Spheroidal Vibrations Research Articles

Analysis of Free Vibrations in Homogeneous Isotropic Rigidly Fixed Thermoelastic Circumferential Spherical Curved Plates

The 1st and 2nd class vibrations of rigidly fixed thermoelastic spherical curved plates have been presented in this article. After mathematical modeling, the obtained equations have been solved by applying normal mode analysis. The uncoupled equation has been considered as 1st class vibrations named toroidal vibrations, remains independent of temperature variations. The coupled system of equations which is considered as 2nd class vibrations named spheroidal vibrations. The series solution of matrix Fröbenius method has been applied to coupled system of ordinary differential equations to obtain potentials and temperature. Fixed point iteration numerical technique has been applied to analytical results with the assistance of MATLAB software tools. Computational techniques have been applied to analytical results and numerically generated data has been represented graphically for frequencies, dissipation factor, displacements and temperature. HIGHLIGHTS The 1st and 2nd class vibrations of rigidly fixed thermoelastic spherical curved plates have been presented in this article The uncoupled equation has been considered as 1st class vibrations named toroidal vibrations, remains independent of temperature variations The coupled system of equations which is considered as 2nd class vibrations named spheroidal vibrations The series solution of matrix Fröbenius method has been applied to couple system of ordinary differential equations to obtain potentials and temperature Computational techniques have been applied to analytical results and numerically generated data has been represented graphically for frequencies, dissipation factor, displacements and temperature GRAPHICAL ABSTRACT

Exciton interaction with acoustic phonons in PbS nanocrystals

The interaction between an exciton and acoustic phonons via the deformation potential in PbS nanocrystals is calculated in the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model. It is shown that the anisotropy of the deformation potential leads to nonzero interaction of an exciton with spheroidal phonons with the total angular momentum $j=2$. The size dependence of the Huang-Rhys factors for the fundamental breathing mode and its overtones as well as for spheroidal vibrations with the total angular momentum $j=2$ is evaluated.

A note on the spheroidal modes vibration of an elastic sphere in linear viscoelastic fluid

Vibration characteristics of elastic nanostructures embedded in fluid medium have been used for biological and mechanical sensing, and also to investigate materials mechanical properties. An analytical approach based on the exact theory has been developed in this paper, to establish a new accurate and simple generalized frequency equation to predict spheroidal vibration of an elastic nanosphere, in a compressible viscoelastic fluid using linear Maxwell fluid model. To demonstrate the accuracy of the present approach, a comparison is made with the published theoretical results in the literature in some particular cases, which shows a very good agreement. Thus, the obtained frequency equation can be very useful to interpret the experimental measurements of vibrational dynamics of nanospheres and can serve as benchmark solution in design of liquid sensors.

3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method

This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been uncoupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal motion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal relaxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use.

Laser ultrasonics detection of an embedded crack in a composite spherical particle

Laser ultrasonics was applied to the manufacturing control of the integrity (no failure) of coated spherical particles designed for High Temperature Reactors (HTR). This control is of major importance, since the coating of the nuclear fuel kernel is designed to prevent from the diffusion of fission products outside the particle during reactor operation. The SiC layer composing the coating is particularly important, since this layer must be an impenetrable barrier for fission products. The integrity of the SiC shell (no crack within the shell) can be assessed by the ultrasonic vibration spectrum of the HTR particle, which is significantly changed, compared to the reference spectrum of a defect-free particle. Spheroidal vibration modes of defect-free dummy particles with a zirconium dioxide (ZrO 2) core were observed in the 2–5 MHz range. A theoretical analysis is presented to account for the observed vibration spectra of defect-free or cracked HTR particles.

Vibration Analysis of Homogeneous Transradially Isotropic Generalized Thermoelastic Spheres

The exact free vibration analysis of stress free or rigidly fixed, thermally insulated/isothermal, transradially (spherically) isotropic thermoelastic solid sphere has been presented in context of nonclassical thermoelasticity. The transradially isotropic is also frequently referred as spherically isotropic in the literature. The basic governing equations of linear generalized thermoelastic, transradially isotropic, sphere have been uncoupled and simplified with the help of Helmholtz decomposition theorem. The formal solution of the coupled system of partial differential equations has been obtained by employing matrix Fröbenius method of extended series. The secular equations for the existence of possible modes of vibrations in the sphere have been derived by employing boundary conditions. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations have also been deduced and discussed. It is found that the toroidal motion gets decoupled from the spheroidal one and remains independent of thermal variations and thermal relaxation time. In order to illustrate the analytical development, the numerical solution of secular equations for spheroidal motion (S-mode) is carried out with respect of magnesium and solid helium spheres. The lowest frequency and damping factor of vibrational modes have been computed with the help of MATLAB programming and the results are presented graphically. The study may find applications in aerospace, navigation, geophysics tribology, and other industries where spherical structures are in frequent use.

