Introduction
Recently, pneumatic springs have become very widely used on trans-regional trains, high-speed trains [3, 16]. The advantage of pneumatic spring as an element of spring suspension is that it can be used to realize large static deflections in the se-condary spring suspension (up to 300 mm) and to obtain a noise reduction in the passenger compartments. In addition, for lines with raised platforms it is necessary to maintain a constant floor height of the car regardless of the number of passengers in the car. This is provided by the operation of a le-velling valve and pneumatic springs [16]. It is ne-cessary to consider the generalized structure of an equivalent mechanical model, adopted on the basis of pneumatic spring tests (Fig. 1) [3].
The ability to perceive high horizontal and diagonal movements, as well as torsional resistance, makes pneumatic suspension systems an attractive solution for all types of bogies [3, 16].
In general, the pneumatic suspension system (Fig. 2) on the car has the form of the pneumatic cylinder 1 and the auxiliary reservoir 4 connected to each other [17, 19, 22]. To regulate the dissipative and elastic properties of the pneumatic suspension system, the throttling element 2 is located in the connection fitting 3.
To calculate the properties of the pneumatic system shown in Figure 1, there is a significant number of mechanical models.
Fig.
1.
Generalized structure of equivalent
mechanical
system
Fig. 2. Diagram of pneumatic spring with reservoir
This
model is based on the analysis of the graphs obtained during the
pneumatic spring tests. The model consists of elements with
instantaneous contacts, represented in the same way as in the first
model, in the form of a connection between elastic and dissipative
elements. Herewith, in the model, each of the elastic elements with
a force
and dissipative ones with a force
enables a physical interpretation, which allows us to propose a
corresponding approximating expression for it. In particular, the
elements with forces
,
,
model the elastic properties of the main and additional capacity of
air suspension. The elastic element with a force
simulates the change in the effective area of the pneumatic cylinder
when it deforms and the elasticity of the rubber-cord shell. The law
of change is determined by the results of a static calibration of
the pneumatic springs and approximated by a piecewise-linear
relationship.
The
elements
and
simulate
losses due to air throttling. At the same time, the element
works mainly at low velocities of deformation of the pneumatic
cylinder (subcritical mode of air flow through the throttle
opening); the element
is switched on at the deformation rate of the pneumatic spring when
the supercritical mode of air flow through the throttle occurs.
However, the awkwardness of the model and a significant number of parameters limit its wide application in theoretical calculations.
Another structural scheme for the calculation of pneumatic spring is presented in [6] (Fig. 3).
Fig. 3. Mechanical model of pneumatic spring
The
presented model of the pneumatic spring takes into account the
static stiffness of the pneumatic cylinder
,
the effect of changing the ratio of air volumes in the pneumatic
cylinder and the auxiliary reservoir to the stiffness of the
pneumatic spring
and the damping coefficient of the equivalent viscous friction of
the pneumatic spring
.
The defining parameters of this pneumatic spring scheme are: the load-bearing capacity of the spring, the parameters of the throttling device and air.
This model gives fairly accurate results in the case of small oscillations.
The
more complex and frequently used model is the Nishimura pneumatic
spring model (Fig. 4). The Nishimura pneumatic spring model was
developed more than 40 years ago [9, 10, 15, 18]. The model takes
into account the changes in the stiffness of the pneumatic cylinder
and the re-servoir
,
depending on the change in air vo-lumes in them,
is a change in the level of stiffness of the pneumatic cylinder. The
model can take into account both linear and quadratic changes in the
viscosity index
.
That allows using it both for modelling the stiffness of a pneumatic
balloon and for determining the resistance to air flow through a
connecting armature.
Fig. 4. Model of Nishimura pneumatic spring
VAMPIRE
model is an extension of the Nishimura model [9, 18, 19]
with quadratic attenuation. In addition, inertial effect
and auxiliary stiffness
are introduced (Fig. 5).
Fig. 5 Model of VAMPIRE pneumatic spring
Berg pneumatic springs model, often found in the literature under the name GENSYS, is three-dimensional and can describe transverse, longitudinal and vertical oscillations [9, 11, 14, 18, 20, 24]. It is worth mentioning that the model describes several parameters of the spring: elasticity, friction and viscosity. The vertical model has a nonlinear attenuation, which depends on the rate of change in pressure in the pneumatic cylinder.
Fig. 6. Model of GENSYS (Berg) pneumatic spring
There is another model of pneumatic springs Bouc-Wen [12, 13, 18, 21], which includes not only elastic and dissipative elements describing the state of gas in the system, but also a block that simulates changes in the condition of the rubber-cord shell of a pneumatic cylinder.
Fig.
