Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

Abstract

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).

Higher-order theories for the free vibrations of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method

“…Furthermore, these characteristics have been emphasized mostly by the use of composite materials, such as laminated composites, in order to obtain the best engineering properties with respect to conventional materials. Several applications in literature [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80] have demonstrated the advantages that can be obtained in many dynamic problems, based especially on the free vibration analysis of beams, plates and shells. However, the heterogeneity of the material properties that characterizes the laminate and the shear stresses between the laminae can cause the phenomenon of the delamination.…”

Convergence Investigation for the Exponential Matrix and Mathematical Layers in the Static Analysis of Multilayered Composite Structures

“…A simple method of reducing its order to the first one is described in [3,31] and it may be summarized as a redoubling of the involved variables. In the first author's past works [20,21], a detailed description of this approach can be found. The compact form of the reduced system of first order partial differential equations is:…”

“…Details about their determination can be found in the first author's past works [20][21][22][23][24][25][26].…”

“…In the present 3D solution, the equilibrium equations are written in orthogonal mixed curvilinear coordinates. The first author's past works extensively described all the details of the proposed 3D formulation used for the free vibration analysis of one-layered and multilayered spherical, cylindrical and flat one-layered panels [20,21]. The same idea was extended to the static problem of shells subjected to harmonic loads on the top and bottom surfaces in [22].…”

A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature

“…Theories on shells have been developed a lot through the years. In particular, the physical three-dimensional (3D) problem [32][33][34][35] can be simplified using different approximations which can lead to two-dimensional (2D) models. As a function of the hypothesis of the theory under study the solutions can closely fit the original 3D shell or plate.…”