Vibrations Of Shallow Shells Research Articles

Dynamic Instability of Shallow Shells Interacting with a Three-Dimensional Potential Gas Flow

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

On nonlinear vibration analysis of shallow shells ? A new approach

A method for the analysis of nonlinear vibration of shallow shells of arbitrary shape is presented. The method is based upon the concept of constant deflection contours on the surface of the shallow shell. The constant deflection contour method has previously been found to be a simple tool for the study of linear vibration analysis of shallow shells of arbitrary shape. A new approach has been made here to utilize this concept to study the large amplitude vibration of shallow shells in conjunction with the Galerkin method. A number of illustrative examples are included to demonstrate the accuracy of the proposed method.

Influence of reinforcement and elastic foundation on the vibrations of shallow shells with rectangular planform

A method for determining the natural frequencies and modes of ribbed shallow shells with rectangular planform on an elastic foundation is developed. The method takes into account the discrete arrangement of the ribs. A shallow spherical shell with square planform is considered as example to analyze the effect of the number of ribs and the Winkler and Pasternak coefficients of subgrade reaction on the natural frequencies and modes. It is recommended to take into account the discrete arrangement of ribs when they are few

Free vibrations of a shallow shell in fluid under geometrically nonlinear deformation

The method of calculating the nonlinear vibrations of shallow shells in fluid is proposed. Normal modes and natural frequencies of linear vibrations in fluid are specified, then the nonlinear vibrations of a shell are expanded in terms of the normal modes of the system. The Shaw–Pierre normal nonlinear modes are determined to get the shell model with the finite number of degrees of freedom.

Nonlinear modes of cylindrical panels with complex boundaries. R-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.

Analysis of nonlinear chaotic vibrations of shallow shells of revolution by using the wavelet transform

In the present paper, we study complex vibrations of flexible axisymmetric shallow shells under the action of transverse sing-alternating pressure. In addition to the traditional methods of nonlinear dynamics, we for the first time use the wavelet transform to analyze the transition from harmonic to chaotic vibrations. We analyze the use of Gauss-type wavelets (the order of derivatives varies from m = 1 to m = 8) and also the use of the Morlet wavelet (both real and complex). We conclude that the use of the complex Morlet wavelet is preferable to that of the Gaussian wavelets: the more zero moments a wavelet has, the better it describes the complex vibrations of flexible shallow shells.

Nonlinear vibrations of shallow shells with complex boundary: R-functions method and experiments

Geometrically nonlinear vibrations of shallow circular cylindrical panels with complex shape of the boundary are considered. The R-functions theory and variational methods are used to study the problem. The R-functions method (RFM) allows constructing in analytical form the sequence of basis functions satisfying the given boundary conditions in case of complex shape of the boundary. The problem is reduced to a single second-order differential equation with quadratic and cubic nonlinear terms. The method developed has been initially applied to study free vibrations of shallow circular cylindrical panels with rectangular base for different boundary conditions: (i) clamped edges, (ii) in-plane immovable simply supported edges, (iii) classically simply supported edges, and (iv) in-plane free simply supported edges. Then, the same approach is applied to a shell with complex shape of the boundary. Experiments have been conducted on an aluminum panel with complex shape of the boundary in order to identify the nonlinear response of the fundamental mode; these experimental results have been compared to numerical results.

Moderately large vibrations of doubly curved shallow open shells composed of thick layers

This paper addresses nonlinear flexural vibrations of shallow shells composed of three thick layers with different shear flexibility, which are symmetrically arranged with respect to the middle surface. The considered shell structures of polygonal planform are hard hinged simply supported (i.e. all in-plane rotations and the bending moment vanish) with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first-order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. Numerical results of rectangular shallow shells in nonlinear steady-state vibration are presented for various ratios of shell rise to thickness, and non-dimensional load amplitude.

Forced nonlinear vibration of shallow shell and application to technical diagnosis

This paper presents direct and inverse problems of forced nonlinear vibration of shallow shell. From that the method of establishment and solution of technical diagnosis for shallow shell under the vibrated load is proposed. The diagnosis criterion is the minimum of the difference between the square of experimental and theoretical data. The diagnosed parameters are the external loads and the reactions arose at the boundaries.

Vibration Analysis of Laminated Shallow Shells with Non-uniform Curvature

Although there are a large number of technical papers on vibration of shallow shells, all the previous papers have dealt with shallow shells with uniform curvature to avoid difficulty arising in the variable curvature analysis. Recent products, however, require various surface designs of thin panels from the viewpoint of industrial design, for example, in the fender and door panel designs of commercial vehicles. The present study proposes an analytical method to deal with vibration of shallow shells with non-uniform curvature. An interpolating function is introduced to represent the required surface shape and the corresponding curvature is derived as a function of the position (x, y). The obtained curvature is substituted into the total potential energy of the shell, and the process is shown to derive a frequency equation in the Ritz method. Numerical examples show the effects of non-uniform curvature on the natural frequencies and mode shapes.

