# Mechanical Fault Prognosis through Spectral Analysis of Vibration Signals
## 4 Measurement and Analysis  
In industrial production, water pumps are a common and important industrial equipment. Water pumps are widely used in many fields, such as nuclear power, agriculture, and chemical fields. In this paper, we selected the pump of a water company as an application example. First, a standard vibration signal source generates vibration signals with adjustable frequency and amplitude within a certain range. The sensors are connected to the vibration signal source and the monitoring software runs on the Raspberry Pi. The GUI (Graphical User Interface) is shown in Figure 3. The vibration test bench output is set so that the vibration acceleration varies by  
\[a(t)=(1+0.5\cos(30\pi t))\cos(100\pi t+0.5\sin(20\pi t))\, \tag{24}\]  
where 1 volt corresponds to 1 gravitational acceleration, i.e., \(9.8\,\mathrm{m/s^{2}}\). The real-time wave sampled from the DAU and its corresponding spectrum are shown in Figure 3.  
The left panel in Figure 3 shows the analysis results of all sensor signals, including the vibration acceleration/velocity/displacement signals of each sensor. The time-domain statistics include the maximum, minimum and effective values as a whole, in the low and the high frequency bands, as well as at the frequency at the largest vibration amplitude. As a result, the statistics of each sensor have 28 parameters. The statistics of all sensors are displayed by the QTreeView control in the left panel of the interface. At the same time, the real-time waveform, power spectrum, envelope spectrum, cepstrum, and historical peak of the vibration signal are displayed on the right panel. The blue curve shows the signal waveform corresponding to Equation (24). Moreover, from Equation (24), the envelope of the signal is  
\[b(t)=1+0.5\cos(30\pi t)\, \tag{25}\]  
and its waveform is shown in red color. The amplitude spectrum of the signal (blue color) shows that the main energy of the signal is concentrated around the carrier frequency (herein 50 Hz). In contrast, the envelope spectrum in red color indicates that the energy is concentrated at lower frequencies (herein 15 Hz). The envelope spectrum contains the modulation information of the original signal, and thus, is an important tool for bearing fault diagnosis. Figure 3: Real-time monitoring GUI for Raspberry Pi-based vibration test platform. Second, we use a computer to simulate the DAU for complex vibration signals that cannot be generated by the vibration test bench. As shown in Figure 1, the Raspberry Pi is connected to the DAU through a 5G router. We follow the protocol of the DAU to generate complex vibration signals. For our proposed algorithm, the results are exactly the same whether the vibration signal originates from a real DAU or a simulated DAU. To verify cepstrum performance, a computer was used instead of DAU to simulate the low-frequency gear shaft rotational vibration signal \(a_{1}(t)\) and the high-frequency gear mesh vibration signal \(a_{2}(t)\):  
\[ \begin{cases}a_{1}(t)=1.0\cos(10\pi t)+0.7\cos(20\pi t)+0.5\cos(40\pi t)\\ a_{2}(t)=2.0\cos(100\pi t)+1.5\cos(200\pi t)+2.5\cos(400\pi t)\end{cases} . \tag{26}\]  
These two signals are amplitude modulated, that is, \(a(t)=a_{1}(t)a_{2}(t)\), and quantized (simulation of 16-bit analog-to-digital converter) before being fed to the Raspberry Pi via the network. As can be seen from the spectrum subgraph on the right of Figure 4, in the vicinity of 50, 100, and 200 Hz, the \(a_{2}(t)\) signal are offset by 5, 10, and 20 Hz by the \(a_{1}(t)\) signal, respectively. Eighteen spectral lines appear in the spectrogram, which makes it difficult to identify the presence of the low-frequency gear shaft rotational vibration signals. The left subgraph is the cepstrum computed by (23). The three peaks, at 0.06 s, 0.12 s, and 0.20 s, correspond to frequencies of 17 Hz, 8 Hz, and 5 Hz respectively, which approach the low-frequency components of the modulated signal. These low-frequency components appear as side bands in the right subgraph.