On Vibrations of Elastic Spherical Shells

1962 ◽  
Vol 29 (1) ◽  
pp. 65-72 ◽  
Author(s):  
P. M. Naghdi ◽  
A. Kalnins
Keyword(s):  
Middle Surface ◽  
Free Edge ◽  
Free Vibrations ◽  
Numerical Results ◽  
Mode Shapes ◽  
Normal Displacement ◽  
Spherical Shells ◽  
Hemispherical Shell ◽  
Basic Equations ◽  
Circumferential Wave

This investigation is concerned with axisymmetric as well as asymmetric vibrations of thin elastic spherical shells. First, with the limitation to torsionless axisymmetric motion, the basic equations for spherical shells of the classical bending theory of Love’s first approximation are reduced to a system of two coupled differential equations in normal displacement of the middle surface and a stress function; this system of equations is applied to free vibrations of a hemispherical shell with a free edge and numerical results are obtained for the lowest natural frequency as a function of the thickness of the shell. The remainder of the paper is, in the main, devoted to a study of asymmetric vibrations of a hemispherical shell with a free edge according to the extensional theory. Numerical results for natural frequencies (of the four lowest circumferential wave numbers) and mode shapes are given and the results are compared with the prediction of Rayleigh’s inextensional theory.

Sensor Electromechanics and Distributed Signal Analysis of Piezo(Electric)-Elastic Spherical Shells Based on the Bending Approximation

2001 ◽  
Author(s):  
H. S. Tzou ◽  
P. Smithmaitrie
Keyword(s):  
Piezoelectric Sensor ◽  
Free Edge ◽  
Open Circuit ◽  
Mode Shapes ◽  
Shells Of Revolution ◽  
Spherical Shells ◽  
Distributed Sensor ◽  
Hemispherical Shell ◽  
Spatially Distributed ◽  
Strain Components

Abstract Spatially distributed modal voltages and sensing signal generations of a distributed piezoelectric sensor layer laminated on spherical shells of revolution are investigated in this study. The generic sensing signal equation is derived based on the direct piezoelectric effect, the Gauss theory, the open-circuit assumption, the Maxwell equation, and also the generic double-curvature thin shell theory. Due to difficulties in analytical solution procedures, assumed mode shape functions based on the bending approximation theory are used in the modal signal expressions and analyses. Spatially distributed electromechanical characteristics resulting from various meridional and circumferential membrane/bending strain components are evaluated and major signal sources are identified. Analytical results suggest that the spatially distributed modal voltages clearly illustrate the distinct modal behavior, similar to mode shapes. The major signal source of a free-edge hemispherical shell is the circumferential bending component. Accordingly, circumferential layout of distributed sensor strips would provide effective monitoring and diagnosis of free hemispheric shells.

scholarly journals A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mariame Nassit ◽  
Abderrahmane El Harif ◽  
Hassan Berbia ◽  
Mourad Taha Janan
Keyword(s):  
Analytical Method ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Current Method ◽  
Free Edge ◽  
Free Vibrations ◽  
Thin Shells ◽  
Element Calculation ◽  
Spherical Shells ◽  
Clamped Edge

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

VIBRATION AND BUCKLING OF SS-F-SS-F RECTANGULAR PLATES LOADED BY IN-PLANE MOMENTS

2001 ◽  
Vol 01 (04) ◽  
pp. 527-543 ◽  
Author(s):  
JAE-HOON KANG ◽  
ARTHUR W. LEISSA
Keyword(s):  
Beam Theory ◽  
Free Edge ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
One Dimensional ◽  
Simply Supported ◽  
Free Vibration Frequency ◽  
Edge Conditions

This paper presents exact solutions for the free vibrations and buckling of rectangular plates having two opposite, simply supported edges subjected to linearly varying normal stresses causing pure in-plane moments, the other two edges being free. Assuming displacement functions which are sinusoidal in the direction of loading (x), the simply supported edge conditions are satisfied exactly. With this the differential equation of motion for the plate is reduced to an ordinary one having variable coefficients (in y). This equation is solved exactly by assuming power series in y and obtaining its proper coefficients (the method of Frobenius). Applying the free edge boundary conditions at y=0, b yields a fourth order characteristic determinant for the critical buckling moments and vibration frequencies. Convergence of the series is studied carefully. Numerical results are obtained for the critical buckling moments and some of their associated mode shapes. Comparisons are made with known results from less accurate one-dimensional beam theory. Free vibration frequency and mode shape results are also presented. Because the buckling and frequency parameters depend upon the Poisson's ratio (ν), results are shown for 0≤ν≤0.5, valid for isotropic materials.

