## 3 Digital Monitoring System
### Structure and Parameters of the Digital Monitoring System  
This section presents issues related to determining the structure and parameters of the digital monitoring system used in the diagnostic device of the analyzed real machine component. The operating driving shaft is the real object, the analysis of which requires monitoring its rotational motion. Therefore, the physical model of a digital monitoring system is a discrete torsional vibrating system. Such a model consists of non-deformable discs with one degree of freedom and elastic components constituting connections between weightless shaft sections, transmitting only torques. The structure of the digital monitoring system is implied by the analyzed real system and the number of installed sensors monitoring it. Within the analyzed shaft, three critical areas were identified that should be monitored and diagnosed (Figure 3a), which correspond to the number of areas between the components cooperating with the shaft: bearings, gear wheel and clutch. On this basis, the arrangement of four sensors (S1, S2, S3 and S4) detecting angular displacement at the ends of these areas was assumed (Figure 3b). In the analyzed case, the test was carried out on the driving shaft on which four measuring points were installed. Therefore, a single-axis discrete physical model with four degrees of freedom is the structure of the digital monitoring system (Figure 4). The adopted physical model is a free system (the inertial components are not connected to the support via springs), indicating the possibility of rotational motion as one rigid body. This causes the frequency of first natural vibrations to be zero. For the system structure adopted in this way, its dynamic parameters should be determined to meet specific dynamic properties, in the form of the first four frequencies of torsional vibrations of the analyzed object (taking into account the first zero value). For the purposes of estimating the parameters of the discrete model, the forms and values of the natural vibrations frequencies of the analyzed component were determined (Figure 5). The Pre/Post module based on the finite element method of the PLM Siemens NX 2306 software was used to determine the form and value of the shaft's own vibration frequency. The following basic parameters  
Figure 4: Structure of a discrete physical model with four degrees of freedom. Figure 3: Separated monitoring and diagnostic areas with the arrangement of sensors. (**a**) Critical areas of the analyzed shaft; (**b**) The arrangement of sensors detecting angular displacement at the ends of critical areas. were adopted in the analysis in relation to the finite element mesh: type of finite element--CTETRA (10--ten nodes), size of finite element--2.65 mm. Limiting the input data only to natural frequencies makes it impossible to use the synthesis apparatus [94; 95] directly to determine the required dynamic parameters. At the same time, the use of the method of discretization of a continuous system, in relation to the adopted number of degrees of freedom, is subject to high calculation inaccuracy (due to the natural frequencies of the analyzed object). For this purpose, a combined method was suggested to determine the dynamic parameters of a discrete physical model. In the first step, the mass moments of inertia of non-deformable discs with one degree of freedom are determined. The rigid finite element method (RFEM) is used for this purpose, on the basis of which the system is divided into RFEMs (Figure 6), and on this basis, the mass moments of inertia relative to the main central axes of inertia of each RFEM are determined. Each mass moment of inertia of non-deformable discs with one degree of freedom, for such a division of the system (Figure 6), is as follows: \(\mathrm{I_{1}}=11.26\cdot 10^{-6}\) kgm\({}^{2}\), \(\mathrm{I_{2}}=53.93\cdot 10^{-6}\) kgm\({}^{2}\), \(\mathrm{I_{3}}=71.31\cdot 10^{-6}\) kgm\({}^{2}\), \(\mathrm{I_{4}}=21.34\cdot 10^{-6}\) kgm\({}^{2}\).