## 1 Introduction  
Rotating-disk-like structures are a common element in rotating machines (such as circular saws, wafer cutting machines, disk brakes, grinding wheels) and turbomachinery components (blisks, centrifugal fans, gas turbines, hydraulic turbines). These structures are excited by different kinds of dynamic loads and therefore a resonance can occur if the excited frequency coincides with one of the natural frequencies of the structure and the excitation shape matches the mode shape of the structure [1; 2]. For this reason, in this kind of rotating structure, it is of paramount importance to predict and determine the natural frequencies and mode shapes of both rotating and stationary frames. There are many types of rotating structures used in real engineering cases that can be assimilated to a disk-type or disk with blades structure. Some examples of real applications are shown in Figure 1. Figure 1a shows a simple disk, which is basically a circular thin plate. Figure 1b shows a bladed-diskor blisk, which consists of a central ring with attached blades. Figure 1c is the combination of a simple disk with blades attached to one side. In Figure 1d, a structure that can be considered as an assembly disk-blades-disk is shown. Each type has different mode-shapes and, therefore, their structural response (natural frequencies and mode shapes) has to be studied separately. The natural frequencies and mode shapes of simple disks (Figure 1a) have been analyzed in many studies [7; 8; 9]. These studies use the governing structural equation of a thin circular plate in order to deduce the natural frequencies and mode shapes of the structure. In [8; 9], the characterization of the mode shapes of disk-like structures is defined by the number of nodal diameters (diameters with no deformation) and nodal circles (entire circles with no deformation). This characterization has also been used in many researches on disk-like structures [10; 11; 12; 13; 14; 15]. The effect of the rotation for disk-like structures on the natural frequencies and mode shapes can be analyzed from the rotating and stationary frames. When analyzing the structural response from the rotating frame, for simple rotating thin disks, effects such as gyroscopic, centrifugal forces, and shear modify the structural equation at high speeds [16; 17; 18]. Therefore, for low rotating speeds, when such effects are not relevant, the structural response of the rotating structure analyzed from the rotating reference frame is almost identical to the structural response of the non-rotating structure. For the same structure rotating at a low rotating speed, when analyzing the structural response from the stationary frame, the following can be observed: Standing mode shapes (observed from the rotating frame) are separated into two travelling modes (stationary frame). The frequency shift between both reference frames depends on the number of nodal diameters of the mode and the rotating speed of the disk [17; 19]. Bladed-disks (or blisk), which consist of a rotor disk and blades (Figure 1b), are widely used in the field of stationary gas turbines and in aircraft engines. This kind of structure is said to have cyclic symmetry [20; 21; 22; 23], which means that they are composed of identical sectors (each sector contains a part of the disk and the blade). A discussion of the natural frequencies and mode shapes of this kind of structures can be found in [20; 21] for \"tuned\" bladed disks, which means that the condition of cyclic symmetry is accomplished. This mode shape has characteristics of single blade modes (also known as clamped modes) and disk modes. The effects of small differences in the N sectors (mistuning of the bladed-disk) is also discussed in the aforementioned references. The effects of rotation on the natural frequencies are discussed in [23] for the rotating and stationary frames. For disk-blades configurations (Figure 1c), there is less literature on the structural response (natural frequencies and mode shapes) and associated dynamic problems.