse wave signals. This study thus demonstrates the feasibility and potential of the ML-based strategy for clinical practice while offering an effective and robust tool for diagnosing and intervening ventricular remodeling.

UHV AC/DC power grid geomagnetically induced currents harmonic characteristics analysis

The power grid’s geomagnetically induced currents (GIC) during geomagnetic storms, are associated with a broad and abrupt harmonic impact, with the potential for grid paralysis and extensive power outages in severe scenarios. Therefore, the harmonic characteristics of geomagnetically induced currents in China’s ultra-high voltage (UHV) AC/DC power grid are analyzed in this paper. Firstly, a harmonic equivalent model of transformer GIC is established, and the injection of harmonics into the UHV AC/DC power grid during GIC events is quantified.Secondly, a harmonic state-space model is established by combining the 12-pulse converter switching function theory with the exponential Fourier series. The harmonic transmission and dynamic coupling characteristics of GIC in the UHV AC/DC power grid are analyzed using the harmonic equivalent model of the main transformer GIC and the harmonic equivalent model of the converter transformer GIC. Finally, a harmonic simulation model of the UHV AC/DC power grid under GIC is constructed using PSCAD, and the distribution of harmonics in the UHV AC/DC power grid under GIC is calculated. The calculations indicate that with 30 A injected GIC in the sending end transformers, the 2nd harmonic current on the sending end network side alone is 61.45 A, and the 3rd harmonic voltage on the valve side is 8.57 kV. The characteristics of harmonic transmission in the power grid result in a 2nd harmonic current of 34.87 A on the receiving end network side. Simultaneously, the total harmonic distortion of the AC power grid at the sending end is 2.35 %, surpassing the permissible limit established by the national standard. Consequently, this issue should be prioritized by the relevant power authorities.

Cosmic microwave background in a non-Archimedean world

The universe begins its expansion from dimensions below the Planck scale, where the uncertainty principle reigns. If at these dimensions the Archimedean geometry is useless, the p-adic non-Archimedean world creates an environment conducive to phenomenological modeling. The standard description of the cosmological density contrast is made by the Fourier decomposition of plane waves in the field of p-adic numbers Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. The coupling of the primordial perturbations is done by coupling the phases. Since the phases are the stars of this article, we represent them in the hue-saturation-brightness color space both in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} and in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. If the primordial perturbations interact weakly with each other, with their growth they become important enough from an energetic point of view to gravitationally interact with each other. We identify the coupling using the phase gradient operator, and extract the coupled points from the spectrum. We use pseudoentropy for the quantitative analysis of the information generated by the phases.

Improved bridge modal identification from vibration measurements using a hybrid empirical Fourier decomposition

Bridge health monitoring has been a prominent focus within the global engineering community. Bridge owners, stakeholders, and engineers face the formidable tasks of ensuring efficient monitoring, conducting reliable data analysis, interpreting data logically, and making timely decisions. With the increasing global infrastructure deficit, there is an ever-increasing need to develop reliable and economical bridge monitoring solutions. In this paper, a bridge condition assessment technique is proposed that can utilize the vibration data collected from the instrumented sensors and provide reliable system identification results. The proposed method develops a hybrid approach by integrating the Natural Excitation Technique (NExT) and Empirical Fourier Decomposition (EFD) to analyze ambient bridge vibration data and determine the modal parameters of the bridge. First, NExT is formulated to determine the cross-correlation functions of the bridge measurements, and then EFD is explored to decompose the signals into their monocomponents to identify the bridge modal parameters. The proposed methodology can overcome mode mixing and perform modal identification of a system with closely spaced frequencies and low energy modes. The estimated modal parameters such as bridge frequencies, mode shapes, and damping ratio are used for condition assessment of numerical, experimental and full-scale structures, including a short-span steel bridge located in Ontario, Canada. The results demonstrate that the proposed methodology can provide accurate and robust estimates of bridge modal parameters. Future research is reserved for real-time implementation of the proposed methodology for a wide range of civil structures.

Three-Dimensional Phase Resolved Functional Lung Magnetic Resonance Imaging.

