Trigonometric Series Research Articles

Hardy–Littlewood theorems for trigonometric series with general monotone coefficients

Hardy–Littlewood theorems for trigonometric series with general monotone coefficients

Flat bending shape stability of the beams with variable section width

The paper proposes a methodology for calculating lateral buckling of beams of variable rectangular cross section based on the energy approach. The technique is considered on the example of a cantilever beam of variable width with two sections under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized secular equation.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Instability of Surface Quasigeostrophic Spatially Periodic Flows

AbstractThe surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.

New analytical solutions for vibration problem of thick plates

Exact solutions for the vibrations and stability problems in the mechanics of solids are sufficiently rare. For rectangular thick plates, exact solutions in the form of trigonometric series were constructed only for the case when all or two opposite sides of the plate are simply-supported. The discussion about the possibility of constructing exact solutions continues to this day. As a rule, an approximate solution is constructed in the analytical form on the basis of a variational approach. It should be noted, that as frequency increases, the number of basic functions involved in the solution also has to be increased, consequently the solutions of such type are inefficient for describing a structural element within the framework of such methods as the Continuous Element method, the Spectral Element method and the Dynamic Stiffness method. In the present research, for the first time, the analytical solutions for the vibration problem of thick orthotropic plates are obtained. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in a series, permitting the derivation of an infinite system of linear algebraic equations. Cases of practically important boundary conditions of completely free sides and fully clamped sides are considered. It should be noted that the presented analytical solutions for FFFF and CCCC boundary allows to completely describe the structural element in the form of a plate by means of a dynamic stiffness matrix and their use for modeling more complex structures, also in the framework of methods such as Continuous Element method, Spectral Element method and Dynamic Stiffness method. The obtained results could also be applied in projecting constructions, in developing new devices and in the optimization of their parametersin projecting of constructions, in developing new devices and in the optimization of their parameters.

Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series

A summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on $\RR_+$ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as the corresponding orthogonal polynomials on $\RR_+$, are also considered. Finally, in order to illustrate the efficiency of the presented summation/integration method two numerical examples are included.

A Computational Technique for Free Vibration Response of a 2-DOF Parametric System

In this paper, a closed-form solution is presented for the free vibration response of a 2-DOF parametric system by a technique with special trigonometric series. The technique extends the concept of modulation feedback to the problem of multiple degrees of freedom (M-DOF) parametric system using an equivalent dynamic system. The characteristic equation is derived by the harmonic balance in a parametric equation with the principal oscillation frequencies computed. The computation of modes and splitting modes is carried out by operation on a mixed matrix. The arbitrary constants in the general solution are determined by the initial conditions. A real-life example of a coupled inverted-pendulum system is provided to illustrate the applicability of the proposed approach. An approaching error is adopted to assess the computational errors both using the proposed technique and the Runge–Kutta algorithm. Of interest is that the time-periodic coefficient results in the changeability of the mode in the 2-DOF parametric system, while exciting the splitting mode in the harmonic combination resonance. The proposed technique is numerically demonstrated to possess not only high approaching accuracy, but is also capable of revealing some new phenomena in the parametric system, such as characteristics of mode and slitting mode, etc. It proves to be useful for future research and engineering applications of parametric systems.

Convergence of lacunary SU(1, 1)-valued trigonometric products

This note attempts to study lacunary trigonometric products with values in the matrix group \begin{document}$ \rm{SU}(1,1) $\end{document} in analogy with lacunary trigonometric series. The central questions are the characterization of their convergence in an appropriately defined \begin{document}$ \rm{L}^p $\end{document} -metric and the characterization of their convergence almost everywhere. These can be interpreted as nonlinear analogues of the classical results by Zygmund and Kolmogorov.

Calculation of temperature oscillations in thermal layer of regular solids at unsteady coefficient of heat transfer

The temperature oscillations in thermal layer of regular solids, caused by periodic oscillations of the ambient temperature and heat transfer coefficient, are analysed. A halfspace, a plate, a space with cylindrical channel, a cylinder, a space with spherical cavity are considered as solids. Exact solutions of the corresponding cyclic problems of heat conduction in the complex and real forms of trigonometric Fourier series are obtained. Methods of simplified solution of heat conduction problem based on boundary condition modification are considered. Comparison of exact solutions with approximate ones is carried out and the admissibility of the proposed approximate solutions is shown.

