## 2 Research Methods
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**No. of**} & \multicolumn{3}{c}{**Vibration Velocity**} & \multicolumn{3}{c}{**Vibration Acceleration**} \\ \cline{2-7}
**Measuring** & **Vertical** & **Horizontal** & **Axial** & **Vertical** & **Horizontal** & **Axial** \\
**Points** & **Direction** & **Direction** & **Direction** & **Direction** & **Direction** & **Direction** \\ \hline \hline \multirow{8}{*}{14 (hit)} & \(195\) & & & 195 & 293 \\  & \(195\) & 195 & & 488 & 977 & 781 \\  & \(488\) & 391 & 293 & 879 & 1660 & 1270 \\  & \(781\) & 879 & 684 & 1172 & 2148 & 1465 \\  & \(1172\) & 1660 & 1270 & 2148 & 2441 & 1458 \\  & \(2148\) & 2148 & 1465 & 3223 & 2930 & 2148 \\  & \(3125\) & & & 3418 & 3223 & 2930 \\  & & & & & 3418 & 3223 \\ \hline \multirow{8}{*}{14 (response)} & \(195\) & 391 & 195 & 195 & 391 & 195 \\  & \(391\) & 684 & 488 & 391 & 684 & 488 \\ \cline{1-1}  & \(586\) & 1465 & 879 & 586 & 1465 & 879 \\ \cline{1-1}  & \(879\) & 2441 & 1270 & 879 & 2441 & 1270 \\ \cline{1-1}  & \(1172\) & & & 1172 & 2832 & 1270 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 2: Stand model in the form of six discrete masses with elastic-dissipative bonds: 1—Turbine; 2—Spacer; 3—Measurement system; 4—Multiplier; 5—Generator; 6—Frame with foundation. forms for the following parameters: generalized coordinates of displacements \(\mathbf{q}\), inertial forces F0(q) with inertial parameter A, and elastic forces with quasielastic coefficient C, without taking into account dissipative forces (no losses), which will reduce the amount of computation and obtain modal frequencies of structural elements. The condition of free oscillations of the stand's elements system is determined from the equality of the kinetic and potential energies. For free oscillations, the kinetic and potential energies of the system are written accordingly:  
\[ \begin{array}{l}T=\frac{1}{2}\mathbf{A}\dot{\mathbf{q}}^{2}\\ U=\frac{1}{2}\mathbf{C}\mathbf{q}^{2}\end{array}  \tag{1}\]  
where \(T\) and \(U\) are the kinetic and potential vibrational energies in the system; A is the matrix of inertial coefficients; \(\mathbf{C}\) is the matrix of quasielastic coefficients; \(\mathbf{q}=(\xi,\eta,\zeta,\alpha,\beta,\) and \(\gamma)\) is the vector characterizing the movement of the body during oscillations in the directions: \(\xi,\) and \(\eta,\) which are the indices of longitudinal and torsional vibrations along the axis of rotation of the turbine; \(\zeta,\)\(\alpha\) are the indices of longitudinal and torsional vibrations in the vertical plane to the axis of rotation of the turbine; \(\beta,\)\(\gamma,\) and \(\alpha\) are the indices of longitudinal and torsional vibrations in the horizontal plane to the axis of rotation of the turbine. The differential equation of free oscillations in matrix form will have the form:  
\[\mathbf{A}\dot{\mathbf{q}}+\mathbf{C}\mathbf{q}=0, \tag{2}\]  
The modal (free) frequency f of longitudinal vibrations of any element of the stand, in any direction, as an elastic system is written:  
\[\omega=\left(\frac{c}{a}\right)^{1/2}=\frac{1}{\left(af\right)^{1/2}}, \tag{3}\]  
where \(f=\mathrm{s}^{-1}\) is the unit compliance corresponding to the quasi-elastic coefficient \(c\); \(a\) is the size of the element in the direction of propagation of the elastic wave. For the stand considered as a system of discrete masses of its elements connected in a series with a common frame with a foundation (shown in Figure 2), the equations of eigenfrequencies will be determined by the following matrix:  
\[\left(