Figure 1.1.9-9 shows a displaced plot of the axisymmetric model at the end of Step 2. The spring has undergone an upward displacement along its axis because the downward load being applied to it is slightly smaller than the axial reaction force at the end of the previous step. Once again, comparison between the results from both models reveals almost identical solutions.
Linearized stiffnesses for the airspring are obtained from the linear perturbation steps. The stiffness is computed by dividing the relevant reaction force at the rigid body reference node by the appropriate displacement, which gives the airspring's axial stiffness under variable cavity pressure conditions as 826 kN/m for the axisymmetric model. From the results of Step 4 the axial stiffness under "fixed"
Loading
The airspring is first pressurized to 506.6 ´ 103 kPa (5 atms) while holding the upper disk fixed. This
pressure is applied by prescribing degree of freedom 8 at the cavity reference node using the
*BOUNDARY option. In this case the air volume is adjusted automatically to fill the cavity.
In the next step the *BOUNDARY, OP=NEW option is used to remove the boundary condition on the pressure degree of freedom, thus sealing the cavity with the current air volume. In addition, during this step in the ABAQUS/Standard analysis the boundary condition on the vertical displacement degree of freedom of the rigid body reference node is removed, and in its place a downward load of 150 kN is applied.
The next two steps in the ABAQUS/Standard axisymmetric model analysis are linear perturbation steps to test the axial stiffness of the airspring with the cavity pressure allowed to vary (closed cavity conditions) and with it fixed. The three-dimensional ABAQUS/Standard analysis contains three linear perturbation steps, all under variable cavity pressure (closed cavity) conditions: the first to test the axial stiffness of the airspring, the second to test its lateral stiffness, and the third to test the rotational stiffness for rocking motion in the symmetry plane.
The ABAQUS/Explicit axisymmetric analysis concludes with a nonlinear step in which the airspring is subjected to a downward displacement of 75 mm. The ABAQUS/Standard axisymmetric analysis concludes with a nonlinear step in which the airspring is compressed by increasing the downward load to 240.0 kN. The three-dimensional analyses conclude with a nonlinear step in which the airspring is subjected to a lateral displacement of 20 mm.
Results and discussion
Figure 1.1.9-7 and Figure 1.1.9-8 show displaced plots of the axisymmetric shell model at the end of the pressurization step, Step 1. It is of interest to compare the results from this model with those from the 180° model to validate the material model that was used for the rebar reinforcements in the axisymmetric model. A close look at the nodal displacements reveals that the deformation is
practically identical for the two models. Moreover, the axial reaction force at the rigid body reference node is 156 kN for the axisymmetric model and 155 kN for the 180° model (after multiplication by a factor of 2). The cavity volume predicted by the axisymmetric model is 8.22 ´ 10-2 m3 versus 8.34 ´
10-2 m3 for the 180° model (again, after multiplication by a factor of 2).
ABAQUS/Standard results
Figure 1.1.9-9 shows a displaced plot of the axisymmetric model at the end of Step 2. The spring has undergone an upward displacement along its axis because the downward load being applied to it is slightly smaller than the axial reaction force at the end of the previous step. Once again, comparison between the results from both models reveals almost identical solutions.
Linearized stiffnesses for the airspring are obtained from the linear perturbation steps. The stiffness is computed by dividing the relevant reaction force at the rigid body reference node by the appropriate displacement, which gives the airspring's axial stiffness under variable cavity pressure conditions as 826 kN/m for the axisymmetric model. From the results of Step 4 the axial stiffness under "fixed"
cavity pressure conditions is 134 kN/m. The difference in axial stiffness between these two cases (a factor of 6) is the result of differences in cavity pressure experienced during axial compression. Under variable cavity pressure conditions, a fixed mass of fluid (air) is contained in a cavity whose volume is decreasing; thus, the cavity pressure increases. Under "fixed" cavity pressure conditions, the pressure is prescribed as a constant value for the step. For the 180° model the predicted stiffnesses are as follows: the axial stiffness is 821 kN/m, the lateral stiffness is 3.31 MN/m, and the rotational stiffness is 273 kN/m.
Figure 1.1.9-10 shows a series of displaced plots associated with the compression of the axisymmetric airspring model during Step 5. Figure 1.1.9-11 shows the load-deflection curve corresponding to this deformation. The response of the airspring is slightly nonlinear; consequently, there is good agreement between the axial stiffness obtained with the linear perturbation analysis (Step 3) and that obtained from the slope of the load-displacement curve. Figure 1.1.9-12 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 5, which shows that the gauge pressure in the cavity increases by approximately 50% during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-13 shows a plot of cavity volume versus the
downward displacement of the rigid body in Step 5. The corresponding results for the axisymmetric membrane model are in good agreement with the above results.
