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Top1. Introduction
Vehicle dynamics control systems (VDCS) exist on the most modern vehicles and play important roles in vehicle ride, stability, and safety. For example, Anti-lock brake system (ABS) is used to allow the vehicle to follow the desired steering angle while intense braking is applied (Yu et al., 2002; Morteza, et al, 2015). In addition, the ABS helps reducing the stopping distance of a vehicle compared to the conventional braking system. The Active suspension control system (ASC) is used to improve the quality of the vehicle ride and reduce the vertical acceleration (Yue et al., 1988; Alleyne and Hedrick, 1995; Jongsang, et al., 2015). From the view of vehicle transportation safety, nowadays, occupant safety becomes one of the most important research areas and the automotive industry increased their efforts to enhance the safety of vehicles. Seat belts, airbags, and advanced driver assistant systems (ADAS) are used to prevent a vehicle crash or mitigate vehicle collision when a crash occurs.
To evaluate the crashworthiness, real crash tests or vehicle modelling are carried out. Due to the complexity and the high cost of crash tests, vehicle modelling is commonly used in the first stage of development. Vehicle modelling can be mainly classified as finite element and mathematical modelling. Finite element models of vehicles are increasingly used in preliminary design analysis, component design, and roadside hardware design (Belytschko, 1992). However, finite element modelling is also costly and slow in its simulation analysis. Mathematical modelling produces very quick results and it can be accurately used for unlimited numbers of different types of vehicles in case of vehicle-to-barrier crash tests (Kamal, 1970).
Using mathematical models in crash simulation is useful at the first design concept because rapid analysis is required at this stage. In addition, the well-known advantage of mathematical modelling provides a quick simulation analysis compared with FE models. Vehicle crash structures are designed to be able to absorb the crash energy and control vehicle deformations, therefore simple mathematical models are used to represent the vehicle front structure (Emori, 1968). In this model, the vehicle mass is represented as a lumped mass and the vehicle structure is represented as a spring in a simple model to simulate a frontal and rear-end vehicle collision processes. Also, other analyses and simulations of vehicle-to-barrier impact using a simple mass spring model were established by Kamal (1970) and widely extended by Elmarakbi and Zu (2005, 2007) to include smart-front structures. To achieve enhanced occupant safety, the crash energy management system was explored by Khattab (2010). This study, using a simple lumped-parameter model, discussed the applicability of providing variable energy-absorbing properties as a function of the impact speed.