HYDROELASTIC COUPLED VIBRATIONS IN A CYLINDRICAL CONTAINER WITH A MEMBRANE BOTTOM, CONTAINING LIQUID WITH SURFACE TENSION

A linear free hydroelastic vibration analysis of a frictionless liquid with a free surface contained in a cylindrical tank with a flexible bottom has been performed. The side-wall has been treated as rigid and the effect of surface tension taken into consideration. The container bottom was treated as a membrane, while for the free liquid surface the effect of two contact line conditions has been investigated. One edge condition was that of a slipping contact line, while the other one treated the contact line as fixed, ie., an anchored edge. The vibration characteristics of a membrane-liquid coupled system have been investigated for various system parameters, i.e., membrane tension parameter T, liquid surface tension parameter σ, material density parameter ρ, liquid height ratio ?₀ and vibration mode numbers m and n. The degree of coupling between a membrane and a liquid was represented with vibration mode diagrams as well as with frequency diagrams. For axisymmetric coupled vibrations with anchored edge condition, vibration mode exchanging of both a membrane and liquid free surface with membrane bottom tension parameter T has been investigated. An interesting phenomenon which is only observed for a flexible bottom container and an anchored edge free surface condition is presented.     

Whispering gallery modes in a hemispherical isotropic dielectric resonator with a perfectly conducting planar surface

The eigenmodes of a hemispherical dielectric resonator placed on a perfectly conducting planar surface are studied. The resonator is shown to support independent E-and H-type oscillations. For E and H oscillations, the sums n + m of the polar and azimuthal indices are, respectively, even and odd. Accordingly, E and H oscillations exhibit, respectively, n-and (n + 1)-fold frequency degeneracy in azimuthal index m. The distributions of the whispering gallery mode fields on the surface of the hemisphere and on the external conducting plane are obtained. The energy distribution among the eigenmodes with polar index n=36 is studied. The polar angle that corresponds to the maximum of the whispering gallery mode field and the width of the energy localization region in terms of the polar angle are calculated. The number of field variations in the spherical coordinates are correlated with the indices of the resonator’s eigenmodes. The eigenfrequencies and Q factors of the resonator are plotted versus the permittivity of the isotropic environment in the 8-mm wavelength range.     

An actuator for the n=2 circumferential mode of a pipe

The axial propagating wave of the n=2 circumferential mode in a pipe can cause large vibration when the wave cuts on and may cause fatigue damage or transmit structure-borne noise along the pipe. For this reason, a control system (passive or active) is desirable. In this paper, a PZT modal actuator for the n=2 mode, which could be used in an active control system, is described. It is constructed from a set of PZT elements bonded to the pipe. By arranging them in form of the n=2 mode, only the motion of that particular mode is generated and is proportional to the applied voltage. With two PZT modal actuators, which are in forms of sine and cosine functions, the orientation of the propagating wave of the n=2 mode can be modified to any angle. In this paper a theoretical model for the actuator is developed and is validated by some experimental work.     

Collective excitations in neutron-rich nuclei within the model of a Fermi liquid drop

We discuss a new mechanism of splitting of giant multipole resonances (GMR) in spherical neutron-rich nuclei. This mechanism is associated with the basic properties of an asymmetric drop of nuclear Fermi liquid. In addition to well-known isospin shell-model predictions, our approach can be used to describe the GMR splitting phenomenon in the wide nuclear-mass region A ～ 40–240. For the dipole isovector modes, the splitting energy, the relative strength of resonance peaks, and the contribution to the energy-weighted sum rules are in agreement with experimental data for the integrated cross sections for photonuclear (γ, n) and (γ, p) reactions.     

On nonlinear vibrations of a charged drop in the third-order approximation in the amplitude of initial single-mode excitation

Nonlinear axisymmetric motions of the free surface of a charged drop of an ideal liquid under the single-mode initial deformation of its equilibrium shape is investigated in the third-order approximation in the initial perturbation amplitude. An analytical expression for the drop shape generatrix is derived. Nonlinear corrections to the vibration frequencies for the initial perturbation of an arbitrary mode are found for the first time. The effect of vibration nonlinearity on the instability of the drop against its self-charge is studied.     

