# Modal Verification and Strength Analysis of Bladed Rotors of Turbine in Rated Working Conditions
## 2 Materials and Methods
### Analysis of Aerodynamic Force of TRT Rotor  
The TRT system uses blast furnace top pressure to push the rotor blade, and the shaft drives the generator to generate electricity. When the rotor is in operation, the blast furnace gas flows across the flow channel between stationary blades, and the blast furnace gas is transferred to the moving blade. In this working condition, the moving blade moves among different flow channels due to the rotation of the shaft, so the external force on the moving blade is a periodic function. In actuation conditions, the aerodynamic force on the moving blade is a periodic function, and the aerodynamic load change can be expressed as follows. \[F_{A}=F_{A0}+F_{A1}\sin(\omega_{b}t+\theta_{1})+F_{A2}\sin(\omega_{b}t+\theta _{2})+\ldots+F_{Am}\sin(\omega_{b}t+\theta_{m}) \tag{7}\]  
where \(F_{A}\) is the external force on the moving blade; \(F_{A0}\)-\(F_{Am}\) are the magnitude of aerodynamic force flowing through the flow channel between different stationary blades when the rotor is in operation; \(\omega_{b}\) is the frequency of the moving blade passing through the flow channel when the rotor is in operation; and \(\theta_{1}\)-\(\theta_{m}\) are the phase difference of aerodynamic force between flow channels. According to Equation (7), the aerodynamic external force of various moving blades in different time conditions can be calculated as long as the magnitude of the aerodynamic force, the frequency of the rotor blade passing through the flow channel, and the phase difference of aerodynamic force between flow channels can be obtained. The phase difference of aerodynamic force between flow channels has resulted from unequal numbers of moving blades and flow channels of the TRT rotor. Therefore, the  
Figure 3: Schematic diagram of a part of rotor blades. phase difference is related to the geometry of the moving blade and flow channel. The law of phase difference repetition can be obtained by calculating the greatest common factor between the number of moving blades and the number of stationary blades, and the phase difference is calculated in the minimum unit. The minimum phase difference \(\theta_{min}\) is calculated as the following equation:  
\[\theta_{min}=\frac{360}{N_{s}}\times\frac{\mathrm{g}cf(N_{d},N_{s})}{N_{d}} \tag{8}\]  
where \(N_{d}\) is the number of moving blades; \(N_{s}\) is the number of stationary blades; \(\mathrm{g}\mathrm{c}\mathrm{f}\mathrm{(}N_{d},N_{s}\mathrm{)}\) is the greatest common factor of moving blades and the number of stationary blades. After the angle of minimum phase difference and the law of phase difference repetition are obtained, the phase difference of each blade can be calculated by the following equation. \[\theta_{m}=\theta_{min}\times N_{o} \tag{9}\]  
where \(\theta_{m}\) is the aerodynamic phase difference of the m-th moving blade; \(\theta_{min}\) is the minimum phase difference; \(N_{o}\) is the repetition number of the phase difference. To obtain the ordinal number between the blade and flow channel, the following equation is used for computation. \[B_{m}=mod\bigg{(}m,\ \frac{\left[\mathrm{g}cf(N_{d},\ N_{s})\right]}{N_{d}} \bigg{)} \tag{10}\]  
\[B_{m}=0,\ N_{o}=\frac{N_{d}}{\mathrm{g}cf(N_{d},N_{s})}-1 \tag{11}\]  
Equation (11) uses the geometrical relationship between the moving blade and flow channel to calculate the ordinal number between the moving blade and flow channel.