Vibration of a two layered ring on periodic radial supports

Natural frequencies of a two layered elastic ring, on equi-spaced, identical radial supports, are obtained by using a wave approach. Two types of support conditions are investigated. With the outer layer viscoelastic, the theory of forced damped normal modes is used to obtain the resonant frequencies and the modal loss factors of the structure. Results presented show the effect of the thickness ratio on the resonant frequency and the modal loss factor. The effect of rotational constraints at the supports is also reported.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Wave propagation and vibration response of a periodically supported pipe conveying fluid

Propagation of free harmonic waves, in a periodically supported infinite pipe, has been studied. The presence of the Coriolis term in the equation of motion renders the phase velocity different for the positive and the negative going waves. Hence no classical normal modes (in the sense of standing modes) exist. Natural frequencies of a periodically supported finite pipe have been obtained by using the wave approach. The response of the infinite pipe to a convected harmonic pressure field has also been obtained. Owing to the difference in the phase velocities of the positive and the negative going free waves, the coincidence frequency depends on the direction of the convected loading. The static buckling or the divergence instability of such pipes has also been considered from the wave approach.     

Optical phonon modes in a free-standing quantum wire with ring geometry

The confined longitudinal-optical (LO) phonon and surface-optical (SO) phonon modes of a free-standing quantum wire with ring geometry are discussed within the dielectric continuum (DC) approximation. Two branches of SO phonon modes have been investigated. The frequencies of the SO phonons are found to be dispersed and radius dependent for small size systems. When the wave vector qz→∞, the frequencies of each SO modes converge to the frequency values of the single planar heterostructure.     

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.     

Forced axisymmetric response of circular plates

Forced axisymmetric response of circular plates is considered on the basis of an improved theory which takes into account the effect of rotatory inertia and shear deformation. Explicit solutions are obtained for distributed, ring and concentrated impulsive loadings with half-sine, triangular, blast and rectangular pulse shapes. Natural frequencies are presented for the first 20 modes of four plate models with clamped and supported edges. Numerical results are presented to illustrate the differences between the improved and classical theories, and the effects of edge conditions, loading conditions, pulse shapes and pulse durations.     

Bending wave propagation along a symmetric groove in an elastic plate

Normal bending waves propagating along a symmetric groove in an infinite elastic plate are considered. The characteristic equations in wave numbers of these modes are obtained for symmetric and antisymmetric modes of vibration. Critical frequencies and eigenfunctions of the problem are determined. Results of numerical calculations for the wave numbers, critical frequencies, and modes of vibration are presented.     

Vibrations of initially stressed cylinders of variable thickness

Natural frequencies and buckling loads for cylindrical shells having linearly varying thickness are obtained by using a segmentation technique. The present results for free vibration of a cylinder compare very well with those obtained previously. The effect of the thickness variation on the frequencies of a cylindrical shell is studied. Frequencies are also calculated for a cylinder of variable thickness under axial compression and a relationship between the frequency and axial compression is obtained for a particular wave number.     

Energy concentration at the center of large aspect ratio rectangular waveguides at high frequencies

Waveguides in non-destructive evaluation (NDE) applications are commonly of a regular geometry (e.g., circular and ring cross section) for which analytical solutions exist. In this paper, wave propagation in infinitely long strips of large rectangular aspect ratio is discussed. Due to the finite width of strips, a large number of modes exist within the structure. This complicates the analysis and usually discourages the use of strip waveguides in NDE sensors. However, it is shown that among the many modes of a strip, there are some with very desirable properties. This is highlighted by the example of two guided wave modes of a large aspect ratio rectangular strip whose dispersion characteristics approach those of the fundamental modes of an infinitely wide plate at high frequencies. The energy of these modes concentrates in the central region of the strip and decays toward the edges so that the strip waveguide can easily be mechanically attached to other components without influencing the wave propagation. Dispersion curves and mode shapes were derived by using a semianalytical finite element technique and are presented over a range of frequencies. It is shown that selective excitation of both modes is possible in practice and the experimental setup is described.     

Tkachenko modes of vortex lattices in rapidly rotating bose-einstein condensates

We calculate the in-plane modes of the vortex lattice in a rotating Bose condensate, from the slowly rotating to mean-field quantum Hall limits. The Tkachenko mode frequency, linear in wave vector k for lattice rotational velocities Omega much smaller than the lowest sound wave frequency in a finite system, becomes quadratic in k in the opposite limit. The system also supports an inertial mode of frequency >or=2omega. The calculated frequencies are in excellent agreement with recent observations of Tkachenko modes by Phys. Rev. Lett., 91, 100402 (2003)].     

Ring resonator of surface modes based on photonic crystals

We design a compact ring resonator of surface modes based on photonic crystals (PCs). The structure is formed by sandwiching a surface mode ring waveguide (SMRW) into two parallel surface mode waveguide (SMW) based on two dimensional (2D) PCs. The SMRW is created on the surface of a circular photonic crystal (CPC) structure, where the wave propagates with high transmission efficiency. As a fundamental mode is introduced in the input SMW, at certain frequencies, the SMRW modes are enhanced because of resonance and the light-waves are coupled to the output SMW. It is demonstrated by the simulation results that the surface mode ring resonator has a low radiation loss with a very small size because of the good wave-guiding of surface mode based on PCs, and can be used in the future wavelength division multiplex (WDM) optics communication systems.     

