DEPENDENCE OF AIR SPRING PARAMETERS ON THROTTLE RESISTANCE

Dep. «Car and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 373 15 19, e-mail reidemeister@mail.ru, ORCID 0000-0001-7490-7180 Dep. «Car and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 373 15 19, e-mail nastaci@yahoo.com, ORCID 0000-0001-8811-7243


Introduction
Air springs are the most progressive elastic elements of running gear, which are used in passenger car bogies.Their main advantage is the ability to maintain the position of the body at a certain level relative to the rail heads, regardless of the load, due to automatic adjustment of air pressure within the spring [4,6,7].Furthermore, they have good noise and vibration reduction properties, providing comfort of passengers [9,12,16].
Ability to take up high horizontal and diagonal displacements as well as the torsional strength make the air suspension systems an attractive solution for use on all bogies [1,4,7].
There are several types of air spring systems [4,8,10,14] (air damper and air spring with one or two extra reservoirs) and it is convenient to de-scribe their dynamic properties with the help of the mechanical equivalent model.
To date, the most common for use on passenger cars is a pneumatic system design, consisting of an air spring and an auxiliary reservoir.The reservoir is needed to reduce the vertical stiffness [4,7,13].As an auxiliary reservoir the internal cavity of the bogie frame truss is used; besides the auxiliary reservoirs may be located in the car body or in the space under the body.
As a rule, the air springs are limited by size according to the condition of their location on the bogie frame, so they are separated from auxiliary reservoirs and communicate with the latter by connecting pipes.The pipes are equipped with connecting elements, which have a calibrated vent holes.When overflowing from the cylinder into the auxiliary reservoir the air has to overcome an air resistance of the connecting element, causing the spring to get the damping characteristics along with the elastic ones.

Purpose
The purpose of this work is to research and analyse the influence of throttle element pneumatic resistance on elastic and damping parameters of air spring.To obtain the dependence of air spring parameters on throttle element pneumatic resistance value.
The changes in the connecting element flow capacity largely affect the overall pneumatic system.For the analysis of the newly constructed or improved existing air suspension system the issue of the connecting element capacity, along with the capacity of the auxiliary reservoir and the air spring casing, is an extremely important parameter.

Methodology
To analyse the dependence of elastic and damping properties of air suspension system (hereinafter -the spring) on the connecting element parameters we consider the spring (Fig. 1) comprising the following components: rubber-cord casing cylinder (1), reservoir (4), piping (3) and connecting element (2).

Fig. 1. Diagram of air spring with reservoir
The spring is considered as a dynamic system with three phase coordinates (cylinder pressure, auxiliary reservoir pressure, cylinder air mass).The process of air condition change inside the cylinder (reservoir) is adiabatic; the mass air flow through the connecting element depends on the difference of cylinder and reservoir pressure.[15].
The spring has the following parameters: cylinder volume V 1 and reservoir volume V 2, support surface area S (we assume that it is independent of cambering of spring), air mass m.The total pressure in cylinder and reservoir is denoted by р 1 and р 2, respectively.They differ from the excessive one by the atmospheric pressure value р а .
The set of equations describing the system operation is as follows: where z -cambering of spring; 1 m -air cylinder mass; 2 m -reservoir air mass; γ -polytropic ex- ponent; f -function defining the mass air flow through the connecting element.
We assume that the spring operation proceeds at the ambient temperature equal to t=25 °C and the pressure equal to p a = 1 atm.
Equations ( 1) are interdependent, which allows approximate representation of the air spring as the elastic element with stiffness С and the viscous friction element with viscosity β (Fig. 2) mounted in parallel [5].
Let us consider cinematic excitation of this system, when its deformation is described by the expression sin 2 z a f = ⋅ π (a -body oscillation amplitude, f -disturbing frequency).
The maximum strength occurs when z a = (the signs are not significant), it is equal to: and the work done by an external power source per full oscillation cycle -to the viscous friction element operation, i.e.Expressions ( 2) and ( 3) are used for determining the equivalent stiffness and viscosity of the air spring.For this purpose we integrate the equation (1) when sin 2 z a f = ⋅ π , which will allow determining the spring overpressure p 1 as a function of time t.The force exerted by the external source on the spring supporting surface is equal to: and the work of this force per oscillation period is equal to: Having determined max max ( ) and А , we find the equivalent coefficients: The research is carried out at a variable amplitude of oscillations (a = 0.005m, a = 0.010m, a = 0.020 m).
Taking into account the dependence of changes in gas (liquid) flow value on the section resistance and the vessel pressure employed in the fluid dynamics [2,11,14], we choose the dependence of pressure difference changes ( 12) p p − , but for the research completeness and based on various types of connecting elements, we do not neglect the following dependencies: Calculations are carried out with different flow capacity of the connecting element: from virtually open connecting element, which equalizes the cylinder and reservoir pressure, to almost completely closed one, which shuts off the cylinder from the reservoir [3].
Under these assumptions, we determine the dependence of cylinder overpressure p 1 on time t.The obtained data on pressures allow drawing the conclusions about the work done by the system, its equivalent coefficients of stiffness and viscosity under various operating conditions.

