The tightening process of bolted joints shows a highly dynamic behavior, which depends on numerous factors and can therefore be challenging to be adequately quantitatively reproduced. The presented model solves this problem by combining a multi-body model of the joint with the sophisticated LuGre friction model. This allows for a detailed simulation while simultaneously reducing the computational work in comparison to a standard finite element simulation. We demonstrate that, with a constant tightening angular velocity, the progression of the tightening torque and the preload force can be described by three constants. The model further allows to implement custom torque sources, screw types, and materials to further extend its capabilities. In this contribution, we focus on the basic relationships of acting torques on the joint and friction-induced vibrations during the tightening process. Furthermore, effects of typical geometric and material changes on the stick-slip frequency are demonstrated and discussed.