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Dynamic performance of air spring based on SIMPACK and Matlab Simulik joint simulation.

1. Introduction

Air spring was born in the middle of 19th century, early in 1947, air spring was used in the Pullman Car of United States for the first time. In 1850, John Lewis carried out a lot of research work of on air spring, and apply for air spring patent: "Pneumatic Springs for Railroad Cars, Locomotive, Burden-Cars, Bumpers & c" (Li, 2012), as for its superior performance, air spring was widely used in automotive, aerospace and other fields by many European countries at first. In 1910, George Bancroft got an air spring patent, which is related to the use of air spring on the car suspension mechanism (Qi, 2013). In 1930, the United States Firestone Tire Company used air spring in cars for the first time, and it cooperated with General Motors to use air spring in passenger cars and carried out the relevant tests, the test results show that air spring has good damping performance and nonlinear characteristics. Subsequently, the air spring began to be used in various fields, and it has been widely used in cars. In 1955, the vehicle power laboratory of Japan National Railway Technology Research Institute made a deep research on the air spring and provides a valuable resource for the development and design of air spring (Li, 2012). In the same year, the vehicle power laboratory tested the vehicle with air spring and got the test data. In recent years, some Japanese passenger cars and trucks are equipped with a variety of air springs. In 1975, George WJackson applied for a patent of air spring with control valve, which can effectively control the air flow situation in the vehicle suspension system Yang, 2012). In the 1990s, with the increasing demands of people on the speed of railway vehicles, such as trains, bullet trains and high-speed railways, and with the increasing demands of people on the comfort of cars traveling on the roads, the speed of railway vehicles is becoming more and more fast, air spring is widely used in these areas and it has got a great development.

In 1993, William E. Richeson obtained a patent of air spring with a hydraulic pneumatic mechanism (Chen, 2012). In 1995, John EAmold applied for a patent of air spring with a piston, and the air spring has the characteristics of easy assembly, as well as other characteristics, such as ensuring low-frequency vibration. In the past few years, the scholars in domestic and overseas have applied for a lot of patents on air springs.

Since 1990, with the rapid development of computer technology, scholars began to use computer simulation software, such as SIMPACK and Matlab Simulik, to study the air spring. In 1998, Alf Homeyer used the finite element method to carry on the optimized design to the air spring. In recent years, based on SIMPACK and Matlab Simulik software, many foreign scholars analyze the dynamic characteristics of air spring (Chen, 2013). In 2002, based on Matlab Simulik modeling, MALIN PRESTHUS analyzed the influence factors of air spring's vertical stiffness and damping (Facchinettia, 2014). In 2010, Alan Facchinetti, Laura Mazzola and others established the simulation analysis model of air spring, and the air spring parameter and locomotive dynamics performance were analyzed and compared to find out the influence of air spring parameter on it (Docquiern, 2015). In 2011, H.LIU and J.C.LEE analyzed the influence of the orifice's parameter and the additional air chamber's volume on the dynamic stiffness and damping of the air spring.

From the development situation at home and abroad, the theoretical research and simulation application of air spring are becoming more and more mature, but the key technology also needs to study continuously and makes breakthrough, in particular, the production and manufacture of China's air spring are mainly based on experience, so it needs to develop the research work. Therefore, this paper proposes the fluid solid coupling method, and then carries out joint simulation for the dynamic performance of air spring combined with the software SIMPACK and Matlab Simulik, which can overcome the nonlinear stiffness and hysteresis characteristics.

2. Analysis of air spring model

Nonlinear model of air spring has been established by scholars at home and abroad, and they carried out a large number of experiments. Based on the Matlab Simulik platform, a nonlinear model is established in the vertical direction of the mechanical structure of the air spring (Qin, 2013), then, based on the established nonlinear model, the nonlinear dynamical theory of air spring under different excitation (0.05Hz, 0.5Hz and 5Hz different frequency sinusoidal excitation) is analyzed and introduced. The purpose of this study is to based on the developing of general air thermodynamic equation, the spring model's change under without adiabatic or isothermal conditional assumptions and the change with air quality. Analysis and development of the model is one of the important parts to be revealed, because the stiffness of the air spring directly affects the lag of dynamic performance. Therefore, the research purposes of current researches are to improve the dynamic stiffness of air spring, and to establish an optimal nonlinear model for this, in addition, the stability also is an important indicator to evaluate the air spring model.

