## 2 Methodology
Thus, the frequency characteristic of the considered system is represented as the product (34) of the frequency characteristics of two blocks, one of which determines the output signal at the section \(\overline{x}=0\) (block \(\Phi_{0}\)); and the other--block \(\widetilde{\Phi}\)--depends on the spatial coordinate, \(\overline{x}\). If several forces act on the rail, then the deflection, \(\omega_{ij}\), under the force, \(P_{i}\), with the abscissa, \(\overline{x}_{j}\), is equal to  
\[\omega_{ij}=\Phi_{0}\widetilde{\Phi}\left(\overline{x}_{i}-\overline{x}_{j}, \overline{i\overline{\omega}}\right)P_{j} \tag{37}\]  
Equation (37), taking into account (23) and (36), can be written as  
\[\omega_{ij}=\frac{\sum_{n=0}^{M_{1}}b_{ijn}\left(\overline{i\overline{\omega} }\right)^{n}}{\sum_{n=0}^{M}a_{n}\left(\overline{i\overline{\omega}}\right)^ {n}}P_{j} \tag{38}\]  
Since the block \(\widetilde{\Phi}\) must be inertial, in the fractional-rational function (38), the degree of the polynomial in the denominator must be higher than that in the numerator, i.e., \(M>M_{1}\), which is assumed in the calculation. In expression (38), the \(b_{n}\) coefficients are given additional indices, \(ij\), indicating their dependence on the abscissas, \(\overline{x}_{i}\) and \(\overline{x}_{j}\). Obviously, \(b_{ijn}=b_{jin}\) since \(\widetilde{\Phi}\left(\overline{x}_{i}-\overline{x}_{j},\ \overline{i \overline{\omega}}\right)=\widetilde{\Phi}\left(\overline{x}_{j}-\overline{x} _{i},\ \overline{i\overline{\omega}}\right)\), and also \(b_{i\bar{i}n}=0\) for \(n\
eq 0\), and \(b_{i\bar{i}n}=1\) for \(n=0\). There is no fundamental difficulty in considering the case when the asymmetry of the function \(\widetilde{\Phi}\left(\overline{x}_{i}-\overline{x}_{j},\ \overline{i \overline{\omega}}\right)\) must be taken into account. In this case, the matrix of coefficients, \(a_{ijn}\), is not symmetric with respect to the indices, \(ij\), thus leading to an increase in the amount of computation. By summing Equation (38) over the index, \(j\), one can obtain an expression for the total deflection of the rail at the section with abscissa, \(\overline{x}_{i}\), under the action of all forces. \[\omega_{i}=\sum_{j=1}^{\mathrm{S}}\omega_{ij}=\sum_{j=1}^{\mathrm{S}}\frac{b_ {ijw}\left(i\overline{\omega}\right)^{n}}{a_{n}\left(i\overline{\omega} \right)^{n}}P_{j} \tag{39}\]  
Or  
\[\sum_{n=0}^{M}a_{n}\left(i\overline{\omega}\right)^{n}\omega_{i}=\sum_{j=1}^{ \mathrm{S}}\sum_{n=0}^{M_{1}}b_{ijn}\left(i\overline{\omega}\right)^{n}P_{j} \big{(}i=\overline{1.5}\big{)} \tag{40}\]  
Expression (40) allows us to write the following differential relationships between the inputs, \(P_{j}\), and the outputs, \(\omega_{i}\):  
\[a_{M}\frac{d^{M}\omega_{i}}{dt^{M}}+a_{M-1}\frac{d^{M-1}\omega_{i}}{dt^{M-1}}+ \cdots+a_{0}\omega_{i}=\sum_{j=0}^{\mathrm{S}}\left(b_{ijM_{1}}\frac{d^{M_{1}} P_{j}}{dt^{M_{1}}}+b_{ijM_{1}-1}\frac{d^{M_{1}-1}P_{j}}{dt^{M_{1}-1}}+\cdots+b_{ ij\bar{\omega}}P_{j}\right)\left(i=\overline{1.5}\right)l \tag{41}\]  
Thus, the system with distributed parameters is reduced to a system with lumped parameters (41). In conclusion of the conducted study, let us determine the equivalent (reduced) parameters of the rail during its vibrations in the horizontal plane. In this case, two maxima of the frequency response function (FRF) occur in the absence of damping. However, if damping in the foundation is taken into account, the second maximum disappears, and the FRF of the system, as in the case of vertical vibrations, becomes close to that of an oscillator.