## 4 Diagnostic Interpretation of F-F Images
### Motor with a Defective Spring That Resets the Bearing Slackness  
The spring washer erasing the bearing slack (axial pressure) does not fulfill its task due to the too-deep seating of the bearings in the bearing shields. As a result, there is a play between the rolling elements and the inner and outer raceways. Figure 11 shows the spectrum of vibrations of the motor's outer casing without pre-compensated bearing clearances. The dominant frequency components are low frequency range components at multiples of the supply voltage frequency \(f_{1}\), rotational frequency \(f_{\text{s}}\) and bearing frequency \(f_{\text{ep}}\). In the frequency range 4-6 kHz and 7.5-12 kHz, there are significant resonant vibrations induced by totally random impacts of the rolling elements on the inner and outer rolling surface of the bearing.  
The F-F image with characteristic frequency \(f_{\text{P}}=2f_{\text{s}}\) (Figure 12) reveals the occurrence of the dominating chaotic vibration process in the resonance range. The random nature of the vibrations is evidenced by the lack of any pattern. In the low frequency range, apart of the vertical pattern is indexed by frequencies \(f_{\text{s}}\) and \(2f_{\text{s}}\) related to the toothed rotor frequencies; also, another new frequency component appears.  
Figure 10: Resonance range vertical pattern intensity of F-F image presented in Figure 9.  
Frequency components with frequencies \(f_{eps}\) being a result of the modulation of the rotational frequency \(f_{s}\) and its higher harmonics, and the bearing frequency \(f_{ep}\) and its higher harmonics:  
\[f_{eps(n_{ep},n_{s})}=n_{ep}f_{ep}\pm n_{s}f_{s} \tag{11}\]  
where:  
\(n_{ep}=1,2,3,\ldots,\) order of the harmonic component of vibrations caused by rolling of the rolling elements along the outer race of the bearing;  
\(n_{s}=1,2,3,\ldots,\) order of the harmonic component of the rotor's rotational frequency;  
form the elements of the image pattern that is present in the low frequency range. Identified frequency components and their mutual position on the F-F image reduced to the low frequency range are presented in Figure 13.  
Figure 11: Frequency spectrum of vibration acceleration of the outer casing of the motor with a bearing slackness. The dominant frequency components of vibrations are marked.  
Figure 12: F-F image of the frequency spectrum of the vibration acceleration of the motor’s outer casing with the bearing slackness (\(f_{\text{P}}=2f_{s}\)).  
The position of identified frequency components, limited to the ones with the highest amplitudes, are marked with yellow dots. Successive elements of the column are distant by the frequency \(2f_{s}\), while the frequency difference between two frequency components linked by diagonal lines is equal to \(f_{ep}\). The frequency distance between two columns containing the abovementioned frequency components takes the value of the smallest difference between \(f_{ep}\) frequency and multiples of \(2f_{s}\) frequency (in this case, \(4f_{s}\)-\(f_{ep}\)).  
The reference pattern indicates the possible position of the above-discussed frequency components consists of columns with positions denoted by sidebands with frequency \(4f_{s}\)-\(f_{ep}\) and its multiples around frequency \(f_{s}\) and its multiples. The reference pattern is marked by yellow lines. Frequency components occupy only some positions in the reference pattern. They appear only at positions that are coincident with the reference pattern and slant lines linking the frequency components distant by \(f_{ep}\) frequency. Their distribution is unique to individual electro-mechanical construction.