2.1. Linear model

The linear model of the air spring can be deduced by the method of omitting the high-order infinitesimal linearization according to the equation of gas flow direction, the orifice or the pipeline gas mass flow linearization equation. The linear model is shown in Figure 1.

[FIGURE 1 OMITTED]

As shown in Figure 1, 1Krepresents the stiffness of the air spring; the additional 2K represents the stiffness of the air chamber; 3Krepresents the effective area change rate stiffness; 4Krepresents the vertical angle of series connection rubber pile; 2d represents orifice or equivalent damping of connecting pipe; 3drepresents the rubber damping of air spring; 4drepresents the vertical damping of the series connection rubber pile; 0z represents the total deformation of the air spring;

The formulas for each stiffness and damping value are shown in formula (1)--(6):

[k.sub.1] = [np.sub.0][A.sub.e.sup.2] (1)

[k.sub.2] = [[V.sub.10]/[V.sub.20]] [k.sub.1] (2)

[k.sub.3] = ([p.sub.0] - [p.sub.at]) d[A.sub.e]/s[Z.sub.0] (3)

[d.sub.2] = C[[rho].sub.0][A.sup.2.sub.e] (4)

[d.sub.3] = [d.sub.a] (5)

[d.sub.4] = [d.sub.r] (6)

In the formula, n is the gas variable index; [p.sub.0] is the internal gas pressure when the air spring is balanced; [A.sub.e] is the effective area of the air spring body; [V.sub.10] is the initial volume of the gas in air spring body; [V.sub.20] is the initial volume of the gas in the additional air chamber; [p.sub.at] is the atmospheric pressure; Cis the drag coefficient between the air spring body and the additional air chamber; [[rho].sub.0] is the initial gas density inside the air spring body; [d.sub.a] is the damping of the air spring body capsule; [d.sub.r] is the damping of the series connection rubber pile.

2.2. Nonlinear Model

The core problem of establishing nonlinear model is how to establish the accurate sub-models of orifice, connecting pipe, series connection rubber pile, air spring body and additional air chamber. In addition, how to consider the multiple physical parameters of the air spring into the modeling is also a key issue of modeling. As the contact area with the wall is small when the air flows into the orifice, and the flow rate is fast, so this is regarded as one-dimensional isentropic flow (Liu, 2011), according to the Bernoulli equation, it can be deduced the flow rate at the orifice shrinkage is shown in formula (7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In the formula, [p.sub.u] is the absolute pressure of the gas in the upstream gas chamber of the orifice; [p.sub.d] is the absolute pressure of the gas in the gas chamber downstream of the orifice; [mu] is the Poisson's ratio of the gas; and [[rho].sub.u] is the density of the gas in the upstream gas chamber of the orifice.

According to the adiabatic equation, the ideal gas state equation and formula (7), the theoretical general mass flow of the orifice can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In the formula, A is the area of the cross-section at the orifice shrinkage; R is the molar gas constant; and [T.sub.u] is the temperature of the gas in the upstream gas chamber.

Taking into account the flow loss of the gas through the orifice and the saturation characteristics of the orifice mass flow, the actual mass flow of the gas in the orifice is obtained.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

In the formula, [C.sub.q] is the flow coefficient of the orifice.

According to the Helmholtz effect, the gas in the connecting pipe can be modeled as a mass with fixed quality and volume, then the connecting pipe model can adopt the the air mass block oscillation model. According to the Helmholtz equation and the Bernoulli equation, the equation of the state of the gas mass in the connecting pipeline can be obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[[xi].sub.p] = [lambda] [L.sub.p]/[d.sub.p] + [[sigma].sub.pd] - [[sigma].sub.od] (11)

In the formula, [m.sub.p] is the mass of the gas mass in the connecting pipe, y is the displacement of the gas mass in the connecting pipe, [[rho].sub.p] is the density of the gas in the connecting pipe, [[xi].sub.p] is the total pressure drop coefficient of the connecting pipe, [A.sub.p] is the area of the cross-section of the connecting pipe; [p.sub.pu] is the absolute pressure of the gas in the air chamber upstream of the connecting pipe; [p.sub.ad] is the absolute pressure of the gas in the air chamber downstream of the connecting pipe; g is the acceleration of gravity; [h.sub.pu] is the centroid height of the gas in the upstream gas chamber of the connecting pipeline; [h.sub.pd] [lambda] is the centroid height of the gas in the downstream gas chamber of the connecting pipeline; [lambda] is the friction coefficient of the connecting pipe wall; [L.sub.p] is the length of the connecting pipe; [d.sub.p] is the diameter of the connecting pipe; [[sigma].sub.pd] is the pipe diameter drag coefficient of the connecting pipe; [[sigma].sub.od] is the damping transfer coefficient of the orifice.

