Analysis and control of the dynamical response of a higher order drifting oscillator. (English) Zbl 1402.93125
Summary: This paper studies a position feedback control strategy for controlling a higher order drifting oscillator which could be used in modelling vibro-impact drilling. Special attention is given to two control issues, eliminating bistability and suppressing chaos, which may cause inefficient and unstable drilling. Numerical continuation methods implemented via the continuation platform COCO are adopted to investigate the dynamical response of the system. Our analyses show that the proposed controller is capable of eliminating coexisting attractors and mitigating chaotic behaviour of the system, providing that its feedback control gain is chosen properly. Our investigations also reveal that, when the slider’s property modelling the drilled formation changes, the rate of penetration for the controlled drilling can be significantly improved.
MSC:
| 93B52 | Feedback control |
| 94C30 | Applications of design theory to circuits and networks |
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34H10 | Chaos control for problems involving ordinary differential equations |
Keywords:
vibro-impact drilling; position feedback control; progression optimization; bistability; chaos controlSoftware:
COCOReferences:
| [1] | Samuel, GR, Percussion drilling. Is it a lost technique? A review. In \(SPE 35240, The Permian Basin Oil & Gas Recovery Conf., Midland, TX, 27-29 March 1996\). Richardson, TX: Society of Petroleum Engineers, (1996) |
| [2] | Rabia, H., A unified prediction model for percussive and rotary drilling, Min. Sci. Technol., 2, 207-216, (1985) · doi:10.1016/S0167-9031(85)90149-5 |
| [3] | Wiercigroch, M., Resonance enhanced drilling: method and apparatus. Patent no. WO2007141550, (2007) |
| [4] | Pavlovskaia, E.; Wiercigroch, M.; Grebogi, C., Modeling of an impact system with a drift, Phys. Rev. E, 64, 056224, (2001) · doi:10.1103/PhysRevE.64.056224 |
| [5] | Pavlovskaia, E.; Wiercigroch, M., Periodic solution finder for an impact oscillator with a drift, J. Sound Vib., 267, 893-911, (2003) · doi:10.1016/S0022-460X(03)00193-7 |
| [6] | Luo, GW; Lv, XH, Dynamics of a plastic impact system with oscillatory and progressive motions, Int. J. Nonlinear Mech., 43, 100-110, (2008) · Zbl 1203.74114 · doi:10.1016/j.ijnonlinmec.2007.10.008 |
| [7] | Nguyen, VD; Woo, KC; Pavlovskaia, E., Experimental study and mathematical modelling of a new of vibro-impact moling device, Int. J. Nonlinear Mech., 43, 542-550, (2008) · Zbl 1203.74008 · doi:10.1016/j.ijnonlinmec.2007.10.003 |
| [8] | Ajibose, OK; Wiercigroch, M.; Pavlovskaia, E.; Akisanya, AR, Global and local dynamics of drifting oscillator for different contact force models, Int. J. Nonlinear Mech., 45, 850-858, (2010) · doi:10.1016/j.ijnonlinmec.2009.11.017 |
| [9] | Cao, Q.; Wiercigroch, M.; Pavlovskaia, E.; Yang, S., Bifurcations and the penetrating rate analysis of a model for percussive drilling, Acta Mech. Sin., 26, 467-475, (2010) · Zbl 1269.70032 · doi:10.1007/s10409-010-0346-3 |
| [10] | Aguiar, RR; Weber, HI, Mathematical modeling and experimental investigation of an embedded vibro-impact system, Nonlinear Dyn., 65, 317-334, (2011) · doi:10.1007/s11071-010-9894-0 |
| [11] | Pavlovskaia, E.; Wiercigroch, M., Modelling of vibro-impact system driven by beat frequency, Int. J. Mech. Sci., 45, 623-641, (2003) · Zbl 1045.70517 · doi:10.1016/S0020-7403(03)00113-9 |
| [12] | Pavlovskaia, E.; Hendry, D.; Wiercigroch, M., Modelling of high frequency vibro-impact drilling, Int. J. Mech. Sci., 91, 110-119, (2015) · doi:10.1016/j.ijmecsci.2013.08.009 |
| [13] | Cavanough, GL; Kochanek, M.; Cunningham, JB; Gipps, ID, A self-optimizing control system for hard rock percussive drilling, IEEE/ASME Trans. Mech., 13, 153-157, (2008) · doi:10.1109/TMECH.2008.918477 |
| [14] | de Souza, SLT; Caldas, IL, Controlling chaotic orbits in mechanical systems with impacts, Chaos Solitons Fractals, 19, 171-178, (2004) · Zbl 1086.37045 · doi:10.1016/S0960-0779(03)00129-2 |
| [15] | Dankowicz, H.; Jerrelind, J., Control of near-grazing dynamics in impact oscillators, Proc. R. Soc. A, 461, 3365-3380, (2005) · Zbl 1334.70046 · doi:10.1098/rspa.2005.1516 |
