# Static and Dynamic Analysis of Non-Pneumatic Tires Based on Experimental and Numerical Methods
## 1 Introduction
[10] studied the hexagonal honeycomb structure developing three models varying the geometric dimensions of the hexagon. They noted that even small changes in honeycomb's geometry strongly affect the vertical displacement values; in particular, the increase in the spoke's thickness causes a reduction in spokes' deformation. [PERSON] et al. [11] developed a 3D FE model of a commercially available NPT for designing the proper radial spoke. Further, a test campaign on specimens taken from the NPT was developed to obtain the hyperelastic constants that describe the material behavior. The numerical simulations were aimed at selecting the optimal thickness to replicate the radial characteristics of the tire. For the same NPT, [PERSON] et al. [12] studied the possibility of using 3D printing technique to build the complex shape of spoke. They compared the mechanical properties of 3D printing material with the one of the actual spokes. They found that the performances of the 3D printed spoke do not differ significantly from the ones of the original NPT, so the 3D printing technique can be used in future developments. [PERSON] et al. [13] developed a new concept named \"MEW\", i.e., mechanical elastic wheel, whose peculiarity was represented by the use of mechanical hinges for the spokes. In their work, they performed numerical and experimental studies with the aim of investigate the wheel's overall stiffness and its relationship with the number of hinges used. An unconventional spokes design was proposed by [PERSON] et al. [14]. Studying the kangaroo's lower limb structure, they proposed spokes having the same shape and proportions of the kangaroo's thighs and calves and developed numerical simulation to compare the overall stiffness of the NPT designed with a pneumatic tire showing that the former has better load-bearing characteristics, better handling performance, and better comfort. Besides the comparative tests and innovative designs of specific elements of the NPT, researchers also focused on the overall performances. [PERSON] et al. [15] estimated the energy consumption of a pneumatic and non-pneumatic tire, chosen for size and application, evaluating experimentally the rolling resistance of both. Results showed that for the smaller values of load, the NPT is characterized by lower or quite the same rolling resistance. [PERSON] et al. [16] studied the effect of geometric and material parameters of non-pneumatic tire elements on overall performances. In particular, the effects of design variables such as the thickness of spokes, shear band, and the material of spokes on radial stiffness and rolling resistance were analyzed. [PERSON] et al. [17], using the finite element analysis (FEA), investigated both static and dynamic behavior of NPTs with different honeycomb spoke structures to find the optimal geometric characteristics that minimize the rolling resistance, a key factor for the fuel consumption. All the aforementioned works used the FEA, a powerful tool for engineers since it can be used to design NPTs [18] by predicting the performance parameters upon different testing conditions with varying sizes, geometries, and materials of the NPT itself. With FEA, tires can be analyzed in static condition, whose main purpose is the evaluation of the radial stiffness and stress-deformation distribution, and dynamic condition, in which the tire is rolling and the interaction forces between the tire and the road are evaluated. In recent years, to perform dynamic simulation, transient FEA has been extensively applied using an explicit FE code whose computational cost is very high [19]. However, a steady state rolling analysis, which uses an implicit time integration method, is becoming widely preferred due to its effectiveness and efficiency [20]. To the author's knowledge, no studies about the dynamic behavior of a non-pneumatic tire applying the steady state analysis to an NPT with plate spoke structures and detailed tread pattern have been conducted.