# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 1 Introduction
Papers with high-order formulations can be found in the works of [PERSON] [12], [PERSON] [8], [PERSON] [13], [PERSON] & [PERSON] [14], and [PERSON] [2]. Among the finite element formulations based on these theories is the seven-parameter formulation with the drilling degree of freedom, the seventh parameter being the thickness stretch (see [PERSON] & [PERSON] [11]). In addition, seven-parameter formulations were developed with the intention to eliminate the phenomenon of Poisson locking presented in six-parameter formulations (i.e., three displacements and three rotations). The Poisson locking arises when a constant state of normal strain through the thickness is assumed, and this phenomenon is due to the mathematical model and not to finite element model discretization [15]; in other words, a mesh refinement does not alleviate the locking. There are numerous works related to shells in the literature. Finite element formulations to perform different types of analyses (e.g., static, dynamic and vibration or wave propagation analysis) can also be found. These finite element models can be classified as follows [14]: facet-shell elements, 2D elements based on shell theories, 3D elasticity elements, and continuum or degenerated elements. Often, these formulations are restricted to model elements with particular geometries, such as shallow shells, doubly curved shells, circular cylindrical shells, and so on. In addition, in general, formulations for dynamic analysis, free vibration, or transient response are developed for specific cases of materials (e.g., laminated shells and functionally graded materials). Examples related to these formulations can be found in the works of [PERSON] et al. [16], [PERSON] et al. [17], [PERSON] & [PERSON] [18], [PERSON] et al. [19], [PERSON] et al. [20], [PERSON] & [PERSON] [21], [PERSON] et al. [22], among others. A review for plates and shells fabricated with FGM can be found in the work of [PERSON] [3]. [PERSON] et al. [23] presented a dynamic analysis model by means of degenerated shell elements, where the discretization of the 3D elasticity equations is made in terms of mid-surface nodal variables. In this formulation, five parameters were used to define the shell kinematic; however, they do not present any analysis to determine natural frequencies. [PERSON] et al. [24] reported a free vibration analysis of isotropic, open conical panels using the \(hp\) version of the finite element method (i.e., a refinement in the mesh size \(h\) and the order \(p\) of the polynomial interpolation functions) based on Kirchhoff-Love thin shell equations, and they assumed cones of uniform thickness. They presented natural frequencies and mode shapes, and compared their results with those obtained using a commercial finite element code and experimental results. [PERSON] & [PERSON] [25] determined natural frequencies for plates and shells by means of a nine-node degenerated element with six degreesof freedom, which was developed by them based on assumptions of the FSDT, and using the ANS approach to mitigate the locking phenomena. In the work of [PERSON] & [PERSON] [26], a similar formulation to the one presented by these authors in 2001 [25] is used, but this time it is applied to laminated composite plates and shells. Recently, [PERSON] et al. [27] reported a vibration analysis of thin shallow shells with the spectral interpolation functions and subjected to impact loads. [PERSON] et al. [28] presented formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells; they considered nonuniform thickness. [PERSON] & [PERSON] [29] presented a finite element with six degrees of freedom and spectral interpolation functions for dynamic analysis of isotropic shell-like structures; the transverse deformation was considered and the structures studied were of uniform thickness.