# Kohonen Mapping of the Space of Vibration Parameters of an Intact and Damaged Wheel Rim Structure
## Appendix A  
In the task under consideration, the shaft of the balancer of length L\({}_{\text{val}}\) was divided into \(n=10\) equal elementary elements of length 1. Also, the entire mass of the system was divided into \(n=10\) equal masses m\({}_{\text{n}}\) elementary shafts. The wheel rim, in turn, was divided into 4 equal segments (rim sections-so-called 'quarters') of equal masses. The decomposition of the individual elements into elementary elements is only exemplary and only serves to make the model easier to understand. Finally, in the final model, the number of elementary elements of each structure element is freely defined by the user. According to the adopted method of describing the rim reference points, the beginning of the first area of the wheel rim has a valve hole. The direction of the subsequent wheel rim segments is marked clockwise when looking from the front to the wheel rim. It is assumed that each wheel rim segment is connected to its rim segment by means of a single fillet weld. Corresponding to the assumed pattern of concentrated masses, the shaft is fixed in resistance, which happens to be a third and a tenth of the mass. We consider the fixation absolutely rough, and the resistances themselves have some susceptibility (magnitude, backward roughness). It is then possible to propose for consideration the next diagram, shown in Figure 11. The complements, compared to the previous diagram, are shown in red. Then, transformations counting masses and elastic resistance must be introduced into the system of differential Equation (2):  
\[\left\{ \begin{array}{l}m_{a}\frac{d^{2}x_{x}}{dt^{2}}+D\frac{dx_{x}}{dt}+C_{1 a}(x_{a}-x_{1})+C_{2a}(x_{a}-x_{b})-C_{2d}(x_{d}-x_{a})=F(t);\\ m_{b}\frac{d^{2}x_{0}}{dt^{2}}+D\frac{dx_{0}}{dt}+C_{1b}(x_{b}-x_{1})+C_{2b}(x_ {b}-x_{c})-C_{2a}(x_{a}-x_{b})=F(t);\\ m_{c}\frac{d^{2}x_{c}}{dt^{2}}+D\frac{dx_{c}}{dt}+C_{1c}(x_{c}-x_{1})+C_{2c}(x_ {c}-x_{d})-C_{2b}(x_{b}-x_{c})=F(t);\\ m_{d}\frac{d^{2}x_{d}}{dt^{2}}+D\frac{dx_{d}}{dt}+C_{1d}(x_{d}-x_{1})+C_{2d}(x _{d}-x_{c})-C_{2c}(x_{c}-x_{d})=F(t);\\ m_{1}\frac{d^{2}x_{1}}{dt^{2}}+D\frac{dx_{1}}{dt}-C_{1a}(x_{a}-x_{1})-C_{1b}(x _{b}-x_{1})-C_{1c}(x_{c}-x_{1})-C_{1d}(x_{d}-x_{1})+C(x_{1}-x_{2})=0;\\ m_{2}\frac{d^{2}x_{2}}{dt^{2}}+D\frac{dx_{2}}{dt}+C(x_{2}-x_{3})-C(x_{1}-x_{2}) =0;\\ (m_{3}+m_{on})\frac{d^{2}x_{3}}{dt^{2}}+D\frac{dx_{3}}{dt}+C(x_{3}-x_{4})+C_{N }x_{3}-C(x_{2}-x_{3})=0;\\ m_{4}\frac{d^{2}x_{3}}{dt^{2}}+D\frac{dx_{4}}{dt}+C(x_{4}-x_{5})-C(x_{3}-x_{4} )=0;\\ \cdots\\ (m_{10}+m_{on})\frac{d^{2}x_{10}}{dt^{2}}+D\frac{dx_{10}}{dt}+C_{10}x_{10}-C( x_{9}-x_{10})=0.\end{array} \right. \tag{11}\]  
where C\({}_{N}\), C\({}_{10}\)--elastic coefficients of resistivity, \(m_{on}\)--mass of resistivity. We will bring the equations described by equation to a normalised form taking into account Newtonian damping, then:\[\left\{ \begin{array}{l}\frac{dx_{a}}{dt}=\dot{x}_{a};\\ \frac{dx_{a}}{dt}=\frac{1}{m_{a}}\big{(}-(C_{1a}+C_{2d}+C_{2a})x_{a}-D\dot{x}_{a }+C_{2a}x_{b}+C_{2d}x_{d}+C_{1a}x_{1}\big{)}+\frac{F(t)}{m_{a}};\\ \frac{dx_{b}}{dt}=\dot{x}_{b};\\ \frac{dx_{b}}{dt}=\frac{1}{m_{b}}\big{(}-(C_{1b}+C_{2a}+C_{2b})x_{b}-D\dot{x}_{ b}+C_{2b}x_{c}+C_{2a}x_{a}+C_{1b}x_{1}\big{)}+\frac{F(t)}{m_{b}};\\ \frac{dx_{c}}{dt}=\dot{x}_{c};\\ \frac{dx_{c}}{dt}=\dot{x}_{d};\\ \frac{dx_{d}}{dt}=\dot{x}_{i};\\ \frac{dx_{i}}{dt}=\dot{x}_{1};\\ \frac{dx_{i}}{dt}=\frac{1}{m}\big{(}-(C_{1a}+C_{1b}+C_{1c}+C_{1d})x_{1}-D\dot{ x}_{1}+C_{2a}x_{a}+C_{2b}x_{b}+C_{2c}x_{c}+C_{2d}x_{d}\big{)};\\ \frac{dx_{2}}{dt}=\dot{x}_{2};\\ \frac{dx_{2}}{dt}=\frac{1}{m}\big{(}-2 Cx_{2}-D\dot{x}_{2}+Cx_{1}+Cx_{3}\big{)}; \\ \frac{dx_{10}}{dt}=\dot{x}_{10};\\ \frac{dx_{10}}{dt}=\frac{1}{m}\big{(}Cx_{9}-(C+C_{10})x_{10}-D\dot{x}_{10} \big{)}.\end{array} \right. \tag{10}\]  
The parameters of the shaft elements (size and mass) are determined as follows: \(C=\frac{EF}{l}\approx 9\frac{E}{L_{val}}\frac{md_{ad}^{2}}{4}\ m\approx\rho \frac{md_{ad}^{2}}{4}\frac{L_{val}}{10}\).