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The drilling process is considered one of the most expensive and complex in the oil and gas industry. This paper is focused on the dynamics of a drill-string in which stick-slip oscillations can happen, reducing op- eration performance and increasing the costs. The main concern of this work is to analyze the passage from a soft to a harder rock layer in the system response and how uncertainties influence this response. A one degree- of-freedom model is proposed where the top speed is prescribed, the drilling mud is represented by a viscous friction and the bit-rock interaction is represented by a non-linear model. First, the response of the system is analyzed when there is no soil transition, i.e., a sin- gle rock layer is considered. The transition between a soft to a harder rock layer is then included and the de- terministic and stochastic responses of the system are assessed. The results show that the different scenarios can significantly change the system response.