Double Orthogonal Series Research Articles

Complete guided wave modes in piezoelectric cylindrical structures with fan-shaped cross section using the modified double orthogonal polynomial series method

The complete guided wave modes are made up of propagative wave modes and evanescent wave modes whose amplitudes exhibit an exponential or damped attenuation with propagation distance. The solutions for evanescent modes are complex and very difficult to be solved. The conventional method usually needs an iterative search procedure to find the complex roots. Accordingly, the modified double orthogonal polynomial series method, an analytical treatment, is proposed to investigate the complex guided waves in piezoelectric cylindrical structures with fan-shaped cross section. By the present method, the real, imaginary and complex solutions are simultaneously obtained without iterative process. Compared with available reference results, the correctness of the proposed method is confirmed. The displacement and electric potential distributions are illustrated, and the piezoelectric effect is detailed. Results reveal that there are some evanescent wave modes, which have much higher velocities than the propagative modes and simultaneously have very low attenuation. The piezoelectric effect can be adjusted by changing the geometric size of the cross section.

Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section

For the purpose of the design and optimization of piezoelectric transducers, the modified double orthogonal polynomial series method is proposed to investigate guided waves in functionally graded piezoelectric(FGP) cylindrical structures with sectorial cross-section. The real, imaginary and complex solutions are obtained simultaneously without iterative process. The real solutions represent propagative waves; the imaginary and complex solutions are evanescent waves. The boundary conditions are incorporated into the constitutive equations by virtue of the Heaviside function. Subsequently, the amplitudes are expanded into the double orthogonal polynomial series, and the motion equations are converted into a matrix eigenvalue problem about complex wavenumber. Numerical comparison with available reference result confirms the validity of the present method. Dispersion curves and the Poynting vector distributions are illustrated. The influences of angular measure, radius-thickness ratio and graded index on dispersion curves are analyzed. Results show that there exist some evanescent guided wave modes that have higher velocities than that of the propagative wave modes and simultaneously have low attenuation at high frequencies. These results can be utilized to improve the performance of transducers.

Guided wave propagation in functionally graded cylindrical structures with sector cross-sections

The differential equations governing guided waves in functionally graded cylindrical structures with sector cross-sections are solved by introducing the double orthogonal polynomial series method into the cylindrical coordinate system, and the wave characteristics are subsequently investigated. The validity of the present method is confirmed by way of the comparison with available references, and the convergence is discussed. The corresponding phase velocity dispersion curves, displacement distributions and Poynting vectors are illustrated. The influences of the variation in the radius to thickness ratio, angular measure and gradient index on the guided wave characteristics are discussed, which can be used as significant guidance on ultrasonic guided wave non-destructive testing for functionally graded cylindrical structures with sector cross-sections.

Wave propagation in functionally graded piezoelectric-piezomagnetic rectangular bars

Abstract In recent years, guided wave propagation in piezoelectric-piezomagnetic composite (PPC) structures are paid much attention for the design and optimization of PPC transducers. Previous investigations are mainly limited in horizontally infinite flat plates, axially infinite hollow cylinders, and so on. They are all one-dimensional model structures, i.e. structures having variable displacement field in only one direction, and the other two displacement fields are both constant. In this paper, a double orthogonal polynomial series approach is proposed to solve the guided wave in two-dimensional model structures, namely, a functionally graded piezoelectric-piezomagnetic (FGPP) bar with a rectangular cross section. The validity of the double polynomial approach is illustrated by the comparison with the available reference results for a pure elastic homogeneous rectangular bar. The guided wave characteristics, including dispersion curves and mechanical displacement distributions, are discussed by calculating various FGPP rectangular bars.

Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series

Let (X,F, μ) be a finite positive measure space and {ϕj,k(x): j, k = 1, 2,...} be a double orthonormal system of real-valued functions on X. We investigate the pointwise convergence of the double orthogonal series (2.1) in Pringsheim’s sense and in the regular sense introduced by Hardy, as well as its Cesaro (C, 1, 1) summability and its strong Cesaro |C, 1, 1| summability. In our main theorem (Theorem 2 in Section 3 below) we extend a previous result of Borgen [2] from single to double orthogonal series. The key ingredient of our proof is the extension of the familiar Kronecker lemma from single to double sequences of numbers (see in [8, Theorem 1]. As an application of our Theorem 2, we are able to conclude the a.e. statistical convergence of the double orthogonal series (2.1) under a weaker condition than (2.5) in the Rademacher–Menshov theorem (see Theorem 3 in the last Section 6).

Pointwise convergence and Cesàro summability of double orthogonal series

Pointwise convergence and Cesàro summability of double orthogonal series

Wave propagation in functionally graded piezoelectric rods with rectangular cross-section

For the purpose of design and optimization of functionally graded piezoelectric material (FGPM) transducers, wave propagation in FGPM structures has received much attention in the past twenty years. But research focused essentially on semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2D) FGPM structure, namely, an FGPM rod with a rectangular cross-section. The dispersion curves and electric potential distributions are illustrated.

Order of magnitude for Hausdorff summability of double orthogonal series

In [3], we established a necessary and sufficient condition for absolute Hausdorff summability of a double orthogonal series. In this paper, we estimate the order of magnitude for Hausdorff summability of a double orthogonal series under appropriate conditions.