Pulmonary magnetic resonance imaging (MRI) offers a variety of radiation-free techniques tailored to assess regional lung ventilation or its surrogates. These techniques encompass direct measurements, exemplified by hyperpolarized gas MRI and fluorinated gas MRI, as well as indirect measurements facilitated by oxygen-enhanced MRI and proton-based Fourier decomposition (FD) MRI. In recent times, there has been substantial progress in the field of FD MRI, which involved improving spatial/temporal resolution, refining sequence design and postprocessing, and developing a comprehensive whole-lung approach. The two-dimensional (2D) phase-resolved functional lung (PREFUL) MRI stands out as an FD-based approach developed for the comprehensive assessment of regional ventilation and perfusion dynamics, all within a single MR acquisition. Recently, a new advancement has been made with the development of 3D PREFUL to assess dynamic ventilation of the entire lung using 8 min exam with a self-gated sequence. The 3D PREFUL acquisition involves employing a stack-of-stars spoiled-gradient-echo sequence with a golden angle increment. Following the compressed sensing image reconstruction of approximately 40 breathing phases, all the reconstructed respiratory-resolved images undergo registration onto a fixed breathing phase. Subsequently, the ventilation parameters are extracted from the registered images. In a study cohort comprising healthy volunteers and patients with chronic obstructive pulmonary disease, the 3D PREFUL ventilation parameters demonstrated strong correlations with measurements obtained from pulmonary function tests. Additionally, the interscan repeatability of the 3D PREFUL technique was deemed to be acceptable, indicating its reliability for repeated assessments of the same individuals. In summary, 3D PREFUL ventilation MRI provides a whole lung coverage and captures ventilation dynamics with enhanced spatial resolution compared to 2D PREFUL. 3D PREFUL technique offers a cost-effective alternative to hyperpolarized 129Xe MRI, making it an attractive option for patient-friendly evaluation of pulmonary ventilation.

The effect of different edge conditions on the motion of a submerged elastic disc

The present study examines the effects of various edge conditions of a submerged flexible disc on wave propagation for infinite-depth water within the framework of linear water waves theory. Three types of edge conditions (namely Free edge, simply supported edge, and clamped edge) are taken into consideration for the analysis. The governing boundary value problem (BVP) has been solved by reducing it to a two-dimensional hypersingular integral equation. The notion of modal analysis has been adopted to examine the structural response of the disc on the propagation of waves. Later, the two-dimensional hypersingular integral has been transformed into a second kind one-dimensional Fredholm integral equation by applying Fourier series expansion. Finally, the Nystrom technique based on Gauss–Legendre quadrature nodes is used to obtain an approximate solution of the one-dimensional integral equation. The computed solution is used to evaluate the numerical estimates of the physical quantities such as added mass, damping coefficient, hydrodynamic force, and surface elevation for different edge conditions mentioned earlier.

A Fourier series model and parametric study for single person three-way continuous walking load

A Fourier series model and parametric study for single person three-way continuous walking load

Orthonormal rational functions on a semi-infinite interval

In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence and find the associated Sturm-Liouville problems, which bring orthogonality. Then we establish a Hilbert space of functions defined on a semi-infinite interval with an inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. The even subsequence of the irrational function sequence above forms an orthonormal basis for this space. The Fourier expansions of some power functions are deduced. The sequence of non-positive integer power functions forms a non-orthogonal basis for this Hilbert space. We give two examples, one example of Fourier expansion and one example of interpolation as applications.

Poles, shocks, and tygers: The time-reversible Burgers equation.

We construct a formally time-reversible, one-dimensional forced Burgers equationby imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equationand can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equationare equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.

The modeling method for vibration characteristics analysis of composite laminated rotationally stiffened shell.

The composite laminated rotationally stiffened shell is widely applied in aviation, aerospace, ship, machinery and other fields. To investigate the vibration characteristics of composite laminated rotationally stiffened shells with varying elastic boundary conditions, a modeling method of composite laminated rotationally stiffened shells is established. Firstly, the first-order shear deformation theory (FSDT) and the modified Fourier series method are effectively applied to establish the allowable displacement function of the composite laminated rotationally stiffened shell. Secondly, the energy function of composite laminated rotationally stiffened shell is established, and the simulation of complex elastic boundary and coupling boundary is realized by using artificial virtual spring technology. Thirdly, the Rayleigh-Ritz method is used to solve the energy function. Finally, the vibration characteristics of composite laminated rotationally stiffened shells are obtained and analyzed. In the analysis of numerical results, the fast and uniform convergence of analysis modeling and the accuracy of the calculated results are verified. On this basis, the effect of some important parameters such as thickness-to-radius ratio and length-to-radius ratio of shell, boundary spring stiffness values, cone apex angle, thickness and width of laminated beams, number of stiffeners on the vibration characteristics of composite laminated rotationally stiffened shell is studied. In theory, it makes up for the vibration characteristics analysis of composite laminated rotationally stiffened shells. In practical application, it guides the noise reduction design of related structures.

Absolute zeta functions arising from ceiling and floor Puiseux polynomials

For the [Formula: see text]-lift [Formula: see text] of a monoid scheme [Formula: see text] of finite type, Deitmar et al. calculated its absolute zeta function by interpolating [Formula: see text] for all prime powers [Formula: see text] using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence [Formula: see text], which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme [Formula: see text] of finite type over [Formula: see text] by means of a pair of Puiseux polynomials which estimate “[Formula: see text]” for sufficiently large [Formula: see text]. We call them the ceiling and floor Puiseux polynomials of [Formula: see text]. In particular, if [Formula: see text] is an elliptic curve, then our absolute zeta functions of [Formula: see text] do not depend on its isogeny class.

