## 1 Introduction
Among the various approaches of structural optimization processes, the assignment of a variable thickness allows to modify the stiffness of the structure by redistributing the materials within the reference domain without increasing the structural weight. In other words, variable thickness can enhance the mechanical behavior of a structure by increasing the stiffness in those parts of the structure where the stresses are high and reducing the amount of material where it is not needed. Several works concerning variable thickness structure are available in the literature, most of them are related to isotropic materials. A partial bet relevant literature review of this topic is presented here. Since the present paper is focused on the free vibration analysis, only the studies that were performed the same kind of structural analysis are mentioned. The work by [PERSON] [76] must be cited since it presented many investigations concerning isotropic plates characterized by different thickness profiles. In his paper, a complete analysis was performed for various geometries and different boundary conditions. In a similar manner, [PERSON] and [PERSON] [77] computed the natural frequencies of plates with variable thickness by using both first-order and higher-order structural models. On the other hand, a variable kinematic Ritz approach was employed by [PERSON] and [PERSON] [78] to evaluate the vibrational behavior of quadrilateral plates with arbitrary thickness. The exact element method was employed to find the axisymmetric vibration frequencies of isotropic circular and annular plates with variable thickness by [PERSON] and [PERSON] [79]. On the other hand, [PERSON] and [PERSON] [80] solved numerically the same problem by using the generalized differential quadrature method, whereas a finite element method was proposed by [PERSON] et al. [81] for the same purpose. The free vibration analysis of circular sandwich plates made of isotropic layers with a parabolic thickness variation was carried out by [PERSON] and [PERSON] [82]. All the aforementioned papers are related to flat panels. The following works are focused on isotropic shell structures. The free vibrations of circular cylindrical shells with variable thickness were investigated by [PERSON] and [PERSON] [83] and by [PERSON] and [PERSON] [84] by means of an analytical procedure and the Wittrick-Williams algorithm, respectively. The works by [PERSON] and [PERSON] [85,86] and by [PERSON] and [PERSON] [87] represented a great contribution to this topic for the results given on thick spherical and paraboloidal shells of revolution with variable thickness obtained by a three-dimensional analysis. The dynamic stiffness method was employed by [PERSON] and [PERSON] [88] to compute the natural frequencies of thick spherical shell panels with variable thickness for various boundary restraints, taking into account both the effects of transverse shear stresses and rotary inertia. Finally, [PERSON] and [PERSON] [89] studied the free vibrations of orthotropic toroidal shells described by the Sanders-Budiansky equations by means of a semi-analytical differential quadrature method. In this circumstance, the variable thickness profile is defined by a sinusoidal variation and the solutions are obtained for different geometric configurations. In this paper an orthotropic medium was considered. At this point, some works concerning FGM structures with variable thickness can be cited. Firstly, the exact element method was used by [PERSON] and [PERSON] [90] to provide some benchmark solutions to the free vibration problem of FGM annular plates. Several thickness profiles, as well as volume fraction distributions, were considered. The same kind of FGM structures resting on the Pasternak elastic foundation were investigated also by [PERSON] et al. [91] and by [PERSON] et al. [92].