## 4 Diagnostic Interpretation of F-F Images
### Motor with Damaged Bearing Cage  
Figure 6 shows the vibrations spectrum of the outer casing of a motor with slight damage to the bearing cage on the drive shaft side. Local damage to the cage means that two of the nine rolling elements are not kept symmetrical to the others. The dominant frequency components are the low frequency range components. The one is a multiple of the frequency \(f_{c}\) component. Other significant frequency components are multiples of the rotational frequency \(f_{s}\), but at frequencies higher than in the case of an undamaged machine. This points to the simultaneous presence of vibration processes evoked by radial forces and bearings. Amplitudes of structural resonance range frequency components are now higher, which indicates the presence of mechanical shocks that excite resonance vibrations. The F-F image with characteristic frequency \(f_{\text{P}}=2f_{s}\) (Figure 7) reveals that the vibration process is much more complicated; several separated processes coexist. In addition to the weakly marked vertical pattern related to the toothed rotor frequencies (indexed by rotational frequency), there are much more intense oblique patterns in the area of resonant vibrations. Oblique patterns are caused by irregular rolling of the rolling elements of the bearing along the outer race (patterns tilted to the right) and the inner race (patterns tilted to the left). Figure 6: Frequency spectrum of vibration acceleration of the outer casing of the motor with a damaged bearing cage. The dominant frequency components of vibrations are marked. Figure 7: F-F image of the frequency spectrum of the vibration acceleration of the motor’s outer casing with a damaged bearing cage (\(f_{\text{P}}=2f_{s}\)). The positions of the new diagonal patterns are marked with arrows. The F-F images with the characteristic frequencies \(f_{\mathbf{P}}=f_{\varphi}\) (Figure 8) and frequency \(f_{\mathbf{P}}=f_{lp}\) (Figure 9) reveal patterns related to the rolling of the rolling elements along the outer and inner bearing raceways. Vertical patterns present in the sidebands distant by \(f_{c}\) and \(f_{s}\)-\(f_{c}\) frequencies and their higher harmonics from resonance frequencies describe the response of the motor core and outer casing to the mechanical shocks. Broadening and irregularity of vertical patterns point to the partly random process of mechanical shock generation. The strict pattern verticality confirms the commensurability of the mean value of frequencies \(f_{c}\) and \(f_{\varphi}\) (relation (6)) and the frequencies \(f_{c}\)-\(f_{c}\) and \(f_{\varphi}\) (relation (7)). The reference pattern that characterizes the potential position of the discussed image pattern consists of columns in which the position is denoted by multiple appropriate specific bearing frequencies. In Figures 8 and 9, the reference pattern is marked by vertical yellow lines. In the case of the F-F image presented in Figure 8, vertical patterns associated with the rolling of the rolling element on the external race are masked by diagonal patterns associated with vibration processes generated by the rolling of the rolling element on the internal race. This points to a different path of vibration disturbance propagation to the transducer. Determination of the characteristic bearing frequencies \(f_{\varphi}\) and \(f_{lp}\) base on the simple bearing model that utilizes the geometry of bearing components offers results that may differ, to some extent, from experimental values. It is interesting that patterns of comparatively equal intensity are present only in the resonance range of the signal spectrum. Figure 10 presents resonance range vertical pattern  
Figure 8: F-F image of the frequency spectrum of the vibration acceleration of the motor’s outer casing with a damaged bearing cage (\(f_{\mathbf{P}}=f_{\varphi}\)). Figure 9: F-F image of the frequency spectrum of the vibration acceleration of the motor’s outer casing with a damaged bearing cage (\(f_{\mathbf{P}}=f_{lp}\)).