Non-polynomial zig-zag unified shear deformation formulation solved via boundary discontinuous method for laminated sandwich plates

Analytical solutions of mechanical problems with refined theories and challenge boundary conditions are still a topic of research due to intrinsic complexity. In this paper, a generalized refined theory is formulated to obtain analytical solutions for the static problem of cross-ply laminated and sandwich plates with several boundary conditions. These theories are implemented in the framework of the Carrera unified formulation (CUF) based on the equivalent-single-layer (ESL) approach. The displacement field of ESL-based models are expanded over the thickness direction by using non-polynomial terms, e.g. trigonometric, hyperbolic and the so-called zig-zag (ZZ) Murakami’s functions. The principle of virtual displacements (PVD) statement is used to derive the strong form of the governing equations in terms of displacements variables. These equations are solved by the boundary-discontinuous double Fourier series approach which provides highly accurate analytical solutions. This work gives a detailed comparison of results obtained using the presented ZZ models highlighting the differences with 3D FEM results. Furthermore, the influence of non-polynomial and ZZ terms in each ESL model is investigated. The superior robustness of the presented methodology compared to past works is demonstrated, making the proposed results suitable as a benchmark for validating new theories and finite elements. The findings can be also useful to train artificial intelligence (AI) models that can allow to the development of faster digital twin structure technology.

Comprehensive Characterisation of a Low-Frequency-Vibration Energy Harvester.

In this paper, we describe a measurement procedure to fully characterise a novel vibration energy harvester operating in the ultra-low-frequency range. The procedure, which is more thorough than those usually found in the literature, comprises three main stages: modelling, experimental characterisation and parameter identification. Modelling is accomplished in two alternative ways, a physical model (white box) and a mixed one (black box), which model the magnetic interaction via Fourier series. The experimental measurements include not only the input (acceleration)-output (energy) response but also the (internal) dynamic behaviour of the system, making use of a synchronised image processing and signal acquisition system. The identification procedure, based on maximum likelihood, estimates all the relevant parameters to characterise the system to simulate its behaviour and helps to optimise its performance. While the method is custom-designed for a particular harvester, the comprehensive approach and most of its procedures can be applied to similar harvesters.

Calibration Optimization of Kinematics and Dynamics for Delta Robot Driven by Integrated Joints in Machining Task

For the application of Delta robots with a 3-R(RPaR) configuration in machining tasks, this paper constructed a 54-parameter kinematic error model and a simplified dynamic model incorporating an integrated joint’s position error and friction, respectively. Utilizing Singular Value Decomposition (SVD) of the Linear Model Coefficient Matrix (LMCM) and the coefficient chart, a criterion for identifiability of error components is established. For good identification results, the optimal measurement surface with Fourier series form is obtained using a combination of the Hook–Jeeves Direct Search Algorithm (DSA) and Inner Point Method (IPM). The friction coefficients and other dynamic parameters are obtained through fitting the integrated joint torque-angle pairs measured along specific trajectories using nonlinear least squares regression. The validation of the calibration process is conducted through simulations and experiments. The calibration results provide a foundation for the precise control of integrated joints and the high-precision motion of robots.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

Price Prediction of Staples with Simultaneous Fourier Series Estimator as a Contribution to Solve the National Food Problem

This study provides an overview related to the prediction of prices for staples that are widely consumed by the Indonesian people and often experience price fluctuations at certain times, namely cooking oil, chili, and chicken eggs. Based on the Market Monitoring System and Staples of the Ministry of Trade, the national average price of cooking oil as of January 3, 2022, reached USD 1.17per liter. The average price of chicken eggs reached USD 1,98 per kilogram, an increase of 20.16% at the same time. In addition, based on the National Strategic Food Price Information Center, the average price of chili in traditional markets throughout the province rose by USD 0.19 or about 3.16% and reached USD 6.29 per kilogram. The prediction of staple food prices in this study was formed using a nonparametric regression model of the Fourier series estimator. This regression model has high flexibility in forming the model and estimating the fluctuation data pattern, which is not known in the form of time-series data. The data in this study is sourced from the National Strategic Food Price Information Center website with the response variables used being the daily prices of cooking oil, chicken eggs, and chili from September 1, 2021, to December 31, 2021, as training data. In addition, the same commodity data from January 1, 2022, to January 10, 2022, is used as testing data. The best model from the estimation results with the Fourier series is a model with an oscillation parameter of 25 which contains a sine cosine basis with a coefficient of determination of 83.05%and an MSE value of 1.628. The selected model also has a very good performance predicting the prices of the three staples with a MAPE value of 1.082%.

Analytic solution for two dimensional beam problems: Pure displacement boundary conditions

The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.