## 2 Research Methods
\begin{array}{cccc}1-\omega^{2}\sum\limits_{k=1}^{n}f_{1k}A_{k1}&- \omega^{2}\sum\limits_{k=1}^{n}f_{1k}A_{k2}&\dots&-\omega^{2}\sum\limits_{k=1 }^{n}f_{1k}A_{k6}\\ -\omega^{2}\sum\limits_{k=1}^{n}f_{2k}A_{k1}&1-\omega^{2}\sum\limits_{k=1}^{n }f_{2k}A_{k2}&\dots&-\omega^{2}\sum\limits_{k=1}^{n}f_{2k}A_{k6}\\ \dots&\dots&\dots&\dots\\ -\omega^{2}\sum\limits_{k=1}^{n}f_{6k}A_{k1}&-\omega^{2}\sum\limits_{k=1}^{n }f_{6k}A_{k2}&\dots&1-\omega^{2}\sum\limits_{k=1}^{n}f_{6k}A_{k6}\end{array} \right) \tag{4}\]  
where \(k\) is the number of the body in the stand scheme (according to Figure 2); \(n\) is the number of stand elements. The general solution of the equation of free (modal) oscillations with the eigenfrequency \(\omega_{a}\) and the natural form \(v_{a}\) can be written in the form:  
\[q(t)=\sum\limits_{a=1}^{n}v_{a}(C_{a}\cos\omega_{a}t+D_{a}\sin\omega_{a}t), \tag{5}\]  
where \(C_{a}\) and \(D_{a}\) are the oscillation amplitudes upon expansion in a Fourier series. The solution to the differential equation of free oscillations is the expression:  
\[q(t)=v\sin(\omega t+\chi), \tag{6}\]where \(\mathbf{v}\) is a vector (column matrix) that characterizes the relationship between the generalized coordinates, and these are also eigenmodes. \(\chi\) is the phase angle of the shift between them, while \(\omega\) and \(v\) must satisfy the matrix expression:  
\[\mathbf{(C-\omega^{2}A)v}=0. \tag{7}\]  
The solution of the differential equation of free vibrations has the form:  
\[\left( \begin{array}{ccccc}C_{11}-\omega^{2}A_{11}&C_{12}-\omega^{2}A_{12}&C _{13}-\omega^{2}A_{13}&C_{14}-\omega^{2}A_{14}&C_{15}-\omega^{2}A_{15}&C_{16}- \omega^{2}A_{16}\\ C_{21}-\omega^{2}A_{21}&C_{22}-\omega^{2}A_{22}&C_{23}-\omega^{2}A_{23}&C_{24}- \omega^{2}A_{24}&C_{25}-\omega^{2}A_{251}&C_{26}-\omega^{2}A_{26}\\ C_{31}-\omega^{2}A_{31}&C_{32}-\omega^{2}A_{32}&C_{33}-\omega^{2}A_{33}&C_{34}- \omega^{2}A_{34}&C_{35}-\omega^{2}A_{35}&C_{36}-\omega^{2}A_{36}\\ C_{41}-\omega^{2}A_{41}&C_{42}-\omega^{2}A_{42}&C_{43}-\omega^{2}A_{43}&C_{44}- \omega^{2}A_{44}&C_{45}-\omega^{2}A_{45}&C_{46}-\omega^{2}A_{46}\\ C_{51}-\omega^{2}A_{51}&C_{52}-\omega^{2}A_{52}&C_{53}-\omega^{2}A_{53}&C_{54}- \omega^{2}A_{54}&C_{55}-\omega^{2}A_{55}&C_{56}-\omega^{2}A_{56}\\ C_{61}-\omega^{2}A_{61}&C_{62}-\omega^{2}A_{62}&C_{63}-\omega^{2}A_{63}&C_{64}- \omega^{2}A_{64}&C_{65}-\omega^{2}A_{65}&C_{66}-\omega^{2}A_{66}\\ \end{array} \right) \tag{8}\]  
where the serial number of matrix elements corresponds to the number of stand elements, according to Figure 2.  
To simplify the solution of this complex system of equations, it is necessary to divide it into six subsystems (partial systems). For example, Formula (9) considers only one coordinate of the stand's wave processes formation and makes it possible to determine the eigenfrequencies of the stand's elements in a given direction. For this, it is necessary to find the partial frequencies of each element of the system. These frequencies will coincide with the eigenfrequencies of the stand's elements system at: A12 = A21 = 0; C12 = C21 = 0:  
\[\left\{ \begin{array}{l}\omega_{1}=\left(\frac{C_{11}}{A_{11}}\right)^{1/2} \\ \omega_{\Pi}=\left(\frac{C_{22}}{A_{22}}\right)^{1/2}\\ \omega_{\Pi\Pi}=\left(\frac{C_{33}}{A_{33}}\right)^{1/2}\\ \omega_{\Pi V}=\left(\frac{C_{44}}{A_{44}}\right)^{1/2}\\ \omega_{\rm V}=\left(\frac{C_{56}}{A_{55}}\right)^{1/2}\\ \omega_{\rm VI}=\left(\frac{C_{66}}{A_{66}}\right)^{1/2}\\ \end{array} \right. \tag{9}\]  
where \(\omega_{\rm I}\), \(\omega_{\Pi}\), \(\omega_{\rm III}\), \(\omega_{\rm IV}\), \(\omega_{\rm V}\), and \(\omega_{\rm VI}\) are the partial frequencies of the components of the stand's element base, respectively. Using this method, an algorithm was developed to find the eigenfrequencies of the stand's elements by analyzing the four subsystems and approximating the change in stiffness from the action of the twist torque of the measuring spring.