Development of a two-dimensional model of a compliant non-pneumatic tire

An analytical model for a compliant non-pneumatic tire on frictionless, rigid ground is presented. The tire model consists of a thin flexible annular band and spokes that connect the band to a rigid hub. The annular band is modeled using curved beam theory that takes into account deformations due to bending, shearing and circumferential extension. The effect of the spokes, which are distributed continuously in the model and act as linear springs, is accounted for only in tension, which introduces a nonlinear response. The quasi-static, two-dimensional analysis focuses on how the contact patch, vertical tire stiffness and rolling resistance are affected by the stiffness properties of the band and the spokes. A Fourier series representation of the shear strain in the annular band and the complex modulus of the material were used to predict rolling resistance due to steady state rolling. From the analysis point of view, when the wheel is loaded at its hub, the following three distinct regions develop: (1) a support region where the hub hangs by the spokes from the upper part of the flexible band, (2) a free surface region where the spokes buckle and have no effect, and (3) a contact region where the flexible band is supported by the ground without the effect of the spokes. The angular bounds of these three regions are determined by the spoke angle and the contact angle, which are respectively the angle at which the spokes start to engage in tension and the angle that defines the edge of contact. Closed-form expressions of contact stress, stress-resultants and displacements at the centroids of the cross-sections of the flexible band are expressed in terms of these angles, which must be determined numerically. A thorough parametric analysis of quantities of interest for the tire is presented, which can be used to help support the optimal and rational design of compliant non-pneumatic tires. The model was validated by comparison with two computational models using the commercial finite element software ABAQUS and by experimental rolling resistance data.