# Kohonen Mapping of the Space of Vibration Parameters of an Intact and Damaged Wheel Rim Structure
## 2 Research Methodology
Therefore, to limit data for calculation and analysis, wheel rims with a mass of up to 15 kg were identified with the use of the front sensor only. For heavier wheel rims, signals from both acceleration sensors were collected for the analysis. Figure 1: General view of the ZRTOK station for identification of the technical condition of wheel rims at the production stage, with highlights of its executive and measurement components. 1—the tested wheel rim; 2—framework with shaft for mounting the tested wheel rim; 3—wheel rim pressure nut; 4a, 4b—sensor of vibration acceleration; 5—shaft rotation angle sensor (encoder); 6—vibration inductor; 7—computing unit. In order to justify the location of the vibration amplitude sensor during the development of the diagnostic methodology, modelling of the ZRTOK station processes was carried out. The general structural scheme of the diagnostic station referred to as ZRTOK is presented at Figure 2--the physical structure of the diagnostic station system, specifying its main structural elements (shaft, supports, wheel rim). Movement of single masses of the dynamic system is described with the following differential equation:  
\[m\frac{d^{2}x}{dt^{2}}+D\frac{dx}{dt}+Cx=F(t) \tag{1}\]  
where: \(m\)--mass, \(x\)--displacement, \(D\)--damping factor, \(C\)--stiffness coefficient, \(F(t)\)--exciting force. Therefore, for the scheme depicted in Figure 2, the structure can be described with differential equations:  
\[\left\{ \begin{array}{l}m_{a}\frac{d^{2}x_{a}}{dt^{2}}+D\frac{dx_{a}}{dt}+C_{1 a}(x_{a}-x_{1})+C_{2a}(x_{a}-x_{b})-C_{2d}(x_{d}-x_{a})=F(t);\\ m_{b}\frac{d^{2}x_{b}}{dt^{2}}+D\frac{dx_{b}}{dt}+C_{1b}(x_{b}-x_{1})+C_{2b}(x_ {b}-x_{c})-C_{2a}(x_{a}-x_{b})=F(t);\\ \ldots\\ m_{y}\frac{d^{2}x_{y}}{dt^{2}}+D\frac{dx_{c}}{dt}+C_{1y}(x_{y}-x_{1})+C_{2y} (x_{y}-x_{x})-C_{2x}(x_{x}-x_{y})=F(t);\\ m_{z}\frac{d^{2}x_{z}}{dt^{2}}+D\frac{dx_{d}}{dt}+C_{1z}(x_{z}-x_{1})+C_{2z}(x _{z}-x_{a})-C_{2y}(x_{y}-x_{z})=F(t);\\ m_{1}\frac{dx_{3}}{dt^{2}}+D\frac{dx_{1}}{dt}-C_{1a}(x_{a}-x_{1})-C_{1b}(x_{b}- x_{1})-C_{1c}(x_{c}-x_{1})-\\ C_{1d}(x_{d}-x_{1})+C(x_{1}-x_{2})=0;\\ m_{2}\frac{d^{2}x_{2}}{dt^{2}}+D\frac{dx_{2}}{dt}+C(x_{2}-x_{3})-C(x_{1}-x_{2} )=0;\\ \ldots\\ m_{i}\frac{d^{2}x_{i}}{dt^{2}}+D\frac{dx_{i}}{dt}+C_{i}(x_{i}-x_{i+1})-C_{N}x _{i}-C_{i-1}(x_{i-1}-x_{i})=0;\\ \ldots\\ m_{k}\frac{d^{2}x_{k}}{dt^{2}}+D\frac{dx_{k}}{dt}-C_{i-1}(x_{i-1}-x_{i})=0.\\ \end{array} \right. \tag{2}\]  
where: 1, 2, 3...--are the sections of the elementary mass of the ZTORK shaft division (division into 10 parts), a, b, c, d--are the indicators of the elementary parts of the wheel rim (0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), 270\({}^{\circ}\)). An elaboration of the proposed approach can be found in Appendix A. The shaft of the post, of the length of Lval, was decomposed to \(n=10\) equal elements of the length equal to l. In addition, the entire mass of the shaft was divided into \(n=10\) equal masses mn. The wheel rim was divided into 4 equal segments (rim sections-the so-called \"quarters\") of equal masses (ma, mb, mc, md). The description of the wheel rim model distinguishes the wheel centre and the hoop. Both of those elements are joined to each other with a defined joint (e.g., welded joint). It was assumed that the beginning of the first elementary area of the wheel rim is indicated by the valve hole. The direction of subsequent wheel rim segments is clockwise. It was also assumed that each wheel centre segment is joined with its hoop segment with one fillet weld. Figure 2: Scheme of the physical structure of the diagnostic station. Decomposition into elementary elements is used for illustration purposes to provide clearer explanation of the model. In the final model, the number of elements making up each of the element of the structure can be freely defined by the user before starting the simulation process. Comparing calculated frequency response and measured frequency response spectra obtained, it can be claimed that the spectra are similar to each other (over-lap).