# Influence of Aging Time on Vertical Static Stiffness of Air Spring
## 4 Numerical Simulation  
The finite element model of the air spring was established using Abaqus software. Since the structure of the air spring and the load it bears are symmetric, only 1/2 of the air spring model was built for the simulation, to simplify the calculation. The rubber airbag thickness of the air spring is 6 mm and can be regarded as a thin wall structure, simplified to a shell structure, and discrete using a four-node shell unit S4R; a rebar unit can be used to simulate the curtain of the air spring. The upper and lower cover plates are made of metal materials. They are set as analytical rigid bodies, and the attributes of the mesh are given as r3d3 and r3d4. The conical emergency spring is mainly composed of a layer spring and conical spring, through step coordination and screw material. The emergency spring is regarded as a whole part during modeling. When the modeling is complete, the metal and rubber parts are divided into areas, and different material attributes are given to each segmentation area. The grid division is shown in Figure 4. The cord py of the air spring is located in the middle layer of the airbag, and is wrapped by two layers of rubber material inside and outside. The main function of the cord layer is to restrain the deformation of the air spring rubber airbag, and the paving form is cross arranged. In the Abaqus finite element simulation software, the rebar element model can be used to simulate this structure. The parameters that the rebar model can define include the Young's modulus of rubber, area of the cord, cord spacing, cord layout angle, etc.  
The symmetry constraints are set at the symmetric surface of the model. The reference point at the center of the upper cover surface is defined to establish coupling constraints with the upper cover surface. The auxiliary spring base is completely secured. Abaqus is used to simulate the static mechanical characteristics of the air spring. Abaqus/Standard is used for the calculation analysis, and the simulation process is divided into two analysis steps. The first is to simulate the air filling process of the capsule. First, fixed constraints are placed on the upper cover plate and the auxiliary spring reference point, and then gas is filled into the capsule to compare the simulation results with the test results, so that the maximum air pressure is 0.612 MPa in the actual test. The second analysis step simulates the process of loading of external force. The constraint of the auxiliary spring base is maintained, the vertical constraint of the upper cover plate is released, and the vertical displacement load is applied to the upper cover plate. Through two-step simulation, the reaction and displacement at the reference point can reveal the vertical stiffness of the air spring.  
The rubber and curtain parameters obtained with the different aging times were fed into the air spring finite element model, and the air spring vertical static stiffness was obtained, as shown in Figure 5. As the figure reveals, aging increases the air spring vertical stiffness. The 12-day increase was from 249.1 to 289.7 N/mm, and the relative error was between 251.5 N/mm and 277.4 N/mm. Therefore, the simulation analysis model and simulation results are reasonable and credible.  
The vertical static stiffness obtained from the simulations and experiments increased by 16.3% and 10.3% before and after aging, respectively. The vertical static stiffness of the air spring increased faster than that obtained in the experiment. The main reason for this phenomenon is that, compared with the air spring in the middle of the rubber, the rubber and curtain materials are more fully exposed to the hot-air environment, and the rubber and cord material, under the same aging conditions, using the material parameters of the spring aging results, have a faster aging rate.  
Figure 4: Finite element model of air spring.  
Figure 5: Variation curve of vertical static stiffness of air spring with aging time.