Figure 1.1.9-14 shows the displaced plot of the 180° model at the end of Step 6, in which a lateral displacement was applied to the airspring. Figure 1.1.9-15 shows the load-deflection curve
corresponding to this deformation. Once again, good agreement is found between the lateral stiffness predicted from the linear perturbation analysis (Step 4) and that obtained from the slope of the load-displacement curve.
ABAQUS/Explicit results
Figure 1.1.9-16 shows a series of displaced plots associated with the compression of the axisymmetric model during the second step. Figure 1.1.9-17 shows the load-deflection curve corresponding to this deformation. Although the displacement of the rigid body was applied over a short enough time period to cause significant inertial effects in the model, there is still good agreement between the slope of the load-displacement curve in this example and the slope of the load-displacement curve for the same analysis performed statically in ABAQUS/Standard. Figure 1.1.9-18 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 2, which shows that the gauge pressure in the cavity increases by approximately 50 percent during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-19 shows a plot of cavity volume versus the downward displacement of the rigid body in Step 2.
Figure 1.1.9-20 shows the displaced plot of the 180° model at the end of Step 2, in which a lateral displacement was applied to the airspring. Figure 1.1.9-21 shows the load-deflection curve
corresponding to this deformation. Although there is a significant amount of noise that results from the contact conditions and the coarseness of the mesh, the load-deflection curve shows good agreement between the analysis performed quasi-statically in ABAQUS/Explicit and the same analysis performed statically in ABAQUS/Standard. The load versus displacement curve shown has been smoothed to cavity pressure conditions is 134 kN/m. The difference in axial stiffness between these two cases (a factor of 6) is the result of differences in cavity pressure experienced during axial compression. Under variable cavity pressure conditions, a fixed mass of fluid (air) is contained in a cavity whose volume is decreasing; thus, the cavity pressure increases. Under "fixed" cavity pressure conditions, the pressure is prescribed as a constant value for the step. For the 180° model the predicted stiffnesses are as follows: the axial stiffness is 821 kN/m, the lateral stiffness is 3.31 MN/m, and the rotational stiffness is 273 kN/m.
Figure 1.1.9-10 shows a series of displaced plots associated with the compression of the axisymmetric airspring model during Step 5. Figure 1.1.9-11 shows the load-deflection curve corresponding to this deformation. The response of the airspring is slightly nonlinear; consequently, there is good agreement between the axial stiffness obtained with the linear perturbation analysis (Step 3) and that obtained from the slope of the load-displacement curve. Figure 1.1.9-12 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 5, which shows that the gauge pressure in the cavity increases by approximately 50% during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-13 shows a plot of cavity volume versus the
downward displacement of the rigid body in Step 5. The corresponding results for the axisymmetric membrane model are in good agreement with the above results.
Figure 1.1.9-14 shows the displaced plot of the 180° model at the end of Step 6, in which a lateral displacement was applied to the airspring. Figure 1.1.9-15 shows the load-deflection curve
corresponding to this deformation. Once again, good agreement is found between the lateral stiffness predicted from the linear perturbation analysis (Step 4) and that obtained from the slope of the load-displacement curve.
ABAQUS/Explicit results
Figure 1.1.9-16 shows a series of displaced plots associated with the compression of the axisymmetric model during the second step. Figure 1.1.9-17 shows the load-deflection curve corresponding to this deformation. Although the displacement of the rigid body was applied over a short enough time period to cause significant inertial effects in the model, there is still good agreement between the slope of the load-displacement curve in this example and the slope of the load-displacement curve for the same analysis performed statically in ABAQUS/Standard. Figure 1.1.9-18 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 2, which shows that the gauge pressure in the cavity increases by approximately 50 percent during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-19 shows a plot of cavity volume versus the downward displacement of the rigid body in Step 2.
Figure 1.1.9-20 shows the displaced plot of the 180° model at the end of Step 2, in which a lateral displacement was applied to the airspring. Figure 1.1.9-21 shows the load-deflection curve
corresponding to this deformation. Although there is a significant amount of noise that results from the contact conditions and the coarseness of the mesh, the load-deflection curve shows good agreement between the analysis performed quasi-statically in ABAQUS/Explicit and the same analysis performed statically in ABAQUS/Standard. The load versus displacement curve shown has been smoothed to
eliminate some of the noise.
Input files
hydrofluidairspring_3d_shell.inp
Three-dimensional ABAQUS/Standard model using shell elements.
hydrofluidairspring_axisymm.inp
Axisymmetric ABAQUS/Standard model.
airspring_s4r.inp
Three-dimensional ABAQUS/Explicit model using shell elements.
airspring_sax1.inp
Axisymmetric ABAQUS/Explicit model.
hydrofluidairspring_3d_mem.inp
Three-dimensional ABAQUS/Standard model using membrane elements.
hydrofluidairspring_max1.inp
ABAQUS/Standard analysis using MAX1 elements with rebars.
hydrofluidairspring_mgax1.inp
ABAQUS/Standard analysis using MGAX1 elements with rebars.
