Form Of Fourier Series Research Articles

Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.

Совершенствование моделей циклов Кондратьева в экономической динамике

The modeling of Kondratiev cycles in economic dynamics in the form of Fourier series has been improved. Models have been developed that allow, by varying the reference points and the duration of the periods, to obtain estimates of these parameters with high accuracy and give estimates of accuracy in the form of dispersion. On the basis of the obtained models and algorithms, estimates of the beginnings of the formation of the duration of the periods of Kondratiev cycles in developed countries were obtained and models of the productivity of technological structures in the developed countries of the modern world were built. The productivity of the fourth technological order was determined by the per capita income of 2,500 US dollars in 1990. The productivity of the fifth technological order was determined by the per capita income of 12,200 US dollars in 1990. The productivity of the sixth technological order was determined by the per capita income of 25,300 US dollars in 1990. Modeling made it possible to predict the end of the downward wave of the modern Kondratiev cycle in world economic dynamics, the systemic global economic crisis associated with the transition to the dominance of the sixth technological order in 2049–2055. Intensification of crises associated with business cycles, financial and investment cycles associated with the downward half-wave of the Kondratiev cycle is predicted starting from 2024-2027.

Dynamic response of a transversely isotropic and multilayered poroelastic medium subjected to a moving load

The dynamic response of a transversely isotropic and multilayered poroelastic medium over an impermeable rigid base due to a surface moving strip load with constant speed is investigated. First, the moving load and induced response of any field quantity are expanded in the form of Fourier series. Then, based on the Biot's poroelastodynamic theory, the governing equations in terms of the displacement and pore pressure components are decoupled via a potential-function method, and the corresponding general solutions in each layer are derived. A recursive relation for the field quantities is established by utilizing the powerful dual variable and position method along with the continuity conditions on the layer interface. By virtue of the boundary conditions, semi-analytical solutions for the transversely isotropic and multilayered poroelastic medium with different surface hydraulic conditions are finally derived. After verifying the accuracy of the developed solutions by comparison with existing solutions, the effects of surface hydraulic condition, material anisotropy, layer thickness, material layering and load speed on the dynamic response are analyzed in detail. Results show that the effects of material anisotropy and layering are significant for the accurate assessment of the dynamic response induced by the moving strip load over a transversely isotropic and poroelastic medium. • Analytical method for anisotropic layered poroelastic media under moving load. • Governing equations in terms of Fourier expansion and potential-function method. • Layered poroelastic media via the stable dual variable and position method. • Effect of anisotropy, layering, and loading speed on the dynamic response.

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.

An Original 2-D Analytical Model for Investigating Coupled Vibrations of Finite Piezoelectric Resonators.

This article deals with 2-D modeling of coupled vibrations of finite piezoelectric resonators. A general solution for all the physical quantities in the Cartesian and cylindrical coordinate systems is deduced from the governing equations by expansion in series summation of trigonometric functions of thickness coordinate and trigonometric or Bessel functions of the lateral one. The essential difference between this model and the earlier ones is that instead of expressing mainly in the thickness coordinate and integration through the thickness, the solutions are expressed in the form of double Fourier series augmented by single Fourier or Fourier-Bessel series, which contributes to better satisfy the mechanical and electrical boundary conditions. The dynamic stiffness matrix of the system is developed. Electrical impedances of a typical piezoelectric parallelepiped under stress-free and symmetrical loading conditions and its frequency spectrum for different width-to-thickness ratios are calculated using our model as well as by the finite element method. A comparison shows an excellent agreement. Finally, theoretical and measured electrical impedances of a piezoelectric parallelepiped and a piezoelectric disk are compared and discussed. The 2-D theoretical model proposed here is shown to be accurate and efficient for coupled vibration analysis of piezoelectric resonators and is applicable for any set of finite dimensions and crystal symmetry.

Approximate resolving equations of mathematical model of a curved thin-walled cylinder

The resolving equations for mathematical model of the stress-strain state of a curved thin-walled cylinder were derived. This model is based on Koiter’s–Vlasov’s theory of moment shells. A method was proposed for approximate solution of a mathematical model equations on the basis of a sequential asymptotic expansion of unknown functions into a small parameter series and representation of the expansion coefficients in the form of Fourier series. Using this method, a one-dimensional statement of the problem was obtained. Limitations on parameters in the shell equations are indicated for which such problem transformation is possible. For the mathematical model of the curved thin-walled cylinder one-dimensional equations in two different formulations were obtained. Conditions determining applicability limits of the constructed one-dimensional mathematical models were proved. Numerical experiments were carried out and it was found that the constructed one-dimensional mathematical model approximates the original problem with high accuracy. From an applied point of view, curved thin-walled cylinder simulates a pipeline section.

