Algebraic-trigonometric Polynomials Research Articles

Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions, asymptotic formulas involving concrete simplest trigonometric and algebraic-trigonometric polynomials were first obtained.

Vibration Analysis for Piezoceramic Circular Plates with V-Notches. Part 1: Theory

AbstractIn this paper the transverse vibration characteristics of piezoceramic circular plates with V-notches are investigated theoretically through use of the Ritz's method incorporated with the defined equivalent constants. The Ritz's method is employed with two sets of admissible displacement functions, algebraic-trigonometric polynomials and corner functions, to guarantee convergence sufficiently and represent the stress singularity, respectively. Moreover, the equivalent constants derived by comparing the characteristic equations of transverse vibration between isotropic and piezoceramic disks are applied to suspend the electrical field consideration regarding the piezoelectricity. With the aid of theoretical analysis, the non-dimensional frequency parameters of transverse vibration modes for completely free V-notching circular plates are exhibited; in addition, the frequency variations depending on various notch angles and depths are explored. Numerical calculations using the finite element method (FEM) are performed and the results are compared with the theoretical analysis. It is shown that the resonant frequencies predicted by theoretical analysis and calculated by FEM are in good agreement.

Approximating smooth functions using algebraic-trigonometric polynomials

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form , where is an algebraic polynomial of degree and is a trigonometric polynomial of order . The precise order of approximation by such polynomials in the classes and an upper bound for similar approximations in the class with are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.Bibliography: 13 titles.

On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates \(({r,\theta , \varphi})\) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate \({\varphi}\) around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and \({\varphi }\) toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and \({\varphi}\) directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.

V노치 또는 예리한 균열을 가지는 직사각형 평판의 굽힘 진동

This paper reports the first known free vibration data for thin rectangular plates with V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained, and (2) corner functions which account for the bending moment singularities at the sharp reentrant corner of the Y-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondirectional frequencies for rectangular plates having the V-notch. In this paper, accurate frequencies and normalized contours of vibratory transverse displacement are presented for various notched plates, so that the effect of corner stress singularities may be understood.

Vibration of Plates with Constrained V-Notches or Cracks

This paper reports the first known free vibration solutions for thin circular plates with V-notches having various edge conditions. The classical Ritz method is employed with 2 sets of admissible functions assumed for the transverse vibratory displacements. These sets include: 1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and 2) corner functions to account for the bending moment singularities at the sharp re-entrant corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondimensional frequencies for circular plates having a free circumferential edge and various combinations edge conditions of the V-notch. Accurate (5 significant figures) frequencies are presented for clamped-free, clamped-hinged, and hinged-free notches for the spectra of notch angles, causing a re-entrant vertex corner of the radial edges. For very small notch angles, a clamped-free, clamped-hinged, or hinged-free radial crack ensues. One general observation is that, for the range of notch angles considered, there is a substantial increase in the first 6 frequencies as the notch depth increases. The frequency increase with increasing notch depth is more pronounced in the higher modes than the lower ones, and is quite substantial for segmental plates with notch angles equal to 180 deg. A large reduction in frequencies is also observed as the notch angle decreases at a constant notch depth. A new database of accurate frequencies and mode shapes for sectorial, semicircular and segmental plates is presented with which future solutions drawn from alternative numerical procedures and finite element and boundary element techniques may be compared. Normalized contours of the transverse vibratory displacement are shown for plates with various notch depths and notch angles of 5, 30, 60, 90, and 180 deg.

A THREE-DIMENSIONAL ANALYSIS OF THE SPHEROIDAL AND TOROIDAL ELASTIC VIBRATIONS OF THICK-WALLED SPHERICAL BODIES OF REVOLUTION

This paper addresses the spheroidal (i.e. coupled bending-stretching) and toroidal (i.e. torsional or equivoluminal) elastic vibrations of thick-walled, spherical bodies of revolution by means of the three-dimensional theory of elasticity in curvilinear (spherical) co-ordinates. Stationary values of the dynamical energies of the spherical body are obtained by the Ritz method using a complete set of algebraic-trigonometric polynomials to approximate the radial, meridional, and circumferential displacements. Extensive convergence studies of non-dimensional frequencies are presented for the spheroidal and toroidal modes of thin-walled spherical bodies of revolution. Results include all possible 3-D modes, i.e. radial stretching, combined bending-stretching, pure torsion, and shear deformable flexure through the wall thickness (including thickness-shear, thickness-stretch, and thickness-twist). It is shown that the assumed displacement polynomials yield a strictly upper-bound convergence to exact solutions of the title problem, as a sufficient number of terms is retained. Since the effects of transverse shear and rotary inertia are inherent to the present 3-D formulation, an examination is made of the variation of non-dimensional frequencies with non-dimensional wall thickness, h/R ranging from thin-walled (h/R=0⋅05) to thick-walled (h/R=0⋅5) spherical bodies. The findings confirm that the variation of the spheroidal frequencies increases with increasing h/R and mode number, whereas the variation of the toroidal frequencies decreases with increasing h/R and mode number. This work offers some accurate 3-D reference data for the title problem with which refined solutions drawn from thin and thick shell theories and sophisticated finite element techniques may be compared. © 1997 by John Wiley & Sons, Ltd.

