Vibration Of Sphere Research Articles

Energy level statistics of the U(5) and O(6) symmetries in the interacting boson model

We study the energy level statistics of the states in U(5) and O(6) dynamical symmetries of the interacting boson model and the high spin states with backbending in U(5) symmetry. In the calculations, the degeneracy resulting from the additional quantum number is eliminated manually. The calculated results indicate that the finite boson number $N$ effect is prominent. When $N$ has a value close to a realistic one, increasing the interaction strength of subgroup O(5) makes the statistics vary from Poisson-type to GOE-type and further recover to Poisson-type. However, in the case of $N \to \infty$, they all tend to be Poisson-type. The fluctuation property of the energy levels with backbending in high spin states in U(5) symmetry involves a signal of shape phase transition between spherical vibration and axial rotation.

Effects of anharmonicity and electronic coupling on photoinduced electron transfer in mixed valence compounds

We develop a semigroup model of electron transfer (ET) dynamics in mixed valence compounds. This model is useful for investigating the effects of anharmonicity in inner sphere nuclear modes, as well as the dependence of the electronic dynamics on the nature of the electronic coupling. Two effective “subsystem” nuclear vibrations are treated explicitly in the model, to account for the rapid electronic energy gap fluctuations induced by the inner sphere vibrations. The essentially Markovian effects of the remaining “bath” modes are approximated by semigroups. We find that including the anharmonicity in inner sphere vibrations leads to a very small increase in the rate of ET. This effect is due to the change in reactant and product vibronic states when anharmonicity is included, as well as the rapid electronic dephasing induced by the bath. An assumption of strong electronic coupling is found to be sufficient to explain experimentally observed ET rates, but the possible role of conical intersections in ultrafast ET reactions is also noted.

Foam rheology in the wet and dry limits

Understanding the rheological behavior of wet foams is important as a basic problem in fluid physics, and as a practical problem in many industries. This lecture will describe research into the wet and dry limits of foam rheology by a relatively new experimental technique in which foam drops are acoustically levitated and driven into motion. In the dry limit, the drops behave as viscoelastic solids. The effective moduli of the foam are estimated by observing the resonances of the drops and matching them to an analytical model for viscoelastic sphere vibrations. Analytical explorations of the wet limit have proceeded by considering the dynamics of a single bubble in a volume of liquid determined by the foam’s void fraction. The linearized result is a wave equation, from which the natural frequencies and mode shapes of wet foam drops are determined. Relationships between this wave equation and those of classical effective medium theories will be described. [Work supported by NASA.]

A PARAMETRIC STUDY ON THE AXISYMMETRIC MODES OF VIBRATION OF MULTI-LAYERED SPHERICAL SHELLS WITH LIQUID CORES OF RELEVANCE TO HEAD IMPACT MODELLING

The free vibration of spheres composed of inviscid compressible liquid cores surrounded by spherical layers of linear elastic, homogeneous and isotropic materials are studied using three-dimensional elasticity equations. The exact three-dimensional equations are first derived for an N -layered sphere with a liquid core and an extensive parametric study is then presented for the first few natural frequencies of the spheroidal modes of vibration. Non-dimensional frequency parameters are compared with values obtained using lower order membrane and shell theories. It is shown that for a remarkably wide range of geometric and material parameters, which encompasses values typical for the human head, the first ovalling mode of a fluid-filled shell behaves like a membrane filled with incompressible fluid and a simple closed-form expression is derived which closely approximates the natural frequencies obtained using the exact three-dimensional equations.

Equivalent potentials of the geometric collective model and interacting boson model

We investigate the achievable degree of agreement between spectroscopic predictions of the geometric collective model (GCM) and the interacting boson model-1 (IBM-1). First, the GCM potential-energy surfaces are made locally equivalent with the IBM-1 semiclassical potentials close to the minimum. The GCM kinetic term is then optimized to reproduce some of the IBM-1 energy ratios. Differences between the spectra, as predicted by the two models, are discussed for the transitional region from the prolate axially symmetric rotor to the spherical vibrator.

The orienting response of Lake Michigan mottled sculpin is mediated by canal neuromasts.

