Solid Elastic Cylinder Research Articles

Asymptotic Analysis of the Modes of Wave Propagation in a Solid Cylinder

An asymptotic method is used to analyze the free vibrations of a solid elastic cylinder of circular cross section. In this method, the displacement components and the frequencies are sought as power series of the dimensionless wavenumber ε, where ε = 2π × RADIUS/WAVELENGTH. By substituting the expansions in the displacement equations of motion and the boundary conditions, and by collecting terms of the same order εn, a system of coupled second-order inhomogeneous ordinary differential equations is obtained with the radial variable as independent variable. Subsequent integration yields the coefficients of εn for the displacement components and the frequencies for all modes and for the whole range of frequencies, but in a range of real-valued dimensionless wavenumbers 0 < ε < 1.

Stresses in long cylinder due to rotating line source of heat.

The subject of this paper is the determination of stresses in an infinite solid elastic circular cylinder due to a moving line source of heat. This source is assumed to be parallel to the axis of the cylinder and is rotating on its surface with constant angular velocity. The problem is two-dimensional, since in the cylinder there is a state of plane strain. By using the Laplace transform method, the temperature distribution in the cylinder has been presented in a form of a sum of six components. Then stresses occurring in the cylinder due to each temperature component have been calculated. The sum of these components represents the total thermal stresses in the cylinder. A simple numerical example has been calculated.

Band of Dynamic Pressure on a Solid Elastic Cylinder

Band of Dynamic Pressure on a Solid Elastic Cylinder

Experimental Confirmation of Circumferential Waves on Solid Cylinders

Following the theory describing circumferential waves (presented in the preceding paper), an experimental investigation was undertaken to observe and measure these waves. The Rayleigh-type surface wave has been observed on elastic solid cylinders immersed in water. This wave was excited by an acoustical pulse of 10-μsec duration projected onto the cylinder at the excitation angle given by sin−1 (c0/c), where c is the velocity of the circumferential wave and c0 is the velocity of sound in water, and has been observed to make several circumnavigations of the cylinder. The waves can be detected since they continuously radiate acoustical energy into the water and have been observed on pure aluminum, 304 stainless steel, and various aluminum-alloy cylinders. Since the surface curvature gives rise to dispersion, cylinders of 3.5- to 6-in. diam were used and the frequency of the incident pulse was varied from 0.15 to 3.5 MHz. This corresponds to a radius-to-wavelength ratio of 10 to 400. Experimental and theoretical results will be presented and shown to compare favorably. Inhomogeneities in brass and in aluminum alloys are found to correlate with poor propagation characteristics for some of the cylinders. Examples will be shown.

Observation of Circumferential Waves on Solid Aluminum Cylinders

We report on an experiment demonstrating the existence of Rayleigh circumferential waves on solid elastic cylinders. Underwater sound pulses are shown to propagate around a solid aluminum cylinder in an experiment that utilizes transmitters and receivers with narrow beams directed at the cylinder at the critical angles. The theoretical prediction of the accompanying letter by Doolittle et al. is verified.

Axisymmetric Vibrations of a Solid Elastic Cylinder Encased in a Rigid Container

This paper describes the effects of end boundary conditions on the vibrational characteristics of solid propellent rocket motors. Previously, in the literature, solutions have been based on those obtained for an infinitely long cylinder. These solutions yield only certain sets of possible end boundary conditions for a finite cylinder, but not those considered here (i.e., fixed on all boundaries). The present approach consists of choosing a series of functions with unknown coefficients. Each term satisfies the governing differential equation and the boundary conditions on the axial displacement. The boundary conditions on the radial displacement are approximated by an orthogonalization procedure. This method yields an eigenvalue matrix the coefficients of which are transcendental functions of the frequency. The accuracy of the final solution depends on how well the radial displacement boundary conditions are satisfied. It is found that the procedure converges, and that sufficient accuracy is achieved, through the use of 20 terms in the series. The range of application of the simpler method, based on an infinite cylinder, is discussed by comparing results obtained from both methods.