3301 曲率が一様でない偏平シェルの解析法(OS14 曲面構造の解析と設計I)

Although there are a large number of technical papers on vibration of shallow shells, almost all the papers have dealt with shallow shells with uniform curvature to avoid difficulty in the analysis. Recent products, however, require various shape design of thin panels from the viewpoint of industrial design, for example, in fender and door panel design in commercial automobiles. In the present study, an analysis method is proposed to deal with vibration of shallow shells with non-uniform curvature. The interpolating function is determined to represent the required shape and the corresponding curvature is derived as the function of position (x,y). The curvature is substituted into the total potential energy of the shell, and the process is shown in the Ritz method to derive a frequency equation.

Investigation of free vibrations of multilayer shallow shells and plates of complex shape in plan

We propose an algorithm for the solution of the problem of free vibrations of multilayer shallow shells and plates based on the variational methods and the theory of R-functions. By using this algorithm, we solve the problems posed for shallow shells (spherical and cylindrical) and plates of complex shape in plan. We present the results of investigation of the dependences of the natural frequencies of vibrations of shallow shells and plates with different number of layers on the angle of rotation of the principal directions of each layer.

Numerical Analysis for the Nonlinear Free Vibration of Shallow Shells

A variational approach for the nonlinear free vibration of shallow shells having a quadrilateral boundary is presented in this paper. Natural coordinates ξ and η are used to map the prescribed geometry in the x-y plane. Displacement fields corresponding to u, v, w, β1, and β2 are expressed in terms of the product of two algebraic functions, the form of which is so chosen that the displacement boundary condition can be imposed by manipulating the coefficients. In arriving at the stiffness matrix, no simplification is applied to the nonlinear strains and the variation of the complete energy equation is considered. For the plate problems numerical results are obtained and compared with approximate analytical results by other researchers. Numerical results for the shallow shells are also presented and their characteristics are found to be significantly different from the results for the plates.

On a one-dimensional version of the dynamical Marguerre-Vlasov system

A one-dimensional version of the so-called Marguerre-Vlasov system of equations describing the vibrations of shallow shells is considered. The system depends on a parameter ∈→0 in a singular way and undergoes the effect of damping mechanisms. We show that the system converges to a nonlinear beam equation while the energy decays exponentially uniformly (on ∈→0) as time goes to infinity.

A note on analysis of the natural vibrations of shallow shells with cuts on the surface

A new approach is proposed to study the dynamic behavior of shells with an arbitrary planform weakened by surface cuts (cracks.) The approach is based on the method of R-functions. The computer simulation is carried out using the POLE-SHELL problem-oriented system, which implements the method of R-functions and variational methods. Numerical results are presented

On the Uniqueness Theorem for Generalized Solutions of Initial-Boundary Problems for the Marguerre—Vlasov Vibrations of Shallow Shells with Clamped Boundary Conditions

The uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre—Vlasov vibrations of shallow shells with clamped boundary conditions is proved. A unique method developed by the author, based upon a nonstandard treatment of smoothing operators, is applied instead of using an enclosure theorem at the critical values of indices.

Vibrations of shallow shells that are circular in plan and have local supports

The paper deals with vibrations of doubly curved shallow shells that are circular in plan and are reinforced by local rod-type supporting elements or cylindrical shells. The coefficients of the frequency equation are found by using numerical-analytical methods to solve boundary-value problems for fixed values of frequency. The natural frequencies and modes of vibration of a system composed of a shell and elastic supports are determined in the course of solving the problem. It is shown that it is possible to also account for reactive moments and shearing forces, in addition to the normal reactions of an elastic support. The potential of the approach which is developed is illustrated by the solution of specific problems.

Vibration of Shallow Shells: A Review With Bibliography

This review article documents recent developments in the free vibration analysis of thin, moderately thick, and thick shallow shells. An introductory review of the studies in Kirchhoff-Love classical thin shell theory is given. The development of studies in moderately thick shells incorporating the effects of transverse shear deformation and rotary inertia is detailed. This review article mainly focuses on research advances in vibration studies since the 1970s using the classical Kirchhoff-Love, first-order, and higher-order theories. The validity and range of applicability of these theories are examined. There are 163 references listed at the end of the article.

On the free vibration of shallow shells

A finite element method for free vibration analysis of generalized shallow shells is described. Results for natural frequencies of corner-supported hyperbolic paraboloidal, cylindrical and spherical shells are presented and compared with those of other investigators to establish the correctness of the method. Natural frequencies of reinforced concrete clamped elliptic paraboloid, cylindrical and conoidal shells of the same thickness and material properties are presented and discussed to illustrate the application of the method to anticlastic conoidal shells.

Large amplitude free vibration of thick shallow shells supported by shear diaphragms

The effect of thickness and curvature upon the large amplitude vibration of shallow shells is studied in this paper. For this purpose, the non-linear governing equations for thick shallow shells which have principle curvatures and rectangular planform are derived by Hamilton's principle using first order shear deformation theory. Applying Galerkin's procedure and eliminating variables except for transverse displacement, the governing equations are reduced to an elliptic ordinary differential equation in time. The period of vibration for the shell is calculated by integrating the equation using a Gauss-Legendre integration method. The present method is applied to a shallow shell which has a rectangular boundary supported by shear diaphragms.