Findings
We obtained the curves for three laws of pressure variation of air suspension stiffness and viscosity characteristics, as well as determined the system operation depending on the connecting element flow capacity (Fig. 3-5.):As can be seen from the above curves, the system indicates the lowest peak value of the work performed in case of pressure difference change curves ( 12) p p − .Likewise in case of the curve ( 1 2) p p − the maximum system performance level is achieved at the beginning of the opening cycle of the connecting element, that allows achieving the system peak performance for a shorter period of time than that of systems with pressure change dependences ( 12) It is also natural that the maximum work performed by the system falls to the oscillation amplitude of 0.020 m, whereas when the oscillation amplitude is 0.0050 m, the changes in the system operation in case of decreased flow capacity of the connecting element are insignificant.
Analyzing the formula (3), it can be concluded that the more work that the system makes, the higher the pressure needs to be maintained in the cylinder for its high-quality work, which in turn can lead to complication in spring supply system.However, the fundamental parameters for spring operation are stiffness and viscosity, not its work.Therefore, it is advisable to analyze these figures for all three dependences of pressure change.
For the above-mentioned law of the spring support surface motion, the phase shift between p 1 and z is virtually absent, which makes it possible to estimate the stiffness spring by the formula (4).Whereas the equivalent viscosity coefficient is determined as the viscosity coefficient of the hydraulic damper, which absorbs per one oscillation cycle the same energy as the air spring by the formula (5).
We obtain the following curves of spring stiffness and viscosity coefficient dependence on the parameters characterizing the pneumatic resistance of the element that connects the cylinder with the auxiliary reservoir (Fig. 6-11):  12) p p − By increasing the viscosity coefficient, the system smooth operation and, consequently, the level of passenger comfort is improved.The curves show that early achievement of maximum viscosity is observed for ( 12) p p − , however, this figure is slightly lower than for (р 1-р 2) 2 and (р 1-р 2).

Originality and practical value
The designed scheme allows determining the optimal parameters of elastic and damping properties of the pneumatic system as function of the connecting element air resistance.The practical value is that the ability to predict the parameters of elastic and damping properties of the pneumatic system as function of the throttle element air resistance will improve the running performance of carriages, the comfort of passenger transportation and reduce the wear of the rolling stock and the track caused by interaction of carriage and rails.

Conclusions
Analysing the above curves, it can be concluded that the system stiffness coefficient increases significantly with increased resistance created by the connecting element, herewith the system viscosity tends to zero, which adversely affects the damping of the system.In its turn, the system viscosity is the highest at the mean resistance value of the flow generated by the connecting element.Moreover, the system reaches the maximum viscosity at the mean value of the stiffness coefficient that proves the beneficial effect on the system damping quality of the presence of the element with variable pneumatic resistance, which connects the cylinder with the auxiliary reservoir.