# Modal Verification and Strength Analysis of Bladed Rotors of Turbine in Rated Working Conditions
## 3 Results and Discussion
### Modal Verification of TRT Rotor
#### 3.1.1 Theoretical Modal Analysis of TRT Rotor  
The finite element model of the TRT rotor is established in commercial software. The finite element type and its parameters are shown in Figure 4 and Table 1. After completing the establishment of the finite element model, the material of the model of the TRT rotor is chosen as SUS630, and the corresponding material parameters are shown in Table 2. In order to obtain the mode shapes of the TRT rotor, the boundary condition of the TRT rotor is set as a free-free condition, and an impulse excitation is applied to the TRT rotor to obtain the natural frequencies and mode shapes of the structural system. Through the Theoretical modal analysis, all the mode shapes in the bandwidth of interest can be obtained, and the position of the measurement point of the blades can be selected based on mode shapes of a structure. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Finite Element Model Information**} \\ \hline Numbers of nodes of finite element model & 1,500,225 \\ \hline Numbers of elements of finite element model & 1,565,132 \\ \hline Element type & solid hexahedral \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Information of finite element model. The blade is a curved structure, so the coordinate direction of each degree of freedom of measurement in the experiment is different to define, and the global coordinate system (global coordinate system) cannot be used for data acquisition. To make the coordinates set in the experimental modal analysis consistent with those in finite element analysis, we redefine a local coordinate system for each degree of freedom of measurement on the blade in the finite element model, as shown in Figure 5, and the analysis result can then be represented in the local coordinate system. Note also that in Figure 5, the red lines are the X\({}_{\text{n}}\)-axis directions of the local coordinate systems, which are the definition of the Y\({}_{\text{n}}\)- and Zn- directions in Cartesian coordinates; the green lines are the Y\({}_{\text{n}}\)-axis directions of the local coordinate systems, which are defined along the tangent direction of the blade surfaces; the blue lines are the Zn-axis directions of the local coordinate systems, defined along the normal directions of the blade surfaces. According to the theoretical modal analysis, the mode shapes of the TRT rotor mainly belong to the bending modes of the shaft, and its typical mode shape is shown in Figure 6. The mode shapes of the blades belonging to the combination of the bending and torsion modes are shown in Figure 7. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Mechanical Property**} \\ \hline Material type & SU5630 \\ \hline Density (kg/m\({}^{3}\)) & 7750 \\ \hline Young’s modulus (GPa) & 197 \\ \hline Poisson ratio & 0.27 \\ \hline Yield stress (MPa) & 724 \\ \hline Ultimate stress (MPa) & 930 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters of TRT rotor. Figure 4: Finite element model of TRT rotor. Figure 5: Schematic diagram of measuring point on the blade in the local coordinate systems. Figure 6: A typical mode shape of TRT rotor. Figure 7: A typical mode shape of TRT blades. To obtain the modal information on the blades fully, each blade is arranged with nine degrees of freedoms. To be able to arrange accurately that the degree of freedom of measurement is in the position to be observed on the actual structure, we use six plastic ropes to position and mark the positioning lines, and install the marked plastic ropes on the TRT rotor blades, and then quickly locate the position of the degree of freedom of measurement. This skill not only effectively reduces the difference in measurement positions in between modal testing and the finite element model, but also can effectively carry out the arrangement procedure of measurement points before performing modal testing. The arrangement of the degree of freedom of measurement is shown in Figure 10.  
In this study, the bandwidth of interest is 1-2000 Hz.