## 2 Computational Modelling of Tuned Bladed Wheels  
Essential information about the vibration properties of a particular tuned rotor can be provided by a suitable computational model based on the finite element method. Based upon a FE dynamic model of an impeller, the equation of motion in a rotating coordinate system is:  
\[M\ddot{\mathbf{y}}+(\mathbf{G}+\mathbf{K})\dot{\mathbf{y}}+\mathbf{C}\mathbf{y}\ =\mathbf{f} \tag{1}\]  
where \(\mathbf{M}\)--mass matrix, \(\mathbf{G}\)--gyroscopic matrix, \(\mathbf{K}\)--structural damping matrix and \(\mathbf{C}\)--stiffness matrix. Vector \(\mathbf{y}\) represents displacements and rotations at nodes and \(\mathbf{f}\) the external forces. The stiffness matrix contains centrifugal stiffness and softening effects resulting from the static analysis under centrifugal, thermal and gas loads. In a non-rotating state, the bladed rotor system eigenvalues and eigenmodes can be found by solving the generalized eigenvalue problem. Due to the cyclic periodicity of the ideal rotor geometry, the system matrices contain a block cyclic structure. Therefore, the resulting eigenvalues contain pairs of double modes. In the case of bladed discs, complicated mode shapes often arise, the character of which is often difficult to specify only on the basis of a geometric shape or visualization by an animation program. The distribution of the specific strain energy in the mechanical structure of the impeller can be used for an objective assessment of the type of mode. Mode shapes having the specific strain energy contained in the blades more than 90% can be considered blade dominated. Mode shapes in which the value of the specific strain energy in the blades lies between 90% and 65% can be considered mixed. For specific strain energy values in the blades less than 65%, these are disc dominated mode shapes. Strain energy is defined as the energy stored in a body due to deformation. Using the [PERSON]'s stress-strain law  
\[\sigma=\mathbf{D}\mathbf{\epsilon}, \tag{2}\]  
the strain energy of the whole impeller structure is given by the volume integral  
\[U=\frac{1}{2}\int_{V}\sigma^{T}\varepsilon dV=\frac{1}{2}\int_{V}\bar{\varepsilon }^{T}\mathbf{D}\varepsilon dV. \tag{3}\]  
Within the discretization of the FE model is the integral reduced to the strain energy sum of all elements of the impeller. The energy distribution of the considered mode shape in the blade or in the disc or shaft body of the impeller structure is available to determine the dominant character of the eigenmode. The rotationally cyclic symmetric impeller structure has similar mode shapes as a simple circular disc, but the number of nodal diameters \(ND\) is not unlimited. These so-called cyclic symmetric modes are characterized by the number of nodal diameters, their maximum number is:  
\[ND\ =\ 0,\ 1\ldots\ \frac{N}{2},\ if\ N\ is\ even \tag{4}\]  
\[ND=0,\ 1\ldots\ \frac{N-1}{2},\ if\ N\ is\ odd \tag{5}\]  
The eigenfrequencies of the mentioned mode shapes exist in pairs except of \(ND=0\) and \(ND=N/2\). These double modes have identical eigenfrequencies and only differ in the angular alignment of the nodal lines. When creating finite element models of an impeller, the use of rotational cyclic symmetry is effective. In an ideal impeller, all blade sectors are geometrically identical and they have the same mechanical properties, so that their stiffness matrices are also identical. Due to the fact that the impeller is a rotationally cyclic symmetric elastic body, a computational numerical modal analysis can be performed for only one segment, and the modal analysis results for the whole structure can subsequently be obtained by using exact mathematical relations [28]. Figure 2 shows the finite elementmodel of one segment and the FE-mesh of the whole turbine impeller. The appropriate type and size of the element used for the computational FE-model were carefully determined by a series of comparative calculations.