The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

/pdf/numerical-and-exact-models-for-free-vibration-analysis-of-4dgqzmylyr.pdf

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

A 3D free vibration analysis of multilayered structures is proposed. An exact solution is developed for the differential equations of equilibrium written in general orthogonal curvilinear coordinates. The equations consider a geometry for shells without simplifications and allow the analysis of spherical shell panels, cylindrical shell panels, cylindrical closed shells and plates. The method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structures. Results are given for multilayered composite and sandwich plates and shells. A free vibration analysis is proposed for a number of vibration modes, thickness ratios, imposed wave numbers, geometries and multilayer configurations embedding isotropic and orthotropic composite materials. These results can also be used as reference solutions for plate and shell 2D models developed for the analysis of multilayered structures.

/pdf/an-exact-3d-solution-for-free-vibrations-of-multilayered-8wolwjbiex.pdf

An exact 3d solution for free vibrations of multilayered cross-ply composite and sandwich plates and shells

Abstract The paper proposes a comparison between classical two-dimensional (2D) finite elements (FEs) and an exact three-dimensional (3D) solution for the free vibration analysis of one-layered and multilayered isotropic, composite and sandwich plates and cylinders. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the finite element method and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 2D finite element results are obtained by means of a well-known commercial FE code. The differences between 2D FE solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders

The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of
equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).

/pdf/three-dimensional-exact-free-vibration-analysis-of-spherical-5u281poecf.pdf

Three-dimensional exact free vibration analysis of spherical, cylindrical, and flat one-layered panels

This new work proposes an exact three-dimensional static analysis of plates and shells. One-layered and multilayered isotropic, orthotropic, sandwich and composite structures are investigated in terms of displacements and in-plane and out-of-plane stresses through the thickness direction. Proposed structures are completely simply-supported and a transverse normal load is applied. The proposed method is based on the 3D equilibrium equations written using general orthogonal curvilinear coordinates which are valid for spherical shells. Cylindrical shell, cylinder and plate results are obtained as particular cases of 3D spherical shell equations. All the considered structures are analyzed without any geometrical approximation. The exact solution is possible because of simply-supported boundary conditions and harmonic form for applied loads. The shell solution is based on a layer-wise approach and the second order differential equations are solved using the redouble of variables and the exponential matrix method. A preliminary validation of the model is made using reference results in the literature. Thereafter, the proposed exact 3D shell solution is employed with confidence to provide results for one-layered and multilayered plates, cylinders, cylindrical shell panels and spherical shell panels. All these geometries are analyzed via a unified and general solution, and the obtained results can be used to validate future numerical methods proposed for plates and shells (e.g., the finite element method or the differential quadrature method). Proposed results allow to remark substantial features about the thickness of the structures, their geometry, the zigzag effects of displacements, the interlaminar continuity of displacements and transverse stresses, and boundary loading conditions for stresses.

/pdf/exact-three-dimensional-static-analysis-of-single-and-multi-4ttehmjkhd.pdf

Exact three-dimensional static analysis of single- and multi-layered plates and shells

In this article, the constitutive relations and the governing equations are derived for nonlocal thermoelastic solid in the presence of diffusion. The free vibration of a thermoelastic diffusive cy...

Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion

The present study focuses on the analysis of free vibrations of axisymmetric functionally graded hollow spheres. The material is assumed to be graded in radial direction with a simple power law. Matrix Frobenious method of extended power series is employed to derive the analytical solutions for displacement, temperature, and stresses. The dispersion relations for the existence of various types of possible modes of vibrations in the considered hollowsphere are derived in a compact form. In order to explore the characteristics of vibrations, the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software. The numerical results have been presented graphically for polymethyl methecrylate materials in respect of natural frequencies, frequency shift, inverse quality factor, displacement, temperature change, and radial stress. 
Open image in new window

Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres

This paper concentrates on the study of the three-dimensional free vibrations in a homogenous isotropic, viscothermoelastic hollow sphere whose surfaces are subjected to stress free, thermally insulated or isothermal boundary conditions. The use of governing partial differential equations is solved into a coupled system of ordinary differential equations. The equation for toroidal motion gets decoupled from rest of the motion and remains unaffected due to thermal variations. Matrix Fr?benious method of extended power series is employed to obtain the solution. The secular equations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived in the com- pact form. The special cases of spheroidal and toroidal modes of vibrations of a hollow sphere have also been deduced and discussed. In order to explore the characteristics of vibrations the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software tools. The computer simulated results have been presented graphically for copper material.

https://file.scirp.org/pdf/OJA20120100007_28540139.pdf

Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere

The analysis of free vibrations of axisymmetric functionally graded isotropic viscothermoelastic hollow cylinder has been investigated in the radial direction. The material of viscothermoelastic cylinder has been assumed to be graded in the radial direction due to simple exponent law. The governing partial differential equations are transformed into ordinary differential equations with th
e help of time harmonic vibrations. The extended power series solution of matrix Frobenius method has been applied to derive the analytical solutions for stresses, displacement and temperature change. The frequency equations of various types of modes of vibrations have been derived in a compact form. To investigate the numerical features of vibrations, the analytical results of frequency equations have been further solved with the help of fixed-point numerical iteration technique using MATLAB software tools. The polymethyl methacrylate material has been used for numerical computations. The numerical results have been presented for frequency shift, natural frequencies, thermoelastic damping, stresses, displacement and variation of temperature. With the increase in the values of grading index, variation of vibrations in field functions go on decreasing. The numerical results show alternate variations in homogenous materials in contrast to inhomogeneous materials. The behaviour of deformation, temperature change, frequency shift and thermoelastic damping have been monitored (increase or decrease) with the help of grading index (in-homogeneity parameter).

Analysis of Free Vibrations of Axisymmetric Functionally Graded Generalized Viscothermoelastic Cylinder Using Series Solution

In this paper an exact analysis of homogenous rigidly fixed vibrations of viscothermoelastic hollow sphere is presented. The basic governing partial differential equations have been reduced to ordinary differential equations by using Helmholtz decomposition equations. The uncoupled equation is taken for first class vibrations and remains independent of temperature variations, while coupled system of equations are taken for second class vibrations. Matrix Frobenious method of extended power series has been applied in the coupled system of differential equations to get displacements and temperature. Numerical results have been presented, giving lowest frequency, dissipation factor, displacements and temperature change.

Dinesh Kumar Sharma

Papers

Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion

Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres

Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere

Analysis of Free Vibrations of Axisymmetric Functionally Graded Generalized Viscothermoelastic Cylinder Using Series Solution

Free vibration analysis of a rigidly fixed viscothermoelastic hollow sphere