Instantaneous axial force of a high-order Bessel vortex beam of acoustic waves incident upon a rigid movable sphere

The present investigation examines the instantaneous force resulting from the interaction of an acoustical high-order Bessel vortex beam (HOBVB) with a rigid sphere. The rigid sphere case is important in fluid dynamics applications because it perfectly simulates the interaction of instantaneous sound waves in a reduced gravity environment with a levitated spherical liquid soft drop in air. Here, a closed-form solution for the instantaneous force involving the total pressure field as well as the Bessel beam parameters is obtained for the case of progressive, stationary and quasi-stationary waves. Instantaneous force examples for progressive waves are computed for both a fixed and a movable rigid sphere. The results show how the instantaneous force per unit cross-sectional surface and unit pressure varies versus the dimensionless frequency ka (k is the wave number in the fluid medium and a is the sphere’s radius), the half-cone angle β and the order m of the HOBVB. It is demonstrated here that the instantaneous force is determined only for (m, n) = (0, 1) (where n is the partial-wave number), and vanishes for m > 0 because of symmetry. In addition, the instantaneous force and normalized amplitude velocity results are computed and compared with those of a rigid immovable (fixed) sphere. It is shown that they differ significantly for ka values below 5. The proposed analysis may be of interest in the analysis of instantaneous forces on spherical particles for particle manipulation, filtering, trapping and drug delivery. The presented solutions may also serve as a method for comparison to other solutions obtained by strictly numerical or asymptotic approaches.     

On nonlinear resonant four-mode interaction between capillary vibrations of a charged drop

Energy transfer from higher modes of capillary vibrations of an incompressible liquid charged drop to the lowest fundamental mode under four-mode resonance is studied. The resonance appears when the problem of nonlinear axisymmetric capillary vibration of a drop is solved in the third-order approximation in amplitude of the multimode initial deformation of the equilibrium shape of the drop. Although the resonant interaction mentioned above builds up the fundamental mode even in the first order of smallness, its amplitude turns out to be comparable to a quadratic (in small parameter) correction arising from nonresonant nonlinear interaction, since the associated numerical coefficients are small.     

On the characterization of the elastic properties of asymmetric single-walled carbon nanotubes

In order to characterize asymmetric single-walled carbon nanotubes, an algorithm has been developed based on numerical simulation to relate the physical geometry to the elastic properties of asymmetric single-walled carbon nanotubes (SWCNTs). A large number of finite element results for the stiffness of asymmetric SWCNTs has been used to develop a best surface fitting function to define the relationship between the geometry of SWCNTs and their stiffness. However, since the stiffness of asymmetric nanotubes depends upon the configuration parameters, n and m, it was impossible to define any diameter dependency. Based on the maximum reaction force concept and in order to account for the hidden mechanical behavior of asymmetric SWCNTs, the chiral factor (CF) has been employed in this study. The proposed CF converts any asymmetric geometry (n and m) into a value between 0 and 1. A group of the SWCNTs with the same applied boundary condition (n+m=30) and different range of the CF was also used for studying of the shear contribution. The chiral factor dependency, which is developed in this study, is applicable for characterising and selecting asymmetric SWCNTs in the design of advanced nanomaterials. Furthermore, the equation which is calculated in this study can be useful for finding the best criteria for selecting asymmetric SWCNTs.     

On a presence of SimHn clusters in a-Si:H/c-Si structures

Dominant aim of the paper was to verify the existence of the Si_mH_n clusters in a-Si:H layers. Thin layers were deposited by plasma-enhanced chemical vapor deposition (PECVD) on both glass and crystalline silicon substrates. Their IR and structural properties were investigated by Fourier transform infrared spectroscopy (FTIR) and X-ray diffraction at grazing incidence angle (XRDGI). We have found that the layer probably consists of larger structurally ordered parts corresponding to Si_mH_n clusters and separated groups of (Si-H_x)_N. The ordered parts could be identified as some of Si_mH_n clusters ranging from (10, 16) to (84, 64) represented by corresponding vibration frequencies in three following IR regions: 600-750, 830-900 and 2080-2180 cm⁻¹. XRDGI measurement indicates that diffraction maximum at around 2Θ = 28° can be attributed to an existing Si_mH_n cluster.     

