A system for estimating the effective longitudinal behavior of a drill bit fastened to the end of a drill string and driven into rotation in a well by surface driving means, wherein a computing unit comprises means for a non-linear physical modeling of the drilling process based on general mechanics equations and represented by a state matrix in specific modes, characterized in that parameters of said modeling means are identified by taking into account parameters of said well and of said string, in that the computing unit comprises means designed for linearization of said model about a working point, means for reducing said linearized model so as to keep only the pertinent modes of the state matrix of said model, means for real-time computation of the vibratory longitudinal displacement of the drill bit or of the stress applied to the bit, by means of the modeling means once linearized and reduced, and means for measuring at least one parameter linked with the displacement of the string at the surface.