Fourier Series Approximation of Tensor Green’s Function in Biaxial Anisotropic Media

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green’s function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green’s function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.

A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

A complex Fourier series solution for buckling of VAT composite laminates with elastically restrained edges

Variable Angle Tow (VAT) composites have general anisotropic characteristics, and their fiber orientation angle varies continuously along an arbitrary direction. Therefore, the investigation of their buckling performance is quite challenging. Different from the commonly used numerical methods, a new method called complex Fourier series solution is proposed to study the buckling of VAT laminates with elastically restrained edges. First the prebuckling analysis is achieved by introducing Airy’s stress function and expanding it into a complex Fourier series form. And the corresponding mid-plane internal forces with non-uniform distribution are determined. Based on classical laminated plate theory, the buckling analysis of VAT laminate is carried out. The transverse displacement is also expressed as a complex Fourier series. And the critical buckling load and buckling mode shape of VAT laminate are obtained. The accuracy of the presented complex Fourier series solution is verified by comparing with the existing results. The influence of fiber orientation angles, boundary stiffness and boundary conditions on the critical buckling load and buckling mode shape is investigated in numerical examples. Results show that boundary stiffness as well as fiber path have great influence on the critical buckling load and buckling mode shape of VAT laminate. By adjusting the fiber orientation angles and the translational stiffness of the elastically restrained boundary can effectively improve the critical buckling load of VAT laminate. This paper provides a new method for the analysis of general anisotropic laminates with in-plane variable stiffness and the present complex Fourier series solution can also be used as a benchmark for the verification of other numerical results.

Vibroacoustic analysis of a thin laminated composite plate with surface-boned piezoelectric patches and subjected to general boundary conditions

In this paper, a semi-analytical model is proposed to deal with the vibroacoustic problems of laminated composite plates with surfaced-boned piezoelectric patches and subjected to general boundary condition using the modified Fourier series method. Based on Kirchhoff plate theory, the dynamic equation of the laminated composite plate is derived using Hamilton's principle. In order to satisfy general boundary conditions, the displacement solution of the plate is expressed in the form of two-dimensional Fourier series and serval auxiliary functions. The acoustic response of the laminated composite plate due to a harmonic concentrated force is obtained with the Rayleigh integral. Besides, the mass and stiffness contribution of the piezoelectric patch are taken into consideration in the present study. Through enough convergent studies and comparative studies, the convergence, accuracy and universality of the proposed method are validated. The developed semi-analytical model can be used for efficient and accurate analysis and design of laminated composite plates equipped with shunted piezoelectric patches. Finally, the effects of the resistor and inductor shunt damping circuits on the vibration and acoustic response is discussed.

Nonlocal Problem with Multipoint Perturbations of Dirichlet Conditions for Even-Order Partial Differential Equations with Constant Coefficients

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

Dynamic response of vertical shaft Pelton turbine unit for forced vibration

This research work was carried out to model the excitation force imparted by water jet in the form of Fourier series and determine the forced response of Pelton turbine unite of Khulelhani-I hydropower analytically by developing mathematical mode. Amplitude of forced vibration form analytical was compared with simulation result.&#x0D; The mathematical model was developed by calculating the kinetic energy of disk and potential energy of both disk and shaft. Hamilton’s principle was used to determine equation of motion and then Galerkin method was used to determine response of the system. Fourier analysis was done to obtain the function in its exact form. The developed methodologies were followed to find the analytical solution of Kulekhani-I unit of 3100 kW rated at 600rpm. A rigid disk (runner and bucket assembly) was situated along end of flexible shaft with fixed support at the shaft.&#x0D; First five Fourier components are to be considered in analysis for meaningful representation of forcing function. The amplitude of vibration of Pelton turbine unit with single nozzle in Y-direction (the direction of water jet) obtained by analytical method was closed with that obtained from the ANSYS simulation.

Feature-based characterization and extraction of ripple errors over the large square aperture.

Freeform surfaces play an important role in modern optical systems with compactness and better performance. The fabrication tools tend to impart a structured signature on optical surfaces, called ripple errors, during the freeform surface manufacturing process. The description and extraction of ripple errors for freeform surface fabrication and testing have attracted extensive attention. In this paper, we develop a fast and accurate method to describe ripple errors for the large aperture based on Fourier model coupling. The polynomial expression is transformed into Fourier series form and surface errors are reconstructed by frequency feature extraction combining with the least square method. The high accuracy and efficiency of the proposed method for representing and filtering ripple errors consuming little computer memory are demonstrated using real experimental data. The proposed method offers a robust and powerful tool not only suitable for surface error characterization but also for image filtering and analysis.

