# Kohonen Mapping of the Space of Vibration Parameters of an Intact and Damaged Wheel Rim Structure
## Appendix A
Where: \(E\)--Yung's modulus, \(\rho\)--density of the shaft material, \(L_{val}\)--length of the shaft, \(d_{val}\)--diameter of the shaft. The following parameters were adopted for the elementary areas of the wheel rim: \(m_{a}=m_{b}=m_{c}=m_{d}=\frac{m_{disk}}{4}\) description of elementary masses  
\(C_{1a}=C_{1b}=C_{1c}=C_{1d}=1.1\)-\(10^{7}\) N/m--is the elasticity coefficient between the individual segments of the wheel rim, while \(C_{2a}=C_{2b}=C_{2c}=C_{2d}=4.9\)-\(10^{7}\) N/m is the elasticity coefficient between the rim segments and the segments of the wheel rim and the shaft on which the wheel rim is fixed (see Figure 11). The defined elasticity parameters of the system are taken as an estimate to enable the mathematical solution of the equation to be built. In reality, these values must be verified on a real object (tester prototype) or determined experimentally. The choice of these coefficient values definitely influences the qualitative and quantitative parameters of the model with respect to the real object, the so-called model verification. The assumption was made that the elasticity coefficients of the wheel disc (\(C_{1a},C_{1b},C_{1c},C_{1d}\)) are a function of the displacement of the elementary elements of the shaft (\(x_{a},x_{b},x_{c},x_{d}\)). Such a description of the presented model was led to the non-linear forms of Equation (3). The indicated dependence of elasticity on displacement can be described by Equation (10):  
The proposed dynamic system model in the form of differential Equation (11) can be considered in the form of a mathematical notation  
\[\left\{ \begin{array}{l}\dot{X}(t)=A(t)X(t)+B(t)U(t);\\ Y(t)=C(t)X(t)+D(t)U(t).\end{array} \right. \tag{11}\]  
Where: \(X(t)\)--dynamic system state vector; \(Y(t)\)--output vector; \(U(t)\)--input vector; \(A(t)\)--dynamic system eigenmatrix; \(B(t)\)--action matrix; \(C(t)\)--output matrix; \(D(t)\)--simple, relationship matrix. The structural diagram of the model in the state domain is shown in Figure 12.  
For the case under consideration, the system's eigenmatrix is a variable matrix of dimension A-28 \(\times\) 28 and can be described by the normalized equation:  
A(1,i)=[0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(2,i)=[\(-\)(C1a+C2a+C2d)/ma\(,-\)D/ma\(,\)C2a/ma\(,\)0,0,C2d/ma\(,\)0,C1a/ma\(,\)0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(3,i)=[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(4,i)=[C2a/mb,\(-\)(C1b+C2b+C2a)/mb\(,-\)D/mb\(,\)C2b/mb\(,\)0,0,0,C1b/mb\(,\)0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(5,i)=[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(6,i)=[0,0,C2b/mc,0\(-\)(C1c+C2c+C2b)/mc\(,-\)D/mc\(,\)C2c/mc\(,\)0,C1c/mc\(,\)0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(7,i)=[0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(8,i)=[C2d/md,0,0,0,C2c/md,0\(-\)(C1d+C2d+C2c)/md,\(-\)D/md\(,\)C1d/md\(,\)0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(9,i)=[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];  
A(10,i)=[C1a/m,0C1b/m,0C1c/m,0C1d/m,0\(-\)(C1a+C1b+C1c+C1d)/m,D/m,C/m,  
So, the model described by Equation (A3) simplifies, its output vector equals the state vector:  
\[\left\{ \begin{array}{l}\dot{X}(t)=A(t)\cdot X(t)+B\cdot U(t);\\ y(t)=x(t).\end{array} \right.\] (A4)  
Thus, the result of Equation (A4) is a matrix multiplication, thus:  
\[ \begin{array}{l}\dot{X}(t)=A(t)\cdot X(t)+B\cdot U(t);\\ X(t+\Delta t)=X(t)+\dot{X}(t)\cdot\Delta t.\end{array} \] (A5)  
The result of such a decision is a vector-matrix \(X\), the even rows containing the dynamic elements of the system (velocity) \(\frac{dx_{k}}{dt}(t)\): and the odd rows their displacements \(x_{k}(t)\). Due to the fact that the obtained equation of state of the system under consideration is described by a non-linear equation, solving it using the methods of classical mathematics is impossible.