Damping of flexural vibrations in circular plates with tapered central holes

The paper develops a numerical approach to the calculation of mobilities for a circular plate with a tapered central hole of power-law profile. The exact solution of the corresponding flexural wave equation that exists for m=2 has been used in the process of the numerical solution of the corresponding boundary problem. Note that this value of m belongs to the power-law range m≥2 associated with zero reflection of quasi-plane waves from a tapered hole in geometrical acoustics approximation. Two cases of added damping in the central hole area have been considered: a thin absorbing layer and a constrained layer. Cross and point mobilities have been calculated for both these cases. The obtained results for point and cross mobilities show a substantial suppression of resonant peaks (up to 17 dB), in comparison with the cases of a plate with an uncovered hole of the same power-law profile and of a reference circular plate of constant thickness covered or uncovered by a thin absorbing layer. Further theoretical and experimental research is needed to examine applications of the obtained numerical results to more practical situations, e.g. to rectangular plates or other structures with arbitrary locations of tapered holes.     

Extensional vibrations of composite circular plates of moderate uniform thickness

Mathematical treatment of such vibrations of plates which do not involve any transverse displacement of points in the middle plane goes back to Cauchy and Poisson. Present-day considerations are based upon the well-known approximate theory worked out by J. H. Michell (1900). Attention has been paid especially to isotropic circular plates, but solutions of problems relating to composite plates have not yet been obtained. The case of a uniform circular plate consisting of any finite number of homogeneous and isotropic concentric parts may serve as an example.     

Transverse vibrations of circular plates with stepped thickness

The study of the dynamic behaviour of circular plates with stepped thickness is of interest in view of their use in the construction of high frequency transducers. A simple analytical approach which allows for the prediction of their natural frequencies is proposed in the present Note.     

Large amplitude vibrations of circular plates with varying thickness

A simple finite element formulation is presented to evaluate the large amplitude vibration frequencies of orthotropic circular plates with linearly varying thicknesses. Period ratios are presented in tables and figures for different values of the orthotropy and taper parameters.     

Free vibrations of randomly inhomogeneous plates

Consider a large collection of elastic rectangular plates with random inhomogeneities, but otherwise indistinguishable in any overall sense. An expression is obtained for the natural frequency, μ, of such plates, vibrating freely under simply supported boundary conditions, in the form μ = μ⁽⁰⁾ + ?μ⁽¹⁾ + ?²μ⁽²⁾ + … where μ⁽⁰⁾ is the natural frequency of a homogeneous comparison plate, ? is a small real parameter measuring the degree of inhomogeneity, and the coefficients μ⁽¹⁾, μ⁽²⁾, …, are given explicitly. By constructing geometrically a correlation function for a special type of composite plate, μ⁽¹⁾ is computed and hence, to first order in ?, the variance of μ. The paper concludes with a theorem linking the mean and variance of μ to the volume concentration and geometry of the inclusions in the plate.     

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.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

On forced vibrations of composite circular membranes

The problem of the free vibrations of circular membranes consisting of any finite number of concentric parts from different materials has been solved quite generally in our former paper [1]. The present considerations are devoted to some new questions in the field of the forced vibrations of composite circular membranes.     

Transverse vibrations of circular plates of linearly varying thicknesses

The title problem is solved using very simple polynomial co-ordinate functions and a variational approach.Rather general boundary conditions are assumed at the edge support. It is shown that the approach is valid for axi- and antisymmetric modal configurations.     

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.