Lake Michigan mottled sculpin, Cottus bairdi, exhibit a naturally occurring and unconditioned orienting response that can be triggered by both live prey and chemically inert vibrating spheres, even in blinded animals. CoCl(2)-induced reductions of the orienting response demonstrate that the lateral line is required for this behavior in the absence of non-mechanosensory cues (such as vision), but shed no light on the relative contributions of superficial and canal neuromasts to this behavior. To determine the relative roles of these two subsystems, we measured the frequency with which mottled sculpin oriented towards a small vibrating sphere before and after two treatments: (i) immersion of fish in a solution of gentamicin, an aminoglycoside antibiotic that damages hair cells in canal, but not superficial, neuromasts; and (ii) scraping the skin of the fish, which damages the superficial, but not the canal, neuromasts. To ensure that both superficial and canal neuromasts were adequately stimulated, we tested at different vibration frequencies (10 and 50 Hz) near or at the best frequency for each type of neuromast. At both test frequencies, response rates before treatment were greater than 70 % and were significantly greater than 'spontaneous' response frequencies measured in the absence of sphere vibration. Response rates fell to spontaneous levels after 1 day of gentamicin treatment and did not return to pre-treatment levels for 10-15 days. In contrast, response rates stayed approximately the same after superficial neuromasts had been damaged by skin abrasion. Scanning electron microscopy confirmed hair cell damage (loss of apical cilia) in canal, but not superficial, neuromasts of gentamicin-treated animals after as little as 24 h of treatment. The sensory epithelium of canal neuromasts gradually returned to normal, following a time course similar to behavioral loss and recovery of the orienting response, whereas that of superficial neuromasts appeared normal throughout the entire period. This study shows that the orienting response of the mottled sculpin is mediated by canal neuromasts.

Shift of the 21+state of10Bein ternary cold fission of252Cf

Recent experimental data indicate that in the ternary cold fission of ${}^{252}\mathrm{Cf}$ the energy of the first excited state of the accompanying light cluster ${}^{10}\mathrm{Be}$ is decreased by an amount ranging between $\ensuremath{\approx}$ 6 and 26 keV. A model is proposed to calculate the shift of the vibrational ${2}_{1}^{+}$ state in ${}^{10}\mathrm{Be}$ when its heavy companions are the even-even nuclei ${}^{146}\mathrm{Ba}$ and ${}^{96}\mathrm{Sr}.$ The stiffness parameters of the $\ensuremath{\beta}$ vibrations are calculated within the self-consistent Hartree-Fock method with BCS pairing correlations taken into account, and their change is determined by the interaction of the light cluster with the heavy fragments. The results point to a dependence of the shift magnitude and signature on the relative distance between the three clusters and their mutual orientation. Eventually it is the anharmonic perturbation of the spherical vibrator which is responsible for obtaining a negative energy shift of the ${2}_{1}^{+}$ state.

On the self-excited vibrations of bluff bodies suspended with a narrow clearance in a vertical pipe

The results of an experimental study of the flow past a sphere suspended with a narrow clearance in a circular cylindrical pipe and allowed only lateral movement, are presented. The study was limited to the case of intense vibrations accompanied by impacts between the sphere and the pipe wall. The experiments were conducted in steel and copper pipes at Reynolds numbers varying from 104 to 105 and several values of the sphere mass and the clearance between the sphere and the pipe wall. The self-excited vibration frequency was measured by a newly developed method based on registering the frequency only in the resonance modes which set in when the sphere and pipe vibration eigenfrequencies coincide. The flow was photographed and filmed at various values of the water flow rate. The major nondimensional parameters characterizing the phenomenon considered were revealed, and a universal relation between them and the nondimensional self-excited vibration frequency was established with acceptable accuracy.