On the modal density and damping of cylindrical pipes

Measurements of the resonant frequencies and quality factors of a series of long, small diameter cylindrical pipes are presented. The dependence of the modal densities, calculated from these measurements, on pipe length, wall thickness and pipe material is in close agreement with theoretical statistical predictions. Measurements of the damping of steel pipes for several different end conditions are also presented. Different, but always well-ordered, variation of modal quality factor with mode order is found in each case. For free-free ends the modal quality factors are large (>1000) and determined by internal material damping, except for modes for which the acoustic radiation damping is large; the effect of radiation damping is most important for those modes which have high wave speeds at low frequencies. The quality factors for rigid end conditions are similar to those for free ends, except for translational modes of low axial order which are damped by vibration of the end supports. For end conditions which allow relative motion between the pipe and the end supports, there is considerable additional damping, probably ascribable to gas pumping in the joints.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

The long wave modes of the Cu2O lattice

The symmetry coordinates for the long wave modes of the Cu₂O lattice are derived explicitly; the selection rules for the first order optical effects (absorption and combinatory scattering) are given. It is pointed out that owing to macroscopic field effects degeneracies cannot be ascertained on symmetry grounds alone, thus nine distinct frequencies are obtained instead of seven. A tentative estimate of the frequencies on a simple ionic model has been made, giving for one of the optically active modes λ=16.5 μ, in close agreement with the observed position of the strongest absorption peak. The tentative nature of such an estimate is stressed, and effects due to electrical polarisation of the ions and possible bond angle distortion forces are briefly discussed.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

Three-Dimensional Scattering of an Incident Plane Shear Horizontal Guided Wave by a Partly through-Thickness Hole in a Plate

We investigate the three-dimensional(3D) scattering problem of an incident plane shear horizontal wave by a partly through-thickness hole in an isotropic plate,in which the Lamb wave modes are also included due to the mode conversions by the scattering obstacle in the 3D problem.An analytical model is presented such that the wave fields are expanded in all of propagating and evanescent SH modes and Lamb modes,and the scattered far-fields of three fundamental guided wave modes are analyzed numerically for different sizes of the holes and frequencies.The numerical results are verified by comparing with those obtained by using the approximate Poisson/Mindlin plate model for small hole radius and low frequency.It is also found that the scattering patterns are different from those of the SO wave incidence.Our work is useful for quantitative evaluation of the plate-like structure by ultrasonic guided waves.     

Receptivity to free-stream disturbance waves for hypersonic flow over a blunt cone

A high-order shock-fitting finite difference scheme is studied and used to do direction numerical simulation (DNS) of hypersonic unsteady flow over a blunt cone with fast acoustic waves in the free stream, and the receptivity problem in the blunt cone hypersonic boundary layers is studied. The results show that the acoustic waves are the strongest disturbance in the blunt cone hypersonic boundary layers. The wave modes of disturbance in the blunt cone boundary layers are first, second, and third modes which are generated and propagated downstream along the wall. The results also show that as the frequency decreases, the amplitudes of wave modes of disturbance increase, but there is a critical value. When frequency is over the critial value, the amplitudes decrease. Because of the discontinuity of curvature along the blunt cone body, the maximum amplitudes as a function of frequencies are not monotone. Supported by the National Natural Science Foundation of China (Grant Nos. 10632050 and 10502052)     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

环形ZnO薄膜谐振器的横模抑制与测试分析

采用Tiersten方程研究了环形ZnO薄膜谐振器中横模寄生问题, 获得了环(圆)形薄膜谐振器的横模振动方程, 求得横模位移场解和频率色散方程; 然后采用电磁学模式合成理论进行分析, 发现环形薄膜谐振器横模频率与环形电极的内外径之比a/b有关, 振动模式可由圆形薄膜谐振器横模模式合成得到, 通过控制a/b能够抑制横模模式数和调控基膜频率. 采用外差激光干涉仪和网络矢量分析仪测量并比较了同批次的圆形和环形薄膜谐振器的上电极横模振动图样和电阻抗曲线. 振动图样显示环形薄膜谐振器振动模式可由半径为a和半径为b的圆形薄膜谐振器振动模式合成, 仅存在节圆数大于0的横模振动, 等于0的横模模式被抑制; 电阻抗曲线显示当a/b为0.436时, 环形薄膜谐振器的基频(约1217 MHz)和圆形的(0, 1)模式频率相等. 测量数据验证了模式合成理论的分析结果正确性, 为薄膜谐振器的横模抑制研究提供了理论基础和新方法.     

Some approximations in the linear dynamic equations of thin cylinders

Theoretical analysis is performed on the linear dynamic equations of thin cylindrical shells to find the error committed by making the Donnell assumption and the neglect of in-plane inertia. At first, the effect of these approximations is studied on a shell with classical simply supported boundary condition. The same approximations are then investigated for other boundary conditions from a consistent approximate solution of the eigenvalue problem. The Donnell assumption is valed at frequencies high compared with the ring frequencies, for finite length thin shells. The error in the eigenfrequencies from omitting tangential inertia is appreciable for modes with large circumferential and axial wave lengths, independent of shell thickness and boundary conditions.