In our paper we consider a model of half of automotive for which the stiffness has a polynomial expression. Different degrees were considered for the polynomials which describe the stiffness. In the most general considered case the polynomials are taken as sums of a first degree term and a term of arbitrary degree. The linear stiffness is a particular case in this situation. We determine the equations of motion and the equilibrium positions. We consider the horizontal position of equilibrium in the situation in which the center of weight is not situated at the middle of the automotive. Depending on the polynomial expressions of the stiffness, the automotive may have one or more equilibrium positions. We present the conditions from which one may determine the stability of these positions of equilibrium. We also determine some conditions for which the equilibrium position is unique. A numerical example will highlight the theory. Some results obtained by considered the damping are also presented.