Boundary Value Problems for a Fourth Order Partial Differential Equation with an Unknown Right-hand Part

In this article in a rectangular domain inverse boundary value problems with local and non-local boundary conditions for a fourth order partial differential equation were formulated and investigated. The solutions of the inverse problems were obtained in the form of Fourier series. It was proved the theorem on uniqueness and existence of the solution of the inverse problems under consideration.

Integro-differential equation with fractional order Caputo operators and spectral parameters

The questions of unique solvability of a boundary value problem for an integro-differential equation with two Caputo fractional operators and spectral parameters are considered. The fractional Caputo operator’s order is bigger than one and smaller than two. Using the method of Fourier series, countable systems of ordinary fractional integro-differential equations with degenerate kernel is obtained. Further, is used a method of degenerate kernel. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system the regular and irregular values of the spectral parameters were calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. In proofing of the convergence of Fourier series the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. We also studied the continuous dependence of the solution of the problem on small parameter for regular values of spectral parameters. The results obtained in this paper are formulated as a theorem.

Selecting the Forecasting Model of Heat Energy Delivery

To solve the problems of saving thermal energy, instead of the currently used cumbersome methods or complex mathematical models, for example, in the form of partial differential equations, it is proposed to use simpler models of time series for the supply and production of thermal energy. The data of a specific heat supply organization on a city scale and the website of the Ministry of Energy of the Russian Federation were used as initial data. The proposed approach is based on a comparison of various mathematical models for predicting the production and supply of heat energy according to the criterion of adequacy (average relative error and coefficient of determination). The results of research carried out using the Matlab software package are presented. A substantiated comparison of the possibility of using different models is carried out. Models have been developed that take into account the seasonal component and higher accuracy. The possibility of using the following time series models is investigated: regression models with a seasonal component, triple exponential smoothing Holt–Winters, multiplicative seasonal autoregression model of the integrated moving average SARIMA (seasonal autoregression integrated moving average), and nonparametric model SSA (singular spectrum analysis). Multiplicative models containing a trend in the form of a linear or autoregression model with the calculation of the seasonality index, as well as additive models with a trend in the form of a linear or autoregression model with a seasonal component in the form of a Fourier series or sum of sines, were compared. In the Holt–Winters model, fminsearch, a function of the Matlab software package, was used to determine the tuning parameters. In the multiplicative SARIMA model, the orders of autoregression and moving average were fitted according to the autocorrelation and partial autocorrelation function. The nonparametric SSA model was used for comparison with the listed parametric models. An opportunity has been obtained to make reasonable forecasts of the production and supply of heat energy It is shown that the best results (average relative error 6.52%) were obtained using the SARIMA multiplicative model.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Периодические решения уравнения Эйлера-Бернулли колебаний балки с жестко заделанными концами

The problem of time-periodic solutions of the quasilinear equation of forced vibrations of an I-beam with fixed ends is investigated. The nonlinear term and the right-hand side of the equation are time-periodic functions. In the paper we study the case when the time period is comparable to the length of the beam. The solution is sought in the form of Fourier series. To construct the corresponding orthonormal system, we study the Sturm-Liouville problem on eigenfunctions and eigenvalues. A transcendental equation that satisfies the eigenvalues of the Sturm-Liouville problem is investigated. The eigenvalue asymptotics is derived from it, which is used to justify the smoothness of the solution of the Euler-Bernoulli equation. The uniform boundedness of the eigenfunctions of the Sturm-Liouville problem is proved and estimates for their derivatives are obtained. Invertibility conditions for the differential operator of the Euler-Bernoulli equation are obtained. The complete continuity of the resolvent of this operator on the complement to the spectrum is proved. The existence and regularity of solutions to the corresponding linear problem are proved by studying double sums of Fourier series. The problem of periodic solutions of the quasilinear equation of beam vibrations is investigated. We consider the case when, for sufficiently large modulo values of the argument, the ratio of the nonlinear term to the argument does not coincide with the eigenvalues of the differential operator. We prove a theorem on the existence of a generalized periodic solution for which the boundary conditions are met in the classical sense. Using the topological methods, we found the periodic solution as a fixed point of the corresponding operator.