Generalized dynamic model of hydrodynamic vibration dampener subject to viscous damping

. Use as an elastic body of the working fluid improves the efficiency of the hydraulic damper, reliability, and durability, in this case, the issues of improving the reliability of existing locomotives hydraulic dampers by upgrading separate the components for major repairs with the extension of the useful life are relevant. To increase the efficiency of damping dynamic vibrations and reduce the load on the support units and running gear elements, a modernized design of a hydro friction damper with improved damping properties was developed, for which an application was filed for the Patent of the Republic of Uzbekistan for Invention No. IAP 2021 0002.


Introduction
In terms of self-financing and connection with the transition to a market economy, one of the most important tasks for our Republic's economic and political independence is to increase freight and passenger traffic, which entails an increase in the fleet of vehicles, including locomotives and high-speed trains. Their use is associated with an application in vehicle suspension systems is complex curved profile and different types of hydraulic dampers and vibration absorbers, which leads to an increase in smoothness of rolling stock and to improve the strength and elastic-dissipative properties of the suspension to reduce the dynamic effects on the path and the transported goods.
Analysis of research on the calculation of hydraulic vibration dampers of vehicles has shown that almost all spring suspension designs of modern electric locomotives, electric trains, metro cars, and trams use hydraulic dampers various types. Hydraulic vibration dampers served as the basis for the creation of new types of adjustable suspensions with automatically controlled parameters speed and high-speed electric rolling stock (for example, in France, Germany, Switzerland, Japan, Spain, China, Russia, and Uzbekistan) [1][2][3]. To increase the speed of movement and the capacity of railways, it is of great importance to improve the dynamic qualities of electric rolling stock, which depend on the correct choice and stability of the hydraulic vibration dampers and the parameters of the spring suspension. These indicators depend on: the interaction forces between wheel and rail, the stability of the wheels on the rail, and the smooth running locomotive.

Materials and Methods
To substantiate the created perspective design of the vibration dampener, a generalized dynamic model of the hydro friction vibration dampener was developed, taking into account the viscous damping [4].
The generalized model under study consists of three coaxially arranged elastic cylindrical shells, one of which is filled with an ideal compressible fluid moving at a constant velocity V and having a pulsating internal pressure (a variable pressure field in time and along the length of the P l (x, t)) [4,5].  [4] and the acting forces, (where it is indicated: 1 is working cylinder, 2 is outer casing (movable), 3 is piston, 4 are throttle channels, 5 is fixed casing part, 6 are bellows, 7 is rod) Let us turn to the equations of the shell, taking into account the propagation of waves in its material. We take here a variant of the linearized theory of Kirchhoff -shells, considering the deflections of the shells as small U 1 , U 2 , W 1 , W 2 -in contrast with the thickness of the shell [6,7]. Circular cylindrical shells have external radii R 1 , R 2 , wall thickness h 1 , h 2 , length -L 1 , L 2 , elastic pinching act in the inner shell 1 at the ends, pulsating pressure waves in the liquid R j (x, t).
Between the inner shell 2 and the outer shell 3, which is rigidly fixed, there is a friction force expressed by the formula: There is an external pulsed dynamic load on the outside [8]: The OX axis is compatible with the longitudinal axis of the shell L 1 . Displacement of the middle surface of the shell 1 in directions forming denote by U 1 , and the displacement of the middle surface of the shell 2 is rigidly fixed relative to the outer shell U 2 . Finally, the radial displacement, respectively -W 1 , W 2 [8].
Divide equations (5) and (6) by  1 h 1 and  2 h 2 respectively, and transform them by entering the notation: As a result of solving this equation, using the Fourier method and the Laplace transform, we get [10,11,12]: for equation (5) (11) The general form of the solution of equations (5) and (6) will be according to the Fourier method [11]: Where k = 1, 2, ...N = 5 is the number of harmonics under vibrations of 2 shells (respectively) under dynamic external loading [13].
The obtained system of equations (7) and (8) takes into account the forces of viscous friction in the details of the proposed damper with the coefficient of energy absorption dynamic loads arising from irregularities in the track [14]. It allows to assess the influence of the geometrical dimensions and parameters of the structure on the parameters of the bodywork of the car electric rolling stock and the details of the damping [15,16,17].
Numerical studies were performed in the Mathcad 15 programming environment and are presented in Figures 2, 3, and 4.

Conclusions
This mechanical model allows us to describe the dynamic processes occurring in the system and calculate the dampener parameters taking into account the pre-set dynamic characteristics, which are important when calculating and designing new vibration dampeners for rolling stock, as well as when upgrading existing ones [4,18]. The resulting mathematical model also allows us to evaluate the influence of structural, power, mass parameters and track irregularities on the process of vibrations of the car body in the