Alfvén node-free vibrations of white dwarf in the model of solid star with toroidal magnetic field

In the context of the white dwarf asteroseismology, we investigate vibrational properties of a non-convective solid star with an axisymmetric purely toroidal intrinsic magnetic field of two different shapes. Focus is laid on regime of node-free global Lorentz-force-driven vibrations about symmetry axis at which material displacements have one and the same form as those for nodeless spheroidal and torsional vibrations restored by Hooke's force of elastic shear stresses. Particular attention is given to the even-parity poloidal Alfven modes whose frequency spectra are computed in analytic form showing how the purely toroidal magnetic fields completely buried beneath the star surface can manifest itself in seismic vibrations of non-magnetic white dwarfs. The obtained spectral formulae are discussed in juxtaposition with those for Alfven modes in the solid star model with the poloidal, homogeneous internal and dipolar external, magnetic field whose inferences are relevant to Alfven vibrations in magnetic white dwarfs.

Coupled analysis of high and low frequency resonant ultrasound spectroscopy: Application to the detection of defects in ceramic balls

A coupled analysis of high and low frequency resonant ultrasound spectroscopy of spheroidal modes is presented in this paper. Experimentally, by using an ultrasonic probe for the excitation (piezoelectric transducer) and a heterodyne optic probe for the receiver (interferometer), it was possible to take spectroscopic measurements of spheroidal vibrations over a large frequency range of 100 kHz-45 MHz in a continuous regime. This wide analysis range enabled variations in velocity due to the presence of defects to be differentiated from the inherent characteristics of the balls and consequently, it offers the possibility of detecting cracks independently of production variations. This kind of defect is difficult to detect because the C-shaped surface crack is very small and narrow (500 x 5 microm(2)), and its depth does not exceed 50 microm. The proposed methodology can excite spheroidal vibrations in the ceramic balls and detect such vibrations over a large frequency range. On the one hand, low frequency resonances are used in order to estimate the elastic coefficients of the balls according to various inspection depths. This method has the advantage of providing highly accurate evaluations of the elastic coefficients over a wide frequency range. On the other hand, high frequency vibrations are considered because they are similar to the surface waves propagating in the surface zone of the ceramic balls and consequently can be used to detect C-crack defects.

High frequency ultrasonic detection of C-crack defects in silicon nitride bearing balls

A non-destructive testing method for silicon nitride bearing balls based on ultrasonic resonance spectroscopy is proposed here. Through the theoretical study of their elastic vibrations, it is possible to characterize the balls using a vibration mode that is similar to surface wave propagation. The study of the influence of C-crack defects on the resonances of Rayleigh modes is presented here. These C-cracks are typically formed by impacts between balls during finishing or handling. They are frequently found on the surface of silicon nitride bearing balls and these C-cracks decrease the rolling contact fatigue life considerably. This kind of defect is difficult to detect because the C-shaped surface crack is very small and narrow (500 μm × 5 μm), and its depth does not exceed 50 μm. The proposed methodology can both excite spheroidal vibrations in the ceramic balls and detect such vibrations over a large frequency range. In particular, high frequency vibrations are considered because these are similar to the surface waves propagating in the cortical zone of the ceramic balls and consequently they can be used to detect C-crack defects.

