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Time series biomechanical data inform our understanding of normal gait mechanics and pathomechanics. This study examines the utility of different quantitative methods to distinguish vertical ground reaction forces (VGRFs) from experimentally distinct gait strategies. The goals of this study are to compare measures of VGRF data—using the shape factor method and a Fourier series-based analysis—to (1) describe how these methods reflect and distinguish gait patterns and (2) determine which Fourier series coefficients discriminate normal walking, with a relatively stiff-legged gait, from compliant walking, using deep knee flexion and limited vertical oscillation. This study includes a reanalysis of previously presented VGRF data. We applied the shape factor method and fit 3- to 8-term Fourier series to zero-padded VGRF data. We compared VGRF renderings using Euclidean L2 distances and correlations stratified by gait strategy. Euclidean L2 distances improved with additional harmonics, with limited improvement after the seventh term. Euclidean L2 distances were greater in shape factor versus Fourier series renderings. In the 8 harmonic model, amplitudes of 9 Fourier coefficients—which contribute to VGRF features including peak and local minimum amplitudes and limb loading rates—were different between normal and compliant walking. The results suggest that Fourier series-based methods distinguish between gait strategies. [ABSTRACT FROM AUTHOR]

This research aims to explore the free vibration behavior of functionally graded porous (FGP) plates resting on a Kerr-type elastic foundation. This investigation employs an innovative trigonometric shear deformation (ITSD) theory with five variables. The study encompasses various plate configurations, including homogeneous FGP plates, hard-core FGP sandwich plates, and soft-core FGP sandwich plates with both regular and irregular pore structures. The ITSD theory naturally addresses shear stress concerns at the outer surfaces, while also considering the thickness stretching effect, all without the need for correction factors. To formulate the governing equations for the free vibration of such plates on an elastic foundation, Hamilton’s principle is employed. The Navier double trigonometric series approach is then utilized to solve this problem. To validate the plate theory and methodologies used in this work, a comparative analysis is conducted with existing studies. Additionally, comprehensive parametric simulations are employed to examine the impact of different factors, such as geometric properties, material characteristics, sandwich schemes, and parameters of the Kerr-type foundation, on the dimensionless natural frequencies of simply supported rectangular FGP plates. [ABSTRACT FROM AUTHOR]

This paper implements the stochastic filters for state and parameter estimation of nonuniform transmission lines (NTL). In general, transmission line (TL) problem is a continuous time and space problem. By taking the line loading noise into account, the TL equations become a stochastic partial differential equation (PDE) rather than a simple set of coupled finite stochastic differential equations (SDE). By transforming the spatial variables into the Fourier domain, the stochastic PDE can be transformed into an infinite sequence of SDE. After truncation to a finite set of Fourier series coefficients, it becomes a finite set of coupled linear SDE, which is the required domain in which extended Kalman filter (EKF) and unscented Kalman filter (UKF) can be applied. For state space equation of EKF and UKF, the voltage and current of periodic NTL are expanded into an infinite set of spatial harmonics. In this way, the voltage and current measurement of NTL become an eigenvalue problem. The NTL is considered as cascade of small infinite NTL and the four distributed primary parameters of the periodic NTL are expressed using Fourier series expansion. Finally, the Kalman filter (KF)-based state estimation and the EKF- and UKF-based parameter estimation have been compared with recursive least squares (RLS) method. The simulation results present the superiority of the method. [ABSTRACT FROM AUTHOR]

This paper establishes a robust stability criterion in the radially symmetric heat equation that admits heat sources belonging to a set of bounded functions. The robust stability criterion is determined by extending a definition of stability under constant-acting perturbations that was originally established for systems of ordinary differential equations. It is assumed that heat sources admit a Fourier series representation whose coefficients are bounded and piecewise continuous functions. The robust stability criterion obtained is useful to conclude that the solution of the heat equation, as well as its first partial derivatives with respect to the radial axis and with respect to time, are bounded by a constant whose value is initially established. The results obtained are illustrated numerically. [ABSTRACT FROM AUTHOR]

A non‐iterative stress‐displacement solution is proposed in this paper to investigate surrounding rock mass with multiple non‐circular deep tunnels. Compared with the Schwarz alternating method, the proposed method only spends one‐time matrix operation, showing considerable efficiency and accuracy of the matrix solution. First, a matrix solution is formulated to obtain the conformal mapping function for the non‐circular tunnel. Then, in consideration of multiple boundary conditions, the generalized complex variable theory and fast Fourier transform (FFT) algorithm are used to formulate a matrix solution of potential functions. The stress and displacement fields around multiple non‐circular tunnels can be further determined from the matrix solution. A series of numerical examples are conducted to verify the proposed method, perform the convergency study, and discuss the effects of tunnel geometries, distances, and arrangements. The convergency study shows that the more accurate conformal mapping function requires more sampling points in the FFT. In addition, the high stress zone would be amplified if the arrangement of multiple tunnels makes the high stress zone adjacent or overlapped. [ABSTRACT FROM AUTHOR]

Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second‐order elliptic problems on open non‐convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition, and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π$$ \pi $$ where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained for harmonic and/or biharmonic problems on a family of convex planar sectors. [ABSTRACT FROM AUTHOR]

In engineering practice, dynamic vibration absorber is an effective method to control vibration of beam structure. The mathematical model of beam coupled with spring–mass systems under elastic edge constraints is established in this paper. In addition to Fourier cosine series, supplementary sine functions are introduced to represent the displacement function of the beam. The spring–mass system is considered by increasing degrees of freedom when constructing the mass matrix and stiffness matrix. The efficiency of solving equivalent mass is greatly improved in the comparison with FEM, which is important when designing dynamic vibration absorber. Subsequently, we conduct parametric analysis of the control effect about the dynamic vibration absorber based on the current mathematical model. The procedure of designing distributed dynamic vibration absorber of beam structure under various boundary condition is also presented. To verify the validity of the current mathematical model and design procedure, a cantilever beam experiment was conducted. The findings presented in the current research may be useful for researchers in the process of multi-line spectrum vibration control of beam structure. [ABSTRACT FROM AUTHOR]

Torsional buckling analysis of composite anisogrid conical shells, with equiangular helical ribs, was investigated based on an equivalent variable stiffness continuum model. The buckling analysis is complicated by the variable geometry and stiffness of the lattice conical shell along the meridian. The smearing approach and the gradual function of the lattice cells were used to build the corresponding continuum model with varying stiffnesses. Recurrence relations were constructed from linearized buckling equations of the equivalent conical shell using the power series method and Fourier decomposition. The excellent precision and tremendous efficiency of the analytical solution was validated compared with numerical results through many design parameters of the lattice conical shell, such as the thickness of the ribs, the radius at the small end, and the height of the cone. Finally, a parametric analysis was used to discuss the influence of parameters on the critical buckling torsional moment. It is found that increasing rib thickness, end radius, the height of the cone, and strengthening boundary constraints can improve the critical buckling torsional moment. [ABSTRACT FROM AUTHOR]

AbstractThis article extends recent asymptotic theory developed for the Hodrick Prescott (HP) filter and boosted HP (bHP) filter to long-range dependent time series that have fractional Brownian motion (fBM) limit processes after suitable standardization. Under general conditions, it is shown that the asymptotic form of the HP filter is a smooth curve, analogous to the finding in Phillips and Jin for integrated time series and series with deterministic drifts. Boosting the filter using the iterative procedure suggested in Phillips and Shi leads under well-defined rate conditions to a consistent estimate of the fBM limit process or the fBM limit process with an accompanying deterministic drift when that is present. A stopping criterion is used to automate the boosting algorithm, giving a data-determined method for practical implementation. The theory is illustrated in simulations and two real data examples that highlight the differences between simple HP filtering and the use of boosting. The analysis is assisted by employing a uniformly and almost surely convergent trigonometric series representation of fBM. [ABSTRACT FROM AUTHOR]

AbstractThis article uses Carrera unified formulation (CUF) to analyze the dynamics characteristics of composite shaft with consideration of thermal effects, axial loads, and rotational speeds. The shaft radial displacement is described by Taylor expansion terms, and the axial displacement is described by improved Fourier series. Hamilton’s principle is used to obtain the control equations. The calculation results indicate that: The increase of temperature can reduce composite shaft natural frequency, and the influence is not obvious when the layer angle is small; The effect of axial load on composite shaft natural frequency is roughly linear, but the influence varies under different boundaries, length-radius ratio, layer orders; Due to the gyroscopic effect, the natural frequency of the composite shaft exhibits two states under the action of rotation speed: forward and backward mode; In forward mode, the influence of temperature and axial load on the composite shaft still follows the previous conclusion; In the backward mode, when the rotational speed exceeds the critical speed, the influence of temperature and axial load on the composite shaft shows the opposite conclusion. These findings are of great significance for the practical application of composite shaft in engineering. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

Tracking errors are caused by the amplitude attenuation and phase lag of the servo system to the command. Based on the mechanism, this study proposes the method of command correction which magnifies the amplitude and advances the phase of the command according to the amplitude attenuation and phase lag of the servo system. Firstly, the command is decomposed into stable components and unstable components using Fourier series fitting and Hilbert-Huang transform. Then, the frequency-domain characteristics of the servo system are used to calculate the amplitude attenuation and phase lag of each component. Finally, the stable components and unstable components of the command are corrected by amplifying the amplitude and advancing the phase to compensate for the attenuation and lag caused by the servo system. The experimental results show that the corrected command reduced the average tracking error of the X-axis of the butterfly-shaped trajectory by 66.58 % and the maximum tracking error by 72.28 %. Moreover, the contour error of the butterfly-shaped trajectory was reduced significantly. The proposed method is suitable for the tracking errors and contour error controlling of a trajectory with a drastic curvature change. • The commands are modified in the frequency domain, thereby offsetting command attenuation and lag by the servo system. • By combining Fourier series and Hilbert-Huang frequency analysis techniques, reasonable commands decomposition is realized. • Stable components and highly variable frequency components of the commands are processed separately in this study. [ABSTRACT FROM AUTHOR]