Aeroelastic stability of a shell supported by a cylinder with a hole of linearly varying radius

The aeroelastic stability of a shell supported by an elastic cylinder of varying thickness with the outer surface exposed to supersonic gas flow is studied. The supporting cylinder is considered as an elastic foundation, whose bedding value is determined from the equations of the three-dimensional theory of elasticity. The solution is obtained by the BubnovGalerkin method in the form of a trigonometric series in the axial coordinate. The characteristic equation is approximated by the Lagrange polynomial, whose stability is investigated with the use of the Routh-Hurwitz criterion. The dependences of the critical flow velocity on the length of the cylinder section with the tapered hole are shown for various values of the axial force, the damping decrement, and the shell thickness.

On weighted integrability of functions defined by trigonometric series with p-bounded variation coefficients

In this paper we introduce new classes of p-bounded variation sequences and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients belonging to these classes. This is a generalization of the results obtained by the first author [J. Inequal. Appl. 2010:1–19, 2010] and Dyachenko and Tikhonov [Stud. Math. 193(3):285–306, 2009].

Study of the correlations between main shocks and aftershocks and aftershock synthesis method

To consider the influence of aftershocks in engineering design, the correlations between main shocks and aftershocks should be examined, and an aftershock simulation method with main shock ground motions needs to be developed. In this study, the data on the sequences of main shock-aftershock ground motions and other related parameters were collected. Using these data, correlations between the magnitude, frequency, duration and energy of the main shock-aftershock ground motions were investigated. The results showed that the magnitude of the aftershock can be larger than that of the main shock. The shapes of the Fourier amplitude spectra of main shocks and aftershocks were similar; however, the predominant frequency and high-frequency components of the aftershock tended to be larger. Considering the magnitude difference between the main shock and the aftershock, the correlation of durations was explored. Additionally, a new concept, the duration ratio, was defined to describe the concentration of seismic energy release, and main shock energy was strongly positively correlated with the energy attenuated during the main shock-aftershock sequence. Finally, based on these results regarding correlation, an aftershock synthesis using recorded main shock ground motions was constructed with the trigonometric series method for seismic design, and some examples are given to analyze the rationality of this synthetic method.

Exact spectral element model for rectangular membranes subjected to transverse vibrations

This study proposes an exact frequency-domain spectral element model for a rectangular membrane subjected to arbitrary boundary conditions. The frequency-domain general solution of a finite rectangular membrane element is first assumed in the trigonometric Fourier series with the spatial coordinates x and y. After all the trigonometric Fourier series terms that vanish at the four boundary edges are eliminated, the general solution is expressed in the spectral (i.e., discrete Fourier transform) forms with respect to the spatial coordinates x and y. By using the projection method, the spatial-domain spectral coefficients of the general solution are related to the spatial cosine/sine series coefficients of the functions specified for the geometric and natural boundary conditions. Lastly, the spectral element matrix (i.e., dynamic stiffness matrix) is formulated from the force–displacement relationships. Owing to the spectral representation of the general solution in both temporal and spatial coordinates, the well-known fast Fourier transform algorithm can be applied for efficiently obtaining the dynamic responses and wave propagations in a membrane, in both the time and spatial domains. The high accuracy of the proposed spectral element model for the rectangular membranes is verified via comparison with existing solution techniques, such as the exact theory, modal analysis method, and finite element method.

Aeroelastic stability of a cylindrical shell of linearly varying thickness

For the first time, the aeroelastic stability equations of a composite cylindrical shell of linearly varying thickness are obtained on the basis of the bending theory of orthotropic shells for a shell subjected to axial forces and supersonic gas flow. The solution of the equations is assumed of the form of a trigonometric series in the axial coordinate. The problem is reduced to an infinite system of algebraic equations by the Bubnov-Galerkin method. The obtained characteristic equation is approximated by the Lagrange polynomial, whose stability is investigated with the use of the Routh-Hurwitz criterion. As a numerical example, the effect of the thickness gradient, structural damping and axial force on the critical velocity for a composite shell of linearly varying thickness in supersonic gas flow is shown. The refinement in the value of the critical velocity resulting from the use of the suggested model is about 35% as compared to the results for a shell of averaged constant thickness. This indicates the relevance of this model for aircraft weight optimization. The suggested approach expands the range of problems to be solved and allows for the analysis of the aeroelastic stability for orthotropic cylindrical shells of linearly varying thickness in supersonic gas flow.