Elasticity of nuclear medium as a principal macrodynamical promoter of electric pygmy dipole resonance

Motivated by arguments of the nuclear core-layer model formulated in [S.I. Bastrukov, J.A. Maruhn, Z. Phys. A 335 (1990) 139], the macroscopic excitation mechanism of the electric pygmy dipole resonance (PDR) is considered as owing its origin to perturbation-induced effective decomposition of a nucleus into two spherical domains–undisturbed inner region treated as a static core and dynamical layer undergoing elastic shear vibrations. The elastic restoring force is central to the excitation mechanism under consideration and has the same physical meaning as in macroscopic model of nuclear giant resonances involving distortions of the Fermi-sphere providing unified description of isoscalar giant electric and magnetic resonances of multipole degree ℓ⩾2 in terms of two fundamental vibrational modes in an elastic sphere, to wit, as spheroidal (electric) and torsional (magnetic) modes of shear elastic oscillations of the nodeless field of material displacements excited in the entire nucleus volume. In the present Letter focus is placed on the emergence of dipole overtone in the frequency spectrum of spheroidal elastic vibrations as Goldstone soft mode. To emphasis this feature of dipole resonant excitation imprinted in the core-layer model we regain spectral equation for the frequency of spheroidal elastic vibrations trapped in the finite-depth layer, derived in the above paper, but using canonical equation of an elastic continuous medium. The obtained analytic equations for the frequency of dipole vibrational state in question and its excitation strength lead to the following estimates for the PDR energy centroid EPDR(E1)=[31±1]A−1/3 MeV and the total excitation probability BPDR(E1)=[1.85±0.05]10−3Z2A−2/3e2 fm2 throughout the nuclear chart exhibiting fundamental character of this soft dipole mode of nuclear resonant response.

Comment on “Infrared and Raman selection rules for elastic vibrations of spherical nanoparticles”

The Raman selection rule for the acoustic vibrations of spherical nanoparticles is revised, showing that the symmetric $l=0$ and quadrupolar $l=2$ spheroidal vibrations are active. The relevant quantity is not the symmetry of the displacements itself but the symmetry of the strain tensor.

Wide band resonant ultrasound spectroscopy of spheroidal modes for high accuracy estimation of Poisson coefficient of balls

An original inversion method specifically adapted to the estimation of Poisson coefficient of balls by using their resonance spectra is described. From the study of their elastic vibrations, it is possible to accurately characterize the balls. The proposed methodology can create both spheroidal modes in the balls and detect such vibrations over a large frequency range. Experimentally, by using both an ultrasonic probe for the emission (piezoelectric transducer) and a heterodyne optic probe for the reception (interferometer), it was possible to take spectroscopic measurements of spheroidal vibrations over a large frequency range (100 kHz-45 MHz) in a continuous regime. This method, which uses ratios between wave resonance frequencies, allows the Poisson coefficient to be determined independently of Young's modulus and the ball's radius and density. This has the advantage of providing highly accurate estimations of Poisson coefficient (+/-4.3 x 10(-4)) over a wide frequency range.

EIGENFREQUENCIES OF NODELESS ELASTIC VIBRATIONS LOCKED IN THE CRUST OF QUAKING NEUTRON STAR

The Newtonian solid-mechanical theory of non-compressional spheroidal and torsional nodeless elastic vibrations in the homogenous crust model of a quaking neutron star is developed and applied to the modal classification of the quasi-periodic oscillations (QPOs) of X-ray luminosity in the aftermath of giant flares in SGR 1806-20 and SGR 1900+14. Particular attention is given to the low-frequency QPOs in the data for SGR 1806-20 whose physical origin has been called into question. Our calculations suggest that unspecified QPOs are due to nodeless dipole torsional and dipole spheroidal elastic shear vibrations.

Raman scattering of acoustical modes of silicon nanoparticles embedded in silica matrix

AbstractThe Raman scattering from acoustical phonons of silicon quantum dots in glass matrix was investigated. Two peaks that correspond to symmetric and quadrupolar spheroidal vibrations were found. A model calculation for in‐ and off‐resonance scattering conditions was used, which considered the homogeneous broadening due to interaction with matrix and the inhomogeneous broadening due to particle size distribution. A strong dependence of the light‐to‐vibration coupling coefficient on the particles size was needed for fitting the Raman data. This result suggests that resonance with electronic transitions of the silicon nanoparticles is important for excitation at 514.5 nm. The size distribution obtained from the Raman data is in agreement with the results of high‐resolution transmission electron microscopy. Copyright © 2006 John Wiley & Sons, Ltd.