Simple formulae for Lebesgue constants arising in the classical Fourier series approximation are obtained. Both even and odd cases are addressed, extending Fejér's results. Asymptotic formulae are also obtained. [ABSTRACT FROM AUTHOR]

We analyze the temperature time series of the EPICA Dome C ice cores in Antarctica and of the Greenland project, Summit, with durations of 800 000 and 248 000 years, respectively, with a recent mathematical tool defined through the Fourier phases of the series, known as the J -index. This data driven index can differentiate between purely random dynamics and dynamics with a deterministic component. It is sensitive to nonlinear components and robust to the presence of noise. Our J -index data analysis shows that both Greenland and Antarctica climatic fluctuations possess deterministic traits and suggests the presence of an underlying nonlinear dynamics. Furthermore, in both regions, it reveals the simultaneous occurrence of an important global event known as the "Pelukian transgression." For Antarctica, it also detects the marine isotopic stage 11. Additionally, our calculation of the time series Hurst exponents and our detrended fluctuation analysis show the presence of long-range persistent correlations for Antarctica and anti-persistent correlations for Greenland. For the latter case, our fractal dimension determinations are indicative of a more complex climatic dynamics in Greenland with respect to Antarctica. Our results are encouraging for further development of climate variability deterministic models for these regions. [ABSTRACT FROM AUTHOR]

We perform two-dimensional particle resolved direct numerical simulations of the steady cross flow past a pair of interacting circular and non-circular cylinders with the cut-cell method, a sub-class of non-body-conforming methods that provides a sharp description of the boundary, is strictly mass and momentum conservative, and can be easily extended to adaptive grids. We use hierarchically refined Cartesian meshes where we place a reference cylinder (i) at the center of the domain and vary the location of a neighboring cylinder (j). We consider a large parameter space defined by the radius of curvature 2 / ζ i and 2 / ζ j ranging from 1 to 0, the angles of incidence α i and α j at values of 0 ° and 45 ° , the center-to-center gap ratio G ranging from 1.5 to 20, the alignment angle θ measured between the free-stream flow and the line connecting the centers of the cylinders, ranging from 0 ° to 360 ° , and finally the Reynolds number Re varied from 1 to 20. Specifically, we investigate the force and flow disturbances introduced by the neighboring cylinder on the reference cylinder, with a focus on the normalized hydrodynamic drag and lift coefficients and the associated prevailing flow regimes. Our study highlights the substantial impact of both the gap ratio G and the alignment angle θ in delineating distinct flow regimes, each exhibiting distinctive flow characteristics and consistent trends in pressure distributions and variations of the normalized drag and lift coefficients. Generally, the flow and force disturbances become more pronounced when significant interactions between the cylinders occur, whether due to proximity, wake interference, or both. We identify a critical threshold for G, beyond which the flow and force disturbances induced by the neighboring cylinder markedly diminish, except in scenarios dominated by significant wake interactions. Our investigation shows that the documented trends in the flow and force variations exhibit remarkable similarity at Re of 10 and 20, but expectedly deviate at R e = 1. Finally, we propose an empirical model to predict the hydrodynamic disturbances between two circular cylinders based on the modulation of the drag C d , i and lift C l , i coefficients. Leveraging the periodic nature of C d , i and C l , i as a function of the relative angular alignment θ of the neighboring cylinder, we use Fourier series expansions demonstrating accurate reconstruction of the data across a wide parameter space. Furthermore, our model exhibits promising predictive capabilities when applied to unexplored parameter ranges, encompassing scenarios involving non-circular cylinders and interpolated regions of Re and G. [ABSTRACT FROM AUTHOR]

Based on the three-dimensional (3D) linear elasticity theory of piezoelectric semiconductor (PS) structures, inspired by the variational principle and the Mindlin plate theory, a two-dimensional (2D) higher-order theory and equations for thin-film devices are established for a rectangular coordinate system, in which Newton's law (i.e., stress equation of motion), Gauss's law (i.e., charge equation of electrostatics), Continuity equations (i.e., conservation of charge for holes and electrons), drift–diffusion theory for currents in semiconductors, and unavoidable thermo-deformation-polarization-carrier coupling response in external stimulus field environment are all considered. As a typical application of these equations, the static characteristic analysis of electromechanical fields for the extensional deformation of a PS thin-film device with thermal field excitations is carried out by utilizing established zeroth-order equations and the double trigonometric series solution method. It is revealed that the extensional deformations, electric potential, electron and hole concentration perturbations, and their current densities can be controlled actively via artificially tuning thermal fields of external stimuli. Especially, a higher temperature rise can induce a deeper potential well and a higher potential barrier, which can play a vital role in driving effectively motions and redistributions of electrons and holes. Overall, the derived 2D equations as well as the quantitative results provide us some useful guidelines for investigating the thermal regulation behavior of PS thin-film devices. [ABSTRACT FROM AUTHOR]

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum. [ABSTRACT FROM AUTHOR]