| [16] | de Souza, SLT; Caldas, IL; Viana, RL, Damping control law for a chaotic impact oscillator, Chaos Solitons Fractals, 32, 745-750, (2007) · doi:10.1016/j.chaos.2005.11.046 |
| [17] | de Souza, SLT; Caldas, IL; Viana, RL; Balthazar, JM, Control and chaos for vibro-impact and non-ideal oscillators, J. Theor. Appl. Mech., 46, 641-664, (2008) |
| [18] | Luo, GW; Lv, XH, Controlling bifurcation and chaos of a plastic impact oscillator, Nonlinear Anal.: Real World Appl., 10, 2047-2061, (2009) · Zbl 1163.34343 · doi:10.1016/j.nonrwa.2008.03.010 |
| [19] | Ing, J.; Pavlovskaia, E.; Wiercigroch, M.; Banerjee, S., Experimental study of impact oscillator with one-sided elastic constraint, Phil. Trans. R. Soc. A, 366, 679-704, (2008) · Zbl 1153.74302 · doi:10.1098/rsta.2007.2122 |
| [20] | Ing, J.; Pavlovskaia, E.; Wiercigroch, M.; Banerjee, S., Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing, Physica D, 239, 312-321, (2010) · Zbl 1183.37091 · doi:10.1016/j.physd.2009.11.009 |
| [21] | Liu, Y.; Wiercigroch, M.; Ing, J.; Pavlovskaia, E., Intermittent control of co-existing attractors, Phil. Trans. R. Soc. A, 371, 0120428, (2013) · Zbl 1327.93209 · doi:10.1098/rsta.2012.0428 |
| [22] | Veldman, DWM; Fey, RHB; Zwart, H., Impulsive steering between coexisting stable periodic solutions with an application to vibrating plates, J. Comput. Nonlinear Dyn., 12, 011013, (2016) · doi:10.1115/1.4034273 |
| [23] | Hosseinloo, AH; Slotine, JJ; Turitsyn, K., Robust and adaptive control of coexisting attractors in nonlinear vibratory energy harvesters, J. Vib. Control, (2017) · doi:10.1177/1077546316688992 |
| [24] | Liu, Y.; Páez Chávez, J., Controlling coexisting attractors of an impacting system via linear augmentation, Physica D, 348, 1-11, (2017) · Zbl 1376.93050 · doi:10.1016/j.physd.2017.02.018 |
| [25] | Liu, Y.; Pavlovskaia, E.; Hendry, D.; Wiercigroch, M., Vibro-impact responses of capsule system with various friction models, Int. J. Mech. Sci., 72, 39-54, (2013) · doi:10.1016/j.ijmecsci.2013.03.009 |
| [26] | Liu, Y.; Pavlovskaia, E.; Wiercigroch, M.; Peng, ZK, Forward and backward motion control of a vibro-impact capsule system, Int. J. Nonlinear Mech., 70, 30-46, (2015) · doi:10.1016/j.ijnonlinmec.2014.10.009 |
| [27] | Liu, Y.; Páez Chávez, J., Controlling multistability in a vibro-impact capsule system, Nonlinear Dyn., 88, 1289-1304, (2017) · doi:10.1007/s11071-016-3310-3 |
| [28] | Páez Chávez, J.; Pavlovskaia, EE; Wiercigroch, M., Bifurcation analysis of a piecewise-linear impact oscillator with drift, Nonlinear. Dyn., 77, 213-227, (2014) · doi:10.1007/s11071-014-1285-5 |
| [29] | Pavlovskaia, E.; Wiercigroch, M.; Woo, KC; Rodger, AA, Modelling of ground moling dynamics by an impact oscillator with a frictional slider, Meccanica, 38, 85-97, (2003) · Zbl 1018.70503 · doi:10.1023/A:1022023502199 |
| [30] | Pavlovskaia, E.; Wiercigroch, M., Analytical drift reconstruction in visco-elastic impact oscillators, Chaos, Solitons Fractals, 19, 151-161, (2004) · Zbl 1129.70327 · doi:10.1016/S0960-0779(03)00128-0 |
| [31] | Wiercigroch, M.; Liu, Y., Control method. US Patent no. 20160245064, (2016) |
| [32] | Dankowicz, H.; Schilder, F., \(Recipes for continuation\), pp. 213-245. Computational Science and Engineering. Philadelphia, PA: SIAM, (2013) · Zbl 1277.65037 |
| [33] | Dankowicz, H.; Schilder, F., An extended continuation problem for bifurcation analysis in the presence of constraints, J. Comput. Nonlinear Dyn., 6, 031003, (2011) · doi:10.1115/1.4002684 |
| [34] | Kuznetsov, YA, Elements of applied bifurcation theory. Applied Mathematical Sciences, vol. 112, (2004), Springer · Zbl 1082.37002 |
| [35] | Banerjee, S.; Ing, J.; Pavlovskaia, EE; Wiercigroch, M.; Reddy, R., Invisible grazings and dangerous bifurcations in impacting systems: the problem of narrow-band chaos, Phys. Rev. E, 79, 037201, (2009) · doi:10.1103/PhysRevE.79.037201 |
| [36] | Páez Chávez, J.; Wiercigroch, M., Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model, Commun. Nonlinear Sci. Numer. Simul., 18, 2571-2580, (2013) · Zbl 1304.70012 · doi:10.1016/j.cnsns.2012.12.007 |
| [37] | di Bernardo, M.; Budd, CJ; Champneys, AR; Kowalczyk, P.; Nordmark, A.; Tost, G.; Piiroinen, PT, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50, 629-701, (2008) · Zbl 1168.34006 · doi:10.1137/050625060 |
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