The depth of exploration and production wells during the development of oil and gas fields can exceed 5-10 km, and the insufficient strength and reliability of the drill string often limits the possibility of further increasing labour productivity. In addition, accidents at wells lead to significant material costs. Therefore, the design and operation of drill strings, as well as the implementation of work to eliminate complications during drilling, should be carried out on the basis of a science-based approach, taking into account the latest advances in the field of drill string dynamics and technologies related to work to eliminate stuck problems. Unsteady vibrations of inhomogeneous distributed systems is a very complex problem in the mechanics of a deformable solid and the theory of vibrations. In connection with the rapid development of the extractive industries, solving this problem is of particular importance. This is due to ensuring the strength of drill string structures with increasing power and speed of drilling units and mechanisms. The study of the problem revealed a number of little-studied problems, which include issues of nonlinear interaction of the column with the surrounding soil, accompanied by various types of complications (sticking, pipe ruptures, etc.), wave and oscillatory processes in the elements of the drilling dynamic system (DDS), finding the boundaries of the interaction areas of the column with the walls of the well using low-cost techniques. The purpose of the work is theoretical research of wave and oscillatory processes in the pipes of the drilling assembly and the dynamics of associated drill strings interacting with the well wall. In this work, a homogeneous rod was used as a drill string model. The drill string consists of a drill pipe string (DPC) and a bottom drill string assembly (BDSA), which includes a bit, a downhole motor, wellbore forming elements: calibrators, centralizers, sections of drill collars (DC), the main purpose of which is to creating an axial load on the bit. The DPC consists of sections of drill pipes that are identical in their characteristics (type, outer diameter, wall thickness, joints used, etc.). Therefore, in general, the drill string is a heterogeneous structure.
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