The intensity and structure of the secondary flows have significant influences on the turbine cascade losses. Combining with the extension method of Non-linear Programming by Quadratic Lagrangian (NLPQLP), two endwall parametric methods respectively based on the B-spline surface and Fourier series are used to optimize the aerodynamic losses of a turbine cascade. The optimization processes are based on Computational Fluid Dynamics (CFD) analysis and two final designs are obtained by different methods. The endwall profile generated by the B-spline surface method (EW-B1) appears to relatively reduce the total pressure loss by 16.58% and the secondary kinetic energy coefficient (C SKE ) by 27.08%, while the endwall profile generated by the Fourier series method (EW-F1) relatively reduces the loss by 16.18% but increases C SKE by 8.99%. The radial movement of upper Passage Vortex (PV) and the weakening of the pressure branch of the Horseshoe Vortex (HVps) are confirmed to be the reasons of loss reduction in EW-B1, while the movement of PV is the main reason for EW-F1 to decrease the total pressure loss. In EW-B1, the weakening of HVps has a much more significant effect on the decrease in loss, which is affected by the non-axisymmetric shapes upstream of the vortex trajectory and the drop in C SKE generation near the throat. [ABSTRACT FROM AUTHOR]

Optical fiber grating strain sensors are currently utilized in a variety of structural health monitoring applications. The encapsulated fiber optic sensor is unable to completely detect the strain of the structure, so the strain transfer theory should be established to maximize the strain sensing of fiber. It is required to explore the embedded four-layer fiber optic sensing model to create a more plausible strain transfer error hypothesis. Based on the three-layer fiber optic sensing model, the Goodman assumption and Fourier series approach were presented to study the strain transfer efficiency of the four-layer model in the elastic state. First, the physical quantity to be analyzed is determined, and finally the radius, interlayer bonding coefficient and elastic modulus are selected as the parameters affecting the strain transfer efficiency. The length range of efficiency evaluation is 0–2.5m, and the transfer efficiency under different radii is above 0.90 when the length L≥2.2m. The interlayer bonding coefficient kf between 1×1010N/m3 and 2×1011N/m3 has little impact on the transfer efficiency, and the same is true for ka, so it cannot be considered in practice. When kp is between 2.5×1010N/m3 and 1×1011N/m3 and the length L>2m, the strain transfer coefficient reaches 95%. The influence of elastic modulus on the transfer efficiency is very significant when L≤0.4m. The four-layer model performs similarly to the three-layer model within the paste length range of 1.0m, but has a superior strain transfer effect when the pasted length exceeds 1.0m. As the radius of the protective layer rises, the effect of strain transfer deteriorates. [ABSTRACT FROM AUTHOR]

Under arbitrary boundary conditions, a Jacobi–Ritz time domain semi-analytical method is proposed to study the vibration behaviors of uniform and stepped conical shells. Based on the idea of the differential element method and the first-order shear deformation theory (FSDT), the vibration analysis models of uniform and stepped conical shell structures are established. The axial and circumferential displacement tolerance functions are expressed by the Jacobi polynomial and the Fourier series. The complex boundary conditions are simulated by using artificial spring technology. In light of the Ritz method and the Newmark-β integral method, the free and forced vibration results of the structure can be figured out in the time domain. The numerical results demonstrate that the proposed method has excellent accuracy and remarkable dependability when compared to the finite element method (FEM). Numerical examples are used to examine the forced and free vibration behaviors under various boundary conditions, semi-vertex angles, thicknesses, and load characteristics. [ABSTRACT FROM AUTHOR]

In this study, modified couple stress theory-dependent stability analysis of functionally graded porous nano/microbeams under deformable support conditions, for the first time, is carried out via a solution method based on the Fourier series and Stokes’ transform. The modified couple stress theory is used as a size-dependent elasticity theory including the small-scale effect. The functionally graded material formulation is based on the power-law distribution found in the literature. The governing equation of the problem and especially the force boundary conditions are determined and the appropriate solution technique is selected. Fourier sine series is chosen as the lateral displacement function of functionally graded porous nano/microbeam under axial loads. Accordingly, the Fourier coefficients are found by substituting this function in the governing equation. Then, these coefficients and the Stokes’ transformation are applied to the force boundary conditions to construct general eigenvalue problems for the stability analysis. The results are compared with the rigid boundary conditions and it is shown that an excellent agreement is achieved. Then, several examples are solved for the functionally graded porous nanobeams and the effects of different parameters are analyzed and presented in a series of graphs and tables. [ABSTRACT FROM AUTHOR]

Aiming at the complex characteristics of negative pressure waves in low-pressure pipelines inside of buildings, we proposed an estimation method of pressure fluctuation trends based on the robust Kalman filter and the improved VMD, which can be used for leakage detection. The reconstructed baseline signal can accurately describe the fluctuation trend of the negative pressure wave after the pressure drop, and quantitatively express the characteristic difference between the leakage condition and the gas usage condition. The robust Kalman filter was used to estimate the pressure fluctuations. The parameters of VMD were adaptively calculated based on the WAA and discrete scale space. The trend components contained in the IMFs were separated by a reconstruction based on the Fourier series. Based on the simulation signal, the method can accurately restore the trend component contained in the complex pressure signal. Based on the actual signals, the accuracy of small leakage detection is 96.7% and the accuracy of large leakage detection is 73.3%. [ABSTRACT FROM AUTHOR]

