Double Trigonometric Series Research Articles

Analytic method for solving a class of three-dimensional thermoelasticity problems for layered transversely isotropic plates with variable thicknesses

This paper examines three-dimensional boundary value problems in the theory of heat conduction and thermoelasticity for layered transversely isotropic rectangular plates with variable thicknesses acted on by a nonuniform temperature field. It is assumed that known temperature and heat flux at the surfaces of the plate or temperature of the surrounding medium allow a representation of the solution in terms of double trigonometric series. An approximate analytic method has been developed for solving this class of problems which makes it possible to reduce the initial boundary value problem for a plate of variable thickness to a recurrence sequence of the corresponding problems for plates with constant thicknesses.

Majorant estimates for partial sums of multiple fourier series of functions from Orlicz spaces vanishing on some set

In this paper majorants supn |Sn (x; ϕ)| of partial sums of double trigonometric Fourier series of functions fromL(log+L)3 vanishing on some subsets ofT2=[−π, π)2 are considered. In particular, the following theorem is proved.

Orthotropic pressure vessels subject to local loads

Differential equations for unsymmetrical loading of simply supported orthotropic circular cylindrical shell are obtained in terms of displacement components. Then displacement components are expressed in the form of double trigonometric Fourier series that satisfies the boundary conditions at the end. Uniformly distributed load on rectangular areas is considered, which is again expressed in terms of double trigonometric Fourier series. Expressions are obtained for displacement components, stress-resultants and stress-couples. Numerical results are obtained and plotted for a particular case.

Buckling analysis of a laminated cylindrical shellunder torsion subjected to mixed boundary conditions

A new efficient method is developed in this paper for buckling analysis of a cross-plylaminated cylindrical shell under torsion subjected to mixed boundary conditions. The transverseshear is taken into account by a first-order theory with a shear correction factor of 5/6. The mixedboundary conditions include conditions in forces as well as conditions in displacements, and theseforces and displacements are selected as basic unknowns. The other displacements and forces areexpressed in terms of the basic unknowns by taking inverse of a matrix composed of operators.The equations of buckled equilibrium in terms of the basic unknowns are solved with doubletrigonometric series which satisfy the mixed boundary conditions. Comparison of the obtainednumerical results with those given in the literature based on completely clamped boundaryconditions checks with the fact that the mixed boundary conditions yield appreciably lowerbuckling load and less circumferential wave number than the completely clamped boundaryconditions. The curves in the figures show how the difference in buckling loads between the twokinds of boundary conditions varies when the length and thickness of the shell vary.

Weighted integrability of double trigonometric series

We study the double trigonometric series whose coefficients c j k c_{jk} are such that ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ | c j k | > ∞ . \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty |c_{jk}|>\infty . Then its rectangular partial sums converge uniformly to some f ∈ C ( T 2 ) f\in C(T^2) . We give sufficient conditions for the Lebesgue integrability of { f ( x , y ) − f ( x , 0 ) − f ( 0 , y ) + f ( 0 , 0 ) } ϕ ( x , y ) \{f(x,y)-f(x,0)-f(0,y)+f(0,0)\}\phi (x,y) , where ϕ ( x , y ) = 1 / x y , 1 / x \phi (x,y)=1/xy, 1/x , or 1 / y 1/y . For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.

新しい応力関数を使用するせん断変形を含む平面板のフーリェ級数解

New stress functions introduced by prof. Tosaka and prof. Sueoka for expressing stress components at a point in three dimensional elastic body are used in a theory of plate bending including the effect of transverse shear deformation. Two conditions of compatibility are obtained as governing differential equations expressed by new stress functions. Displacement components and resultant stress components acting at a point of section of the plate obtained by employing double trigonometric series agree with these obtained by using displacement method in expression. It is shown that numerical results of displacements, bending moments and shearing forces for a uniformly loaded rectangular plate with two opposite edges simply supported, the third edge clamped, and the forth edge free, agree with calculated results by employing displacement method.

Generalized localization for the double trigonometric Fourier series and the Walsh-Fourier series of functions in

For an arbitrary open set and an arbitrary function such that on the double Fourier series of with respect to the trigonometric system and the Walsh-Paley system is shown to converge to zero (over rectangles) almost everywhere on . Thus, it is proved that generalized localization almost everywhere holds on arbitrary open subsets of the square for the double trigonometric Fourier series and the Walsh-Fourier series of functions in the class (in the case of summation over rectangles). It is also established that such localization breaks down on arbitrary sets that are not dense in , in the classes for the orthonormal system and an arbitrary function such that as or for , .

