# Static and Dynamic Analysis of Non-Pneumatic Tires Based on Experimental and Numerical Methods
## 4 Dynamic Study: Steady State Analysis  
The phenomenon of rolling structures can be studied in a steady or transient state. Although the transient analysis allows to simulate the rolling, imposing directly the value of acceleration, the computational cost is very high. Therefore, the steady state analysis, using a specific kinematic formulation to describe the phenomenon, shows a less-expensive computational cost, though it needs some mesh requirements. In this paragraph, we introduce the kinematic description at the basis of the steady state transport analysis--a specific procedure of the Abaqus environment [26]--and the guidelines for the creation of a FE model capable of meeting its requirements explained. Only in recent periods, in FE codes, objects with complex geometric shape were allowed to undergo the steady state simulation. [PERSON] et al. [27] showed the approach to submit a pneumatic tire with a detailed tread pattern to a steady state analysis. This analysis was also applied to NPTs. [PERSON] et al. [28] simulated the steady state rolling of the \"MEW\"--the NPT with a slick tread developed by [PERSON]--obtaining a good match between the experimental and FEM analysis results, for what concerns the graph of longitudinal force vs. slip ratio and contact patch. The mathematical formulation behind this analysis forced the construction of a FE model with a methodology different from the one shown in the static analysis and this step was also complicated by the complex geometry of the case study. Figure 12: Contact patch comparison: experimental pressure distribution (**a**), FE contact pressure distribution (**b**), and contact opening plot (**c**). In fact, the model must be generated starting from the rotation of a bidimensional surface or three-dimensional sector along about an axis. This requirement is a consequence of the peculiar kinematic description of the phenomenon that removes the time dependence in favor of a spatially dependent analysis, using specific coordinates. The rolling movement of an axisymmetric body is described in a reference frame that moves with the ground motion of the body and, in this moving frame, the rigid body rotation is described in spatial or Eulerian manner whilst the deformation in a Lagrangian manner, introducing the requirement of continuous lines in the circumferential direction--streamlines that define a static mesh in which the material can flow. In this case, the presence in the circumferential direction of voids between the spokes and the treads can cause nonconvergence issues because, during the rotation, the elements of these components leave the mesh defined in the static step. To overcome this problem, the spokes and the treads cannot be meshed individually, as was made for the static analysis, but together, with the inner and outer ring for the spokes and the shear band for the treads. Thus, all nodes and elements at the interface of these components must be the same and equivalenced without any kind of discontinuity. In this way, the elements and nodes of the inner and outer ring can easily flow inside the streamlines thanks to their cylindrical geometry and can transmit the deformation status to the elements and nodes of the spokes. The same is for the tread blocks. With the previous guidelines, the FE model of the NPT was made, revolving a single sector, shown in Figure 13, along the tire axis. It is recommended to choose the sector with the smallest angle to be as close as possible to the revolution of a bidimensional cross-section. In this case, the sector has a 60\({}^{\circ}\) extension due to the repetition pattern of its elements. In fact, the single pair of spokes has an angular extension of 15\({}^{\circ}\) whilst the tread pattern repeats itself every 20\({}^{\circ}\). The meshing process of the components was the same as the one shown before, aside from some changes in the spokes modeling.