<title>Finite element of a rod in an immovable coordinate system</title>

The Influence of Friction Forces on Longitudinal Wave Propagation in a Stuck Drill String in a Borehole

“…According to Equation (4), the increment of the longitudinal force is determined as follows: (8) The equation of dynamic equilibrium of an infinitely small string element is written as follows: (9) in this case, (10) Taking account of correlations (1)-(3), (5), (6), (8) reduce equation (9) to the form (11) where (12) From correlations (1)-(3), (5), (6)-(8), (10), we obtain the intensity of dry friction forces distribution along the drill string length at arbitrary instant of time (13) The boundary condition of the integration of the equation in partial derivatives (11), which should be fulfilled at the lower end of the drill string, is written as (14) where (15) Similarly, we can write the boundary condition for the upper end of the drill string in the following way: (16) ISSN 0208-7774 T R I B O L O G I A 6/2018 where (17) in this case, (18) So, the analysis of the wave phenomena in a stuck drill string consist in solving the equation in partial derivatives (11) for boundary conditions (14), and (16) taking account of analytical correlations (12), (13), (15), and (17). The pulse loading of a unit of mass m 2 present in form (18). The formulated problem is solved in the following sequence: at first discretization of Equation (11) where the space coordinate is used, owing to this, the mathematical model of dynamic process is written in the form of a non-linear system of normal differential equations; after that, the numerical integration of the obtained system is done applying widely used software.…”

“…Pulse loading of the mechanical system is determined according to (17) and (18). In this case, P s = 425.26 kN; P d max = 400.00 kN; and, Δt = 0.02 s.…”