INTEGRABILITY, MEAN CONVERGENCE, AND PARSEVAL'S FORMULA FOR DOUBLE TRIGONOMETRIC SERIES

Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order $(p, 0)$, $(0, p)$, and $(p, p)$ with the weight $(\overline{|j|}\, \overline{|k|})^{p-1}$ for some $p>1$. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted $L^r$-integrability and weighted $L^r$-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], M\'oricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].

On the three-dimensional thermostressed state of rectangular plates under nonuniform heating

We consider three-dimensional boundary-value problems of the stationary theory of heat conduction and thermoelasticity for rectangular homogenous isotropic plates of arbitrary thickness. It is assumed that the temperature or heat flux density prescribed on the top and bottom surfaces admit a representation in the form of double trigonometric series. A closed-form analytic solution is obtained for the boundary-value problems of thermoelasticity in the case of plates with contacting edges along the lateral faces. Numerical computations are given for three types of boundary-value problems using the software package mathcad PLUS 6.0 for thin and thick plates. We construct the graphs of variation of the temperature, deflection, and normal stresses over the thickness of the plate. Three figures, 1 table. Bibliography: 6 titles.

INTEGRABILITY, MEAN CONVERGENCE, AND PARSEVAL'S FORMULA FOR DOUBLE TRIGONOMETRIC SERIES

Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order $(p, 0)$, $(0, p)$, and $(p, p)$ with the weight $(\overline{|j|}\, \overline{|k|})^{p-1}$ for some $p>1$. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted $L^r$-integrability and weighted $L^r$-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], M\'oricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].

One and two dimensional Cantor-Lebesgue type theorems

Let φ ( n ) \varphi (n) be any function which grows more slowly than exponentially in n , n, i.e., lim sup n → ∞ φ ( n ) 1 / n ≤ 1. \limsup _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1. There is a double trigonometric series whose coefficients grow like φ ( n ) , \varphi (n), and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like φ ( n ) , \varphi (n), and which has the everywhere convergent partial sum subsequence S 2 j . S_{2^j}. For any p > 1 , p>1, there is a one dimensional trigonometric series whose coefficients grow like φ ( n p − 1 p ) , \varphi (n^{\frac {p-1}p}), and which has the everywhere convergent partial sum subsequence S [ j p ] . S_{[j^p]}. All these examples exhibit, in a sense, the worst possible behavior. If m j m_j is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence S m j . S_{m_j}.

Generalized localization and equiconvergence of expansions in double trigonometric series and in the Fourier integral for functions fromL(ln+ L)2

Generalized localization and equiconvergence of expansions in double trigonometric series and in the Fourier integral for functions fromL(ln+ L)2

Delamination buckling and postbuckling of composite cylindrical shells

A higher-order shear deformation theory is developed for accurately evaluating the transverse shear effects in delamination buckling and postbuckling of cylindrical shells under axial compression. The theory assures an accurate description of displacement field and the satisfaction of stress-free boundary conditions for the delamination problem. The governing differential equations of the present theory are obtained by applying the principle of virtual displacement. The Rayleigh-Ritz method is used to solve both linear and nonlinear equations by assuming a double trigonometric series for the displacements. Both linearized buckling analysis and nonlinear postbuckling analysis are performed for axially compressed cylindrical shells with clamped ends. Comparisons made with the classical laminate theory and a first-order theory show significant deviations.

Uniform convergence of double trigonometric series

Uniform convergence of double trigonometric series

5443884 Film-based composite structures for ultralightweight SDI systems

A wide range of engineering structures, such as aircraft fuselages or ship hulls have as the foundation a shell orthogonally strengthened by two sets of stiffeners. Solution of the task related to determining the vibrations of such complicated structures requires an application of special methods which permit accounting for the interaction between the shell and the two sets of discrete stiffeners correctly. The present work proposes an effective method of predicting the vibrations of a finite orthogonally stiffened structure as a part of an infinite one when the edge conditions permit. The prediction method proposed is based on the method of space-harmonic expansions when the shell displacements and forces are presented in the form of special double trigonometric series. The method allows the interconnection of all three components of displacement and rotation of the shell and the stiffeners to be taken into account. The vibration velocity of the construction is determined directly without a need for solving the task of eigen-values first. The vibration shapes are broken into a large number of non-interacting groups of shapes. The solution reduces to a system of equations relating to the generalized reactions at supports. All this allows predictions to be made for large parts of the investigated construction over practically the whole frequency range of sound.

Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion \({\cal H}({\bf R} \times {\bf R}) \subseteq {\cal H}({\bf R}^2),\) where \({\cal H}({\bf R} \times {\bf R})\) is the product Hardy space defined for example by R. Fefferman and \({\cal H}({\bf R}^2)\) is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space \({\cal J}({\bf R} \times {\bf R})\) of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in \(L({\bf R}^2)\) and \({\cal H}({\bf R} \times {\bf R}),\) respectively. In particular, we obtain a broad class of multipliers on \(L({\bf R}^2)\) and \({\cal H}({\bf R}^2),\) respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on \(L({\bf T}^2)\) and \({\cal H}({\bf T}^2),\) respectively.

Elasticity and viscoelasticity in layered shells and plates subject to thermal and force loading

A revised model has been devised for the thermoelastic equilibrium in a multilayer shell of rotation that incorporates the thermal sensitivity of the material and the variable layer thicknesses, as well as the temperature dependence of the transverse tangential stress distribution. The linear theory of inherited elasticity is employed to derive differential equations for the quasistatic equilibrium of a multilayer viscoelastic shell or plate made from thermorheologically complicated materials. The solution is obtained as double trigonometric series for a hinge-supporied viscoelastic cylindrical shell or plate subject to sinusoidal temperature and force loadings.

Three-dimensional bending stress-strain state of rectangular variable-thickness orthotropic plates

A review of studies on the stress-strain state during bending of a refined two-dimensional theories of various order is given. A solution of the problem of bending of a rectangular orthotropic plate of constant thickness within the framework of the three-dimensional theory of elasticity was given. Bending of variable-thickness orthotropic plates was considered. Using the theory of thin plates, Sarkisyan obtained solutions for a freely supported thin plate with a thickness that varies according to a power law and a tightly clamped plate, with a gradually changing thickness under a sinusoidal load and a uniformly distributed load, respectively. Khoma and Chernopiskii studied the bending of freely supported and tightly clamped plates, whose thickness varied linearly in one direction; to obtain the solution they reduced the three-dimensional problem to a two-dimensional problem by expanding the desired function in a Fourier-Legendre series. In this study we consider a three-dimensional boundary-value problem of the bending of a variable-thickness orthotropic plate, with nonplanar faces perpendicular to the load, which admits representation by a double trigonometric series. The method of perturbation of the shape of the boundary for surfaces close to the coordinate planes is used to solve the problem. On the basis of themore » solution built we examine the laws governing the stress-strain state of orthotropic plates with surfaces that periodically curve in one direction as they are bent by localized loads, depending on the amplitude and frequency of the undulation of the curvatures and the degree of load localization.« less

Integrability of Multiple Trigonometric Series and Parseval′s Formula

Let s mn ( x, y) denote the rectangular partial sums of the double trigonometric series with the coefficients c jk . We prove that if the c jk form a null sequence of bounded variation, then the improper Riemann integral of ƒ( x, y)φ( x, y) over [−π, π] × [−π, π] exists and Parseval′s formula holds, where ƒ( x, y) is (in Pringsheim′s sense) the limiting function of s mn ( x, y) and the generalized Fourier series of φ has bounded one-sided partial sums at (0, 0), One of its consequences is that the c jk are the Fourier coefficients of ƒ in the sense of the improper Riemann integral. This implies that if ƒ is Lebesgue integrable, then the double trigonometric series determining ƒ is the Fourier series of ƒ. These results can be extended to any multiple trigonometric series. Our results not only extend the results of Bary ["A Treatise on Trigonometric Series," 1964, p. 656] and Boas [ Duke Math. J. 18 (1951), 787-793], but also generalize Móricz [ J. Math. Anal. Appl. 154 (1991), 452-465; 165 (1992), 419-437]

Integrability and L-convergence of multiple trigonometric series

It is proved that under the following condition, the sum f of the double trigonometric series with coefficients cjk is integrable and the rectangular partial sums smn(f, x, y) converge to f in L1 norm:.