Free vibration of a spherical liquid drop attached to a conical base in zero gravity

Free vibration analysis of a spherical liquid drop attached to a conical base is presented. Assuming the liquid is incompressible and inviscid, and introducing a velocity potential, axisymmetric and asymmetric vibration characteristics are clarified, considering two liquid contact conditions: slipping edge and anchored edge. In the numerical calculations, for a wide range of conical base apex angles, the natural frequency and vibration mode of a liquid drop are presented. From these parameters, the vibration characteristics of a liquid drop attached to a conical base with general apex angle can be easily predicted. When the apex angle tends to 180°, natural frequencies are found to converge to those of a spherical drop under both slipping and anchored edge conditions, except in the axisymmetric mode with meridian mode number m=0 in the anchored edge case and asymmetric mode m=1 and n=1 in the slipping edge case.     

VIBRATION AND STABILITY OF ROTATING POLAR ORTHOTROPIC ANNULAR DISKS SUBJECTED TO A STATIONARY CONCENTRATED TRANSVERSE LOAD

In the present paper, natural frequencies and stability of a spinning polar orthotropic disk subjected to a stationary concentrated transverse load are investigated. The analysis of the free vibration of a spinning disk is performed first to find natural frequencies and corresponding vibration modes. The resulting eigenfunctions obtained from the free vibration are used as deflection functions of the forced vibration of a disk where the load is modelled as a mass-spring-dashpot system fixed in space. By using the Galerkin approximation method, eigenvalues of the whole system are determined. Results show that disks with higher values of modulus ratios or the Poisson ratios have higher natural frequencies, and the stability of the whole system can be improved by raising the value of the modulus ratio or lowering that of the Poisson ratio.     

Exact solution of the asymmetric Mindlin's plate equations applied to a disk

An exact solution is presented for the static and dynamic asymmetric response of a disk governed by Mindlin's plate equations forced by a pressure that varies radially as r^m. The static solution agrees with a modal solution adopting the dynamic Mindlin's plate equations in the limit when excitation frequency vanishes. This solution is useful in sizing magnitude and shape of surface asymmetries on a disk from pressure loading with slight eccentricity and circumferential non-uniformity.     

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.     

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.     

Vibrational response of a moving suspension-slider loading system exciting a rotating flexible disk

To investigate the vibrational response of the magnetic read/write head in hard disk drives this paper models a rotating flexible disk excited by a moving suspension-slider system which is considered to be a mass-dashpot-spring loading system, with the initial unstressed transverse runout integrated into the rotating disk dynamic model. The slider motion on the disk surface is driven by the suspension rotating at a constant speed. By subtracting the steady-state deflection component from the instantaneous deflection response of the rotating disk system, the relative vibration transverse deflection of the slider caused by the motion of the suspension-slider loading system is obtained. The effects of the slider initial and final positions, speed of movement, the disk rotational speed, and the disk mode of the initial transverse runout on the maximum amplitude of the relative vibration deflection are analyzed.     

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.     

Micromagnetic simulation of magnetic vortex cores in circular permalloy disks: Switching behavior in external magnetic field

Switching behaviors of magnetic vortex cores under external magnetic field in submicron circular permalloy disks have been systematically studied by using micromagnetic simulations. Simulation results show that the vortex core is stable in out-of-plane field even when it is located at the edge of the disk. The out-of-plane switching field H_sw is strongly dependent on the thickness of the disk. The core polarity and the vortex chirality can be modulated simultaneously on purpose by using a tilted field far smaller than the out-of-plane switching field H_sw. Moreover, it is found that the core polarities in asymmetric disks do not follow the direction of the z projection of the external saturation field.     

Closed expressions for matrix elements of the trigonometric Pöschl-Teller potential

Analytical matrix elements of the xn (n>0) and m[tan(x)]m^′[cos(x)]dn/dxn operators are derived using the eigenfunctions of the symmetric trigonometric Pöschl-Teller potential. The closed formulas are written in terms of Gauss hypergeometric functions and could be used in variational calculations to describe vibrational energy levels associated with bending modes. Multiprecision computational packages are considered in order to obtain an arbitrary level of precision.     

The Use of a hierarchy scheme for the evaluation of normal modes of vibration

In vibrational analysis it is the common practice to initially approximate a set of force constants rather than to initially approximate a set of normal modes. This does not allow a flexible and easily analyzed computational system. A hierarchy scheme of approximation has been developed. Systematic testing of alternative hierarchies and geometries is essential since a force field may always be evaluated to match exactly observed and calculated frequencies for any set of approximate normal modes. Other physical requirements must therefore be considered to establish the correctness of an answer. A set of N linearly independent internal coordinates of a particular symmetry type are placed in a hierarchy and a set of N orthormal modes are developed in such a way that the nth normal mode does not change the internal coordinates of the n + 1 to Nth members of the hierarchy. Constraints on the form of the force field require iteration from these well-defined initial approximations. The hierarchy scheme offers a much more satisfying physical picture of characteristic group frequencies and lends itself readily to the use of perturbation theory.     