3. Analysis of air spring dynamic performance

3.1. The basic idea of fluid solid coupling

Fluid solid coupling mechanics is a branch of mechanics that is produced by the intersection of fluid mechanics and solid mechanics. As the name suggests, it is a science to study the interaction of the various behavior of deformed solids under the action of the flow field and the influence of solid configuration on flow field. The important characteristic of fluid solid coupling mechanics is the fluid solid interaction: the deformable solid under the action of the fluid load will produce deformation or movement, and in turn the deformation or movement will affect the flow field, so as to change the distribution and size of the fluid load. It is this interaction will produce all kinds of fluid solid coupling phenomenon under different conditions. The three-dimensional schematic diagram is shown in Figure 2:

[FIGURE 2 OMITTED]

The fluid solid coupling problem can be defined by its coupling equation, the domains of which both include the fluid domain and the solid domain. And the unknown variables contain the variables that describe the phenomenon of fluid and the variables that describe the phenomenon of solid, in general, it has the following two characteristics:

1. Both the fluid domain and the solid domain cannot be solved separately.

2. Unable to obviously cut the describe of the independent variables of fluid motion and the independent variables of solid phenomena.

The basic idea of fluid solid coupling is to use the ABAQUS finite element software and STAR-CCM+fluid analysis software to carry out co-simulation, that is, to establish the solid part of the ABAQUS software, its modeling process is similar to the air spring stiffness modeling method, but the density of structural material need to be added additionally. Then to establish the model of the fluid part in the STAR-CCM+software, and divide the grids, define the fluid properties, select control equations, establish boundary conditions, and so on. After the successful operation respectively, then add the statement of the close analysis steps into the INP file of ABAQUS, and set the coupling analysis in STAR-CCM+ at the same time to carry out calculate. Fluid solid coupling includes one-way coupling and two- way coupling. The coupling method used in this paper is a two-way coupling, and it is a "instant" coupling, that is, STAR-CCM+ software will pass the gas flow and the size of pressure in air spring to the air spring in the ABAQUS software, At the same time, in turn ABAQUS will pass the displacement and deformation generated by the air spring to the gas of STAR-CCM+. Repeat this cycle, and finally complete the calculation. In this paper, the more efficient simulation software SIMPACK and MatlabSimulik are used, and the joint simulation of the dynamic characteristics of the air spring is carried out.

After a series of studies, the science researchers found that the gas flow situation in the air spring is very complex, especially when up and down vibration frequency is high of the movement machine, it is very difficult to carry out modeling for the dynamic performance analysis, therefore, in order to explore the gas flow situation in the air spring, a gas partial model is established in MatlabSimulikin this paper. Assuming that there is no heat exchange between the air spring with the additional chamber and the outside air, the inner wall of the air spring system is adiabatic; assuming that the air in the air spring is the ideal gas.

3.2. Analysis of air spring dynamics

Based on the air spring nonlinear model has been established by the MatlabSimulik software, the analysis of the dynamic characteristics of air spring, than is stiffness, becomes very essential. The gas flow essence in air spring is the problem of fluid flow and heat transfer. Fluid flow is one of the most complex physical behaviors, and it is much more difficult to model and simulate compared with the problems of stress analysis in structural design area. However, for any complex fluid flow, mass conservation, energy conservation and momentum conservation are applicable, for gas, it also follows the gas state equation.

In which, the continuity equation is:

[partial derivative][rho]/[partial derivative]t + [partial derivative]([rho]u)/[partial derivative]x + [partial derivative]([rho]v)/[partial derivative]y + [partial derivative]([rho]w)/[partial derivative]z (12)

In the formula, [rho] is the density of the fluid; u, v, ware the vector components of the velocity of fluid in the directions of X-axis, Y-axis, and Z-axis, respectively.

The energy conservation equations are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In the formula, his the entropy, k is the micro-element molecular conductivity, T is the gas temperature, [S.sub.h] is the defined volume source, [k.sub.x], [k.sub.y], and [k.sub.z] are the components of the conductivity in the X-axis, Y-axis, and Z-axis directions caused by the two- stage flow in the flow process.

