Vibration of Plates and Shells

Shells have been the subject of several types of structural investigations in the past due to their attractive property of high load to weight ratio. The broad aim of this study is to study suitability of shells as load measuring devices. This paper discusses a shell device for measuring loads and provides a framework for design studies to maximize load capacity. To accomplish this, we propose a unification of design space and perform a consolidated design of experiments to quantify the relative merits of configurations. The design framework includes both geometrical and material properties to enable generic conclusions and extensibility. A validated finite element model has been used to aid in detailed structural assessment. A significant increase in load capacity has been obtained compared to previously reported configurations paving the way for compact device setup. Nomenclature R Outer Radius of hemispherical shell r Inner radius of hemispherical shell t Thickness of hemispherical shell h Height of hemispherical shell (r+t) r* Radius of flat portion E Young's Modulus Y Yield strength t/R thickness ratio I. INTRODUCTION Shells subjected to compression by a rigid flat plate has been extensively studied in the literature both experimentally and numerically. These studies focused on elastic, elastic-plastic and buckling characteristics of conventional hemispherical shells. A brief overview of key literature is presented below along with motivation for the present study. Reissner[1] investigated the shell-plate contact problem and provide expressions for direct and bending stresses for shallow shells with h/R < 4. Naghdi[2] later extended Reissner's study to include the effect of shear strains. Kalnins[3] devised a multi-segment method for nonlinear analysis of elastic shells. Updike etal[4] investigated the load behavior of an compressed elastic shell, coming up with an analytical formulation for elastic load-deflection and also buckling phenomena wherein the shell deforms with an axisymmetric dimple at the center. Subsequent studies by Shwarz[5] and Kitching[6] focused on load-interference behavior as a function of shell thickness and radius. Interestingly, they were looking at contrasting shell applications, one concerning cornea and other collision of vehicles.

In systems with rotational symmetry, bending modes occur in doubly-degenerate pairs with two independent vibration modes for each repeated natural frequency. In circular plates, the standing waves of two such degenerate bending modes can be superposed with a 1/4 period separation in time to yield a traveling wave response. This is the principle of a traveling wave ultrasonic motor (TWUM), in which a traveling bending wave in a stator drives the rotor through a friction contact. The stator contains teeth to increase the speed at the contact region, and these affect the rotational symmetry of the plate. When systems with rotational symmetry are modified either in their geometry, or by spatially varying their properties or boundary conditions, some mode-pairs split into singlet modes having distinct frequencies. In addition, coupling between some pairs of distinct unperturbed modes also causes quasi-degeneracies in the perturbed modes, which leads their frequency curves to approach and veer away in some regions of the parameter space. This paper discusses the effects of tooth geometry on the behavior of plate modes under free vibration. It investigates mode splitting and quasi-degeneracies and derives analytic expressions to predict these phenomena, using variational methods and a degenerate perturbation scheme for the solution to the plate’s discrete eigenvalue problem; these expressions are confirmed by solving the discrete eigenvalue problem of the plate with teeth.

The delamination is defect which leads to reduce the mechanical properties of composite structure, as a results different mechanical behavior of the structure can be altered. Since the buckling behavior is one the most important mechanical behavior of composite structure for many engineering application. Then, it is necessary to investigate the effect of various delamination parameters, such as size (dimensions of delamination) and position (at x, y, and z-directions) on the buckling characterization of composite plate structure. The present study includes evaluating of the critical buckling load by using two techniques. First, experimental techniques, by testing the plates manufactured samples (including delamination) with buckling test, As well as, evaluating the mechanical properties of materials which are necessary to be used as input data for numerical techniques. Second, numerical technique by using Ansys program as an application of finite element method. A comparison between the results of the two methods are made to show the validity of these techniques. A model of composite plate combined from glass woven reinforcement fiber and epoxy resin materials with six layers is fabricated and tested, the plate is clamped supported from two edge and free supported from other edges. The comparison between the results shows an acceptable agreement between the experimental and numerical analysis where the error is not exceed about (10.7%). Finally, the results show that the delamination leads to reduce the buckling strength of composite plate structure, with the increasing of the delamination size and as the location is approached the point at which the maximum bending moment occurs.

