## 3 Digital Monitoring System
### Structure and Parameters of the Digital Monitoring System
In the next step, after determining the inertial parameters of the model, the elastic parameters of the discrete physical model should be determined. It should be borne in mind that each physical parameter has a significant impact on the distribution of the resonant and anti-resonant zones of the vibrating system. In order to meet the requirements in terms of dynamic properties, in the form of the first four frequencies of torsional vibrations of the analyzed object (taking into account the first zero value), the comparative synthesis method was used to determine the searched parameters [94; 95; 96]. For this purpose, based on the adopted structure of the discrete model of a physical system, dynamic stiffness  
Figure 5: Graphical representation of the first three forms of torsional vibrations of the analyzed shaft along with their values. Figure 6: Scheme of dividing the drive shaft into RFE. matrices are built in the following form (taking into account the numerical parameters of the determined inertial parameters):  
\[\mathbf{Z_{s}(s)}= \begin{bmatrix}\text{I}_{1}\text{s}^{2}+\text{c}_{1}&-\text{c}_ {1}&0&0\\ -\text{c}_{1}&\text{I}_{2}\text{s}^{2}+\text{c}_{1}+\text{c}_{2}&-\text{c}_{2}&0 \\ 0&-\text{c}_{2}&\text{I}_{3}\text{s}^{2}+\text{c}_{2}+\text{c}_{3}&-\text{c}_{3} \\ 0&0&-\text{c}_{3}&\text{I}_{4}\text{s}^{2}+\text{c}_{3}\end{bmatrix}  \tag{19}\]  
where, based on the stiffness matrix Z(s), the following polynomial is determined:  
\[\text{det}(\mathbf{Z_{s}(s)})=\text{A}_{8}\text{s}^{8}+\text{A}_{6}\text{s}^{ 6}+\text{A}_{4}\text{s}^{4}+\text{A}_{2}\text{s}^{2} \tag{20}\]  
where \(\text{A}_{8}=9.23889\cdot 10^{-19}\), \(\text{A}_{6}=9.91746\cdot 10^{-14}\cdot\text{c}_{1}+3.00881\cdot 10^{-14}\cdot \text{c}_{2}+5.62608\cdot 10^{-14}\cdot\text{c}_{3}\), \(\text{A}_{4}=2.91214\cdot 10^{-9}\cdot\text{c}_{1}\text{c}_{2}+6.03929\cdot 10^{-9} \cdot\text{c}_{1}\text{c}_{3}+1.65052\cdot 10^{-9}\cdot\text{c}_{2}\text{c}_{3}\), \(\text{A}_{2}=1.57831\cdot 10^{-4}\text{c}_{1}\text{c}_{2}\text{c}_{3}\). To calculate the values of elastic parameters, the obtained polynomial should be divided by the coefficient \(\text{A}_{8}\) and then compared to the polynomial characterizing the adopted dynamic properties in the form of four natural frequencies of the system. This equation takes the following form:  
\[\frac{\text{det}(\mathbf{Z_{s}(s)})}{\text{A}_{8}}=\text{s}^{2}\overset{\text{ n}}{\underset{\text{i=2}}{\prod}}\Big{(}\text{s}^{2}+\text{\omega}_{\text{b} 1}^{2}\Big{)} \tag{21}\]  
where \(\text{\omega}_{\text{b}1}=64,025.7\text{ rad}/\text{s}\), \(\text{\omega}_{\text{b}2}=102,604\text{ rad}/\text{s}\), \(\text{\omega}_{\text{b}3}=178,442\text{ rad}/\text{s}\). After comparing the coefficients with the same powers of the polynomial of Equation (21), a system of equations is built, on the basis of which the elastic parameters are determined. The dynamic parameters of the searched elastic elements, in the case of the structure shown in Figure 6, are listed in Table 1.  
All obtained sets of elastic parameter solutions meet the adopted requirements regarding the dynamic properties, in the form of the first four frequencies of torsional vibrations of the analyzed object. From this set, a solution was selected whose parameters were closest to the stiffnesses determined by the RFE method. This solution was the second set of parameters listed in the table. As a result of the calculations, a set of all dynamic parameters of a discrete physical model was obtained, constituting the structure of a digital monitoring system used to detect damage to the tested object.