On the effects of a general form of the initial runout on the oscillation frequencies and critical speeds of a spinning disk

This paper is concerned with the effect of the lack of flatness on the dynamics of spinning disks. Of particular interest is the study of the effect of an unsymmetrical initial runout on the oscillation frequencies and the critical speeds of the spinning disks. It is assumed that a spinning disk is initially warped in an initial asymmetric shape. Using Von Karman's plate theory, the equations of motion are derived in a rotating frame. Taking advantage of the orthogonal properties of the eigenfunctions of a stationary disk, the lateral displacement and the stress function are expressed as their summation. Since these eigenfunctions produce an orthonormal space, any shape and level of initial runout can be written as their summation also. Using Galerkin's method, the equations of motion are discretized and a set of coupled linear equations are found taking into account the effect of an initial arbitrary runout. The numerical results are found to have a good agreement with those obtained using ANSYS. It is found that the natural frequencies of the spinning disk calculated in the space fixed frame increases due to an initial runout. When the initial runout is assumed to be asymmetric, it is found that, due to asymmetric stress distributed in the disk, a frequency split between the backward and forward traveling waves of a given preferential mode of a stationary disk occurs. It is also seen that in some cases the initial runout in the form of the (n,m) mode has the least effect on the natural frequencies of the backward traveling waves of the same mode. As such it is observed that in the case the critical speed the (n,m) mode is less sensitive to an initial runout in the form of the (n,m) mode. To verify the accuracy of the numerical predictions, experiments were conducted on two disks, one which was flat and the other that had an initial nonflatness in the form of the (0,2) mode. The numerical and experimental results indicate a close correspondence.     

Analysis of shape perturbations of a drop on a vibrating substrate for different wetting angles

An exact solution is obtained to the problem of axisymmetric normal modes and natural frequencies characterizing surface perturbations of a drop that sits with an arbitrary wetting angle on a substrate and experiences only gravity and surface tension. The resulting mode solutions are used to calculate and analyze different shapes of the perturbed surface for the same drop placed on a vibrating base. The distinctive feature of the present study is the explicit representation of the results in the form of calculated shapes of the surface of a vibrating drop, comparison of the parameters of actual drops with resonance frequencies, and comparison with experimental data.     

On internal mode resonance in a nonlinearly vibrating volumetrically charged dielectric drop

In the approximation quadratic in the amplitude of an arbitrary initial deformation of an equilibrium spherical uniformly (volumetrically) charged drop of a dielectric liquid, an analytical expression for the drop surface generatrix as a function of time is derived in the case when the drop shape executes axisymmetric vibrations. A condition that must be imposed on mode frequencies in order for resonant interaction between modes to take place in the quadratic approximation is found. It is shown that many resonances, rather than one known previously, are realized when the self-charge is insufficient (subcritical) for drop surface instability against self-charge to arise. Nonlinear two-and three-mode resonant interactions are studied.     

Electron—hole liquid in CdS crystals as revealed by non-equilibrium transmission spectroscopy

Transmission measurements in the excitonic region on thin platelets under high N₂ laser excitation showed the disappearance of exciton absorption into a strong absorption continuum. The low energy edge of the latter — about 12 meV below the free exciton A_n=1 — coincides with the high energy edge of the gain spectrum of stimulated emission under similar excitations. These results favour the assumption of electron—hole drop formation with a ground state of the e-h-pair about 12 meV below A_n=1.     

The resonant mechanism of subdivision of a gas bubble in a fluid

Small nonlinear oscillations of a bubble in a fluid at the resonance of the frequencies of the radial mode and an arbitrary deformational mode 2 : 1 are considered. The deformational mode is determined by the associated Legendre polynomial with indices n = 2, 3, ??, m = 0, 1, ??, n. The energy transfer from the radial mode into the Legendre deformational mode is described by the method of invariant normalization. An analogy is established with oscillations of the material point on the string with the frequency ratio of the vertical mode to the horizontal one of 2. During the transfer, the amplitude of the Legendre mode with indices n, m exceeds the amplitude of radial oscillations by a factor of 3n at m = 0. As index m increases, the transfer time increases considerably and the maximal amplitude of the Legendre mode increases insignificantly in this case. From here, it is concluded that the deformational Legendre mode with indices n, m = n has the greatest probability to rise. The considered effect can serve as a mechanism of subdivision of gas bubbles under varying the external pressure in the fluid.     

