## 2 Methodology
In the frequency range of \(0\leq\overline{\omega}\leq 0.7\), the agreement of all characteristics is good; at higher frequencies, there is a divergence, especially between the real frequency characteristics. This divergence is explained by the overestimated value of the parameter \(a_{2}=0.62\) Mg. Therefore, to determine the reduced parameters of the path based on the known distributed parameters, it is recommended to use Formula (31) for investigations in the low-frequency range, not exceeding 0.7\(\omega_{0}\), and for higher frequencies (up to 1.5\(\omega_{0}\)), the following formulas:  
Figure 3:3  
The parameters calculated using Formula (33) are presented in line 12 of Figure 3:4 The conducted study makes it possible to determine the reduced parameters of the track without considering the influence of the adjacent wheelest. The results of solving the problem of wagon vibrations are given in Figure 3:5 If the influence of the adjacent wheelest is to be taken into account, it is necessary to consider the frequency characteristics (19) and (20) after isolating the \(\Phi_{0}\) block. Figure 3:6  
where \(\widetilde{\Phi}(\overline{x},\overline{i\omega})=\frac{\Phi(\overline{x}, \overline{i\omega})}{\Phi_{0}(i\overline{\omega})}=\widetilde{\Phi}_{1}( \overline{x},\overline{i\omega})+i\widetilde{\Phi}_{2}(\overline{x}, \overline{i\omega})\). After carrying out the corresponding derivations, we obtain  
\[ \begin{array}{l}\widetilde{\Phi}_{1}(\overline{x},\overline{i\omega})=\frac {1}{2}\left[e^{-\overline{x}_{1}\overline{x}}(\cos\overline{a}_{2}\overline{ x}+\sin\overline{a}_{2}\overline{x})+e^{-\overline{x}_{2}\overline{x}}(\cos \overline{a}_{1}\overline{x}+\sin\overline{a}_{1}\overline{x})\right]\\ \\ \widetilde{\Phi}_{2}(\overline{x},\overline{i\omega})=\frac{1}{2}\left[e^{- \overline{x}_{1}\overline{x}}(\sin\overline{a}_{2}\overline{x}-\cos \overline{a}_{2}\overline{x})+e^{-\overline{x}_{2}\overline{x}}(\cos \overline{a}_{1}\overline{x}-\sin\overline{a}_{1}\overline{x})\right]\end{array}  \tag{35}\]  
Let us consider the frequency characteristic (35); and the real and imaginary frequency characteristics, \(\widetilde{\Phi}_{1}\) and \(\widetilde{\Phi}_{2}\), for the section \(\overline{x}=1.79\). This characteristic can be approximated in various ways over different segments of the frequency range. In the range of \(0\leq\overline{\omega}\leq 1.5\) for fixed values of \(\overline{x}\), the frequency characteristic, \(\widetilde{\Phi}\), is approximated by a polynomial:  
\[\widetilde{\Phi}=\sum\
olimits_{m=0}^{M_{1}}\overline{b}_{n}(i\overline{ \omega})^{n}=\sum\
olimits_{m=0}^{M_{1}}b_{n}(i\overline{\omega})^{n}\left( \overline{b}_{n}=b_{n}\omega_{0}^{n}\right) \tag{36}\]  
For example, if we take \(M_{1}=3\), then, as before, we construct the squares of the differences between the given (25) and the target (36) frequency characteristics separately for the real and imaginary parts. This leads to systems of algebraic equations analogous to (26), where the \(\overline{a}_{n}\) coefficients are replaced with \(\overline{b}_{n}\), and \(\Phi_{01}^{-1}(\overline{\omega}_{i}),\ \Phi_{02}^{-1}(\overline{\omega}_{i})\) in (27) are replaced with \(\widetilde{\Phi}_{1}(\overline{\omega}_{i}),\ \widetilde{\Phi}_{2}( \overline{\omega}_{i})\). The numerical values of the \(\overline{b}_{n}\) parameters were obtained as follows: \(\overline{b}_{0}=0.1848\), \(\overline{b}_{1}=-0.19452\), \(\overline{b}_{2}=0.11413\), and \(\overline{b}_{3}=0.04966\). The frequency range of \(0\leq\overline{\omega}\leq 1.5\) is of greatest practical interest, as the natural frequencies of wagon vibrations are significantly lower than the upper limit of this range. Therefore, in subsequent calculations, we will consider only this range and assume that the system's frequency characteristic is approximated by expressions (23), (34), and (36).