# Kohonen Mapping of the Space of Vibration Parameters of an Intact and Damaged Wheel Rim Structure
## 3 Research Results and Analysis  
### Spectral Analysis of Vibrations  
Identification of the technical condition of a disc wheel rim (Figure 6) was executed at the ZRTOK wheel rim test station presented in Figure 1.  
Tests were conducted for new \(9\times 15.3\) disc wheel rims, in which the connection of a wheel rim with a wheel disc is executed with a filled weld over the entire circumference of the wheel rim. During verification tests of the presented method, for each kind and type of wheel rim, rated parameters for diagnostic symptoms were described based on vibration and acoustic response. Cycle analysis of vibration amplitude change (Figure 7) shows that it is possible to isolate characteristic areas in the frequency domain, i.e., single-dimension clusters. In each cluster, it is possible to define a diagnostic parameter of \(P_{j}\), which, in turn, may be presented in the form of a mean value of maximum amplitudes per given area in the cluster of frequency change:  
\[P_{j}=\frac{1}{n}{\sum_{i=1}^{n}{\max_{[f_{1},f_{2}]}}}A_{i}(f), \tag{4}\]  
where: \(P_{j}\) is condition parameter discriminant (condition symptom); \(A\) is the identified vibration acceleration amplitude in the reference frequency domain; \(f\), \(i\) are wheel rim position during measurement; \(f_{1},f_{2}\) are vibration frequency range (initial and final) for the  
Figure 5: Methodology flowchart. Figure 6: General view of a wheel rim, size \(9\times 15.3\), utilized in duty vehicles and agricultural machines. purpose of defining condition symptom cluster; \(j\), \(n\) are number of measurement positions of a wheel rim. Example features and diagnostic parameters of the serviceable condition for 9 \(\times\) 15.3 wheel rim are presented in Table 3. The following abbreviations have been used for clarity in the designations of the individual component vibrations: NFS (natural frequency of the shaft), NFWR (natural frequency of the wheel rim), NFH (natural frequency of the hoop). For evaluation of the technical condition of a tested wheel rim, four diagnostic symptoms were highlighted and described with 20 features of the diagnostic signal. A parameter of the condition symptom in the form of NFS (Natural Frequency of Shaft, ZRTOK) was obtained through the definition of maximum vibration amplitude value from the reference range of the identified frequencies (in this case, the range was 90 \(\pm\)1 Hz, i.e., 89-91 Hz) at each position of wheel rim measurements (0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), 270\({}^{\circ}\)). For these  
Figure 7: Spectra of vibration amplitudes for a new 9 \(\times\) 15.3 wheel rim in serviceable condition (**a**) and unserviceable condition (**b**), obtained at the ZRTOK diagnostic station, with selected characteristic frequencies of the station highlighted and the tested wheel rim for four angles of measurements: 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), 270\({}^{\circ}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Designation** & **Condition Parameter, m/s2** & \
\begin{tabular}{c} **Frequency Range, Hz** \\ **[\(f_{1}\), \(f_{2}\), \(f_{3}\)]** \\ \end{tabular} & **Source of Signal Features** &
\begin{tabular}{c} **Serviceable** \\ **Condition Vector** \\ \end{tabular} \\ \hline NFS & P\({}_{1}\)\(\pm\) 10\% & 90 \(\pm\) 1 & ZRTOK & 1 \\ \hline NFWR & P\({}_{2}\)\(\pm\) 10\% & 180 \(\pm\) 1 & Wheel Rim & 1 \\ \hline NFH & P\({}_{3}\)\(\pm\) 10\% & 810 \(\pm\) 10 & Wheel Rim & 1 \\ \hline DR & P\({}_{4}\)\(\frac{P_{1}}{P_{1}}\)\(>\) 1 & - & ZRTOK/Wheel Rim & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Set-up of features and diagnostic parameters of the rated serviceable condition of a 9 \(\times\) 15.3 wheel rim with a condition vector DISCRIMINANT (designation according to Figure 6). four features of the diagnostic signal, the arithmetic mean was calculated, based on which the scattering range of vibration amplitude against the verified measurements was defined. The NFWR and NFH symptoms were defined the same way.