# Electrophysical Method for Identifying the Causes of Excessive Hydro Generator Vibration
## Appendix C Description of Vibration Force Calculation  
The radial component of the linear force \(F_{0}\) is proportional to the induction B in the air gap, which can be represented as the waves of positive \(B_{1}\) and negative \(B_{2}\) sequence:  
\[F_{0}=\frac{B_{1}\cdot B_{2}\cdot l}{2\mu_{0}}\cdot\cos(2\cdot\omega\cdot t- \gamma_{2}) \tag{1}\]where \(l\) refers to the active length of the machine, \(\gamma_{2}\) is a certain phase angle, and \(t\) stands for time. The torsional vibration of stator steel structures is caused by the impact of an alternating electromagnetic moment \(M\) of frequency \(2\omega\). Under imbalanced load conditions, the relative value of the moment \(M\) is approximately equal to the negative sequence current \(i_{2}\) (pu). As a result, the tangential force can be expressed as:  
\[T_{0}=\frac{M_{n}\cdot i_{2}}{2\pi\cdot R_{1}^{2}} \tag{10}\]  
where \(M_{n}\) is the nominal moment of generator and \(R_{1}\) is the core radius. The relationship between tangential forces and displacement is described by the following system of two equations:  
\[\left\{ \begin{array}{l}-4\omega^{2}m_{1}W_{1}+k_{t1}\cdot(W_{1}-W_{2})=\frac {M}{2\pi\cdot R_{1}^{2}}=T_{0}\\ -4\omega^{2}m_{2}W_{2}+k_{t1}\cdot(W_{2}-W_{1})+k_{t2}W_{2}=0\end{array} \right. \tag{11}\]  
Here, \(W_{1}\) and \(W_{2}\) are the amplitudes of tangential displacements of the core and the housing, respectively; \(m_{1}\) and \(m_{2}\) are the linear masses of the core (along with the winding) and the housing, respectively; and \(k_{t1}\) and \(k_{t2}\) are the stiffnesses of the connection between the core and the housing and that between the housing and the foundation, respectively, in the tangential direction. \(T_{0}\) refers to the linear tangential force corresponding to the moment \(M\). Hence, the amplitude of the tangential core displacement is expressed as:  
\[W_{1}=\frac{T_{0}}{-4\omega^{2}m_{1}+\frac{k_{t1}\cdot(k_{t2}-4\omega^{2}m_{2} )}{k_{t1}+k_{t2}-4\omega^{2}m_{2}}} \tag{12}\]  
To estimate the vibration of the end-winding parts of stator winding in the first approximation, it is assumed that there is one degree of freedom for the basket-type end-winding parts, i.e.,  
\[4\omega^{2}m_{0}A-k_{0}(A-W_{1})=0 \tag{13}\]  
where \(m_{0}\) refers to the equivalent mass and \(k_{0}\) stands for the equivalent stiffness. The absolute (\(A\)) and relative (\(A-W_{1}\)) amplitudes of oscillation of the basket-type end-winding parts are expressed as:  
\[A=\frac{W_{1}}{1-4\omega^{2}\cdot\omega_{0}^{-2}} \tag{14}\]  
\[A-W_{1}=\frac{4\omega^{2}\cdot\omega_{0}^{-2}}{1-4\omega^{2}\cdot\omega_{0}^{ -2}}\cdot W_{1} \tag{15}\]  
where \(\omega_{0}=\sqrt{k_{0}/m_{0}}\) is the frequency of natural vibration of the basket-type end-winding parts when these are under kinematic excitation. Under imbalanced load conditions, the radial forces of the magnetic pull and the magnetic vibration excited by these forces are estimated considering that the linear force of the magnetic pull \(F(\varphi,t)\) is equal to:  
\[F(\varphi,t)=\frac{1}{2\mu_{0}}\cdot[B(\varphi,t)]^{2}\cdot l \tag{16}\]  
where \(B(\varphi,t)=B_{1}\cdot\cos(p\cdot\varphi-\omega\cdot t)+B_{2}\cdot\cos(p \cdot\varphi+\omega\cdot t-\gamma_{2})\) is the induction in the air gap presented as the rotating waves of positive (\(B_{1}\)) and negative (\(B_{2}\)) sequences, \(\varphi\) refers to the angular coordinate, \(p\) refers to the number of pole pairs, and \(\mu_{0}\) stands for the vacuum permeability. It is assumed that \(B_{1}=B_{n}\) and \(B_{2}=B_{n}x_{2}i_{2}\), where \(B_{n}\) is the nominal induction in the air gap and \(x_{2}\) refers to the negative sequence reactance expressed in pu.