## 3 Selected Research Results  
### Analysis of Signals in the Time Domain  
The analysis involved the dynamics of the mechanical components of the injection subsystem, which include the nozzle and the injector spring (Figure 3). The equation of the dynamics of movement of the nozzle needle during fuel injection can be described as follows:  
\[m_{w}\frac{d^{2}h}{d\tau^{2}}+A_{S}\frac{dh}{d\tau}+k_{sw}(h+h_{zw0})-p_{w} \big{(}A_{ip}-A_{i}\big{)}-p_{s}A_{i}=0 \tag{4}\]  
where \(A_{S}\) is the flow area through the firing pin seat, \(A_{i}\) is the cross-sectional area of the firing pin, \(A_{ip}\) is the cross-sectional area of the firing pin guide part, \(h\) is the instantaneous stroke (displacement) of the firing pin of the injector, \(h_{zw}\) is the initial tension of the injector spring, \(k_{sw}\) is the spring constant of the injector, \(m_{w}\) is the mass of the moving parts of the injector, \(p_{w}\) is the pressure in the chamber of the injector, \(p_{s}\) is the pressure in the sack of the injector nozzle, and \(\tau\) is time. Figure 3: Cross-section of the injector nozzle including the needle (**a**) and diagram of the injector space (**b**): 1—injector nozzle body, 2—injector needle, 3—injector pin, 4—spring. The first term in Equation (4), \(a=dh^{2}/d\tau^{2}\), is the acceleration of the injector needle. This equation describes the loads acting on the injector nozzle, which initiate and propagate the cracks. Simultaneously, the first term of Equation (4) relates to the mass of the needle, the vibration acceleration, and the amplitude of the vibration acceleration at a certain time \(a=f(\tau)\). Analyzing the diagnostic signals in the time domain allows one to obtain the time waveform of a single call and averaged signals [9]. Here, it is possible to determine the process waveform's beginning, end, and duration. Signals were recorded as time-domain waveforms using the time selection with an opening interval. However, due to numerous worldwide time-domain studies, in this work, more detailed analyses of diagnostic signals were not conducted in this area. By contrast, decimation and windowed analysis of time waveforms are considered innovative. Figure 4a shows the time waveforms for cylinder No. 1 in the suitability state and before damage in No. 5. Figure 4b shows the time waveforms of in-cylinder pressure No. 3 of auxiliary engine No. 2 and in cylinder No. 5 of auxiliary engine No. 3. Due to the abnormal pressure waveforms of the three cylinders of test engine No. 2, caused by the malfunctioning of the indicator valves, they were not measured in the other cylinders. There are also marine engines without indicator valves, for which measuring cylinder pressure waveforms is difficult [31]. Decimation transforms a discrete signal, which involves retaining every _Mth_ sample and discarding the rest. Denoting the input and output signals as \(a_{x}\) and \(a_{y}\), respectively, this decimation can be written as follows:  
\[a_{y}(n)=a_{x}(Mn) \tag{5}\]  
Figure 4: Time waveforms of vibration acceleration signal (**a**), in-cylinder pressure (**b**), decimation (**c**), and the result of windowed analysis (**d**) of the vibration acceleration signal \(a\) for cylinder No. 1 in an upstate and cylinder No. 5 with a working time of 247 h before damage detection. where \(M\) is the decimation factor. The signal after \(M\)-fold decimation contains \(M\) times fewer samples than before, which means it was sampled at \(M\) times lower frequency (Figure 4c). On the other hand, Figure 4d shows another sought-after innovative method of signal analysis: windowed time. The windowed analysis of a signal changes the shape of the waveform in the time domain; when this method was used, equal signal variation was possible for the suitability state and the developing crack. The visualization of signals for the state of suitability before failure and the values of correlation and sensitivity coefficients showed significant differences between the analyses in these domains.