Free Vibration Characteristics of Isotropic and Laminated Orthotropic Spherical Caps (original) (raw)
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Journal of Sound and Vibration, 1996
This paper is concerned with the free vibrations of a stiffened conical thin shell within the context of Donnell-Mushtari theory. A truncated cone with simply supported ends is reinforced by relatively closely spaced elastic stringers and/or rings. The tapered stringers are used to obtain an efficient stiffening. Change in the stringer spacing in the meridional direction is taken into account in the formulation. The stiffening elements are ''smeared out'' along the conical shell to yield a single equivalent orthotropic shell. The resulting orthotropic shell has a kind of inhomogeneity due to the tapered stringers. The equations of motion for the free vibrations of the stiffened conical shell are derived by the use of Hamilton's principle. The differential equations of the stiffened truncated conical shell, together with the boundary conditions, are solved by the use of the collocation method. Solutions are presented to show the influence of geometrical parameters and material properties on the vibration characteristics. The numerical results are compared with certain earlier results.
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An investigation of the generic vibration of prestressed thick orthotropic cylindrical shells was carried out. A set of Fliigge type equations of motion for orthotropic shells has been reduced to an equivalent set of equations containing only non-dimensionalised geometrical parameters, loadings and generalised global constants of orthotropic materials. The influence and importance of the global constants on the vibration of cylindrical shells are studied. Based on the results, the figures shown in this paper are applicable for a large group of materials, making the understanding of the correlation between composite material properties and vibrational behaviour of the shells very easy. The results of this paper show that an increase in vibrating frequency is brought about by (1) increasing generalised rigidity ratio, (2) increasing transverse shear modular ratios, (3) decreasing generalised Poisson's ratio and (4) decreasing principal rigidity ratio. The key parameters to the problems are generalised rigidity ratio and principal rigidity ratio only. NOTATION a an A^D^Fi, A
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Computer Methods in Applied Mechanics and Engineering, 1987
An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction. The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meri~onal direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain ~om~nents on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique. The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.
Vibration Analysis of Orthotropic Thin Cylindrical Shells with Free Ends by the Rayleigh-Ritz Method
Journal of Sound and Vibration, 1996
An analytical model is developed to predict the modal characteristics of thin-walled circular cylindrical laminated shells with free ends. The shell is orthotropic and has mid-plane symmetry. By using Love's first-approximation shell theory, a strain energy functional containing both bending and stretching effects is formulated. The shell vibration mode shapes are then modelled by utilizing characteristic beam functions in the Rayleigh-Ritz variational procedure and the accuracy of the model is verified by test data. With the developed model, inextensional Rayleigh and Love modes can be identified having frequencies close to each other. The contributions to the strain energy due to various elastic properties are also investigated. Results show that the circumferential modulus provides a major portion of the flexural energy of the vibrating structure while the longitudinal and in-plane shear moduli contribute mostly to the stretching energy. It is also observed that reducing the shell thickness would result in a substantial increase in the ratio of the energies associated with the longitudinal and shear moduli, respectively. By rearranging the lamination stacking sequence, shells can be made to be more resilient to bending or twisting with only minor alterations in natural frequencies.
Vietnam Journal of Mechanics, 2022
This study investigates the static and free vibration responses of orthotropic laminated composite spherical shells using various refined shear deformation theories. Displacement-based refined shear deformation theories are presented herein for the analysis of laminated composite spherical shells via unified mathematical formulations. Equations of motion associated with the present theory are derived within the framework of Hamilton's principle. Analytical solutions for the static and free vibration problems of laminated spherical shells are obtained using Navier's technique for the simply supported boundary conditions. Few higher order and classical theories are recovered from the present unified formulation; however, many other theories can be recovered from the present unified formulation. The numerical results are obtained for symmetric as well as anti-symmetric laminated shells. The present results are compared with previously published results and 3-D elasticity solution. From the numerical results, it is concluded that the present theories are in good agreement with other higher order theories and 3-D solutions.
Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness
Journal of Mechanics of Materials and Structures, 2010
A dynamic stiffness method is presented for determining the free vibration frequencies and mode shapes of thick spherical shell segments with variable thickness and different boundary conditions. The analysis uses the equations of the two-dimensional theory of elasticity, in which the effects of both transverse shear stresses and rotary inertia are accounted for. The displacement components are taken to be sinusoidal in time, periodic in the circumferential direction, constant through the thickness, and solved exactly in the meridional direction using the exact element method. The shape functions are derived from the exact solutions for the system of the differential equation of motion with variable coefficients. The dynamic stiffness matrix is derived from the exact shape functions and their derivatives. Highprecision numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and for several combinations of boundary conditions. Comparison is made with results of published research and with two-and three-dimensional finite element analyses.
Composite Structures, 2005
In the present work, a study of free vibrations of functionally graded cylindrical shells made up of isotropic properties is carried out. A semi-analytical axisymmetric finite element model using the 3D linear elastic theory is developed. The 3D equations of motion are reduced to 2D by expanding the displacement field in Fourier series in the circumferential direction, involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with simple benchmark problems and the results show that the frequency characteristics are found to be close to published results of isotropic cylindrical shells. New results are included for FGM shells.
International Journal of Non-Linear Mechanics, 2014
The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell's non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.
Axisymmetric buckling of laminated thick annular spherical cap
Communications in Nonlinear Science and Numerical Simulation, 2005
Axisymmetric buckling analysis is presented for moderately thick laminated shallow annular spherical cap under transverse load. Buckling under central ring load and uniformly distributed transverse load, applied statically or as a step function load is considered. The central circular opening is either free or plugged by a rigid central mass or reinforced by a rigid ring. Annular spherical caps have been analysed for clamped and simple supports with movable and immovable inplane edge conditions. The governing equations of the Marguerre-type, first order shear deformation shallow shell theory (FSDT), formulated in terms of transverse deflection w, the rotation w of the normal to the midsurface and the stress function U, are solved by the orthogonal point collocation method. Typical numerical results for static and dynamic buckling loads for FSDT are compared with the classical lamination theory and the dependence of the effect of the shear deformation on the thickness parameter for various boundary conditions is investigated.