Measuring the mechanical stiffness of an eggshell using resonant frequency analysis

1. A new method was introduced for the measurement of mechanical eggshell characteristics based on the analysis of the vibration response of a chicken egg excited with a non-destructive impact. Four major topics were examined. 2. The 3 dimensional vibration pattern of an intact chicken egg was described for the spherical vibration with the lowest resonant frequency. 3. The resonant frequency of this basic vibration mode and the total egg mass were used to calculate a dynamic stiffness value of the eggshell based on a mathematical mass-spring model. 4. A correlation analysis between this dynamic stiffness value and several common egg and eggshell variable was calculated. Pearson correlation coefficients between the dynamic stiffness and static stiffness, eggshell thickness at the equator, eggshell width and shape index were all significantly different from zero and were 0.71, 0.60, 0.43 and 0.51, respectively. 5. Graphical analysis of the static vs the dynamic eggshell stiffness indicated that eggs with thick shells (>360 mum) had a higher static stiffness value than the dynamic stiffness measurement, whereas the static stiffness of eggs with thin shells (>300 mum) was lower than the dynamic stiffness. 6. The new vibration test method is a promising alternative for evaluating mechanical eggshell properties because of its non-destructive nature and brief measuring time.

The forward projection of harmonic pressure fields using the generalized internal source density method

A previously developed generalized internal source density (GISD) method is extended to address the forward projection of harmonic pressure fields from normal velocity fields on a closed surface of revolution in a fluid. The GISD approach is based on decomposing the normal velocity field on the closed surface into a summation of circumferential orders where the pressure field for each order is associated with an internal linear source distribution of ring sources on the axis of revolution of the surface. The linear source distributions are formulated as integral equations of the first kind which are then solved using singular value decomposition methods. Far-field pressures are simply related to the Fourier transforms of the linear source distributions. Numerical results are presented to illustrate the forward projection of harmonic fields using the GISD method for both Dirichlet and Neumann problems. The numerical results for spherical and finite cylindrical vibrators illustrate the usefulness of the GISD method over a wide range of frequencies for both acoustic radiation and scattering problems.

Theory of Spherical-Wave Debye-Waller Factors in Photoelectron Diffraction Spectra Compared with that in EXAFS

A new approach is applied to the analyses of thermal effects in ARXPS (photoelectron diffraction) spectra, where we include both spherical wave correction and anharmonic vibration effects. A partial summation technique is applied to get a plane wave part and a spherical wave part(dynamical spherical wave effect ). Numerical calculations illustrate the importance of the spherical wave effects in high energy ARXPS spectra. In particular the spherical wave effects play an important role in the small-angle scatterings which predominate in the ARXPS processes. In comparison with the dynamical spherical wave effects in EXAFS those in ARXPS only reduce the intensities whereas the phase is unchanged.

Spherical Surface Vibration Identification by Spherical Holography and Acoustic Intensity

Spherical Surface Vibration Identification by Spherical Holography and Acoustic Intensity

Spherical wave effects on ARXPS Debye—Waller factors

In this work we have applied a new approach developed by us to the analyses of thermal effects in ARXPS spectra, where we include both spherical wave correction and anharmonic vibration effects based on Brouder's Lie group formulas. We apply partial summation technique to get a plane wave part which does not disappear even for large kR limit, and a spherical wave part which disappears for that limit. Numerical calculations illustrate the spherical wave effects in ARXPS spectra.

The G-Band Systematics of Vibrational and Transitional Nuclei with A = 150 to 166

Abstract The vibrational and rotational characteristics of the ground bands of even-even nuclei with 150 ≤ A ≤ 166 (2 < R4< 3) are studied. The spectra of these nuclei (with band crossing angular momentum IC ≥ 12) are analyzed with a cubic polynomial in I. The considered nuclei (150Sm, 152Gd, 154-156Dy, 156Er, 158-160-162Yb and 162,164, 166Hf) lie in the central region between the Z = 50 and Z = 82 major shell closures and span the spherical to the well deformed region. The gradual shape transition from a soft spherical vibrator to a deformed rotor from 150Sm to 166Hf is thus made explicitly apparent from the g-band spectrum analysis in terms of the vibrational, rotational and softness coefficients. The transition at N = 88-90 for the vibrational-rotational ratios and for the kinematic moments of inertia are reproduced.

Radial Vibrations of Spheres

The free and forced radial vibrations of spheres have been studied analytically. Explicit expressions are presented for the natural frequencies and dynamic stresses for various time-varying internal pressures.