7. Model of Bouc-Wen pneumatic spring:
– vertical force from the body,
– non-linear
elastic
stiffness,
– damping component
To
describe the behavior of pneumatic springs, the mechanical model
includes three parallel branches: non-linear elastic stiffness
;
damping component
;
block, which determines the parameters of the pneumatic cylinder
operation depending on the behavior of the rubber-cord shell
Bouc-Wen.
The branch with nonlinearly elastic stiffness is used to simulate the work of air within the pneumatic spring and to describe its geometric parameters, as well as the thermodynamic processes that take place within the pneumatic spring, which can be derived in accordance with the thermodynamic equation.
The branch with a damping component is used to simulate the resistance of the air flow in the pipe of the system «pneumatic cylinder – connecting fittings – auxiliary reservoir».
The friction branch is a block simulating a hysteresis loop and amplitude dependences.
The
widespread model is considered to be that [1, 17] describing the
pneumatic spring as a connection between the parallel installed
elastic element with stiffness
and viscous friction element with viscosity
(Fig. 8).
Fig. 8. Equivalent mechanical system
In the previously mentioned schemes, the role of the throttling device was performed by additi-onally introduced viscosity parameters.
Throttle is local adjustable or non-adjustable resistance, set in the way of the working fluid flow [1].
The main throttle characteristics are the metering and tuning characteristics. The metering cha-racteristic is the relationship between air flow through the throttle and the ratio of pressures before and after the throttle. The metering charac-teristic largely depends on the regime of laminar or turbulent flow. The flow regime is determined from the Reynolds number. When Re < Rcr the flow is considered laminar, when Re > Rcr -turbulent.
According to functional purposes, the throttles are classified into constant and variable ones. According to the principle of operation, there are li-near and quadratic throttles.
Constant throttles are generally an orifice in the plate. They are conventionally divided into throttling plugs and bushings.
The wide class of adjustable throttles include: cylindrical, conical needle, threaded, throttles such as «rotary valve», flapper-nozzle and spool-type throttling valves, as well as the role of a throttle can be performed by throttling valves.
Purpose
The purpose of this work is to determine the dependence of the working medium flow on the capacity of the throttling device, its geometric features and the pressure difference in the pneumatic spring cylinder and in the auxiliary reservoir.
An important task in the course of the work will be to determine the pressure difference when the working medium passes through the throttling element. The necessary part of the work is also to determine the reliability of the results obtained when determining the pressure difference in the throttling device.
Methodology
Calculation of the dependence of the working medium and pressure drop is performed in two ways:
– By numerical simulation of a stationary gas flow through a throttling element;
– Its analytical calculation expression using empirical relationships (control calculation to evaluate the reliability of numerical simulation results).
We selected several different types of throttling devices [4, 5], namely: corrosion-proof throttle valve Aisi 304 (L)/316 (L) DN 40 (Fig. 9); San Marino valves for air, gases and liquids normally open of type «T» (Figure 10) and type «Y» (Figure 11) with G11/4» connecting thread.
Fig.
9. Corrosion-proof throttle valve
Aisi
304 (L)/316 (L) DN 40
Fig.
10. San Marino valve for air, gases and
liquids
normally open of type «T»
Fig. 11. San Marino valve for air, gases and liquids normally open of type «Y»
The diameters of the inlet and outlet openings are 40 mm. The length of the Aisi throttle valve body is 160 mm, San Marino valves for air, gases and liquids normally open of type «T» – 120 mm and type «Y» – 113 mm.
For analysis, we selected three stem positions: open – «0», closed by 50% – «1/2», closed by 75% – «3/4».
In numerical simulation, we consider the motion of the working medium (air) in the cavity of the throttling element represented by the 3D mo-del. The flow is stationary turbulent. The boundary conditions are given as follows (Figure 12). The surface that bounds the cavity is divided into three areas:
– inlet opening – the pressure is determined;
– outlet opening – the mass air flow is determined;
– walls – all motion speed components are zero.
Fig. 12. Boundary conditions
The air movements describe the averaged Navier-Stokes equations with two additional equations of the k-ε model of turbulent flow [25].
As the input data, we set the pressure of the working medium obtained in calculations of the generalized pneumatic spring model in the Matlab Simulink software package [23]. The change in the flow rate of the working medium at the outlet from the throttling device is set in the range from 0.001 kg/s to 0.26 kg/s, also according to the previously obtained data.
The calculation is carried out at the environmental temperature equal to 20°C. The software package allows taking into account the properties of various materials used in the design of throttling devices, such as a bronze body, a brass plate, a steel stem. Therefore, in order to take into account the different degrees of resistance when the working medium rubs against the surface, one of the input parameters indicates the type of material.
During the calculation it is possible to see not only the final result, as a diagram with pressure values, but also to trace the nature of the working medium flow at any point of the throttling device (Fig. 13).
Fig. 13. Distribution of velocities of the working medium in the throttling device
Based on the results of calculations, graphs of the dependence of the pressure drop on the flow rate of the working medium are plotted.