Non-destructive testing of ceramic balls using high frequency ultrasonic resonance spectroscopy

Although ceramic balls are used more and more for bearings in the aerospace and space industries, defects in this type of ceramic material could be dangerous, particularly if such defects are located close to the surface. In this paper, we propose a non-destructive testing method for silicon nitride balls, based on ultrasonic resonance spectroscopy. Through the theoretical study of their elastic vibrations, it is possible to characterize the balls using a vibration mode that is similar to surface wave propagation. The proposed methodology can both excite spheroidal vibrations in the ceramic balls and detect such vibrations over a large frequency range. Studying their resonance spectrums allows the balls’ elastic parameters be characterized. Ours is an original method that can quickly estimate the velocity of surface waves using high frequency resonances, which permits surface and sub-surface areas to be tested specifically. Two applications are described in this paper. Both use velocity measurements to achieve their different goals, the first to differentiate between flawless balls from different manufacturing processes, and the second to detect small defects, such as cracks. Our method is rapid and permits the entire ceramic ball to be tested in an industrial context.

Raman spectra of CdS nanocrystals in Nafion: longitudinal optical and confined acoustic phonon modes

Quantum confinement effects on the longitudinal optical and acoustic phonons in CdS nanocrystals in the strongly confined regime in the polymer matrix Nafion are studied using Raman spectroscopy. The LO-phonon modes show size-dependent asymmetric broadening though the broadening and asymmetry are less than those predicted by the phonon confinement models. Two types of confined acoustic modes corresponding to n=1, l=0 and n=1, l=2 spheroidal vibrations are observed. Softening of the spheroidal modes is observed in the strongly confined regime.

Elastodynamical properties of nuclear matter from the observed activity of neutron stars

The results of studies devoted to describing the large-scale motions of the nuclear matter of neutron stars are reviewed. The approach is based on the idea that stellar nuclear matter is an elastic Fermi continuum possessing the properties of a compensated magneto-active plasma. The fundamental dynamical equations of motion of self-gravitating nuclear matter are taken to be the equations of nuclear elastodynamics, formulated in the macroscopic theory of nuclear collective processes. A variational method is presented for calculating the frequencies of gravitational-elastic spheroidal ( s -mode) and torsional ( t -mode) vibrations of a neutron star, and the star’s stability with respect to linear deformations is studied. The effectiveness of the elastodynamical method is illustrated by calculations of the periods of global gravitational elastic vibrations within the standard homogeneous model, and also for a family of realistic models of neutron stars, constructed on the basis of the relativistic equation of gravitational equilibrium and the nuclear-matter equations of state taking into account the heterophase nature of the nuclear statistical equilibrium. The motions of the magnetized Ae phase of the neutron-star matter are described using the magneto-hydrodynamical approach, which is used to calculate the frequencies of toroidal and poloidal Alfven oscillations. It is emphasized that magneto-plasma vibrations can significantly affect the electromagnetic activity of pulsars.

Stability of the non-radial eigenmodes of a uniformly charged elastic globe

The eigenmotions of an electrically charged isotropic elastic material are modeled by self-consistent system of elastodynamic equations (comprising the continuity equation for the density, the Euler equation for the velocity field of elastic displacements coupled with equation for the stress tensor governing evolution of elastic distortions) and Poisson’s equation of electrostatics. To elucidate the question of how the elastodynamic features of bulk elastic substance can be traced in a spherical sample of finite size, the low-frequency electromagnetic response of the sample is studied in the homogeneous model of a uniformly charged elastic globe. The response is described in terms of long wavelength, essentially non-radial, spheroidal (S-mode) and torsional (T-mode) electro-elastic vibrations. The restoring force is presumed to be dominated by the surface cohesive stresses and the volume disruptive stresses of electric origin acting destructively. It is shown that the continuum model in question leads to explicit form for the frequency of the non-radial eigenmodes. The emphasis is placed on the onset of vibrational instability originating from conflicting tendency of the above surface and volume elastic stresses. The elastodynamic criteria of instability for both spheroidal and torsional vibrations are established in analytic form.

Nondestructive assessment of the grain size distribution of SnO2 nanoparticles by low-frequency Raman spectroscopy

A nondestructive methodology is reported to obtain the grain size distribution of SnO2 nanoparticles, with grain sizes below 10 nm, using low-frequency Raman scattering measurements. The position of the main low-frequency Raman bands depends on the grain size and can be correlated with the spheroidal vibration modes of a spherical elastic body with “rigid” boundaries. The grain size distributions deduced from Raman scattering are in agreement with the direct measurements from transmission electron microscopy, especially for the smaller grains.

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.