## 5 Conclusions  
This work concerned the formulation and formalization of a non-invasive method for continuous monitoring of machine components in real time. The formalized mathematical model presented in this work determines proper coupling of the digital model (discrete vibrating model) with the sensors located on the monitored device. Based on the formulated method, damage to the machine component is detected at an early stage of its development, which is often not possible by using classic non-destructive diagnostics methods. In the case of such a monitoring and diagnostic method, a series of verification tests were carried out using an machine component designed in the PLM Siemens NX program in the form of a gear shaft with bearings, which fully recreated the operating conditions of the real machine component. Four acceleration sensors were placed on the machine component in places between which the minor damage was introduced in relation to the volume of the tested system. Due to the number of sensors used (in the context of the number of monitored areas), a discrete vibrating system with four degrees of freedom was built. The discrete system obtained in this way, called in the work a digital monitoring system, was connected to the tested machine component in accordance with a formalized mathematical model. The application of the DS concept allowed for the acquisition of data obtained from a virtual object and, based on them, the definition of rules allowing for the diagnosis of the machine condition (shaft damage). Moreover, the application of the DS concept for research did not determine the necessity of conducting the inference process in real time. In the case of applying the method presented in this paper to a real object, the DT concept can be applied, where based on real measurements and the developed database of diagnostic rules, it will be possible to automatically infer the condition of the object. Based on the tests, a regularity can be noticed in the case of damage between the sensors. The observed regularity is the deviation of the time characteristics, processed by the digital shadow in relation to the pattern generated in the initial stage of operation of the monitored system. As a result of damage in the tested machine component, deviation in the response of time processes to the inertial elements of the digital monitoring system (CSM) occurs. The CSM inertial elements on which significant differences in deviations in the state characteristics were observed correspond to the sensors located on the machine component between which the damage was simulated. Three areas were monitored, where in the first two, a deviation of the external state characteristics relative to the reference was observed, which indicated the occurrence of damage in this area. Characteristics of the areas between the forces applied to the machine component (sensors S1, S2, S3) and the type of vibrations considered (angular displacements) are the form of the describeddeviation. Based on the test results, it was also observed that for continuous systems, a deviation of the characteristics in all inertial elements of the CSM occurs. In such cases, the deviation values of all characteristics should be compared and the location of the damage should be determined in accordance with the suggested criteria described in Section 4. To generalize the interpretation of the results, the authors planned the tests taking into account the displacements of the tested machine component in various directions during its movement, which will also enable determining the sensitivity of the formulated method due to the type of vibrations of the considered machine component. In addition, the authors also intend to test the impact of the number of sensors on the form of the state characteristics results.