Free Radial Vibrations Research Articles

Exact Size-Dependent Elasticity Solution for Free Radial Vibration of Nano-Sphere

An exact size-dependent elasticity solution based on surface energy theory is used to investigate the free radial vibration behavior of a nanoscale sphere. The Gurtin-Murdoch surface elasticity model is employed to incorporate surface stress terms into pertinent boundary conditions. This leads to frequency equations involving spherical Bessel functions. Extensive numerical calculations have been carried out to illustrate the size effect of the nano-sphere on the first and second dimensionless breathing natural frequencies. The numerical results describe the imperative influence of surface energy and radii ratio on the vibrational characteristic frequency of the nano-sphere. In particular, the surface energy is much more important when the inner radius is smaller than 50 nm.

On the Natural Frequency of Free Vibrations of the Gas Envelope Surrounding a Solid Particle or a Droplet

The behavior of two-phase bubbles containing a solid or a liquid particle in their interior has been studied. A simple formula for the frequency of free radial vibrations of such a bubble has been obtained.

Vibration of a prestressed tube in the presence of plastic zone

The paper is devoted to a problem of radial steady state vibrations for a tube taking into account the inhomogeneous residual stress (RS) field. According to the theory of ideal plasticity the RS field is caused by elastoplastic deformation due to the internal pressure and the further unloading process. Various RS distributions inside both elastic and plastic zones are described. A finding of displacement function is reduced to the numerical solving of the first-order differential equations. Within the framework of the acoustical method, the inverse problem for RS field identification is stated. On the basis of the free radial vibration problem for tube, we derived the formula for a dependency of eigenfrequency on RS distribution by using the linearization method. By using this formula for two different values of eigenfrequencies, we state the system of two equations to identify the plasticity zone radius and RS level. Examples of numerical solution of this system for different values of parameters (RS level and tube thickness) are demonstrated. The estimation accuracy of eigenfrequencies measurement for successful implementation of the proposed approach is given.

An applied model for free radial vibrations of a closed spherical sandwich shell

Free vibrations across the thickness of a closed spherical shell are studied. A finite solution to the one-dimensional (along the radius) wave problem is derived for the shell, which differs from the solution for radial vibrations of a liquid. The solution is used in constructing a model for joint vibrations of three spherical layers. An approximate version of their vibratory motion is suggested, which is based on a shell model with thin stiff layers enveloping a thick soft core layer. The solution to the dynamic problem of free vibrations of the soft core is obtained by the method of separation of variables. The elastic reactions of the thin face layers were taken as boundary conditions, which allowed us to reduce the dynamics of the three layers to a vibratory motion of the midlayer under mixed boundary conditions. A transcendental equation for eigenvalues of the problem on free vibrations is derived and analyzed. A graphic description of the effect of layer thickness and elastic properties on the spectrum of eigenvalues of the mixed boundary problem is represented. Relationships for the main eigenfrequency versus the stiffness of face layers and characteristic geometric parameters of the sandwich shell wall are obtained.

On the radial vibration of a non-homogeneous circular disc of variable thickness

Free radial vibration of a non-homogeneous circular plate of variable thickness has been studied. The thickness at a distancer from the axis through the centre of the plate perpendicular to the middle surface is chosen to be a function ofr. The problem has been solved for a disc with a central hole and also for one having no central hole.

Dynamical Aspects of Viscoelasticity

This paper is concerned with some aspects of the dynamic response of a three-dimensional linear viscoelastic medium to prescribed boundary conditions. The constitutive equations, expressed in operator form, are applied to a three-dimensional continuum and are combined with the equations of motion to yield a system of linear partial differential equations. This system may be expressed in terms of a scalar and vector potential and parameters called “generalized velocities c1* and c2*” (operators describing the properties of the medium). The free radial vibrations of a viscoelastic sphere and spherical shell are worked out for the prescribed boundary conditions. In particular the phase velocity is obtained as a function of the wave number and a complex frequency. The wave number is obtained from the roots of a transcendental equation. The free longitudinal vibrations of a viscoelastic cylinder are determined by solving the equations for the scalar and vector potentials with the appropriate boundary conditions. The phase velocity is obtained for the Voigt and Maxwell models. In addition, asymptotic results are obtained which reduce to either the thin rod or the Pochhammer approximation.

On Nonhomogeneous Elastic Spheres

Radial deformation due to body forces and free radial vibration of nonhomogeneous elastic spheres is investigated. The elastic coefficients and the mass density are taken to vary as the n-th power of the distance from the sphere center. The boundary conditions are general insofar as spherical symmetry is maintained. The stress-displacement relations of Stojanovitch are used and solutions are obtained through the use of a generalized finite Hankel transform.

Radial Vibrations of Thick-Walled Hollow Cylinders

Approximate formulas are given for the natural wavelengths associated with the free radial vibration of a thick-walled, infinitely long, hollow cylinder. For the radial extensional mode, the wavelength is nearly proportional to the nth-order mean radius, where n depends on Poisson's ratio. For the thickness modes, a correction factor is derived which may be applied to the plane-wave solution. The extensional solution is also applied to the case of a thin, flat washer.

On Elastic Continua With Hereditary Characteristics

Abstract In a previous paper (1) the free and forced vibrations of systems of one degree of freedom with hereditary damping characteristics were discussed. In the present paper the classical equations of motion for elastic media are extended on the basis of the general linear stress-strain law involving hereditary damping. These equations are applied to the case of free radial vibrations of a sphere. Furthermore, the free vibrations of strings, the free transverse vibrations of beams, and the free vibrations of rectangular and circular membranes are studied under the assumption of hereditary damping.