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In this paper, we derive analytical expressions describing spectral and mutual spectral densities of random processes by performing analytical Fourier transform of the corresponding correlation functions. On the basis of accepted analytical expressions for correlation functions of differentiable random processes, analytical expressions for spectral density and mutual spectral density components, which is a complex frequency function, are derived. The influence of the accepted analytical expressions on the correlation functions, spectral densities, and mutual spectral density components (real, imaginary, amplitude, and phase) on the parameters is presented. Plots obtained using obtained analytical expressions have showed full convergence with those obtained by direct integration of the analytical expression of the correlation function. The analytical expressions given in this paper may be used for investigating various random processes. In particular, spectral and reciprocal spectral densities can be used for approximation of experimentally received spectral densities, including equivalent geometrical irregularities of a rail track and random fluctuations of a rail vehicle. The parameters of analytical expression obtained by such an approximation can be used for generation of analogous experimental multidimensional random processes in the tasks of mathematical modelling.

Introduction. In the undercarriage of modern locomotives hydraulic vibration dampers are widely used to dampen vertical vibrations that occur in the axlebox stage of the spring suspension. Their resistance force is directly proportional to the deformation rate of the spring set. High levels of vibration frequencies and amplitudes in the axlebox stage lead to a significant increase in the strain rate and dissipative forces, which causes frequent failures of hydraulic vibration dampers. Therefore, issues related to improving the performance of axlebox hydraulic dampers are actual.Materials and methods. In order to evaluate the effectiveness of elastically protected hydraulic vibration damper in axlebox suspension, the authors studied the vibrations of a simplified model of a 2ES5K electric locomotive with a typical layout of the axlebox stage of spring suspension and a circuit with an elastically protected hydraulic vibration damper. The authors used a high-frequency random process as a kinematic perturbation of vibrations. The system of differential equations was solved in the MatLab — Simulink software package by numerical integration using the Runge — Kutta method of the fourth order. As a result of the solution, realisation of stationary and ergodic random processes were obtained. Results. The comparison of the obtained characteristics of random processes of vibrations of these two options showed that the use of an elastically protected axlebox vibration damper significantly reduces the frequency range and amplitudes of dissipative forces, which ensures a reduction in the number of axlebox damper failures and an increase in the number of runs before replacement is required.Discussion and conclusion. The shown efficiency of using an elastically protected axlebox vibration damper allows increasing the reliability of the damper. In order to determine the distance run between overhauls, it is necessary to carry out a controlled operation cycle or test the prototype axlebox assembly with the elastically protected vibration damper and measure the parameters of such assembly.