In this paper, we propose an online adaptation method for Fourier series-based acoustic transfer function (FS-ATF) models for robot audition systems using microphone array signal processing. The ATF represents the characteristics of signal propagation from a sound source to a microphone, which is essential for sound source localization (SSL) and sound source separation (SSS). For practical applications in dynamically changing real-world environments, ATF models must satisfy two criteria: (1) they must be adaptable to changes in the acoustic environment; and (2) they must be lightweight to be suitable for resource-constrained systems, such as robots with limited memory and computational capacity. The proposed method addresses these challenges using Fourier series expansions for interpolation, which reduces the memory footprint of the ATF model and facilitates online adaptation to acoustic environmental changes. The experimental results demonstrate that the proposed online adaptation method both improves the SSL and SSS performance while reducing the size of the ATF model, which represents a significant improvement over existing online ATF adaptation methods. [ABSTRACT FROM AUTHOR]

This paper presents a design approach for a class of rotationally invariant 2D filters of finite impulse response (FIR) type, which may form circular filter banks with imposed specifications. The design is conducted analytically in the frequency domain and starts from a maximally flat low-pass prototype based on a trapezoidal function with specified width and slope. Its trigonometric approximation is derived using the Fourier series expressed analytically, truncated to a number of terms depending on the imposed accuracy. The chosen trapezoidal function leads to significantly smaller ringing oscillations compared to the approximation of an ideal square characteristic. By shifting the LP prototype to various frequencies, the desired filter bank is generated, where the component filters have a specified bandwidth, steepness, and overlap. The 2D circular filter bank results by applying a specific frequency mapping to the factored frequency response of the prototype filter. Thus, the frequency responses of the 2D filter bank components will also result in factored form, which is an advantage in implementation. The circular filter bank is designed in two versions, a uniform and a non-uniform (dyadic) filter bank. The designed filter banks have accurate shapes and relatively low order for the specified parameters. These filter banks are then used in a sub-band image decomposition application. Finally, an efficient implementation of these filters at the system level is proposed based on polyphase decomposition and the block filtering technique with a high degree of parallelism, resulting in a lower computational complexity. [ABSTRACT FROM AUTHOR]

Arch and serpentine structures are two fundamental structural forms with significant applications in various fields. When subjected to compressive loading at both ends, these structures undergo flexural-torsional post-buckling, resulting in complex deformation modes that are challenging to describe using basic functions (e.g., trigonometric functions and polynomial functions), posing significant challenges in finding analytical solutions. In this study, we propose a novel approach to address this issue. By representing the lateral displacement with a trigonometric series expansion and utilizing the equilibrium equation, the angular displacement is expressed in terms of special functions known as Mathieu functions. Furthermore, the energy method is employed to obtain analytical solutions for the flexural-torsional post-buckling deformation components. The theoretical findings are validated through experiments and finite element analysis. Based on the theoretical results, explicit analytical expressions for the maximum principal strain and the bending-torsion ratio of the structures are derived, offering valuable insights for the design of arch and serpentine structures in practical applications. [ABSTRACT FROM AUTHOR]

The poor interfacial properties of polyvinyl alcohol (PVA) fibers and epoxy resin seriously affect the toughening effect of PVA fibers on epoxy resin. To solve this problem, the surface of PVA fibers was modified with 3-aminopropyltriethoxysilane (KH550) and 3-glycidoxypropyltrimethoxysilane (KH560), as well as the surface of PVA fibers with tannic acid (TA) and KH560. The acid-cured epoxy resin toughening mechanism of the modified PVA fibers was systematically investigated by multiscale analysis. The experimental data showed that the epoxy resin performed best when TA and KH560 synergistically modified PVA fibers. The impact strength of the epoxy resin increased by 76.3% and the tensile strength by 20.3%. The toughening mechanism of modified PVA fibers on the epoxy resin was revealed by a series of characterizations such as Fourier transform infrared spectroscopy (FT-IR), scanning electron microscopy (SEM) and dynamic mechanical analysis (DMA). Because KH560 can participate in the curing reaction of the epoxy resin, it greatly enhances the adhesion between the fiber and the resin. The toughening mechanism of TA and KH560 working together on PVA fibers to strengthen epoxy resin is summarized through the analysis of different scales. This also offers theoretical and experimental support for using PVA fibers in composites. [ABSTRACT FROM AUTHOR]

Considering the approximation of , with , by the partial sums of the Fourier series in a system of polynomials orthogonal in the sense of Sobolev and generated by a system of classical Laguerre polynomials, we obtain some estimates for the convergence rate of to . [ABSTRACT FROM AUTHOR]