Wave propagation in graded rings with rectangular cross-sections

In this paper, a double orthogonal polynomial series method is proposed to investigate the guided wave propagation in a two-dimensional (2-D) structure, namely, a FGM ring with a rectangular cross-section. Two kinds of graded rings are considered: material gradient directions being in the radial direction and in the axial direction respectively. Numerical comparison with available reference results for a straightly homogeneous rectangular bar illustrates the validity of the proposed method. The dispersion curves and displacement distributions of various FGM rings, which have different radius to thickness ratios, different material gradient directions and different thickness to height ratios, are calculated to reveal the guided wave characteristics.

Guided wave characteristics in functionally graded piezoelectric rings with rectangular cross-sections

For the purpose of design and optimization of functionally graded piezoelectric material (FGPM) transducers, wave propagation in FGPM structures has received much attention in the past twenty years. But previous research efforts have been focused essentially on semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solving the wave propagation problem in a two-dimensional FGPM structure, namely an FGPM ring with a rectangular cross-section. By numerical comparison with the available reference results for a purely elastic homogeneous rectangular rod, the validity of the extended polynomial approach is illustrated. The dispersion curves and the electric potential distributions of various FGPM rectangular rings with different material gradient directions, different polarization directions, different radius to thickness ratios, and different width to thickness ratios are calculated to reveal the guided wave characteristics.

Guided wave in multilayered piezoelectric–piezomagnetic bars with rectangular cross-sections

For the purpose of design and optimization of multilayered piezoelectric–piezomagnetic material (PPC) transducers, wave propagation in these structures has received much attention in past ten years. But the research objects of previous works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) structure, namely, a layered PPC bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic layered rectangular bar, the validity of the double polynomial approach is illustrated. The dispersion curves and mechanical displacement profiles of various layered PPC rectangular bars are calculated to reveal the wave propagation characteristics.

To the question on the almost everywhere convergence of the Riesz means of double orthogonal series

Sufficient conditions of the classical type ensuring the almost everywhere (a.e.) convergence of the nonnegative-order Riesz means of double orthogonal series are indicated. Analogies of the onedimensional results of Kolmogoroff [7] and Kaczmarz–Zygmund [5, 12] have been obtained for the Cesaro means and those of Zygmund [13] for the Riesz means. These analogies establish the a.e. equiconvergence of the lacunary subsequences of rectangular partial sums and of the entire sequence of Riesz means, generalize the corresponding results of Moricz [9] for the Cesaro a.e. summability by (C, 1, 1), (C, 1, 0), and (C, 0, 1) methods of double orthogonal series, and were announced earlier without proofs in the author’s work [3].

On absolute generalized Norlund summability of double orthogonal series

On absolute generalized Norlund summability of double orthogonal series

Rate of Cesàro summability of double orthogonal series

Unimprovable estimates (in the class of double orthogonal series) are obtained on the rate of almost everywhere summability of orthogonal expansions of square-integrable functions by Cesaro methods of positive order. The conditions imposed on the coefficients are of classical type.

On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

On the order of growth of rectangular partial sums of double orthogonal series

We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems.

On the almost-everywhere convergence of the Riesz means of double orthogonal series

We establish coefficient conditions of the classical type that guarantee the almost-everywhere summability of double orthogonal series by the Riesz methods of nonnegative order. We also prove certain equiconvergence theorems.

Rate of approximation by rectangular partial sums of double orthogonal series

пОлУЧЕНы НЕУлУЧшАЕМ ыЕ (НА клАссЕ ВсЕх ДВОИ Ных ОРтОгОНАльНых РьДОВ) ОцЕНкИ скОРОстИ пРИБ лИжЕНИь пОЧтИ ВсУДУ пРьМОУгОльНыМИ ЧАст НыМИ сУММАМИ ОРтОгОНАльНых РАжлО жЕНИИ В ДВОИНыЕ РьДы Ф УНкцИИ с сУММИРУЕМыМ кВАДРАт ОМ пРИ ОпРЕДЕлЕННых Усл ОВИьх УБыВАНИь кОЁФФ ИцИЕНтОВ клАссИЧЕскОгО тИпА. У стАНАВлИВАЕтсь НЕОБхОДИМОсть УкАжА ННых УслОВИИ НА МНОжЕ стВЕ ВсЕх ДВОИНых ОРтОгОНАльН ых РьДОВ.

Almost Everywhere Convergence of Orthogonal Series Revisited

We deal with single and double orthogonal series and give sufficient conditions which ensure their convergence almost everywhere. Among others, we prove that if ∑ ∞ j = 3 ∑ ∞ k = 3 a 2 jk log j log k log 2 + (1/ a 2 jk ) < ∞, then the series ∑ j ∑ k a jk ψ jk ( x) converges a.e. in Pringsheim′s sense for each double orthonormal system {ψ jk ( x)}. The interrelation between the well-known Rademacher-Menshov (type) theorems and ours are discussed in detail. At the end, we raise three problems concerning the characterization of a.e. convergence of orthogonal series.

On the (C,−1&lt;α&lt;0,−1&lt;β&lt;0)-summability of single and double orthogonal series

On the (C,−1&lt;α&lt;0,−1&lt;β&lt;0)-summability of single and double orthogonal series