Rotatory vibration of sphere of higher order viscoelastic solid

An attempt is made to investigate the rotatory vibration of a sphere of higher order viscoelastic solid considering higher order strain rate and stress rate. The general frequency equation is obtained for this type of vibration of a sphere. As a special case of this analysis, the frequency equations for the first order and the second order viscoelastic solids are derived. It is shown that the classical frequency equation for an isotropic elastic solid follows from this analysis.

Forward and backward projection of axisymmetric pressure fields from shells of revolution using an internal source density method

The forward and backward projection of measured harmonic pressure fields from complex vibrators is a subject of considerable interest in the areas of transducer calibration and structural acoustics. A general methodology for addressing such projection problems for surfaces of revolution using internal line source density distributions along the axis of the body is presented. The focus of the present paper is however on the rotationally symmetric problem. A least-mean-square error method is used to determine an internal line monopole source density distribution along the axis of symmetry of the measurement surface by matching the measured pressure in the field to the pressure field of the source distribution at the surface. The resultant source density distribution can then be used to determine the pressure and velocity fields exterior to the body of revolution; hence, the measured pressure field can be forward and backward projected. Numerical results are presented to illustrate the accuracy of the projection method for the case of spherical, spheroidal, and finite cylindrical vibrators with rotationally symmetric velocities.

The free vibrations of inhomogeneous elastic cylinders and spheres

The variational statement, governing equations and corresponding Ritz approximations are derived in Cartesian, cylindrical and spherical coordinates for the evaluation of the natural frequencies of free vibrations of elastic cylinders and spheres. The formulation can account for orthotropic material symmetry, and can be applied to either solid or hollow geometries. The approximating functions selected for the displacement components depend on the geometry and coordinate system used to describe the problem, and are a combination of power series. Fourier series and spherical harmonics. Representative examples are given in the various coordinate systems for both the cylinder and the sphere. In general, excellent agreement is found with results obtained by other methods.

Sub-barrier fusion of 37Cl with 58,60,62,64Ni.

Near-barrier and sub-barrier fusion excitation functions have been measured for the $^{37}\mathrm{Cl}$${+}^{58,60,62,64}$Ni systems using a recoil velocity spectrometer to identify evaporation residues. A considerable amount of enhancement was observed for all systems studied. The measured excitation functions for $^{37}\mathrm{Cl}$${+}^{58,62,64}$Ni are not in agreement with published results for these systems. The data were analyzed in the context of a coupled-channels model which took into account either the dependence of the fusion cross section on the coupling to surface vibrational modes, or, alternatively, on the orientation of deformed colliding nuclei. Very good agreement was obtained with the experimental data under the assumption that $^{37}\mathrm{Cl}$ exhibits a moderate oblate static deformation, while the light Ni isotopes are spherical vibrators. On the other hand, the data for $^{37}\mathrm{Cl}$${+}^{64}$Ni favor a moderate static oblate deformation for $^{64}\mathrm{Ni}$.

Hot-film anemometry for measuring lateral line stimuli

A hot-film anemometer system has been calibrated and evaluated for the measurement of sinusoidal water motions used in stimulating the mechanosensory lateral line system of a teleost fish. The response of the anemometer system to water motions created by a vibrating sphere was measured over a wide range of frequencies, intensities, and distances from the sphere. The amplitude response of the system to signals along the axis of sphere vibration was found to be linear over a 50-dB range for frequencies from 10-200 Hz, with the lowest end of the dynamic range (between 10(-8) and 10(-9) m) corresponding to physiological measures of best sensitivity in the lateral line system of the mottled sculpin, Cottus bairdi. The measured attenuation of the signal with distance was also linear over this frequency range out to distances of six times the radius ( = 3 mm) of the sphere and followed the predicted falloff rate for a dipolar source. The linear response of the anemometer system over a wide dynamic range encompassing the detection range of the lateral line system, and the match between predicted and measured motions at varying distances from a dipolar source, indicates that hot-film anemometry is a useful technique for measuring low-level, low-frequency signals likely to stimulate the lateral line system and other hydrodynamic detectors.