• A semi-analytical method is adopted for arbitrary fluid-conveying spatial pipeline. • The actual flexible-bracket clamp stiffness is considered. • Virtual springs are used for the straight-curved connection. In-depth research is imperative for complex spatial pipelines with fluid-structure interaction. A comprehensive and universal three-dimensional semi-analytical modeling method is first established for complex fluid-conveying spatial pipelines under the base excitation with arbitrary connections and configurations under different boundary conditions. The virtual spring technology is used to connect the straight and curved pipeline segments with spatial angles and simulate boundary conditions. For a thorough analysis, the actual stiffness of the bracket-clamp system is experimentally measured for the first time. The multi-level coupling dynamic model of part, component, and system levels under actual boundary conditions is established. An efficient method for solving the fluid-structure coupling pipeline system dynamics considering the base excitation is proposed based on energy method. The mid-plane displacement function of the Fourier series is introduced and solved using the principle of modal superposition. Hammer impact modal experiments, ANSYS software and vibration response experiments are performed on spatial pipelines supported by flexible bracket-clamps; the numerical and experimental results support the semi-analytical results. The degenerate model is verified with the simple plane pipelines in other references, and the errors are less than 5 %. The proposed method exhibits broad applicability and generalizability, making it suitable for various configurations and boundary conditions. [ABSTRACT FROM AUTHOR]

This work investigates the influence of couple stresses on intrinsic and extrinsic dislocations within three-dimensional heterogeneous multilayered structures. The materials consist of orthotropic and dissimilar layers with different elastic properties, incorporating the microstructural lengths from a special version of the anisotropic couple-stress elasticity. Double Fourier series expansions are used to describe extended displacements and tractions, considering rotation and couple-stress components and accounting for arbitrary displacement and rotation discontinuities associated with dislocation and disclination loops as well as periodic dislocation and disclination networks at semi-coherent interfaces. The dual-variable and position technique is used to recursively determine the field responses at any point in the multilayered laminate. The application examples focus on dislocation problems with couple stresses, revealing the effects of the core-spreading dislocation technique, free surfaces, anisotropic elasticity, and couple stresses on the field solutions. The analysis also uncovers pressure and von Mises stress amplitudes, non-zero rotation, and non-symmetric stress profiles due to couple stresses in copper/niobium bimaterials. Changes in the sign and magnitude of the Peach-Koehler forces are observed when couple stresses are introduced, which deviate from the classical theory of dislocations. The investigation finds a significant 29% reduction in energy per unit area at a semi-coherent interface between copper and niobium influenced by couple stresses. These results deepen our understanding of how couple stresses affect dislocation behavior in multilayered materials, and provide prospects for future discrete dislocation and disclination dynamics simulations with engineering applications in various nano- and micro-structured materials. • Novel solutions for dislocation and disclination loops in anisotropic composites. • The misfit dislocation network with couple stress is considered. • Solutions are provided by means of double Fourier series expansions. • Couple-stresses lead to a 29% reduction in energy at a semi-coherent Cu/Nb interface. [ABSTRACT FROM AUTHOR]

Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain S$$ S $$, and use the null field equation (NFE) of Green's representation formulation, where the source nodes Q$$ Q $$ are located on a pseudo‐boundary ΓR$$ {\Gamma}_R $$ outside S$$ S $$ but not close to its boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$. Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives uν$$ {u}_{\nu } $$ of the solutions on the boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$ can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise q$$ q $$‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with O(10−4)$$ O\left(1{0}^{-4}\right) $$ can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing. [ABSTRACT FROM AUTHOR]

In this work, a general eigenvalue solution of an arbitrarily constrained nonlocal strain gradient nanobeam made of functionally graded material is presented for the first time for the stability response by the effect of the Winkler foundation. Elastic springs at the ends of the nanobeam are considered in the formulation, which have not been considered in most studies. In order to analyze deformable boundary conditions, linear equation systems are derived in terms of infinite power series by using the Fourier sine series together with the Stokes' transform. The higher-order force boundary conditions are used to obtain a coefficient matrix including different end conditions, power-law index, elastic medium, and small-scale parameters. A general eigenvalue problem of technical interest, associated with nonlocal strain gradient theory, is mathematically evaluated and presented in detail. Parametric results are obtained to investigate the effects of material length scale parameter, Winkler stiffness, power-law index, nonlocal parameter, and elastic springs at the ends. In addition, the effects of the other higher-order elasticity theories simplified from nonlocal strain gradient theory are also investigated and some benchmark results are presented. [ABSTRACT FROM AUTHOR]

In approximation theory, the "direct theorem" is a fundamental concept that relates to how well more straightforward functions or signals can approximate a function or signal from a chosen approximation space. The direct theorem typically provides bounds on the error or the quality of the approximation in terms of various parameters, such as the degree of approximation, the dimension of the approximation space, and the properties of the approximated functions. The specific form of the direct theorem can vary depending on the context and the type of approximation being considered (e.g., polynomial approximation, Fourier series approximation, spline approximation, etc.). The direct theorem aims to establish the mathematical framework for comprehending how well elements from a given space can approximate a function, which is crucial in various fields like numerical analysis, signal processing, and scientific computing. Finally, the direct theorem is a crucial result in approximation theory. It describes how elements from a given approximation space get closer and better at approximating a given function as the number of terms in the approximation sequence grows. It provides essential insights into the behavior of approximations and plays a central role in various mathematical and computational disciplines. In this article, we aim to prove the direct theorem by defining a new appropriate operator for functions Lp ([[0, 2π]d) that uses our new k-mixed modulus of smoothness to estimate the best approximation degree and the new k-mixed difference. Also, to prove the direct theorem, we introduce the angular approximation in Lp ([0, 2π]d to reach the most accurate estimate. [ABSTRACT FROM AUTHOR]

