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The impact of a drill pipe on the wall of a borehole is treated as a time-dependent force acting laterally on the pipe assuming the existence of periodic motion. The impact force is estimated by Hertz theory of impact. The solution obtained must satisfy the constraint conditions to approximate the impact action.
Transverse vibration of drill pipe is investigated here to yield some notion of the stresses induced in the pipe by the motion. The problem is highly non-linear in nature not only due to the non-uniformity of the drill string, inelasticity of the wall of the borehole, nonlinearity of the various forces induced by the presence of the non-Newtonian drilling mud, etc., and also due to the notion itself if the wall is replaced by a spring. From an engineering point of view, a simple model constructed by neglecting all the non-linearities due to materials will reveal some aspects of the effect on stresses due to motion alone.
Some similarities can be observed between the vibration of drill pipe and the simple one-dimensional mass-spring system with clearance if the elastic impact between pipe and wall is replaced by a non-linear spring [see Fig. 1]. In general, there is no assurance of the existence of a simple periodic motion even for the one-dimensional simple system if viscous damping is included and an arbitrary force is applied. However, under certain conditions, a simple periodic motion can be obtained. For instance, if the energy fed into the system is such that it balances the energy dissipation due to damping at any moment, then the system can be regarded as an undamped system. For such a system, a simple periodic motion with fixed mode does exist.
On the assumption that a prolonged periodic motion deserves more attention on account of the fatigue effect on the material, and also that a suitably constructed periodic motion serves as an average of some random or highly complicated notion over a certain range, this paper treats the case of simple periodic motion of the system. A sample section of pipe which vibrates in a fixed mode, between two adjacent nodal points, is isolated from the drill string. This sample section is considered as being simply supported. The impact is considered as a time dependent force acting laterally on the pipe. Thus the impact problem is changed to a forced vibration problem under the assumption of simple periodic motion. However, the solutions obtained from the forced vibration system must satisfy a set of constraint conditions in order to interpret it as the solution of the impact problem.
If the drill string is considered as a uniform cylinder and the borehole is treated as a vertical straight hole, then the sample section and a typical pipe element can be regarded as shown in Fig. 1.
In Fig. 1, the lateral force term FL includes the impact force, the inertia force of the pipe and the inertia force of the fluid contained in the pipe.