Study of Vertical Non-linear Systems with Polynomial Characteristics

In our paper we considered a vertical system of rigid bodies linked one to another and to the ground by non-linear springs. We obtained the differential equations of motion using the Lagrange second order equations and, based on them, we discussed the system of equations used to obtain the equilibrium positions. Considering two of the most used cases, name them the cubic non-linear springs, and the neo-Hookean non-linear springs, we deduced the conditions for the uniqueness of the equilibrium position. We also presented the type of the equilibrium and we studied the small oscillations around the stable equilibrium position. In our paper we studied the stability of the motion, too. The linear case is obtained as a particular case of those considered previous in this paper. Finally, we considered a numerical example, which describes the vertical vibrations of an automobile.