In this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes' transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters. [ABSTRACT FROM AUTHOR]

The control of spoilage yeasts in wines is crucial to avoid organoleptic deviations in wine production. Traditionally, sulfur dioxide (SO2) was used to control them; nevertheless, SO2 influence on human health and its use is criticized. Biocontrol emerges as an alternative in wine pre-fermentation, but there is limited development in its applicability. Managing kinetics is relevant in the microbial interaction process. pH was identified as a factor affecting the interaction kinetics of Wickerhamomyces anomalus killer biocontrol on Zygosaccharomyces rouxii. Mathematical modeling allows insight into offline parameters and the influence of physicochemical factors in the environment. Incorporating submodels that explain manipulable factors (pH), the process can be optimized to achieve the best-desired outcomes. The aim of this study was to model and optimize, using a constant and a variable pH profile, the interaction of killer biocontrol W. anomalus vs. Z. rouxii to reduce the spoilage population in pre-fermentation. The evaluated biocontrol was W. anomalus against the spoilage yeast Z. rouxii in wines. The kinetic interactions of yeasts were studied at different pH levels maintained constant over time. The improved Ramón-Portugal model was adopted using the AMIGO2 toolbox for Matlab. A static optimization of a constant pH profile was performed using the Monte Carlo method, and a dynamic optimization was carried out using a method based on Fourier series and orthogonal polynomials. The model fit with an adjusted R2 of 0.76. Parametric analyses were consistent with the model behavior. Variable vs. constant optimization achieved a lower initial spoilage population peak (99% less) and reached a lower final population (99% less) in a reduced time (100 vs. 140 h). These findings reveal that control with a variable profile would allow an early sequential inoculation of S. cerevisiae. The models explained parameters that are difficult to quantify, such as general inhibitor concentration and toxin concentration. Also, the models indicate higher biocontrol efficiency parameters, such as toxin emission or sensitivity to it, and lower fitness of the contaminant, at pH levels above 3.7 during biocontrol. From a technological standpoint, the study highlights the importance of handling variable profiles in the controller associated with the pH management actuators in the process without incurring additional costs. [ABSTRACT FROM AUTHOR]

Purpose: A time-varying grey Fourier model (TVGFM(1,1,N)) is proposed for the simulation of variable amplitude seasonal fluctuation time series, as the performance of traditional grey models can't catch the time-varying trend well. Design/methodology/approach: The proposed model couples Fourier series and linear time-varying terms as the grey action, to describe the characteristics of variable amplitude and seasonality. The truncated Fourier order N is preselected from the alternative order set by Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal Fourier order is determined by hold-out method to improve the robustness of the proposed model. Initial value correction and the multiple transformation are also studied to improve the precision. Findings: The new model has a broader applicability range as a result of the new grey action, attaining higher fitting and forecasting accuracy. The numerical experiment of a generated monthly time series indicates the proposed model can accurately fit the variable amplitude seasonal sequence, in which the mean absolute percentage error (MAPE) is only 0.01%, and the complex simulations based on Monte-Carlo method testify the validity of the proposed model. The results of monthly electricity consumption in China's primary industry, demonstrate the proposed model catches the time-varying trend and has good performances, where MAPEF and MAPET are below 5%. Moreover, the proposed TVGFM(1,1,N) model is superior to the benchmark models, grey polynomial model (GMP(1,1,N)), grey Fourier model (GFM(1,1,N)), seasonal grey model (SGM(1,1)), seasonal ARIMA model seasonal autoregressive integrated moving average model (SARIMA) and support vector regression (SVR). Originality/value: The parameter estimates and forecasting of the new proposed TVGFM are studied, and the good fitting and forecasting accuracy of time-varying amplitude seasonal fluctuation series are testified by numerical simulations and a case study. [ABSTRACT FROM AUTHOR]

The main objective of the study was to develop a method for identifying the components of a time series disrupted by crisis events, allowing for evaluation, comparison and short-term forecasting of the impact of such situations. The article, using mathematical modelling, analyses and evaluates the impact of the pandemic and war on rail transport using Poland as an example. Three methods of local trend matching were used: locally estimated trend, locally estimated trend with seasonal components, locally estimated trend with locally estimated seasonal components. The study showed that the effects of the crisis are still felt today, but their impact on individual types of transport varied, and what's more, differences were also visible depending on the item being transported. Passengers, despite the introduction of high sanitary standards, severely limited their mobility. The freight transport market turned out to be more resistant to the impact of the pandemic, but both the pandemic and the war in Ukraine affected the volume of goods transported. The article presents a method that allows to identify time series disturbed by crisis events and therefore difficult to describe and make reliable forecasts. The method proposed by the authors is an innovative answer to the problems of identifying very variable series and can also be used in the analysis of other issues in which time series have been disturbed in a sudden and unexpected way. [ABSTRACT FROM AUTHOR]

