## 2 Diametrical Mode Shapes of Rotating Disk-Blades-Disk Like Structures from the Rotating Frame
### Simple Disk Case  
Considering that all the points are moving in phase or in counterphase to each other (real mode shape), the deformation of a disk can be written as [8; 44]:  
\[w(r,\theta,t)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}W_{nm}(r)\cos(n\theta)\, \cos(\omega_{nm}t) \tag{1}\]  
where \(w\) is the transverse deformation (perpendicular to the plane of the disk), \(r\) the radial position, \(\theta\) the angular position and \(t\) time. \(W_{nm}\) is a function that involves the use of Bessel functions [8; 9]. \(n\) is the number of nodal diameters and m the number of nodal circles. Figure 3 shows the generic transverse motion (in \(z\) direction) of a generic disk-like structure in cylindrical coordinates.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Stationary Frame** & **Rotating Frame** \\ \hline Rotating structure has to be axisymmetric around the rotating axis & Rotating structure can be not axisymmetric around the rotating axis \\ \hline Rotating structure can be connected to stationary structures & Only rotating structure can be considered in the analysis \\ \hline More than one rotating structure & Only a single rotating structure \\ \hline Campbell diagrams for computing rotor & Campbell diagrams are not applicable \\ \cline{2-3} critical speeds are possible & Cyclic-symmetric analysis can be used \\ \hline \hline \end{tabular}
\end{table}
Table 1: Restrictions of the numerical analysis of typical commercial codes (ANSYS® [37], MSC NASTRAN® [38]).  
Figure 3: Coordinates for a generic disk-like structure.  
For a thin disk clamped at its center (clamped by the rotating shaft) with high outer to inner radius ratio, the first mode shapes are usually diametrical modes (modes with no nodal circles, i.e., \(m=0\)) [8; 9; 13]. These modes are also usually the most prone to be excited in centrifugal impellers [30; 31; 40], and for this reason will be the modes considered in this paper. Therefore and for simplicity, \(\omega_{n}\) will be used instead of \(\omega_{nm}\). For these kind of modes, the deformation at the periphery is larger than that in the more inner radii. Consequently, the peripheral deformation will be used to characterize the diametrical mode shapes.  
From Equation (1), considering that \(t=0\), all the points pass through their maxima to minimum (\(\cos(\omega_{nm}t)=1\) for \(t=0\)) and considering that there is a local maximum for \(\theta=0\), the deformation on the periphery at this time can be expressed as:  
\[w_{p,n}(r=r_{max},\theta,t=0)=A_{n}\cos(n\theta) \tag{2}\]  
where \(w_{p,n}\) is the deformation on the periphery (p) for the n diametrical mode and \(A_{n}\) the amplitude of vibration of the corresponding mode, which depends on the angular position \(\theta\). From Equation (2), it can be seen that each mode has a single \(\cos(n\theta)\) component, which defines the diametrical mode n.  
For other \(t\), where the points are not passing through its maximum/minimum, \(w_{p,n}\) will be lower than for \(t=0\). Nevertheless, since all the points are moving in phase or in counterphase to each other (same term \(\cos(\omega_{nm}t)\) in Equation (1)), the deformation shape will be proportional to the deformation at \(t=0\) and, therefore, the deformation shape on the periphery at any time is representative of the mode shape (except for \(t=\frac{\pi}{2\omega_{nm}},\frac{3\pi}{2\omega_{nm}}\) where all the points pass through 0). An example of the peripheral deformation of this kind of structure can be seen in Figure 4a.