The mechanical stress-strain behavior of the leaf springs was calculated by FEM analysis. Another elastic leaf spring model was also developed for multi-body vehicle systems of a sport utility vehicle to simulate the axle wind-up under severe braking [10]. A nonlinear FE formulation newsletter subscribe based on the floating frame of reference approach was introduced with a full FE model of leaf springs with contact and friction. When contact and friction are considered, nonlinear model analysis is considered instead of linear analysis. For nonlinear model analysis, various models such as gun control system were optimized through Pareto optimal solution [11] and electrohydrostatic actuator through signal compression method [12]. The nonlinear model is preferred to be solved in dynamic scheme where static analysis could not encounter the friction, material, and geometric nonlinearities.

The most implemented algorithms in dynamic FE analysis (FEA) are the implicit and the explicit schemes. In implicit dynamic simulation, an extension of the Newmark method known as ��-HHT is used as a default time integrator [13]. Mousseau et al. implemented the implicit dynamic schemes to predict the handling performance of a vehicle [14]. This approach is time efficient and yields reasonable results. However, the explicit dynamic method derived from the Newmark scheme was also widely adopted in dynamic analysis [15, 16]. An explicit dynamic simulation for the stamping part of automotive components was performed [17]. The explicit method shows stability of convergence during simulation.

Both the implicit and the explicit methods have their pros and cons. The explicit technique entails a lower cost; however, given a slow case, the solutions are unstable. Given the same condition, the implicit method provides more accurate results [18]. The simulation of the crimping process, which uses both the implicit and the explicit techniques, was conducted by Kugener [19]. The simulation results indicated that the explicit method is superior to the implicit method especially when numerous contacts are considered. Other than the mentioned two schemes, it is worth mentioning that the new developed approach semi-implicit finite difference scheme is implemented to analyze the second law of thermodynamics of fluid [20].The design of a parabolic leaf spring in a bus presents a challenge to engineers given very complex and limited considerations.

Road conditions and the driving behavior of the drivers subject the leaf springs to varying loading conditions, at times severely damaging the leaf springs. Currently, leaf spring designs focus solely on the load-carrying capabilities or relative vertical stiffness. As mentioned in other previous studies, the design of the leaf spring with vertical stiffness only GSK-3 is insufficient when catastrophic failures have the possibility of occurrence.