## 2 Methodology
Having performed the calculations, we get \(m_{re}=\frac{3m_{re}}{2\alpha_{*}},K_{re}=2E_{p}J_{py}\alpha_{*}^{3}+\frac{3} {2}\frac{u_{1}}{\alpha_{*}}=8E_{p}J_{py}\alpha_{*}^{3}=\frac{2u_{1}}{\alpha_{*}}\). As for the viscosity parameter, considering the dissipation function to be proportional to the potential energy of a system with proportionality coefficients, \(\chi_{1}\)--in case of deformation of the bending of the beam; and \(\chi_{2}\)--in case of compression deformation of the base--we express the given coefficient of viscous friction in the form of \(\beta_{re}=\chi_{1}2E_{p}J_{py}\alpha_{*}^{3}+\chi_{2}\frac{3}{2}\frac{u_{1}} {\alpha_{*}}\). If we consider that \(\chi_{1}=\chi_{2}=\chi=\frac{K}{\beta}\), where \(K\) and \(\beta\) are the stiffness and coefficient of viscous friction of the dampers of the crew spring suspension, then we will get the final formulas of this method of reducing the base parameters:  
Table 1.5Taking in the expression for \(\beta_{re}\), that \(\chi_{1}=0\) (friction in the beam is neglected), we get \(\chi_{2}=\frac{f_{ev}}{u_{1}}\). Then,  
Table 1.6  
Another approach is based on selecting parameters \(m_{re}\) and \(\beta_{re}\), such that the frequency characteristics of the system with one degree of freedom and the system with distributed parameters are close. It is recommended to choose the reduced parameters based on the condition of matching the real, \(\Phi_{01}\), and imaginary, \(\Phi_{02}\) parts of the frequency characteristics at the peak of the frequency characteristic module and at \(\omega=0\). Let \(\Phi_{01}\) and \(\Phi_{02}\) (21) be known for the continuous model at frequency \(\omega_{1}\), where the maximum of the frequency response occurs. For the system with one degree of freedom,  
Table 1.7 where Table 1.8 Therefore, Table 1.9 Figure 3:0  
If we equate \(\Phi_{1 re}=\Phi_{01}\) and \(\Phi_{2 re}=\Phi_{02}\), we obtain \(d=2\Phi_{01}\) and \(K_{re}d-\frac{d^{2}}{2}=Figure 3:1  
All the formulas above were derived for the case of a stationary force (\(c=0\)). The influence of the force's velocity on the system's frequency characteristics, as noted earlier, is negligible at the existing speeds of railway vehicles. Nevertheless, let us determine the reduced parameters by taking into account the force's velocity. For the continuous system, let the frequency, \(\omega_{1}\), and the maximum of the frequency response function, \(Q\), be known at some velocity, \(C\). Let \(\omega_{1}^{2}=\omega_{re}^{2}\Big{(}1-\frac{\gamma_{2}^{2}}{4}\Big{)}\) and \(Q=\gamma^{-1}\). From this, we can find that  
Figure 3:2  
Calculations were performed according to Formulas (28)-(32), and the reduced parameters were found for all five methods considered. The results are summarized in Table 1 (lines 7-11). From the analysis of the data in Table 1, it can be seen that, in all variants (lines 7-11), the reduced stiffness is the same; when approximating the inverse frequency characteristic in the range of \(0\leq\overline{\omega}\leq 1.5\), it differs by only \(2.5\%\), and in the range of \(0\leq\overline{\omega}\leq 3\), it is \(28\%\) smaller (for \(M=2\)). The values of the reduced viscous friction coefficient are close to each other in variants \(6,\ 9\div 11\); in the eighth variant, it is 25% lower than the previous ones; and in the third, it is 68% higher. In the seventh variant, its numerical value is 6.5 times greater than in the tenth: it is extremely overestimated. The reduced mass, calculated using Formula (31), is 32% greater than the one computed by the approximation method of the inverse frequency characteristic (26) in the range of 0-1.5 (line 6) and 68% greater in the range of 0-3 (line 3); the difference in the reduced mass in the tenth variant from the one obtained by the method based on the equality of the kinetic energies of the distributed and reduced systems (28) and (29) is 26%.