VIBRATION ANALYSIS OF TWISTED CANTILEVERED CONICAL COMPOSITE SHELLS

The finite element method based on the Hellinger-Reissner principle with independent strain is applied to the vibration problem of cantilevered twisted plates and cylindrical, conical laminated shells. With a small number of elements, the present assumed strain finite element method is validated by convergence tests and numerical tests, comparing with the previous published vibration results for cantilevered conical shell. Computational effort and virtual storage reduce significantly due to good convergence. This study presents the twisting angle effect on vibration characteristics of conical laminated shells. Parameter studies with varying shallowness of cylindrical and conical shells are carried out. As the curvature increases, the fundamental mode shape changes from twisting mode to bending mode. For shells with a large curvature, the fundamental frequency, which is always characterized to bending mode, is almost constant independent of twisting angle. The twisting angle affects greatly twisting frequency and mode shape.     

Derivation of a universal estimate for the stiffness of carbon nanotubes

An algorithm has been developed based on numerical simulation to relate physical geometry to the Young’s modulus of symmetric and asymmetric single-walled carbon nanotubes (SWCNTs). A large number of finite element results for the stiffness of SWCNTs has been categorized into three main classes (i.e., armchair, zigzag and chiral) and the best curve fitting function has been obtained to describe the relation between the geometry of SWCNTs and their stiffness. For two standard configurations of carbon nanotubes (i.e., armchair and zigzag), four equations referring to geometry parameters (n, m) and diameter (d) have been introduced. To find the size dependence of asymmetric nanotubes, three-dimensional surfaces of stiffness (E(n, m)) have been used. However, since the stiffness of asymmetric nanotubes depends upon n and m, it was impossible to define any diameter dependency. To account for the hidden mechanical behavior of asymmetric SWCNTs, a new physical factor (CF) was introduced as the major novelty in this work. The proposed CF converts any asymmetric geometry (n, m) into a value between 0 and 1. The CF for a chiral nanotube can imply the percentage of similarity in its mechanical properties to the two standard symmetric configurations. Based on the CF concept, a new equation is derived to predict the Young’s modulus of asymmetric carbon nanotubes based on the symmetric prediction of standard models. The new physical factor (CF) which is developed in this study can be useful for the characterization of SWCNTs and the selection of all configurations.     

Minimum dissipative relaxed states in toroidal plasmas

Relaxation of toroidal discharges is described by the principle of minimum energy dissipation together with the constraint of conserved global helicity. The resulting Euler-Lagrange equation is solved in toroidal coordinates for an axisymmetric torus by expressing the solutions in terms of Chandrasekhar-Kendall (C-K) eigenfunctions analytically continued in the complex domain. The C-K eigenfunctions are obtained as hypergeometric functions that are solutions of scalar Helmholtz equation in toroidal coordinates in the large aspect-ratio approximation. Equilibria are constructed by assuming the current to vanish at the edge of plasma. For the m=0, n=0 (m and n are the poloidal and toroidal mode numbers respectively) relaxed states, the magnetic field, current, q (safety factor) and pressure profiles are calculated for a given value of aspect-ratio of the torus and for different values of the eigenvalue λ _{r 0}. The new feature of the present model is that solutions allow for both tokamak as well as RFP-like behaviour with increase in the values of λ _{r 0}, which is related directly to volt-sec in the experiment.     

On the calculation of the translational mode amplitude for a drop nonlinearly vibrating in an environment

The second-order amplitudes of the capillary vibration modes of a drop of an ideal incompressible liquid placed in an incompressible ideal medium are calculated. The approximation is quadratic in initial multimode deformation of the equilibrium spherical shape caused by nonlinear interaction. The mathematical statement of the problem is such that the immobility condition for the center-of-mass of the drop is met automatically. When the translational mode amplitude is calculated, a set of hydrodynamic boundary conditions at the interface, rather than the condition of center-of-mass immobility (which is usually applied for simplicity in the problems of drops vibration in a vacuum), should be used.     

Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.     

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.