vibration Article Vibration of Periodic Drill-Strings with Local Sources of Resonance Wael Akl 1 , Hajid Alsupie 2 , Sadok Sassi 3 and Amr M. Baz 4, *   Citation: Akl, W.; Alsupie, H.; Sassi, S.; Baz, A.M. Vibration of Periodic Drill-Strings with Local Sources of Resonance. Vibration 2021, 4, 586–601. https://doi.org/10.3390/ vibration4030034 Academic Editor: Rajamohan Ganesan Received: 15 June 2021 Accepted: 12 July 2021 Published: 17 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 School of Engineering and Applied Sciences, Nile University, Giza, Egypt; waelakl@gmail.com 2 Mechanical Engineering Department, College of Science, Taif University, Taif 21974, Saudi Arabia; haj_a@hotmail.com 3 Department of Mechanical and Industrial Engineering, Qatar University, Doha 2713, Qatar; sadok.sassi@qu.edu.qa 4 Mechanical Engineering Department, University of Maryland, College Park, MD 20742, USA * Correspondence: baz@umd.edu; Tel.: +1-301-405-5216 Abstract: A new class of drill-strings is proposed for attenuating undesirable vibrations to ensure effective operation. The drill-string is provided with passive periodic inserts, which are integrated with sources of local resonance (LR). The inserts make the drill-string act as a low frequency pass mechanical filter for the transmission of vibration along the drill-string. Proper design of the periodic inserts with sources of LR tend to shift these stop bands towards zones of lower frequencies to enable confining the dominant modes of vibration of the drill-string within these bands. In this manner, propagation of the vibration along the drill-string can be completely blocked. A finite element model (FEM) is developed using ANSYS to investigate the bandgap characteristics of the proposed drill-string with sources of LR. The developed FEM accounts for bending, torsional, and axial vibrations of the drill-string in order to demonstrate the effectiveness of the periodic inserts with LR in simultaneous control of these combined modes as compared to conventional solid periodic inserts, which are only limited to controlling bending vibrations. The effect of the design parameters of the periodic inserts with LR on the bandgap characteristics of the drill-string is investigated to establish guidelines of this class of drill-strings. Keywords: drill-string vibration; periodic inserts; inserts with local sources of resonance; finite element analysis; bandgap characteristics; control of multi-mode of vibration 1. Introduction Considerable attention has been devoted, during the past years, to develop a thorough understanding of the complex nature of the vibration of drill-strings to devise effective means for controlling the associated destructive effects as presented, for example, in the comprehensive account of Spanos et al. [1]. Distinct among the exerted efforts are those dealing with modeling the vibration of drill-strings under the influence of combined bending, axial, and torsional modes of vibration [2–8]. The developed models vary from simple continuous system models to the more complex finite element models, which are subjected to various loading and excitation conditions such as axial bit bouncing [9,10], torsional stick-slip [11–15], and whirl vibrations [16]. These models have been utilized and exercised to predict the time and frequency response characteristics, the stability and bifurcation analysis, the limit cycle conditions, equilibrium points, and self-excited vibration [11–15]. In parallel to these attempts, several passive vibration mitigation attempts have been considered to attenuate the vibration of drill-strings. Examples of these attempts include the nonlinear energy sink approach [17], the magneto-rheological damping [18], the tunable vibration absorber [19], and the anti-stalling vibration attenuation devices [20]. More advanced active vibration control approaches have also been considered includ- ing simple control [21], optimal control [22], and robust control approaches [23,24]. Vibration 2021, 4, 586–601. https://doi.org/10.3390/vibration4030034 https://www.mdpi.com/journal/vibration