## 3 Digital Monitoring System
### Mathematical Model of a Digital Monitoring System  
Data from monitoring of real machine components during their operation are verified and then processed by a digital monitoring system. As a result of passing the data through the digital system, response characteristics of the system are obtained, on the basis of which it is possible to observe and locate damage to the monitored object, evidenced by the deviation of the response characteristics of the damaged system from the characteristics of the reference system [93]. Observation of the deviation of the characteristics is possible by using the suggested mathematical model. The suggested method is a non-destructive vibration method, but unlike the classic methods used in diagnostics, it allows the location of damage with high precision at its low values, while the comparison of frequency/amplitude characteristics does not provide satisfactory results. In the next stage of this work, the mathematical model of the digital monitoring system was formalized on the example of a system simulating the reading of data from two sensors (accelerometers). The obtained responses of the monitored system for a system with two degrees of freedom, assuming a failure in the system, are the sum of the general and specific solutions of the differential equations of the object whose natural frequencies are equal to \(\omega_{0}=0\), \(\omega_{12}^{2}=\omega_{1}^{2}+\omega_{2}^{2}=\frac{(\mathrm{c-Ac})}{\mathrm{ m}_{2}}+\frac{(\mathrm{c-Ac})}{\mathrm{m}_{2}}\). The mathematical model for the digital monitoring system takes the form of a system of differential motion equations presented in Equation (1). \[\left\{ \begin{aligned} &\mathrm{m}_{1}\ddot{\mathrm{x}}_{11}+\mathrm{c}( \mathrm{x}_{1}-\mathrm{x}_{2})=\mathrm{F}_{0}\mathrm{sin}\mathrm{u}\\ &\mathrm{m}_{2}\ddot{\mathrm{x}}_{22}+\mathrm{c}(\mathrm{x}_{2}- \mathrm{x}_{1})=0\end{aligned} \right. \tag{1}\]  
where \(\mathrm{x}_{1},\,\mathrm{x}_{2}\) represent the data read out from the sensors monitoring the object, and \(\mathrm{x}_{11},\,\mathrm{x}_{22}\) represent the response of the digital monitoring system. Figure 2: The concept of a digital monitoring system coupled with the monitored object. As a result of the application of the inverse Laplace transform, the response of the digital monitoring system for an object without damage as a function of the time variable takes the following form:  
\[\mathrm{x}_{10}(\mathrm{t})=\frac{\beta\omega_{20}^{2}}{\omega\omega_{120}^{2}} \mathrm{t}-\frac{\beta\omega\left(\omega_{20}^{2}-\omega_{120}^{2}\right)}{ \omega_{120}^{3}(\omega_{120}^{2}-\omega^{2})}\mathrm{sin}\omega_{12}\mathrm{ t}+\frac{\beta\omega\left(\omega^{2}-\omega_{20}^{2}\right)}{\omega^{2}\left( \omega_{120}^{2}-\omega^{2}\right)}\mathrm{sin}\omega\mathrm{t} \tag{11}\]  
\[\mathrm{x}_{20}(\mathrm{t})=\frac{\beta\omega_{20}^{2}}{\omega\omega_{120}^{2}} \mathrm{t}-\frac{\beta\omega\omega_{20}^{2}}{\omega_{120}^{3}(\omega_{120}^{2} -\omega^{2})}\mathrm{sin}\omega_{12}\mathrm{t}+\frac{\beta\omega_{20}^{2}}{ \omega^{2}\left(\omega_{120}^{2}-\omega^{2}\right)}\mathrm{sin}\omega\mathrm{t} \tag{12}\]  
In the case of the determined quantities (\(\mathrm{x}_{10},\ \mathrm{x}_{11},\ \mathrm{x}_{20},\ \mathrm{x}_{22}\)), their components of the obtained sum were compared, in particular the slope coefficient of the linear t function.