Vibrations of circular plates with clamped v-notches or rigidly constrained radial cracks

This study offers the first known free vibration data for thin circular plates with clamped V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and (2) corner functions which account for the bending moment singularities at the sharp corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non-dimensional frequencies for circular plates with clamped notches. Numerical results are obtained for plates having their circular edges completely free. Accurate (five significant figure) frequencies are presented for a wide spectrum of notch angles (0°, 5°, 10°, 30°, 60° and 90°) and depths. For very small notch angles, a rigidly constrained radial crack ensues. Some general findings are that, for the spectrum of notch angles examined, the first six frequencies increase as the notch depth increases, more so in the higher modes than the lower ones. The frequency increase with increasing notch depth is quite substantial for the semi-circular plates, and for segmented plates with sector angles less than 180°. For a constant notch depth, it is found that there is a substantial reduction in the first six frequencies as notch angle decreases. Normalized contours of the transverse vibratory displacement are shown for plates having 90° and 5° notches of various depths ranging from deep to very shallow. The first known frequencies and mode shapes for sectorial, semi-circular and segmented plates with clamped radial edges are also presented as special cases of the title problem.

Vibrations of Sectorial Plates Having Corner Stress Singularities

A procedure is presented for determining the free vibration frequencies and mode shapes of sectorial plates having re-entrant corners (i.e., vertex angles exceeding 180 degrees). No correct results for such problems have been found in the vast literature of plate vibrations. The procedure is applicable to sectorial plates having arbitrary (but continuous) boundary conditions (e.g., clamped, simply supported, or free) along the two radial edges and the circular edge. It is based upon the Ritz method, but utilizes two sets of admissible functions simultaneously. One set consists of algebraic-trigonometric polynomials. The other is the set of corner functions derived by Williams (1952) to deal with the bending stress singularities which may arise at the corner when the vertex angle becomes large. The method is demonstrated for sectorial plates having all edges simply supported, which yields the strongest singularity in a re-entrant corner. Frequencies are compared with those obtained from an analytical solution involving Bessel functions. It is shown that the latter solution is invalid for re-entrant corners. Analytical solutions are also obtained for annular sectorial plates having very small ratios of inner to outer boundary radii. These solutions are found to be consistent with those using polynomials and corner functions. Accurate fundamental frequency data is presented for simply supported sectorial plates having three values of Poisson’s ratio (0, 0.3, 0.5) and the full range of vertex angles (0 < α ≤ 360 deg).

Vibrations Of Circular Plates Having V-notches Or Sharp Radial Cracks

This paper provides the first known free vibration data for circular plates having V-notches. A V-notch has bending moment singularities at its sharp corner due to the transverse vibratory motion. A theoretical analysis is undertaken using two sets of admissible displacement functions, (1) algebraic-trigonometric polynomials and (2) corner functions. These function sets are used with the Ritz method. The first set guarantees convergence to the exact frequencies as sufficient terms are taken. The second set represents the corner singularities exactly, and accelerates convergence greatly. Numerical results are given for non-dimensional frequencies of completely free circular plates having various notch angles and depths. As the notch angle becomes very small, a sharp radial crack ensues. Convergence studies demonstrate the necessity of adding corner functions to achieve accurate frequencies. Extensive, accurate (five significant figure) frequencies are presented for the spectrum of notch angles (0°, 1°, 5°, 10°, 30°, 60° and 90°) and depths. The effect of the Poisson ratio on the frequencies in the case of shallow notches is also investigated. Sharp notches are found to reduce each of the first six frequencies from those of a complete circular plate, whereas large notch angles can increase some of the frequencies. Nodal patterns are shown for plates having 5° notches. The first known frequencies for completely free sectorial, semi-circular and segmented plates are also given as special cases.