## 2 Diametrical Mode Shapes of Rotating Disk-Blades-Disk Like Structures from the Rotating Frame
### Disk-Blades-Disk Structures  
Since there are many factors that can determine the mode shapes of a disk-blades-disk structure (number and geometry of the blades, junctions blade-disk...), there is not a standard analytical form to describe the mode shapes of these structures.  
In this paper, the type of structure disk-blades-disk, which is used for closed impellers in centrifugal compressors, pump-turbines, centrifugal pumps and low specific speed Francis runners [30; 34; 39; 42], is analyzed in detail.  
Considering again the diametrical modes and assuming that these are real modes (points are moving in phase or in counterphase to each other), the deformation at the periphery, when all the points pass through their maxima/minima (supposing this for \(t=0\)), can be written in a general form:  
\[w_{p,i}(r=r_{max},\theta,t=0)=A_{i}(\theta)\text{ for }0\leq\theta\leq 2\pi \tag{3}\]  
Figure 4: \(w_{p,n}\) for different times. (**a**) Mode \(n=2\) of a simple disk structure (one diametrical component). (**b**) Mode with more than one diametrical component.  
In this case, the deformation at the periphery of the mode \(i\) is not pure sinusoidal as for the simple disk case, due to the blades. Nevertheless, for a diametrical mode, again the deformation at the periphery is larger than the inner radii. In this case, the deformation at the periphery is defined by an arbitrary function \(A_{i}\), which depends on the angular position \(\theta\). Once more, for any other time \(t\), the deformation will be lower than that expressed in Equation (3), but it will be proportional.  
Note that \(i\) is used instead of \(n\), since it is assumed that, generally, the mode cannot be defined by a single number of nodal diameter \(n\) (otherwise \(cos(n\theta)\) would be used, as in a simple disk). Nevertheless, no matter how complex the peripheral deformation is, there is a periodicity on the mode shape deformation if the entire periphery is considered (\(A_{i}(\theta~{}=~{}0)=A_{i}(\theta=2\pi)\)), which means that the deformation in Equation (3) can be rewritten using Fourier series in their compact form [45]:  
\[w_{p,i}(r=r_{max},\theta,t=0)=A_{i}(\theta)=\sum_{n=0}^{\infty}A_{n,i}\cdot \cos(n\theta+\varphi_{n}) \tag{4}\]  
As seen in Equation (4), the deformation is expressed as the sum of single components that are characterized by the number of nodal diameters n, with amplitude \(A_{n,i}\) and phase shift between components of \(\varphi_{n}\). Each single term has the same deformation pattern as a single disk diametrical mode shape (Equation (2)). An example of this kind of deformation with more than one diametrical component can be seen in Figure 4b.  
In real engineering applications, these types of structures are clamped to a shaft. The influence of clamping a disk-like structure, on the first diametrical modes with \(n\geq 2\), which are the first diametrical modes appearing disk-blade-disk structures, is almost negligible, especially for \(n>2\)[8]. Many references show that the first modes of a real disk-blade-disk configuration [6; 31; 32; 33; 340] are dominated by disk-modes and therefore the effect of the clamping is also to slightly increase the first diametrical mode shapes without modifying the main characteristics of the mode shape, i.e., the aforementioned characteristics of the diametrical mode shapes are still the same.  
Other effects, such as mistuning, which plays an important role in blade-disk configurations (blisk) for blade dominated modes, are less important for mode shapes that exhibit a shape similar to simple diametrical disk mode shapes [20]. The main effect of a possible mistuning is to separate the doublets in frequency [20], although in many real cases, this separation is less than 1% with respect to the natural frequency value [6]. However, in case the two existing doublets are clearly separated, they will keep the main characteristic of a diametrical mode shape and therefore can be separately considered and discomposed with the following method.