3.1.6 Import of a steady-state rolling tire

Product: ABAQUS/Explicit

This example illustrates the use of the results transfer capability in ABAQUS (“Transferring results between ABAQUS/Explicit and ABAQUS/Standard,” Section 7.7.2 of the ABAQUS Analysis User's Manual) to import results from an ABAQUS/Standard *STEADY STATE TRANSPORT analysis to ABAQUS/Explicit to simulate transient rolling. Examples of loading during transient rolling include impact of a tire with an obstacle or vehicle acceleration. This problem analyzes the impact of a tire with a curb. Because of the small stable time increment necessary for the explicit dynamic procedure and the large time scales involved in simulating quasi-static and steady-state loading, simulating quasi-static inflation loading and steady-state rolling in ABAQUS/Standard provides significant cost savings over performing these simulations in ABAQUS/Explicit. Moreover, the cost to obtain a steady-state rolling simulation in ABAQUS/Explicit increases with rolling speed, whereas in ABAQUS/Standard the cost is independent of the magnitude of the rolling speed.

Problem description and model definition

The model used in this example differs slightly from that used in “Symmetric results transfer for a static tire analysis,” Section 3.1.1. Since only elements common to both ABAQUS/Standard and ABAQUS/Explicit can be imported, reduced-integration solid elements are used in this example and nodes in the bead area are attached to rigid elements representing the rim. A plot of the tire before impact with the curb, a 0.025 m high step, is shown in Figure 3.1.6–1. The large rigid body rotation involved in a transient rolling analysis necessitates the use of the nondefault second-order-accurate kinematic formulation in the explicit dynamic analysis. In addition, the enhanced hourglass control algorithm is used instead of the default integral viscoelastic approach. The *SECTION CONTROLS option is used to specify these nondefault options. To associate the solid elements with a section controls option, it is necessary to specify the CONTROLS parameter on the *SOLID SECTION option at the axisymmetric stage, since section definitions are transferred automatically and cannot be modified during symmetric model generation and import. This parameter is ignored during the ABAQUS/Standard stages of the analysis.

The inflation and footprint preloads are applied in a series of general analysis steps identical to those described in “Symmetric results transfer for a static tire analysis,” Section 3.1.1. The *SYMMETRIC MODEL GENERATION and *SYMMETRIC RESULTS TRANSFER options are used to exploit the symmetric nature of the structure and loading.

The repeated dynamic impact of tire nodes as they come into contact with the road is an unavoidable source of the high frequency noise that can be seen in the reaction force at the rim. Stiffness-proportional damping is used to reduce such high frequency noise in the solution. The tradeoff involved in using stiffness-proportional damping is the adverse impact on the stable time increment. However, in this model the rebar dictate the stable time increment owing to their relatively high stiffness. Thus, it is possible to use a reasonable amount of damping in the matrix material without significant increase in solution cost.

Loading

An inflation load of 200 kPa is applied to the axisymmetric half-tire model in importrolling_axi_half.inp. This load is followed by a footprint load of 1650 N applied to the three-dimensional half-tire model in importrolling_symmetric.inp, and subsequently results are transferred to the full tire model with the complete footprint load of 3300 N.

The rolling analysis involves rolling the tire up to free rolling conditions. As in “Steady-state rolling analysis of a tire,” Section 3.1.2, a translational velocity of 10 km/h is applied with a rotational velocity of 9.023 rad/s, which has been shown in the previous example to be the combination at free rolling conditions. However, unlike “Steady-state rolling analysis of a tire,” Section 3.1.2, inertia loads are accounted for in this example, since the objective is to import results to a transient dynamic analysis in which inertial effects should be considered during impact with the curb. Viscoelastic effects are not accounted for in the rubber material due to limitations with the *IMPORT option.

During the transient dynamic analysis, the tire is moved forward with a prescribed velocity of 10 km/h and the vehicle load is applied to the rim reference node. The tire is allowed to rotate freely about the axle. All other degrees of freedom at the road and rim reference nodes are held fixed.

Results and discussion

Results are imported and the analysis begins at time = 3 sec. Due to differences in formulation between ABAQUS/Explicit and ABAQUS/Standard, oscillations are set up in the solution at the beginning of the import analysis. A plot of the rotational velocity at the rim (Figure 3.1.6–2) shows that as the solution progresses, the oscillations decrease. The rotational velocity begins to converge toward approximately 9 rad/s, which is close to 9.023 rad/s, the free rolling velocity obtained from the previous *STEADY STATE TRANSPORT analysis. Impact with the curb is initiated after almost one full rotation of the tire subsequent to import, at approximately = 3.69 sec, which provides sufficient time for the initial oscillations to subside to acceptable levels. The vertical response of the rim reference node can be seen in Figure 3.1.6–3 and shows oscillations in the tire after impact. The contact patch and the stresses in the belts under steady-state rolling conditions in the transient dynamic solution compare well with the direct steady-state solution from ABAQUS/Standard. The pressure in the footprint can be seen in Figure 3.1.6–4, and the stresses in the belts are plotted in Figure 3.1.6–5. A plot of the shear stress during impact in Figure 3.1.6–6 shows that, as expected, the maximum stress is set up in the shoulder region.