# 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
The Dissipation Rate (DR) was defined as the ratio of an amplitude arithmetic mean of NFWR and NFS. It was assumed that for each such condition symptom there is only one value consistent with a serviceability condition (positive verification result), marked 1; otherwise, the symptom verification value was 0. Thus, a wheel rim is in serviceable condition when a set of all values of diagnostic parameters of the studied wheel rim fall within the range of allowable values, described with the condition vector of \(P_{j}\). Among the diagnostic symptoms used in the diagnostic procedure, one may differentiate two sets of symptoms. The first set is connected with the studied object (NFWR, NFH); the second set is connected with the test station (NFS). The first group of symptoms describes the technical condition of a wheel rim; further, the NFS symptom is connected with the ZRTOK station and is intended for evaluation of the assembly method of a wheel rim at the test station or the quality of the wheel rim's induction during the execution of measurements. In Figure 7, example amplitude and frequency cycles are presented for a tested wheel rim in serviceable (Figure 7a) and unserviceable (Figure 7b) conditions. In the spectrum presented in Figure 7, the station's shaft vibration frequencies are visible, where the tested wheel rim was mounted--marked as NF of Shaft ZRTOK and their subsequent harmonics. With the tested wheel rim, the following frequencies are connected: frequency of natural vibrations of the complete wheel rim, NF of Wheel Rim; and frequency of natural vibrations of the very wheel rim alone, NF of Hoop (Sound of Hoop) and its subsequent harmonics occurring at higher frequencies (not visible in the presented vibration spectrum). By comparing the obtained frequency cycles for individual angles of measurement of the tested wheel rim in serviceable condition (Figure 7a), one may conclude that the cycles are similar to each other (they coincide) in characteristic points (PEAK). In the case of the wheel rim in unserviceable condition (Figure 7b), considerable discord in characteristic points for individual measurement angles may be observed, as well as a decrease in the vibration amplitude values for the registered cycles. This is caused by vibrational energy loss resulting from a lack of structural continuity of the wheel rim. Assuming that the structure under consideration is uniform throughout its volume, its response to a given forcing should also be similar regardless of position (Figure 7a). Subjecting the structure with a defect to the excitation showed discrepancies in the width of peaks in the frequency domain for characteristic points determined on the basis of natural vibrations of individual elements of the considered system (Figure 7b). By analysing cycle forms of self-organising vibrations in the frequency range of NF of Hoop (Sound of Hoop) for both of the cases presented, one may conclude that in the case of the wheel rim in an unserviceable condition, the modulation of NF of Hoop (Sound of Hoop) in a wider frequency range is observed. Since the wheel rim is a symmetrical cylinder element and is mounted at the testing station through the centring opening, the vibrations forced through the energy of a blow make the wheel rim act like a bell--in which mechanical vibrations are translated into acoustic sounds. The sounds are observed in the mechanical vibration domain as a modulation in a narrow range of frequencies (NF of HOOP [Sound of Hoop]). In the case of lack of structural linearity of the wheel rim, the modulation is always affected in a wider range; in acoustic terms, this is identified as a rapidly decaying sound with a broad pitch range. During pre-testing of the method, as well as the diagnostic station, detailed identification of characteristics frequencies was conducted, not only with the experimental method employing the vibration signal but also with the acoustic signal, and the obtained test results were confirmed with analytic and simulation methods [10; 13].