Summary
Stick-slip oscillations in drilling systems are a well-known example of self-excited oscillations established due to nonlinear discontinuous forces resulting from bit-rock and drillstring-wellbore interaction. Since the drillstring typically consists of multiple components with different inertia and/or stiffness, stick-slip oscillations have been observed at different frequencies in experimental data. However, analytical modelling of this phenomena is predominantly limited to the fundamental mode of oscillation, by using a lumped spring-masss model for the drilling system. In this study, we consider a multi degree-of-freedom model subjected to a discontinuous nonlinear force and investigate the establishment of limit cycles as a function of system and forcing parameters. Using numerical root finding algorithms and the method of averaging, we obtain a semi-analytical expression for the limit cycle amplitude(s), as well as the stability of the limit cycles. We observe that limit cycles can be established in any single natural mode of oscillation of the structure, or in a combination of modes depending upon the particular system parameters, which qualitatively match the results observed in numerical simulations of realistic drilling systems. Further, we also observe that in some cases, there can be multiple stable limit cycles that have distinct basins of attraction. These results will help in obtaining a map to predict a priori the nature of stick-slip oscillations that maybe realized for different drillstring configurations and help mitigate vibration-related failures while drilling.
