# Electrophysical Method for Identifying the Causes of Excessive Hydro Generator Vibration
## 4 Evaluation of Torsional and Tangential Vibrations  
A negative impact on the HPP power equipment is due to distortion of both currents and voltages in the 10 kV busbars of the hydropower plant. This is possible when a hydro generator operates under imbalanced load conditions and forces that are uncharacteristic of normal operation emerge. For instance, an alternating electromagnetic moment of double synchronous frequency emerges [31], resulting in torsional vibration of the stator steel structures and the winding and the radial force of the magnetic pull of pole frequency emerging [32; 33]. These forces excite radial core vibration. The vibration forces were evaluated in accordance with the methodology [32] given in Appendix C.  
The estimation of the torsional \(T_{0}\) and radial \(F_{0}\) forces and the double amplitude of vibration (maximum displacement) in the indicated directions \(2W_{1}\) and \(2U_{1}\) for the option of imbalance--the negative sequence current \(i_{2}\)--are presented in Table 1. According to the data of the imbalance factor recording, a level of 2% (0.02 pu), which is actually not exceeded in the power supply system of the smelter, was taken. The linear tangential force \(T_{0}\) is 668 N/m when the imbalance current is 0.02 pu. The amplitude of the radial force \(F_{0}\) is 1040 N/m. As the stiffness values are assumed as a certain interval, the calculation results are presented as a range of variation. As can be seen from the calculations above, the torsional forces and the amplitude of the tangential vibration as a first approximation are inversely proportional to the frequency of imbalanced current components and harmonics. The tangential forces do not produce significant vibration displacement when the 50 Hz negative sequence current is present, which contributes to the forces of double frequency (100 Hz) being produced if the natural frequency of the basket-type end-winding parts is sufficiently well tuned out from 100 Hz. As can be seen, there is an increase in the amplitude of the natural vibration of the end-winding parts at the natural frequency below 100 Hz. In the example considered, the maximum vibration reaches 80 \(\upmu\)m at the natural frequency of 86 Hz. The excitation conditions of the core tangential torsional vibration are far from being resonant: the vibration values are low, within 2 \(\upmu\)m. Thus, the existing load imbalance is safe for the hydro generators. The superimposition of surge current impacts caused by the load of the smelter on the generators can be estimated by introducing the surge current impacts with an equivalent frequency \(\omega_{E}\). Then, the total current \(I(t)\) is expressed as: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameters** & **Symbol** & \begin{tabular}{c} **Natural** \\ **Frequency, Hz** \\ \end{tabular} & **Value, \(\upmu\)m** \\ \hline \multirow{3}{*}{Torsional vibration} & \begin{tabular}{c} Tangential core displacement \\ amplitude \\ \end{tabular} & \(2W_{1}\) & & 1.5–2.2 \\ \cline{2-5}  & \begin{tabular}{c} Absolute amplitude of basket-type \\ end-winding part vibration \\ \end{tabular} & \(A_{1}\) & \begin{tabular}{c} 86 \\ 113 \\ \end{tabular} & \
\begin{tabular}{c} 37.3–80.2 \\ 21.0–45.8 \\ \end{tabular} \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Radial vibration \\ \end{tabular} } & \multirow{2}{*}{Core vibration} & \(2U_{1}\) & & 0.7–0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vibration parameters of the elements of the 38.5 MW SV-655/110-32 type hydro generator when the imbalance factor is 0.02. \[I(t)=I_{p}\cdot\cos(\omega t)\cdot[1+I_{A}\cdot\exp(-t/\tau)\cdot\cos(\omega_{E}t+ \varphi)]=I_{p}\cdot\cos(\omega t)+0.5I_{p}I_{A}\exp(-t/\tau) \tag{1}\]  
where \(I_{p}\) is the amplitude of the operating current and \(I_{A}\) is the amplitude of the surge current. The surge impact is decomposed into positive and negative sequence waves with frequencies that differ by \(2\omega_{E}\) and a significant increase in the amplitude.