Semi-Analytical Approach for Derivation of an Equivalent Modal Friction-Damping Ratio and its Application in a Self-Excited Drilling System

Abstract

In this article, friction-based damping principles and their suitability for dampening self-excited torsional oscillations in drill strings are investigated. Analytical and semi-analytical solutions based on a modally reduced model are derived by approximating the dynamic behavior with the Harmonic Balance method. The results are validated through time-domain simulations, and limitations of the method are shown. The method enables consideration of the damper position that determines the local amplitudes experienced by the damper. In analytical solutions, the damper location is represented by the local value of the mass-normalized mode shape. The analytical approach can be used to calculate and optimize an equivalent damping ratio for every torsional mode based on the parameters and placement of the friction damper. The provided damping can then be used to estimate the stability of critical torsional modes.

A Method for the Design and Optimization of Nonlinear Tuned Damping Concepts to Mitigate Self-Excited Drill String Vibrations Using Multiple Scales Lindstedt-Poincaré

“…In addition, the large but slim design of the drill string with little available installation space leads to low reactive effects of dampers on the torsional dynamics of the drill string. In [8,16,19], a similar reduction method was used to characterize downhole vibrations and in [5,19,22] 'it is used to investigate vibration reduction strategies.…”

“…Assuming that the reactive effect of the damper on the mode is negligible due to the limited installation space in deep drilling and the assumption that no other modes near the considered mode or higher harmonics are excited by the nonlinear forces [22], a modal single degree of freedom (sdof) was derived. The modal reduction of the extended drill string model yields to ..…”

“…Increasing the damping of a system is a common approach to reduce self-excited vibration [20,21]. In [22], the effects of various friction dampers on the stability of critical HFTO modes are analyzed. The derived analytical results show the importance of damper placement regarding the multiple instable modes and multiple damper positions within the BHA.…”

“…Due to the large number of variable parameters such as the critical mode (frequency and mode shape), the position of the damper and the design of the damper, analytical solutions with regard to the effective damping range of the damper are necessary for an effective design and optimization of dampers in drill string dynamics. Therefore, as in [22], analytical and semi-analytical solutions regarding the effective damping range of the dampers must be determined. For nonlinear dampers, various methods to approximate Additionally, a maximum amplitude at the bit (φ bit ) that depends on the revolutions per minute (RPM) in the operating point…”

“…For nonlinear dampers, various methods to approximate nonlinearities exist. The harmonic balance method [23][24][25] is one possibility to approximate nonlinear parts of a differential equation that has been used for drill string dynamic [22]. Especially for tuned systems like nonlinear tuned mass dampers, methods like the Multiple Scales (MS) [26] or Lindstedt-Poincaré (LP) [27] are suitable approximation options.…”