The main objective of the study was to develop a method for identifying the com-ponents of a time series disrupted by crisis events, allowing for evaluation, comparison and short-term forecasting of the impact of such situations. The article, using mathematical modelling, analyses and evaluates the impact of the pandemic and war on rail transport using Poland as an example. Three methods of local trend matching were used: Locally Estimated Trend, Locally Estimated Trend with Seasonal Components, Locally Estimated Trend with Locally Estimated Seasonal Components. The study showed that the effects of the crisis are still felt today, but their impact on in-dividual types of transport varied, and what's more, differences were also visible depending on the item being transported. Passengers, despite the introduction of high sanitary standards, severely limited their mobility. The freight transport market turned out to be more resistant to the impact of the pandemic, but both the pandemic and the war in Ukraine affected the volume of goods transported. The article presents a method that allows to identify time series disturbed by crisis events and therefore difficult to describe and make reliable forecasts. The method proposed by the authors is an innovative answer to the problems of identifying very variable series and can also be used in the analysis of other issues in which time series have been disturbed in a sudden and unexpected way.

This research studies the first part of the failure of a compression member structure due to buckling. This unstable equilibrium collapse, exposes brittle failure which occurs suddenly and therefore should be avoided wherever possible. Unavoidable geometric imperfections due to structural fabrication, will weaken the structure against buckling. The behavior of bar under compression will be closely examined by taking a set of geometric imperfection data synthesized from previously available from the measurement of conical shells. Therefore, the two-dimensional surface imperfection is converted into several one-dimensional imperfection with some probability properties. In order to obtain a comparison tool for different type of imperfections, Fourier analysis is used to convert the imperfection into coefficients of trigonometric function. By examining the coefficients, geometric imperfection patterns introduced by a certain fabrication process are able to be identified. The study successfully demonstrates the applicability of Fourier analysis in representing inherent geometric imperfections as an initial step for conducting probabilistic buckling analysis. Fourier analysis has shown its capability to simultaneously characterize imperfections in two crucial parameters - the magnitude and configuration of the imperfection.

Introduction. Cable structures belong to perspective roof systems. Specialized software packages for structural analysis, however, do not provide optimization tools for obtaining efficient design solutions of the cable structures. Thus, development of improved methods for design and analysis of cable systems is an important task to be solved. An advanced technique for flexible cable analysis is proposed. It includes ordinary operations of summing the coefficients and their products. The technique is applicable for non-commercial mathematical software packages with numerical simulation tools included, thus providing structural optimization capabilities.Materials and methods. The technique proposed is based on the sine-series expansion of the external load and the shape function of the flexible cable. The differential equation of cable equilibrium is thus transformed into the set of algebraic equations. The cable length is expressed in algebraic form by means of the power expressions for the sum of the series.Results. The equation for the flexible cable is derived. It includes cable ordinate, axial stiffness, relative elongation and the external load parameters. The technique for determination of the axial stiffness of the cable is proposed under the operability conditions. The techniques for finding load-induced vertical displacements, as well as the initial cable sag and the undeformed length are given. The length of the cable under load and the ordinate given the cable length are proposed.Conclusions. The technique for flexible cable analysis allows taking into account distributed transverse external loads. For a combined load the coefficients of the series are the sum of the particular load coefficients. The technique is intended for automated structural solution. It allows facilitating the preliminary design stage, thus providing optimal parameters determination and in-depth design study implementation. Further development of the proposed technique encompasses the fields of non-shallow cable analysis, multy-chord cable systems, cable structures with stiffening girders, strutted cable systems and spatial roof structures.

The method of recording electrocardiograms as a non-invasive research method is widely used in modern functional diagnostics. Spectral diagnostic methods based on Fourier transform and wavelet transform are being developed. For the purposes of identification of cardiac rhythm disorders, the method of research selected is spectral (frequency) analysis of short-term ECG recordings, up to one period of heartbeats. Fourier series decomposition of the cardiac signal (ECG) in EDF-format for one period was carried out. It is determined that the maximum accuracy of cardiac signal description is achieved at the number of harmonics equal to half of the number of sampling points of the cardiac signal during the period. The correctness of the script developed for spectral analysis was checked by reconstructing the cardiac signal from its spectrum and comparing it with the original signal. The correlation between the spectrum and the shape of the cardiac signal has been established. The conclusion is made about the applicability of the spectral analysis method for the identification of heart rhythm disorders, as well as about the possibility of using the spectrum of electrical signals of heart contractions as a multidimensional function of the heart state. The direction of further identification of regularities by means of statistical analysis with interpretation of results by specialized specialists is indicated. The theoretical and practical value of this study lies both in determining the areas of application of spectral analysis of the cardiac signal for diagnosis and treatment, and in the practical results obtained, which can be used in the development of an expert system or a specific technical device.