Axisymmetric Vibrations of a Solid Elastic Cylinder Encased in a Rigid Container

The purpose of this study is to investigate the effects of end boundary conditions on the vibrational characteristics of solid-propellant rocket motors. Previously, in the literature, solutions have been obtained based on those of an infinitely long cylinder. These solutions yield only certain sets of possible end boundary conditions for a finite cylinder, but not those considered (i.e., fixed on all boundaries). The approach consists of choosing a series of functions with unknown coefficients. Each term satisfies the governing differential equation and the boundary conditions on the axial displacement. The boundary conditions on the radial displacement are approximated by an orthogonalization procedure. This method yields an eigenvalue matrix whose coefficients are transcendental functions of the frequency. The accuracy of the final solution depends on how well the radial-displacement boundary conditions are satisfied. It is found that the procedure converges, and that sufficient accuracy is achieved, through the use of 20 terms in the series. The limitations of the simpler method, based on an infinite cylinder, are discussed by comparing results obtained from both methods.

Sound Scattering by Elastic Cylindrical Shells

We present the solution for the scattered field corresponding to a plane sound wave incident upon an infinite elastic circular-cylindrical concentric shell, imbedded in a fluid and enclosing another fluid. The results are shown to contain the known special cases of fluid and elastic solid cylinders. Furthermore, limiting expressions for long wavelengths are obtained in special cases.

Axisymmetric, Plane-Strain Dynamic Response of a Thick Orthotropic Shell

The plane-strain, axisymmetric, thickness-stretch modes of thick, isotropic and orthotropic elastic shells embedded in a thin elastic casing are investigated. The steady-state dynamic response of their viscoelastic counterparts is obtained and compared with the response of the elastic shells. Use is made a of a higher-order shell theory and its accuracy for the lower modes is demonstrated. Finally, the effect of orthotropy on the rapidity of the transition from the free vibrations of a plate to that of a solid elastic circular cylinder is investigated.

Experimental Investigation of Acoustic Scattering from Solid Elastic Cylinders of Varying Length

Experimental measurements are presented of backward and general scattering from vertically oriented solid aluminum cylinders in water. Cylinder lengths from 0.15 to 70λ—where λ is the wavelength of the incident sound—are considered. The ka range investigated is from 2 to 22. It is shown that the ratio L/a—where L is the cylinder length, and a is its radius—is the critical parameter in determining the cylinders' scattering behavior. Cylinders for which this ratio is greater than approximately 0.8 display similar backscattering (target strength) and general scattering patterns and echo-pulse forms regardless of incident wavelength; those for which the ratio is less than 0.8 are also similar in these measures but differ from the former group. The immediate conclusion is that two-dimensional analyses are applicable to cylinders appreciably shorter in length than the hitherto accepted value of L > 10λ. It is also shown that a halving of cylinder length yields more nearly a 5-dB rather than the expected 6-dB decrement in the measured target-strength values. [This work was supported by the U. S. Office of Naval Research.]

Elastic Wave Scattering at a Cylindrical Discontinuity in a Solid

This study deals with the scattering of plane compressional and shear elastic waves incident obliquely on an infinitely long nondissipative cylindrical elastic discontinuity in an isotropic solid. Equations whose solutions are the coefficients in series expressions for the scattered waves are derived for the case of scattering at an elastic solid cylinder embedded in a different solid medium. From these equations are obtained those governing scattering at a fluid-filled cylindrical bore, and at an empty bore. In general, when a wave of either the compressional or shear type is incident on an elastic discontinuity, scattered waves of both types are produced, a process known as “mode conversion.” A simple relation is shown to exist between the two far-zone distributions-in-angle, and between the two scattering cross sections for the waves undergoing mode conversion in the scattering of compressional and normally polarized shear waves incident normally upon either a fluid-filled or an empty bore. For an empty bore whose diameter was comparable with a wavelength in the solid, computed distributions-in-angle for the scattering of normally incident compressional and normally polarized shear waves were verified experimentally. Measurements were also made of the scattering of compressional waves at a fluid-filled bore in a solid, and of the far-zone directivity patterns of some compressional and shear wave piezoelectric source crystals radiating into a solid medium.