The mathematical theory of microlocal analysis of hyperbolic partial differential equations shows that the energy density associated to their high-frequency solutions satisfies Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous materials with correlation lengths comparable to the (small) wavelength. The main limitation to date to the existing theory is the consideration of boundary or interface conditions for the energy and power flow densities. This report deals with the radiative transfer regime in a randomly heterogeneous two-plate system. First, we propose an analytical model for the derivation of high-frequency reflection/transmission coefficients for the power flows at the plates junction. These results are used in subsequent computations to solve numerically the radiative transfer equations for this system, including the interface conditions. A transport model is finally proposed for the high-frequency guided waves which could possibly propagate along the junction line.

This paper investigates the free vibration behavior of a rotating functionally graded conical shell, reinforced by an anisogrid lattice structure. The material properties of the shell are assumed to be graded in the thickness direction. The governing equations have been derived based on classical shell theory and considering the effects of centrifugal and Coriolis accelerations as well as initial hoop tension due to shell rotation. The smeared method is also employed to superimpose the stiffness contribution of the stiffeners with those of the shell to obtain the whole structure's equivalent stiffness parameters. The resulting equations, which are the coupled set of three variable coefficient partial differential equations in terms of displacement components, are solved by the Galerkin method for different boundary conditions. The obtained frequencies are compared with the available literature and the finite element software results. Finally, new results are discussed to show the effect of various parameters such as shell geometrical and material properties, stiffeners, rotating speed, and boundary conditions on natural frequencies. ARTICLE HISTORY

The evolution properties of the high-frequency vibrational energy density in slender visco-elastic structures such as Timoshenko beams or thick shells have been derived in previous works. The theory shows that this density satisfies a transport equation, or a so-called radiative transfer equation in the presence of random heterogeneities. The latter can be approached by a diffusion equation after long propagation times. The diffusive characteristics can also be obtained in the course of the derivation. This report presents a Galerkin discoutinuous finite element method to solve numerically the radiative transfer equation which characterizes the energy propagation in a curved Timoshenko beam. Comparisons are made with the analytical solution of a one-dimensional diffusion equation. In view of these results, the relevance of the vibrational conductivity analogy of high-frequency structural-acoustics invoked by some researchers is discussed.

In order to study the free vibration of simply supported circular cylindrical shells, an exact analytical procedure is developed and discussed in detail. Part I presents a general approach for exact analysis of natural frequencies and mode shapes of circular cylindrical shells. The validity of the exact technique is verified using four different shell theories 1) Soedel, 2) Flugge, 3) Morley-Koiter and 4) Donnell. The exact procedure is compared favorably with experimental results and those obtained using a numerical finite element method. A literature review reveals that beam functions are used extensively as an approximation for simply supported boundary conditions. The accuracy of the resonance frequencies obtained using the approximate method are also investigated by comparing results with those of the exact analysis. Part II presents effects of different parameters on mode shapes and natural frequencies of circular cylindrical shells.

Fibre reinforced plates and shells are funding an increasing interest in engineering applications;
in most cases dynamic phenomena need to be taken into account. Consequently effective
and robust computational tools are sought in order to provide reliable results for the analysis of such
structural models. In this paper the laminate hybrid assumed-strain plate element presented in [1],
and used there in a static analysis, has been extended to the dynamic realm. This model is derived
within the framework of the so called First-order Shear Deformation Theory (FSDT) [2], [3]. What
is peculiar in this assumed strain finite element is the direct modelling of the in-plane strain components;
the corresponding stress components are deduced via constitutive law. By enforcing the
equilibrium equations for each lamina, account taken of continuity requirements, the out-of-plane
shear stresses are computed and, finally, constitutive law provides the corresponding strains. The
resulting global strain field depends on a fixed number of parameters, regardless of the total number
of layers. Since the proposed element is not locking prone even in the thin plate limit and provides
an accurate description of inter-laminar stresses, an extension to the dynamic range seems to be
particularly attractive. The same kinematic assumptions will lead to the formulation of a consistent
mass matrix. The element, developed in this way, has been extensively tested for several lamination
sequences and comparison with analytical solutions are presented.

This article is concerned with the numerical solution of the full dynamical von K{\'a}rm{\'a}n plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von K{\'a}rm{\'a}n system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semi-discrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of
plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretisations