This paper investigates the possibility of reproducing the self-sustained oscillation of the tongue tip in alveolar trills. The interest is to study the articulatory and phonatory configurations that are required to produce alveolar trills. Using a realistic geometry of the vocal tract, derived from cineMRI data of a real speaker, the paper studies the mechanical behavior of a lumped two-mass model of the tongue tip. Then, the paper proposes a solution to simulate the incomplete occlusion of the vocal tract during linguopalatal contacts by adding a lateral acoustic waveguide. Finally, the simulation framework is used to study the impact of a set of parameters on the characteristic features of the produced alveolar trills. It shows that the production of trills is favored when the distance between the equilibrium position of the tongue tip and the hard palate in the alveolar zone is less than 1 mm, but without linguopalatal contact, and when the glottis is fully adducted.