# Kohonen Mapping of the Space of Vibration Parameters of an Intact and Damaged Wheel Rim Structure
## 3 Research Results and Analysis
### SOM Results of Wheel Identification  
In order to define a map's space as a finite two-dimensional region, in which nodes (neurons) form a regular hexagonal mesh, a series of preliminary calculations was conducted for several neurons, generating SOMs of \(5\times 5\), \(7\times 7\), \(10\times 10\), \(15\times 15\) and \(50\times 50\). The obtained SOM sample hits are presented in Table 4, and they clearly show the possibilities of verifying wheel structure breaches from the dimension of the neural network. SOM describes mapping based on higher dimension input space, defined by signal sampling frequency from sensors and the number of defined wheel positions (\(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), \(270^{\circ}\)) in the space of a two-dimensional map. Following the training, a map may classify a vector in an input space by finding a node with a weight vector being the closest (metrics of the smallest distance) to the input space vector. The distances may be displayed in greyscale or colour (Table 4). \begin{table}
\begin{tabular}{c c  
If the average distance is large, then the surrounding masses are very different and a light colour is assigned to the weight location. If the average is small, a darker colour is assigned. The obtained maps (Table 4) show that the concentration of various type clusters is more dominant in three zones. The SOMs in Table 4 confirm where the concentration of points is larger (darker regions) and smaller (lighter regions). This representation may help in the visualization of clusters in multi-dimensional spaces or automatically recognize them with the use of simple analysis methods of the tested data. The resulting efficiency of the non-invasive identification method is conditioned by the SOM's ability to locate damage, as well as by the speed of data processing. It may be argued that the best solution is a compromise. An increase in the SOM dimension requires an increase in analysis duration, conditioned by the need to search for weights and connections between nodes (neurons). At the same time, an increase in dimension translates to the reliability and accuracy of the method. To reach a compromise in this study, a series of calculations of neural networks with various numbers of neurons was conducted, as indicated above. For the generation of the SOM, which may be interpreted as a frequency spectrum clustering and vibration acceleration time series, the MATLAB R2014b\({}^{\otimes}\) application was used. The results illustrate the possibility of minimizing the generation time of a SOM (\(5\times 5\)) based on the received frequency spectra down to five seconds. But in this case, it is difficult to read out the wheel's structural condition. SOM for a wheel in serviceable condition and a damaged wheel feature the same characteristics and this means that small signs of structural damages are not recognised. The increase in SOM dimension to \(10\times 10\) requires twice as much time for calculations. In such a case, it is possible to unequivocally identify a breach in the wheel's structure. The defined SOM generation time seems to be acceptable for the identification of objects in real-time, comparable in the function of time with the previously described (Section II) procedure of measurement at the station. Additionally, it must be noted that the SOM for the nominal image of a wheel at \(180^{\circ}\) differs from similar maps obtained at \(0^{\circ}\), \(90^{\circ}\), \(270^{\circ}\) positions. This stems from the presence of a technological opening in the wheel rim, i.e., the cut-out for the wheel valve. This is further proof of the possibility of using SOM to identify a wheel's structure. The self-organising map mirrors a large scatter of weight coefficients, for both damaged and undamaged structures when changes in the vibration amplitude in the function of time are used as input data. It must be noted that the obtained SOMs (Table 4), mirroring a time series, do not allow for the identification of damage.