The equation of state of the standard gas is:

PV = nRT (14)

In the formula, P is the standard gas pressure, Vis the standard gas volume; nis the amount of standard gas material, R is the standard gas constant, in which R value of the air is 287J/(kg x K).

3.3. Experimental results analysis

The sinusoidal motion induced force responses are at 0.05 Hz, 0.5 Hz and 5 Hz (0.05 Hz and 5 Hz data represent lower or higher than the actual value of 200 N respectively). Simulation results do not include the effective area of hysteresis, while the experimental results include it, which is shown in Figure 3(a); The sinusoidal motion induced force responses are at 0.05 Hz, 0.5 Hz and 5 Hz (0.05 Hz and 5 Hz data represent lower or higher than the actual value of 200 N respectively). The simulation results include all the effects other than the change of air quality. The air quality in the air spring has an effect on the response due to the air quality, but no lag occurs, which is shown in Figure 3(b).

From formular (13), the force is defined by the effective area time, and the effective area pressure has a certain influence on the response lag. The change in the effective area will also lead to the change in vertical displacement. In addition, when the vertical displacements are the same, the effective areas change with the pressure, which resulting in the lag effective area. The lag can be seen in Figure 3(a) and Figure 3(b). The simulation results shown in Figure 3(a) do not include the of the effective area lag, but they are included in Figure 3(b). These figures indicate that the lagged effective area increases the force lag response. Finally, the experimental results of the comparison between the simulations confirm the validity of the air spring model.

[FIGURE 3 OMITTED]

The changes of stiffness and frequency are shown in Figure 4, which show the pressure response only having the effect of heat transfer. The total pressure response in equation (14) is determined by the sum of Figure 3 (a) and Figure 3 (b). Thus, the heat transfer at low frequencies can significantly affect the stiffness of the entire air spring, unlike at very high frequencies. More specifically, the heat transfer at low frequencies will result in a significant reduction in the dynamic properties of the air spring due to the change of volume, and will result a significant change in stiffness; at high frequencies, it only reduces stiffness slightly. The negative pressure in Figure 4 occurs because of the above reasons. When the air spring is compressed, because of the pressure of the compression volume increases, the pressure generated by this increases the stiffness of the air spring in the temperature equation and reduces the dynamics performance in the rate equation. Therefore, when the temperature of the air spring increases, and becomes larger than the pressure in the environment, a negative pressure can be generated due to the reduce of the heat transfer. However, the overall pressure increases because the increase of the pressure due to the volume change is greater than the reduction of pressure due to the heat even when the temperature of air spring is greater than the environment. In addition to the pressure change, the effect of the effective area of the change on stiffness, As described above, the outline of the piston of the air spring is manufactured in order to obtain optimum ride comfort, which generates the change of effective area. Therefore, a large change in the effective area can have a significant effect on the stiffness variation. This study in the air used in this experiment shows that most of the stiffness values of spring represent the change of effect effective area varies in the 40% of entire stiffness.

[FIGURE 4 OMITTED]

4. Conclusion

Through SIMPACK and MatlabSimulik joint simulation, this study firstly puts forward linear and nonlinear models of the air spring, and analyzes the dynamic performance of the air spring to develop a general air spring model. This model has good stiffness and lag characteristics, which can connect with a design of a pneumatic system; secondly, it adopts the fluid solid coupling method to calculate the dynamic stiffness of air spring, and it uses the SIMPACK and MatlabSimulik joint simulation to simulate the gas flow in the spring to set the optimization ideas, the model simulation results show that the experimental measurement results and the theoretical results are essentially consistent, the force relative to the displacement and the 0.05Hz, 0.5Hz, and 5Hz pressure excitation verify the validity of the modeling method of air spring and the analysis method of dynamic performance proposed in this study.

Recebido/Submission: 15/06/2016

Aceitacao/Acceptance: 26/09/2016

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Nan Jia (1,2), Junqing Zhan (2), Lishun Li (2), Xinxi Xu (1) *

* Xinxi Xu, ytsdsjn@163.com

(1) Institute of Medical Equipment, Academy of Military Medical Sciences, Tianjin, 300161, China

(2) Military Transportation University, Tianjin, 300161, China
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Author:Jia, Nan; Zhan, Junqing; Li, Lishun; Xu, Xinxi
Publication:RISTI (Revista Iberica de Sistemas e Tecnologias de Informacao)
Date:Dec 1, 2016
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