Dynamics of rods in curved holes

Modelling of rods that are inserted and rotated around its own axis within curved conduits is a relevant industrial problem typically encountered in well construction operations, but is also relevant for various medical applications. In these problems, the rod can undergo large rotations of its cross-section relative to a global Cartesian frame due to the constraints imposed by the curved conduit walls. As a result, modelling these problems typically requires solving large-rotation operators, and small-angle simplifications are not valid globally. Nevertheless, the lateral rotations of the rod cross-section relative to a local frame that follows the curved conduit are generally small, owing to the geometric constraints of the problem. In this article, we propose a mathematical framework derived from the geometrically exact beam theory (GEBT) that describes rod kinematics relative to the curved centerline of the conduit, referred to as the hole. We then develop a finite element (FEM) model from the framework that utilizes small-angle assumptions for the lateral angles relative to the hole centerline. Importantly, the formulation imposes no restrictions on large axial rotation, allowing the rod to rotate freely inside the hole. The proposed approach is verified against a corotational beam model that employs a large rotation formulation, which shows excellent agreement. Finally, a case study from drilling is presented to validate and demonstrate the capabilities of the model.