By recasting the rolling tire system as a large deformation nonlinear eigenvalue problem, this paper develops a methodology capable of handling the critical speed/standing wave response. Based on a moderate/small deformation superposed on large finite element formulation, tangent properties are developed to enable the establishment of the appropriate critical speed eigenvalue problem. The methodology enables the handling of large deformation kinematics material nonlinearity, rotational inertial fields, as well as contact boundary conditions. Employing the model, the results of a benchmark example involving the NASA space shuttle main tires are presented. This includes evaluating the effects of rotation, material stiffness, pressurization and large deformation contact on the critical speed properties.