References
· Dils, M., ``Air Springs vs. Air Cylinders,'' Machine Design, May 7, 1992.
· Fursdon, P. M. T., ``Modelling a Cord Reinforced Component with ABAQUS,'' 6th UK ABAQUS User Group Conference Proceedings, 1990.
Figures
Figure 1.1.9-1 A cord reinforced airspring. eliminate some of the noise.
Input files
hydrofluidairspring_3d_shell.inp
Three-dimensional ABAQUS/Standard model using shell elements.
hydrofluidairspring_axisymm.inp
Axisymmetric ABAQUS/Standard model.
airspring_s4r.inp
Three-dimensional ABAQUS/Explicit model using shell elements.
airspring_sax1.inp
Axisymmetric ABAQUS/Explicit model.
hydrofluidairspring_3d_mem.inp
Three-dimensional ABAQUS/Standard model using membrane elements.
hydrofluidairspring_max1.inp
ABAQUS/Standard analysis using MAX1 elements with rebars.
hydrofluidairspring_mgax1.inp
ABAQUS/Standard analysis using MGAX1 elements with rebars.
References
· Dils, M., ``Air Springs vs. Air Cylinders,'' Machine Design, May 7, 1992.
· Fursdon, P. M. T., ``Modelling a Cord Reinforced Component with ABAQUS,'' 6th UK ABAQUS User Group Conference Proceedings, 1990.
Figures
Figure 1.1.9-2 The airspring model cross-section.
Figure 1.1.9-3 180° model: mesh of the rubber membrane and partial view of the axisymmetric contact master surface.
Figure 1.1.9-4 180° model: mesh of the airspring cavity.
Figure 1.1.9-5 Axisymmetric model: mesh of the rubber membrane and the contact master surface.
Figure 1.1.9-2 The airspring model cross-section.
Figure 1.1.9-3 180° model: mesh of the rubber membrane and partial view of the axisymmetric contact master surface.
Figure 1.1.9-4 180° model: mesh of the airspring cavity.
Figure 1.1.9-6 Axisymmetric model: mesh of the airspring cavity.
Figure 1.1.9-7 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 1.
Figure 1.1.9-8 Axisymmetric ABAQUS/Explicit model: deformed configuration at the end of Step 1.
Figure 1.1.9-6 Axisymmetric model: mesh of the airspring cavity.
Figure 1.1.9-7 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 1.
Figure 1.1.9-9 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 2.
Figure 1.1.9-10 Axisymmetric ABAQUS/Standard model: progressive deformed configurations during Step 5.
Figure 1.1.9-11 Axisymmetric ABAQUS/Standard model: load-displacement curve for Step 5.
Figure 1.1.9-9 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 2.
Figure 1.1.9-10 Axisymmetric ABAQUS/Standard model: progressive deformed configurations during Step 5.
Figure 1.1.9-12 Axisymmetric ABAQUS/Standard model: cavity pressure versus downward displacement in Step 5.
Figure 1.1.9-13 Axisymmetric ABAQUS/Standard model: cavity volume versus downward displacement in Step 5.
Figure 1.1.9-12 Axisymmetric ABAQUS/Standard model: cavity pressure versus downward displacement in Step 5.
Figure 1.1.9-13 Axisymmetric ABAQUS/Standard model: cavity volume versus downward displacement in Step 5.
Figure 1.1.9-14 180° ABAQUS/Standard model: deformed configuration at the end of Step 6.
Figure 1.1.9-15 180° ABAQUS/Standard model: load-displacement curve for Step 6.
Figure 1.1.9-14 180° ABAQUS/Standard model: deformed configuration at the end of Step 6.
Figure 1.1.9-16 Axisymmetric ABAQUS/Explicit model: progressive deformed configurations during Step 2.
Figure 1.1.9-17 Axisymmetric ABAQUS/Explicit model: load-displacement curve for Step 2.
Figure 1.1.9-16 Axisymmetric ABAQUS/Explicit model: progressive deformed configurations during Step 2.
Figure 1.1.9-18 Axisymmetric ABAQUS/Explicit model: cavity pressure versus downward displacement in Step 2.
Figure 1.1.9-19 Axisymmetric ABAQUS/Explicit model: cavity volume versus downward displacement in Step 2.
Figure 1.1.9-18 Axisymmetric ABAQUS/Explicit model: cavity pressure versus downward displacement in Step 2.
Figure 1.1.9-19 Axisymmetric ABAQUS/Explicit model: cavity volume versus downward displacement in Step 2.
Figure 1.1.9-20 180° ABAQUS/Explicit model: deformed configuration at the end of Step 2.
Figure 1.1.9-21 180° ABAQUS/Explicit model: load-displacement for Step 2.
Sample listings
Figure 1.1.9-20 180° ABAQUS/Explicit model: deformed configuration at the end of Step 2.
Figure 1.1.9-21 180° ABAQUS/Explicit model: load-displacement for Step 2.