The acoustic field excited by flexural vibrations of an elastic plate with a circular inclusion

The acoustic field excited by flexural vibrations of a thin elastic plate and the perturbations of this field caused by a homogeneous circular inclusion with other elastic properties are considered. Because the density of air widely differs from that of a metal, this problem can be solved with fair accuracy in two steps: first, by considering the vibrations of the plate in a free space, and, then, by calculating the acoustic field excited by the field of plate’s vertical deflections. The main results of this work are the asymptotic expressions for the far acoustic field excited by each of the Fourier components F _m (r)cosmφ of the flexural wave scattered by the inclusion.     

On flexural vibrations of isotropic elastic circular plate according to Mindlin's theory

This paper presents a numerical solution for an isotropic elastic circular plate. Effects of shear and rotatory inertia are included in a manner analogus to Timoshenko's one dimensional theory of bars. Matrix method has been employed to solve the set of equations obtained by using finite difference approximations. The use of IBM 1620 Computer was made for the computation of results showing thereby that even small machines could be used to handle the problem by the suggested approach.Notation ² the Laplace two dimensional operator - h thickness of plate - a diameter of plate - v Poisson's ratio - E Young's modulus of elasticity - G shear modulus - G K ² E/2(1 +v) - K a constant - density of the plate material - q superimposed normal load per unit plate area - D Eh ³/12(1–v²) - deflection of a point normal to the plane of plate - W amplitude of harmonic deflection - M amplitude of harmonic moment     

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.     

Thermally induced vibrations of a viscoelastic plate

A method for the study of thermally induced vibrations of a viscoelastic plate of arbitrary shape is proposed. The method is based upon the concept of iso-amplitude contour lines on the surface of the plate. It is shown that the time behaviour can be found by assuming a normal mode expansion in terms of the eigenfunctions for the associated elastic plate problem, and the deflection is obtained by using the elastic-viscoelastic analogy. As an illustration of the technique, the vibration of a viscoelastic circular plate under a thermal shock at its centre is discussed, all details of which are illustrated by graphs.     

Axisymmetric vibrations of a circular plate of linearly varying thickness on an elastic foundation according to Mindlin theory

Free axisymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic foundation have been studied on the basis of shear theory [1,2]. The transverse displacement and local rotation are expressed as an infinite series. The frequencies corresponding to the first two modes of vibrations are obtained for a circular plate with clamped and simply supported edge conditions for various values of the taper constant and the foundation modulus. The results have been compared with those of reference [3].     

In-plane vibrations of circular rings

Except in a few cases rings have been treated one-dimensionally as curved beams in analyses to determine their frequencies of vibration. Consequently, for thick rings there are large discrepancies between theoretical and experimental results. In this paper an analysis is carried out in which account is taken of the warping of the cross section. The displacement v in the circumferential direction is assumed to be of the form v₀ + v₁y + v₃y³ + v₅y⁵, where the v_i are functions of the angular co-ordinate θ. The frequencies of free vibration are determined by using the principle of minimum total potential energy. Numerical calculations are made for a ring of rectangular cross section. Results are in good agreement with experimental ones.     

Motions induced by asymmetric vibrations

We discuss theoretically the motions of a coin on a horizontally vibrating plate, with dry friction between the coin and the plate. As first noticed by Daniel and Chaudhury in a different situation (droplets on a plate), when the periodic acceleration γ(t) imposed by the plate is unsymmetrical, the coin can move macroscopically with a certain drift velocity . We analyse here: (a) the vibration threshold below which , and the generic behavior expected just above threshold; (b) the limiting behavior at very high amplitudes, where should become independent of the amplitude; (c) the complications due to small, macroscopic inhomogeneities on the supporting plate.     

圆盘上的绳张力

讨论绕水平定轴转动的圆盘和跨过圆盘的细绳组成的力学系统,在考虑二者的质量及之间的摩擦的情况下,导出了细绳张力的计算式.     

Experiments on large-amplitude vibrations of a circular cylindrical panel

The vibration response of a thin circular cylindrical panel to harmonic excitation in the neighborhood of the first three natural frequencies has been measured for different force levels. The experimental boundary conditions approximate (i) on the curved edges: zero radial, axial and circumferential displacements; all rotations were allowed; (ii) on the straight edges: zero radial and axial displacements; all rotations and circumferential displacements were allowed. The different levels of excitation permitted reconstruction of the relatively strong, softening type non-linearity of the panel.     

