The traffic congestion is growing in urban area these days due to the rapidly increasing number of automobiles. One of the traffic problem solutions is to increase the flow rate of a particular traffic artery. To make the vehicle smaller and narrower might be a good idea. We developed a new conceptual Narrow Tilting Vehicle (NTV) that has one front wheel and two rear wheels. The proposed NTV is very narrow and all three wheels can tilt to improve the stability during turning. We also derive a new switching dynamical model of the NTV. This model considers several states including normal running, temporary running with one rear wheel not on the ground and totally falling down. Based on this model, a simulation platform is established, which is useful to test different control methods and to investigate dangerous driving situations in case hurt drivers in real experiments. The effectiveness of proposed dynamical model is verified through comparison study of simulations and corresponding experiments.