Due to the world increasing energy demands, optimizing the drilling system parameters such as the weight on BIT (WOB), and the structure of drill string and BIT, also vibrations and dynamic behavior of drill strings are of significant interest to researchers and energy industries. Specially, to overcome limitations of drilling operations in oil and gas industry, composite drill strings as high-tech devices are under development. In this research, the fully coupled non-linear axial vibrations of composite drill strings due to the interaction of two common BITs namely; Roller Cone (RC) and Polycrystalline Diamond Compact (PDC) with rock, considering the major non-linear terms, the drill string-wellbore contact, the different weight on BIT (WOB) and the different composite configurations using the finite element method (FEM) and the Lagrangian approach were studied. This study proved that the different configurations of composite drill strings showed specific dynamic behavior at different conditions. Therefore, composite drill string can be designed for particular purposes. Also, the results imply the remarkable effects of weight on BIT (WOB) and type of BITs on the axial vibrations of composite drill strings.