Vibrations Of Shallow Shells Research Articles

Vibration of Shallow Shells with Two Adjacent Edges Clamped and the Others Free∗

ABSTRACT Considerable information is available in the published literature on the free vibration frequencies and mode shapes of rectangular flat plates having two adjacent edges clamped and the other two free. However, no results appear to have been published previously for shallow shells having such edge conditions. The present work uses the Ritz method with displacement components in the form of algebraic polynomials to obtain accurate frequencies. Frequencies are determined for the first eight modes of shallow shells having spherical, cylindrical, and hyperbolic paraboloi-dal curvatures and square planforms. Beginning with the plate, the curvatures are incrementally increased in each case to the limits of shallow

Free and forced vibrations of shallow shells by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic shallow shells of arbitrary geometry and boundary conditions. The integral formulation employs the static fundamental solution of thin elastic flat plates under combined flexural and membrane deformation. This creates not only boundary integrals, but surface integrals as well due to the presence of the inertia forces and the coupling between the flexural and membrane action in the shell. Thus the discretization involves boundary as well as interior elements, both of the quadratic isoparametric type. Both free and forced vibrations are considered. The free vibration problem is reduced to a generalized eigenvalue problem involving matrices with entries independent of frequency. The general forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the time domain response. The effect of the external viscous or internal viscoelastic damping on the response is also taken into account. Numerical examples are presented to illustrate the method and demonstrate its advantages.

Nonlinear forced vibrations of shallow shells on regular polygonal bases

The nonlinear, large amplitude vibrations of elastic, shallow shells when subjected to transient pressure loading are analyzed in a relatively simple manner. The isoamplitude contour lines approach is used to derive the basic equations for shallow shell vibrations. The coupled system of governing equations thus obtained is solved for shells of regular polygonal planform with various types of dynamic loading. Also, an equivalent analysis in which Berger’s approximation is used is presented and a comparison is made of the results obtained by both methods. Some comparison is also made with previously obtained results, as available in the literature. All details are explained by graphs.

Elastic impact of spheres on thin shallow spherical shells

This investigation was concerned with the elastic impact of spheres on thin shallow spherical shells. A non-linear integro-differential equation of the impact process was developed on the basis of Reissner's approximate theory for transverse vibrations of shallow shells and the quasistatic Hertzian impact theory. This equation was numerically integrated and its main results were experimentally verified. Special attention was given to the apparently new observations that the rebound height of a sphere dropped onto a given shell has a minimum value for a critical size of the sphere, and that the contact time differs substantially for sphere sizes greater or less than this critical value.

On the Vibration of Shell Structure

A great deal of fatigue damage due to severe vibration arises at after ship body, because there are some exciting sources. In order to avoid the resonanse between the ship structure and the exciting forces, it is necessary to estimate exactly the natural frequency of ship structure. The side shell structure of after ship body which recieves directly the propeller surface force is not the flat plate as a general rule, therefore it is not easy to calculate the natural vibration of side shell structure.In this paper, a method to calculate the natural vibration of rectangular and trapezoid shallow shell was presented. And the results by present method were compared to the results by the finite element method. The natural vibration of shallow shell can be fairly good estimated by using the present method, but the magnitude of maximum initial deflection which this method can be applied to is about 20% to the side length. It was also clarified that the plate thickness-maximum initial deflection ratio is one of the parameter which decides the natural frequency of the shallow cylindrical shell. According to the calculated results for the trapezoid shell, in the case that the upper side-lower side ratio c/a is about 0.91.0, the natural frequency is almost equal to that of rectangular shell.

Transverse vibrations of shallow shells by the method of constant-deflection contours

A method for the analysis of vibration of shallow shells is presented. The method is based upon the concept of iso-amplitude contour lines on the vibrating surface of the shell. The problem of transient axisymmetric vibrations of shallow elliptical dome is investigated. It is shown that in the limiting case, when the elliptical dome reduces to the spherical dome, Reissner's well-known result is obtained.

Non-linear vibrations of orthotropic doubly-curved shallow shells

The purpose of this study is the evaluation of the effects of curvature, material orthotropy and internal pressure upon the non-linear vibrations of shallow shells. The shells considered here are complete in the circumferential coordinate. A mode shape that leads to a continuous circumferential displacement is used as the basic solution in a Galerkin type procedure. Among the results of interest are the effects of the aforementioned physical parameters on the relative hardness of the shell response, as well as some interesting new perspectives on jump phenomena in non-linear shell vibrations.

Free Vibrations of Spherical Sandwich Shells

In the present study, the problem of axisymmetric vibrations of spherical sandwich shells is considered. In obtaining the governing differential equations, the longitudinal inertia terms u,it and v,it can be neglected. This makes possible the introduction of a stress function F similar to the one in elastostatic problems. The differential equations are written in terms of an arbitrary deformation function φ and the stress function F. For simplicity in the final results, the face sheets and core are considered to be isotropic elastic and of uniform thickness t and c, respectively. In some applications, the core material acts as a heat shield and an insulator. In this paper, the effect of the thermal gradient along the thickness of the sandwich is not considered. Since the core filler is of honeycomb thermoplastic or aluminum foam having low values of shearing modulus, an order analysis of various terms in the differential equations establishes that the problem can be studied in two parts. This results in two sets of differential equations, which are applied in accordance with the physical parameters of the core geometry and material. The final frequency equation |aij| = 0; i, j = 1, ⋯4 and the results are given for two types of boundary conditions (fixed and simply supported). The solution of the frequency equation degenerates to the classical solution for vibration of shallow shells and to that of two layered shells.

The dynamics of a mildly sloping elastic thread

The basic structural element of suspension systems (two-caNe suspension systems, some types of soft shells, suspended coverings, rope bridges, etc.) is a mildly sloping elastic thread. The problem of the vibration of such a thread is of definite interest. It is shown that the results of [2, 4] obtained by analyzing the vibrations of shallow shells and mildly sloping arches can be adapted to cover mildly sloping elastic threads.

Comments on “Axisymmetric Vibrations of Thin Elastic Shells” [C. N. DeSilva and G. E. Tersteeg, J. Acoust. Soc. Am. 36, 666–672 (1964)

In a recent paper, DeSilva and Tersteeg [J. Acoust. Soc. Am. 36, 666–672 (1964)] presented a membrane theory for axisymmetric vibrations of thin shells of revolution, in which the effect of longitudinal inertia was neglected. Numerical comparisons are presented in this discussion to show that the effect of longitudinal inertia is not negligible in axisymmetric vibration of nonshallow shells. For vibrations of shallow shells, the longitudinal inertia effect is small, but the bending effect becomes important.