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This study outlines an approach for speeding up the simulation of the dynamic response of vehicle models that include hysteretic nonlinear tire components. The method proposed replaces the hysteretic nonlinear tire model with a surrogate model that emulates the dynamic response of the actual tire. The approach is demonstrated via a dynamic simulation of a quarter vehicle model. In the proposed methodology, training information generated with a reduced number of harmonic excitations is used to construct the tire hysteretic force emulator using a Neural Network (NN) element. The proposed approach has two stages: a learning stage, followed by an embedding of the learned model into the quarter car model. The learning related main challenge stems from the attempt to capture with the NN element the behavior of a hysteretic element whose response depends on its loading history.