FEM simulation of a sono-reactor accounting for vibrations of the boundaries

The chemical effects of acoustic cavitation are obtained in sono-reactors built-up from a vessel and an ultrasonic source. In this paper, simulations of an existing sono-reactor are carried out, using a linear acoustics model, accounting for the vibrations of the solid walls. The available frequency range of the generator (19-21 kHz) is systematically scanned. Global quantities are plotted as a function of frequency in order to obtain response curves, exhibiting several resonance peaks. In absence of the precise knowledge of the bubbles size distribution and spatial location, the attenuation coefficient of the wave is taken as a variable, but spatially uniform parameter, and its influence is studied. The concepts of acoustic energy, intensity, active power, and source impedance are recalled, along with the general balance equation for acoustic energy, which is used as a convergence check of the simulations. It is shown that the interface between the liquid and the solid walls cannot be correctly represented by the simple approximations of either infinitely soft, or infinitely hard boundaries. Moreover, the liquid-solid coupling allows the cooling jacket to receive a noticeable part of the input power, although it is not in direct contact with the sonotrode. It may therefore undergo cavitation and this feature opens the perspective to design sono-reactors which avoid direct contact between the working liquid and the sonotrode. Besides, the possibility to shift the main pressure antinode far from the sonotrode area by exciting a resonance of the system is examined.     

Free vibrations of circular cylindrical shells

The finite element method is used to predict the dynamic behaviour of circular cylindrical shells in free vibrations. A suitable shape function for the circumferential displacement distribution has been proposed. This reduces the three-dimensional character of the problem to a two-dimensional one. The simultaneous iteration method to determine the eigen-frequencies and eigenvectors is utilised for solving the eigenvalue problem. The accuracy of the method has been checked by verifying the results of known cases. Finally an experimental shell structure containing elastic rings welded at the ends has also been analysed and the experimental results compared with the theoretical ones.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

Thickness vibrations of a piezoelectric plate under biasing fields

Thickness vibrations of a piezoelectric plate under uniform initial thermoelectromechanical fields are investigated in this article. An exact solution is obtained for materials with general anisotropy. A plate of langasite, a relatively new piezoelectric material with strong piezoelectric coupling, is analyzed with imposed uniform acceleration as the biasing field as an example. The results show that the biasing fields have remarkable effect on the free vibrations of piezoelectric plate.     

Non-linear vibrations of a flat plate with initial stresses

Non-linear free vibrations of a simply supported rectangular elastic plate are examined, by using stress equations of free flexural motions of plates with moderately large amplitudes derived by Herrmann. A modal expansion is used for the normal displacement that satisfies the boundary conditions exactly, but the in-plane displacements are satisfied approximately by an averaging technique. Galerkin technique is used to reduce the problem to a system of coupled non-linear ordinary differential equations for the modal amplitudes. These nonlinear differential equations are solved for arbitrary initial conditions by using the multiple-time-scaling technique. Explicit values of the coefficients that appear in the forementioned Galerkin system of equations are given, in terms of non-dimensional parameters characterizing the plate geometry and material properties, for a four-mode case, for which results for specific initial conditions are presented. A comparison of the results with those obtained in previous studies of the problem is presented. In addition, effects of prescribed edge loadings are examined for the four-mode case.     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Free vibrations of a plate with an inner support

The title problem is tackled by using a simple polynomial co-ordinate function and the Rayleigh-Schmidt method. It is assumed that the inner support is parallel to the free edge. When the support coincides with the free edge the frequency equation degenerates properly into the case of a simply supported edge. Numerical results are presented for the situation where two opposite edges are simply supported and the edge parallel to the free edge is either clamped or simply supported.     

Large amplitude vibrations of circular plates carrying a concentrated mass

A finite element analysis for the large amplitude vibrations of circular plates carrying a concentrated mass is presented in this note. The variation of the fundamental frequency with the central amplitude is obtained for simply supported and clamped plates for various positions and various magnitudes of the concentrated mass.