Waveguide properties of a plane ring-shaped plate. I. flexural waves

A new type of the so-called “angular” waves for flexural normal modes propagating in a ringshaped plate is studied. A dispersion equation for wave numbers, an equation for critical frequencies, and expressions for the eigenfunctions of such a waveguide are derived. Solutions to these equations are obtained by numerical methods for various values of parameter d, which represents the relative width of the ring. The solutions are analyzed, and the main properties of dispersion curves are described. Individual normal modes are identified on the basis of the calculation and further analysis of eigenfunctions.     

Dynamic time responses of a flexible spinning disk misaligned with the axis of rotation

Using the finite element method, this study investigates the dynamic time responses of a flexible spinning disk of which axis of rotation is misaligned with the axis of symmetry. The misalignment between the axes of symmetry and rotation is one of major vibration sources in optical disk drives such as CD-ROM, CD-R, CD-RW and DVD drives. Based upon the Kirchhoff plate theory and the von Karman strain theory, three coupled equations of motion for the misaligned disk are obtained: two of the equations are for the in-plane motion while the other is for the out-of-plane motion. After transforming these equations into two weak forms for the in-plane and out-of-plane motions, the weak forms are discretized by using newly defined annular sector finite elements. Applying the generalized-α time integration method to the discretized equations, the time responses and the displacement distributions are computed and then the effects of misalignment on the responses and the distributions are analyzed. The computation results show that the misalignment has an influence on the magnitudes of the in-plane displacements. It is also found that the misalignment results in the amplitude modulation or the beat phenomenon in the time responses of the out-of-plane displacement.     

A critical analysis of the electromagnetic mass shift problem in QCD

The paper analyzes the electromagnetic quark mass shift problem in the framework of QCD and concludes with a physically appealing sum rule for the proton-neutron mass difference, in which the contribution from the divergent part of the photon loop has the right sign, after it has been renormalized. The paper reconciles the conflicting conclusions reached by respectively using the operator product expansion and the Dyson equations. In a recent discussion based on the latter special significance was given to the number of flavours n_f > 11, for which by choosing the ordering of integration and summation naturally dictated by iteration of the Dyson equation, the electromagnetic mass shift comes out to be finite with the correct sign for m_u ? m_d. We show by using dimensional regularization and minimal subtraction for renormalization group improved Feynman graphs one obtains the same result independent of the number of flavours.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Spin eigenfunctions and operators for the Dn groups

An automated procedure for determining symmetry-adapted spin eigenfunctions is developed for the D_n symmetry groups with n equivalent spins for arbitrary size of I and for n=3,…,6. These eigenfunctions are also eigenfunctions of the vector sum of the n equivalent nuclear spins. Generalized hyperfine spin operators are developed that have real matrix elements and that exploit the full symmetry. These spin operators can be combined with operators for other spins and molecular rotation using only real arithmetic.     

Size and formation of a long wavelength mode packet in the FPU oscillator chain at low energy

The formation of a mode packet in the Fermi-Pasta-Ulam (FPU) oscillator chain, starting from a low frequency mode or modes, at low energy, is reexamined. Both the chain with cubic nonlinearity (FPU-α) and quartic nonlinearity (FPU-β) are examined for fixed boundary conditions. Some of the basic equations are reformulated to bring out the important scalings, which are compared with existing numerical data. It is shown, that for a single linear mode initial condition or for a small set of modes with random phases the resulting size of the mode packet scales as n_ef=?^x where n_ef is the fraction of occupied modes, ? is the energy density, and x=1/4,1/2 for the α, β chains, respectively. For other initial conditions, the results can depend on the initial number of modes and their phases. Some of the scaling relations are shown to have counterparts in exact nonlinear periodic solutions on the chains (q-breathers) and the similarities and differences to the results arising from the chaotic dynamics are pointed out.     

A multi-cluster,n-particle generalization of the three-particle faddeev wavefunction equations

Recent work applying certain forms of many-body scattering theory to problems such as molecular potential energy surfaces and equations for nonequilibrium statistical mechanics indicates that a formulation of the theory based directly on multi-cluster, n-particle, wave function components could be of some utility. Such a formulation is derived in this paper using techniques from the Baer-Kouri-Levin-Tobocman and Bencze-Redish-Sloan-Polyzou theories of multi-particle scattering. It is based on components corresponding to the various multi-cluster partitions of an n-particle scattering system and is a generalization of the three-body Faddeev wave function formalism, to which it reduces when n = 3. Except for the full breakup partition, which does not enter the equations, the new components are defined for all possible m-cluster partitions of the n-particles, 2 ≤ m ≤ n ? 1. The sum of all the components yields the solution to the Schrödinger equation for scattering and either the Schrödinger equation solution or an easily identified spurious solution in the case of bound states. Both the two-cluster components and two-cluster transition operators are shown to be solutions of equations involving quantities carrying only two-cluster partition labels. Discussions of the Born term and a multiple scattering representation for the non-rearrangement transition operator and the inclusion of distortion operators in the formalism are also included.