Buckling Problems of Structurally-Anisotropic Composite Panels of Aircraft with Influence of Production Technology

The different approaches were analyzed to investigate the buckling problems of structurally-anisotropic panels made from composite materials. Aircraft composite structure design in the field of production technology is the outlook research trend. New mathematical model relations for the buckling investigation of structurally-anisotropic panels comprising composite materials are presented in this study. The primary scientific novelty of this research is the further development of the theory of thin-walled elastic ribs related to the contact problem for the skin and the rib with an improved rib model. One considers the residual thermal stresses and the preliminary tension of the reinforcing fibers with respect to panel production technology. The mathematical model relations for the pre-critical stressed state investigation of structurally-anisotropic panels made of composite materials are presented. Furthermore, the mathematical model relations for the buckling problem investigation of structurally-anisotropic panels made of composite materials are presented in view of the pre-critical stressed state. The critical force definition of the general bending form of the thin-walled system buckling and the critical force definition of the many-waved torsion buckling are of the most interest in accordance with traditional design practices. In both cases, bending is integral with the plane stress state. Thus, the buckling problem results in the boundary value problem when solving for the eighth order partial derivative equation in the rectangular field. The schematization of the panel as structurally-anisotropic has been proposed as a design model when and the critical forces of total bending form of buckling are determined. For a many-waved torsion buckling study, one should use the generalized functions set. The solution is designed by a double trigonometric series and by unitary trigonometric series. A computer program package is developed using the MATLAB operating environment. The computer program package has been utilized for multi-criteria optimization of the design of structurally-anisotropic aircraft composite panels. The influence of the structure parameters on the level of critical buckling forces for bending and for torsion modes has been analyzed. The results of testing series are presented.

Temperature Fluctuations in a Half-Space under Spatially Inhomogeneous Heat-Transfer Conditions

An unsteady two-dimensional heat-conduction problem without initial conditions is considered for a half-space under boundary conditions of the I, II, and III types. It is assumed that the boundary-value functions (including heat-transfer coefficient) are periodic functions of time and a spatial variable. Cyclic solutions to the problem are found in the form of double trigonometric Fourier series. The application of the obtained dependences is demonstrated with several specific examples. The conditions under which the quasi-one-dimensional solutions can be used are studied.

A Note on the Generalized Cesáro Means of Trigonometric Fourier Series

Different generalized Cesaro summation methods are compared with each other. Analogous of Hardy’s theorem, concerning the order of the partial sums of trigonometric Fourier series, for generalized Cesaro means are obtained.

Dynamic Characteristic Study of Riser with Complex Pre-stress Distribution

Abstract In this study, the dynamic characteristic problem of riser structure with complex pre-stress distribution is investigated. At first, the differential equation of the riser structure with complex pre-stress distribution is derived. The analytical expression of the free vibration of a riser structure with complex pre-stress distribution is discussed by using the orthogonal property of the trigonometric series. A top-tensioned riser (TTR) for example, the influences of the amplitude and direction of complex pre-stress on natural frequency and mode shape characteristics are compared. This study provides a new method for addressing the riser structure response problem with complex loading.

Functions of Minimal Norm with the Given Set of Fourier Coefficients

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

Model assessment of GNSS-based regional TEC modeling: polynomial, trigonometric series, spherical harmonic and multi-surface function

The ground-based GNSS VTEC model can adequately capture the spatiotemporal characteristics of ionosphere activities. However, it is difficult to precisely model VTECs with a unified mathematical model. We dedicate to comprehensively assess the performance of four mathematical models for VTEC modeling, i.e., polynomial, trigonometric series, spherical harmonic and multi-surface function. To capture the varying ionosphere situations, three typical regions in Western Europe, Southeast China, and North America are chosen. To reflect the precision and accuracy from different aspects, four evaluation measures are defined based on the comparison with CODE GIM products, the poster-fitting residuals of VTEC modeling, the comparison with high-precision quasi-VTEC computed with L1–L2 phase observations as well as the analysis of precision and stability of receiver and satellite DCB by-products. The results show that, in small regions, the performances of all models are insensitive to the model orders but sensitive to the ionosphere activity. On the whole, the polynomial and spherical harmonic function are comparable in terms of their performance and computation efficiency; Trigonometric series is instable with systematic biases in large region due to its incapability of describing the spatial ionosphere variations; Multi-surface function outperforms the others thanks to its epoch-wise solution with low computation efficiency. In addition, the annual solutions of satellite and receiver DCBs indicate their sufficient stability, which can then be applied as known for single-epoch ionosphere modeling.