To evaluate the reliability of the results obtained, we perform calculations using the formulas given in [1, 2, 8].
,
where
– hydraulic radius of the channel section, m;
– average velocity of compressed gas flow along the channel
section, m/s;
– gas density,
kg/m3;
– coefficient of dynamic viscosity, determined from the
diagrams [7], Pa s.
,
where
– mass flow rate of gas, kg/s;
– cross-sectional area of the channel, m2.
The density of compressed gas for working pressure is calculated by the formula:
,
where
– atmospheric pressure, kPa;
– overpressure in the system
of pneumatic springs, kPa;
– universal gas constant, J/(mol K);
– temperature, K;
– molar mass of gas, mol.
The pressure loss depends on two coefficients of resistance: the coefficient of resistance ςfr cha-racterizing the frictional losses during the movement of the working fluid and the local loss factor at the input ςin.
The coefficient of resistance, characterizing the friction loss, is determined by the formula:
– at the gas motion, when Rcr < Re < 105, we use the Blasius formula:
,
– at the gas motion, when 105 < Re < 108, we use the formula Nikuradze:
,
The coefficient of local losses at the input ςin is determined depending on the shape and geometric dimensions of the inlet opening [2].
The coefficients of local losses in many cases are determined from tables and graphs obtained experimentally. So the local loss coefficients for the presented throttling elements are selected from the reference tables [2, 8].
The loss of pressure is determined by the formula [2, 8]:
,
According to the obtained data, graphs of dependence of pressure drop on mass flow were also constructed for further comparison with the calculation in the shown complex with the visualization of calculations.
The dependence of the flow rate of the working medium on the type of throttling device will be determined based on the approximation of the graphs of the dependence of the pressure drop on the mass flow:
,
where
– proportionality coefficient.
Findings
The
graphical dependences of the mass flow rate of the working medium G
on the pressure difference ΔP
are shown in Fig. 14–16 (based on the results of numerical
simulation) and in Fig. 17–19
(based on calculations using empirical dependencies). The degree of
closure of the throttling device is denoted by
.
The value of the air flow, obtained by numerical simulation, is greater than the flow rates obtained from semi-empirical formulas. At the same time, they are in good qualitative agreement, and the quantitative difference averages 25%, which can be regarded as confirmation of the reliability of the numerical model.
As
can be seen from the Figures 14–16,
the graphs are well approximated by the function,
which allows us to show the dependence of the working medium flow
rate on the type of throttling device. Calculation using a software
package with visualization of the results gives more accurate data
and visually realizes the process occurring in the throttling device
than the calculation using approximate formulas.
Based on the results of calculations performed with the help of a software package with visualization of the results, we calculated a proportionality coefficient that describes the dependence of the working medium flow on the throttling device capacity and its geometric features for each of the throttling elements considered, with three degrees of closure.
Fig. 14. Graph of
the mass flow rate versus
the
pressure difference of Aisi throttle valve
based
on the mathema-tical
modelling
results
Fig.
15. Graph of the mass flow rate versus
the
pressure difference for San Marino valve
type
«T» based on the mathematical
modelling
results
Fig.
16. Graph of the mass flow rate versus the pressure difference for
San Marino valve type «Y» based
on
the mathematical modelling results
Fig.
17. Graph of the mass flow rate versus the pressure difference for
Aisi throttle valve based on the
calculation
results using empirical dependencies
Fig. 18. Graph of the mass flow rate versus the pressure difference for San Marino valve type «T» based on the calculation results using empirical dependencies
Fig. 19. Graph of the mass flow rate versus the pressure difference for San Marino valve type «Y» based on the calculation results using empirical dependencies
Fig.
20. Graph of the mass flow rate versus
the
type of throttling device
Originality and practical value
The work allows determining the degree of influence of the frictional component on the variation of the pressure difference in the pneumatic cylinder and the auxiliary reservoir of the pneumatic suspension system. Also, the work proposes a method to determine the dependence of the working environment on the capacity of the throttling device and its geometric features.
The practical value lies in the possibility to predict the operating parameters of the pneumatic system depending on the pneumatic resistance of the throttling device will improve the car running characteristics, increase the comfort of passenger transport, and also reduce the wear of the rolling stock and track gauge due to vehicle-track interaction.
Conclusions
Analyzing the above graphs, we can conclude that the use of modern computational complexes with visualization of the results greatly increases the accuracy of calculation and allows analyzing the operation of the system and the state of the working medium in it not only by the final results in digital form, but also observing them directly at the time of flow process.
The obtained data of the pressure drop dependence on the working medium mass flow allowed determining the law that describes the process flowing in the throttling devices. The dependence of the flow rate of the working medium on the capacity of the throttling device and the geometric capabilities of its design is determined.