Free Vibration Analysis of Angle-ply Laminated Shallow Cylindrical Shell with Clamped Edges

2000 ◽  
Vol 123 (2) ◽  
pp. 188-197 ◽  
Author(s):  
Kenji Hosokawa ◽  
Minehiro Murayama ◽  
Toshiyuki Sakata
Keyword(s):  
Cylindrical Shell ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Numerical Approach ◽  
Free Vibrations ◽  
Numerical Results ◽  
Mode Shapes ◽  
Shallow Cylindrical Shell ◽  
Plastic Composite ◽  
Vibration Tests

In a previous paper, the authors proposed a numerical approach for analyzing the free vibrations of a laminated FRP (fiber reinforced plastic) composite plate. In the present paper, this approach is modified for application to a symmetrically laminated shallow cylindrical shell having a rectangular planform. First, the natural frequencies of the shell are calculated for discussion of the convergence and accuracy of the solution. Next, the effects of the curvature ratio and stacking sequence on the natural frequencies and mode shapes of the shell are studied. Furthermore, to justify the numerical results, vibration tests of the clamped symmetrically laminated shallow cylindrical shell having a square planform are carried out. These experimental results are found to agree well with the numerical results computed using the measured material properties of the lamina.

Asymmetric Free Vibrations of Layered Conical Shells

1982 ◽  
Vol 104 (2) ◽  
pp. 453-462 ◽  
Author(s):  
K. Chandrasekaran ◽  
V. Ramamurti
Keyword(s):  
Natural Frequencies ◽  
Middle Surface ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Orthotropic Materials ◽  
Conical Shells ◽  
Shell Models ◽  
Theoretical Predictions ◽  
Parametric Studies ◽  
Ritz Procedure

Asymmetric free vibrations of layered truncated conical shells are studied. Individual layers made of special orthotropic materials and both symmetric and asymmetric stacking with respect to the middle surface are considered. An energy-method based on the Rayleigh-Ritz procedure is employed. The influence of layer arrangements and that of the coupling between bending and stretching on the natural frequencies and mode-shapes are analyzed. Experimental results from tests on two shell models are provided for comparison with theoretical predictions. Numerical results based on extensive parametric studies are presented.

Nonlinear Free Vibrations Analysis of Delaminated Composite Conical Shells

2019 ◽  
Vol 20 (01) ◽  
pp. 2050010 ◽  
Author(s):  
Abbas Kamaloo ◽  
Mohsen Jabbari ◽  
Mehdi Yarmohammad Tooski ◽  
Mehrdad Javadi
Keyword(s):  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Laminated Composite ◽  
Middle Surface ◽  
Free Vibrations ◽  
Wave Number ◽  
Vertex Angle ◽  
Conical Shells ◽  
Circumferential Wave ◽  
Delamination Length

This paper examines the nonlinear free vibration of laminated composite conical shells throughout the circumferential delamination. First, based on the energy method, the governing equation of motion for the shell was derived. To simplify the analysis, the nonlinear partial differential equations were reduced into a system of coupled ordinary differential equations using Galerkin’s method. Consequently, the results were obtained by the numerical methods. Finally, the effects of delamination, variations in the delamination length, conical shells characteristics, materials property and circumferential wave number on the nonlinear response of delaminated composite conical shells were examined. The results show that the presence of delamination leads to increase in the amplitude of oscillations for the shells. Besides, the increase in the delamination length and decrease of the circumferential wave number, number of layers, and half vertex angle of the cone and orthotropy bring about a decrease in the nonlinearity of delaminated composite conical shells. However, an increase of the middle surface radius of the shell leads to a reduction of the nonlinearity as well as an increase of the amplitude.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Earthquake response of linear-elastic arch-frames using exact curved beam formulations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Baran Bozyigit
Keyword(s):  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Opening Angle ◽  
Earthquake Response ◽  
Response Analysis ◽  
Curved Beam ◽  
Curved Beams ◽  
Base Shear ◽  
Content Type

PurposeThis study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.Design/methodology/approachThe dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.FindingsThe displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.Originality/valueThe earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

Analysis of In-Plane Modal Characteristics of an Annular Disk With Multiple Point Supports

2009 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Finite Element ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Element Model ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Multiple Point ◽  
Annular Disk ◽  
Free Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an isotropic, elastic annular disk constrained at some points on the inner and outer boundaries are investigated. The presented study is relevant to various practical problems including disks clamped by bolts along the inner and outer edges or the railway wheel vibrations. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. The boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to realize clamped conditions at discrete points. The frequency parameters for different point constraint conditions are evaluated and compared with those computed from a finite element model to demonstrate the validity of the proposed method. The computed mode shapes are presented for a disk with different point constraints at the inner and outer boundaries to demonstrate the free in-plane vibration behavior of the disk. Results show that addition of point supports causes some of the modes to split into two different frequencies with different mode shapes. The effects of different orientations of multiple point supports on the frequency parameters and mode shapes are also discussed.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

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