## 4 Diagnostic Interpretation of F-F Images
### Healthy Motor  
Figure 1 shows the spectrum of vibrations of the outer casing of a fully operational engine that can be divided into two ranges: low frequency without structural resonances and high frequency dominated by structural resonances. The dominant frequency component is the low frequency range component caused by the radial forces (according to (3)) with the frequency \(z_{f}s\), where \(f_{s}\) is the rotor rotational frequency [14]. Amplitudes of structural resonance range frequency components are relatively low, which indicates a lack of mechanical shocks that may excite resonance vibrations. The F-F image (Figure 2) created with the characteristic frequency equal to the double rotor revolution frequency \(f_{\mathbf{P}}=2f_{s}\) reveals subtle details of the vibration process. Such a selection of the characteristic frequency enables the assessment of the vibration state caused by the fundamental and odd and even harmonics of the radial forces. Noticeable vertical patterns (indexed by rotational frequency) represent the vibration induced by the toothed structure of the rotating rotor. Random vibrations without any regular pattern that appear in the resonance area are imposed over noticeable vertical patterns. The reference pattern connected with the toothed structure of the rotating rotor is created based on the analytical model of radial forces in which Equation (4) (and in consequence, Equation (3)) is presented in the form of a harmonic series:  
\[b_{g}(\alpha,t)=\left[\Lambda_{0}+\sum_{m_{3}=1}^{M_{3}}\Lambda_{m_{3}}\cos(m_ {\lambda}z_{r}(\alpha-\omega_{s}t)-\varphi_{m_{\lambda}})\right]\left[\sum_{m_ {\theta}=1}^{M_{\theta}}\Theta_{m_{\theta}}\cos(m_{\theta}(\alpha-\omega_{1}t )-\varphi_{m_{\theta}})\right] \tag{8}\]  
where:  
\(\Lambda_{0}\), constant air gap conductivity;  
Figure 1: Frequency spectrum of vibration acceleration of the outer casing of a healthy motor. The dominant frequency component of vibrations is marked. \(\Lambda_{m\lambda}\), amplitude of air gap conductivity frequency component \(m_{\lambda}\);  
\(m_{\lambda}=2n-1\);  
\(n=1\), \(2\), \(3\),... ;  
\(\omega_{s}=2\pi f_{\mathrm{s}}\);  
\(\Lambda_{m\theta}\), amplitude of resultant flow frequency component \(m_{\theta}\);  
\(m_{\theta}=6n\pm 1\);  
\(\omega_{1}=2\pi f_{1}\);  
\(f_{1}\), electrical machine supply current frequency. Limiting the highest range of expected vibration frequencies to the few first multiples of \(z_{\mathrm{f}}f_{\mathrm{s}}\), and the discussed frequency components to ones with the highest amplitudes, the reference F-F image pattern is created by the reduced number of frequency components with frequencies:  
\[f_{p_{r}}=nf_{\mathrm{s}}\frac{z_{r}}{p}\pm 2 nf_{1} \tag{9}\]  
The above relation describes the perfectly symmetrical model from the point of view of mechanical and electrical parts of the three-phase motor. The corresponding reference pattern with a limited number of frequency components is marked with black dots in Figure 3. Extended simulations show that the resultant magnetomotive force and air gap dissymmetry lead to the appearance of additional frequency components, with the sidebands being multiples of \(2\langle f_{1}-f_{\mathrm{s}}\rangle\) frequency and odd multiples of \(f_{\mathrm{s}}\) frequency. An exemplary reference pattern with a limited number of sidebands is presented in Figure 3. Sidebands associated with dissymmetry are marked with gray dots. The frequency component position is determined by vertical coordinates (distant by multiples of double slip frequency \(2\langle f_{1}-f_{\mathrm{s}}\rangle\) around \(f_{\mathrm{s}}\) and \(2f_{\mathrm{s}}\) frequencies) and horizontal coordinates (distant by multiples of 1 around natural number \(\mathrm{R_{u}}(m_{\lambda}z_{r}/2)\) where \(\mathrm{R_{u}}\) is the rounded up number operation). It is important to underline that the reference pattern indicates the area of the potential presence of frequency components of real vibration processes.