The Dispersion of Flexural Waves in an Elastic, Circular Cylinder—Part 2

1962 ◽  
Vol 29 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Yih-Hsing Pao
Keyword(s):  
Circular Cylinder ◽  
Frequency Equation ◽  
Flexural Waves

In the first part of this paper, a method was developed for the construction of the branches of Pochhammer’s frequency equation for flexural waves in a circular cylinder. The method was applied to the portions of the branches with real propagation constants. The method is now extended and applied to those portions having imaginary propagation constants.

Dispersion of Flexural Waves in an Elastic, Circular Cylinder

1960 ◽  
Vol 27 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Yih-Hsing Pao ◽  
R. D. Mindlin
Keyword(s):  
Circular Cylinder ◽  
Phase Velocity ◽  
Propagation Constant ◽  
Direct Determination ◽  
Short Wave ◽  
Frequency Equation ◽  
Flexural Waves ◽  
Asymptotic Equations ◽  
Frequency Phase

In this paper it is shown how the branches of Pochhammer’s frequency equation for flexural waves in a circular cylinder may be constructed approximately with the aid of a grid of simpler curves and asymptotic equations for long and short wave lengths. With very little computation, in comparison with that required in the direct determination of the roots of Pochhammer’s equation, a qualitative view is obtained of the relations between frequency, phase velocity, group velocity, and propagation constant, for any branch, as well as some information as to the shapes of the modes.

scholarly journals Effect of Rotation on Wave Propagation in Hollow Poroelastic Circular Cylinder

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
S. M. Abo-Dahab ◽  
A. M. Abd-Alla ◽  
S. Alqosami
Keyword(s):  
Wave Propagation ◽  
Circular Cylinder ◽  
Phase Velocity ◽  
Frequency Equation ◽  
Wave Number ◽  
Elastic Behavior ◽  
Hollow Circular Cylinder ◽  
Poroelastic Material ◽  
Phase Velocity And Attenuation ◽  
Frequency Phase

The objective of this paper is to study the effect of rotation on the wave propagation in an infinite poroelastic hollow circular cylinder. The frequency equation for poroelastic hollow circular cylinder is obtained when the boundaries are stress free and is examined numerically. The frequency, phase velocity, and attenuation coefficient are calculated for a pervious surface for various values of rotation, wave number, and thickness of the cylinder which are presented for nonaxial symmetric vibrations for a pervious surface. The dispersion curves are plotted for the poroelastic elastic behavior of the poroelastic material. Results are discussed for poroelastic material. The results indicate that the effect of rotation, wave number, and thickness on the wave propagation in the hollow poroelastic circular cylinder is very pronounced.

Centrifugal waves

1960 ◽  
Vol 7 (3) ◽  
pp. 340-352 ◽  
Author(s):  
O. M. Phillips
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Circular Cylinder ◽  
Rigid Body ◽  
Water Depth ◽  
Frequency Equation ◽  
Rigid Rotation ◽  
Body Motion ◽  
Small Perturbations ◽  
The Stability

When a hollow circular cylinder with its axis horizontal is partially filled with water and rotated rapidly about its axis, an almost rigid-body motion results with an interior free surface. The emotion is analysed assuming small perturbations to a rigid rotation, and a criterion is found for the stability of the motion. This is confirmed experimentally under varying conditions of water depth and angular velocity of the cylinder. The modes of oscillation (centrifugal waves) of the free surface are examined and a frequency equation deduced. Two particular modes are considered in detail, and satisfactory agreement is found with the frequencies observed.

On the Equivalence of Dispersion Relations Resulting from Rayleigh-Lamb Frequency Equation and the Operator Plate Model

2001 ◽  
Vol 123 (4) ◽  
pp. 417-420 ◽  
Author(s):  
Nikolay A. Losin
Keyword(s):  
Velocity Dispersion ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Dispersion Relations ◽  
Plate Model ◽  
Elastic Plates ◽  
Flexural Waves ◽  
Frequency Spectra ◽  
Infinite Power ◽  
Antisymmetric Vibrations

The Rayleigh-Lamb frequency equations for the free vibrations of an infinite isotropic elastic plate are expanded into the infinite power series and reduced to the polynomial frequency and velocity dispersion relations. The latter are compared to those of the operator plate model developed in [Losin, N. A., 1997, “Asymptotics of Flexural Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 64, No. 2, pp. 336–342; Losin, N. A., 1998, “Asymptotics of Extensional Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 65, No. 4, pp. 1042–1047] for both symmetric and antisymmetric vibrations. As a result of comparative analysis, the equivalence of the corresponding dispersion polynomials is established. The frequency spectra, generated by Rayleigh-Lamb equations, are illustrated graphically and briefly discussed with reference to those published in [Potter, D. S., and Leedham, C. D., 1967, “Normalized Numerical Solution for Rayleigh’s Frequency Equation,” J. Acoust. Soc. Am., 41, No. 1, pp. 148–153].

On a heat flow problem in a hollow circular cylinder

1968 ◽  
Vol 64 (1) ◽  
pp. 193-202
Author(s):  
Nuretti̇n Y. Ölçer
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Eigenvalue Problems ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Frequency Equation ◽  
Hollow Circular Cylinder ◽  
New Ideas ◽  
Finite Integral

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

Stress Wave Propagation in a Bar of Arbitrary Cross Section

1982 ◽  
Vol 49 (1) ◽  
pp. 157-164 ◽  
Author(s):  
K. Nagaya
Keyword(s):  
Wave Propagation ◽  
Cross Section ◽  
Fourier Expansion ◽  
Frequency Equation ◽  
Arbitrary Cross Section ◽  
Stress Wave Propagation ◽  
Flexural Waves ◽  
Elliptical Cross ◽  
Phase Velocities ◽  
Wave Numbers

In this paper a method for solving wave propagation problems of an infinite bar of arbitrary cross section has been presented. The frequency equation for finding phase velocites for longitudinal, torsional, and flexural waves have been obtained by making use of the Fourier expansion collocation method which has been developed by the author on the vibration and dynamic response problems of membranes and plates. As a numerical example, the phase velocities versus wave numbers are calculated for elliptical and truncated elliptical cross-section bars.

Asymptotics of Flexural Waves in Isotropic Elastic Plates

1997 ◽  
Vol 64 (2) ◽  
pp. 336-342 ◽  
Author(s):  
N. A. Losin
Keyword(s):  
Material Parameter ◽  
Velocity Dispersion ◽  
Three Dimensional ◽  
Analytical Form ◽  
Short Wave ◽  
Frequency Equation ◽  
Elastic Plates ◽  
Flexural Waves ◽  
Explicit Formulas ◽  
Longitudinal Waves

The long and short-wave asymptotics of order O(k6h6) for the flexural vibrations of an infinite, isotropic, elastic plate are studied. The differential equation for the flexural motion is derived from the system of three-dimensional dynamic equations of linear elasticity. All coefficients of the differential operator are presented as explicit functions of the material parameter γ = cs2/cL2, the ratio of the velocities squared of the flexural (shear) and extensional (longitudinal) waves. Relatively simple frequency and velocity dispersion equations for the flexural waves are deduced in analytical form from the three-dimensional analog of Rayleigh-Lamb frequency equation for plates. The explicit formulas for the group velocity are also presented. Variations of the velocity and frequency spectrums depending on Poisson’s ratio are illustrated graphically. The results are discussed and compared to those obtained and summarized by R. D. Mindlin (1951, 1960), Tolstoy and Usdin (1953, 1957), and Achenbach (1973).

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