Dry-friction oscillators are mechanical systems with dry friction and stick-slip vibrations. In the context of control theory, the stability analysis of this type of dynamical systems is important since they exhibit nonsmooth bifurcations, or most famously a sliding–grazing bifurcation inducing abrupt chaos. This paper develops a Lyapunov-based framework to study the so-called structural stability of the system, predicting the onset of such unique bifurcations. To achieve this, the nonlinear system is first represented as a nonsmooth Takagi–Sugeno (TS) fuzzy model, and the structural stability is then formulated as linear matrix inequalities (LMI) feasibility problems with less conservative formulation. Solving the resulting LMI problem, the onset of sliding–grazing bifurcation can be accurately predicted.
Issue Section:
Research Papers
References
1.
Filippov
, A. F.
, and Arscott
, F. M.
, 1988
, Differential Equations With Discontinuous Righthand Sides: Control Systems
, Vol. 18
, Springer
, The Netherlands.2.
Leine
, R. I.
, and Nijmeijer
, H.
, 2013
, Dynamics and Bifurcations of Non-Smooth Mechanical Systems
, Vol. 18
, Springer Science & Business
, Berlin-Heidelberg Germany.10.1007/978-3-540-44398-83.
Popp
, K.
, Hinrichs
, N.
, and Oestreich
, M.
, 1995
, “Dynamical Behaviour of a Friction Oscillator With Simultaneous Self and External Excitation
,” Sadhana
, 20
(2–4
), pp. 627
–654
.10.1007/BF028232104.
Hinrichs
, N.
, Oestreich
, M.
, and Popp
, K.
, 1998
, “On the Modelling of Friction Oscillators
,” J. Sound Vib.
, 216
(3
), pp. 435
–459
.10.1006/jsvi.1998.17365.
Galvanetto
, U.
, 1999
, “Nonlinear Dynamics of Multiple Friction Oscillators
,” Comput. Methods Appl. Mech. Eng.
, 178
(3–4
), pp. 291
–306
.10.1016/S0045-7825(99)00021-36.
Di Bernardo
, M.
, Budd
, C. J.
, Champneys
, A. R.
, Kowalczyk
, P.
, Nordmark
, A. B.
, Tost
, G. O.
, and Piiroinen
, P. T.
, 2008
, “Bifurcations in Nonsmooth Dynamical Systems
,” SIAM Rev.
, 50
(4
), pp. 629
–701
.10.1137/0506250607.
Galvanetto
, U.
, 2001
, “Some Discontinuous Bifurcations in a Two-Block Stick-Slip System
,” J. Sound Vib.
, 248
(4
), pp. 653
–669
.10.1006/jsvi.2001.38098.
Dankowicz
, H.
, and Nordmark
, A. B.
, 2000
, “On the Origin and Bifurcations of Stick-Slip Oscillations
,” Phys. D
, 136
(3
), pp. 280
–302
.10.1016/S0167-2789(99)00161-X9.
Di Bernardo
, M.
, 2008
, Piecewise-Smooth Dynamical Systems: Theory and Applications
, Springer
, London, UK.10.
Leine
, R.
, and Van Campen
, D.
, 2002
, “Discontinuous Bifurcations of Periodic Solutions
,” Math. Comput. Modell.
, 36
(3
), pp. 259
–273
.10.1016/S0895-7177(02)00124-311.
Pascal
, M.
, 2008
, “Dynamics of Coupled Oscillators Excited by Dry Friction
,” ASME J. Comput. Nonlinear Dyn.
, 3
(3
), p. 031009
.10.1115/1.290827212.
Nechak
, L.
, Berger
, S.
, and Aubry
, E.
, 2013
, “Wiener–Askey and Wiener–Haar Expansions for the Analysis and Prediction of Limit Cycle Oscillations in Uncertain Nonlinear Dynamic Friction Systems
,” ASME J. Comput. Nonlinear Dyn.
, 9
(2
), p. 021007
.10.1115/1.402485113.
Di Bernardo
, M.
, Kowalczyk
, P.
, and Nordmark
, A.
, 2002
, “Bifurcations of Dynamical Systems With Sliding: Derivation of Normal-Form Mappings
,” Phys. D
, 170
(3
), pp. 175
–205
.10.1016/S0167-2789(02)00547-X14.
Nordmark
, A. B.
, 1997
, “Universal Limit Mapping in Grazing Bifurcations
,” Phys. Rev. E
, 55
(1
), pp. 266
–270
.10.1103/PhysRevE.55.26615.
Kundu
, S.
, Banerjee
, S.
, and Giaouris
, D.
, 2011
, “Vanishing Singularity in Hard Impacting Systems
,” Discrete Contin. Dyn. Syst., Ser. B
, 16
(1
), pp. 319
–332
.10.3934/dcdsb.2011.16.31916.
Dankowicz
, H.
, and Jerrelind
, J.
, 2005
, “Control of Near-Grazing Dynamics in Impact Oscillators
,” Proc. R. Soc. A
, 461
(2063
), pp. 3365
–3380
.10.1098/rspa.2005.151617.
Mehran
, K.
, Zahawi
, B.
, and Giaouris
, D.
, 2012
, “Investigation of the Near-Grazing Behavior in Hard-Impact Oscillators Using Model-Based TS Fuzzy Approach
,” Nonlinear Dyn.
, 69
(3
), pp. 1293
–1309
.10.1007/s11071-012-0348-818.
di Bernardo
, M.
, Kowalczyk
, P.
, and Nordmark
, A.
, 2003
, “Sliding Bifurcations: A Novel Mechanism for the Sudden Onset of Chaos in Dry Friction Oscillators
,” Int. J. Bifurcation Chaos
, 13
(10
), pp. 2935
–2948
.10.1142/S021812740300834X19.
Luo
, A. C.
, and Gegg
, B. C.
, 2006
, “Periodic Motions in a Periodically Forced Oscillator Moving on an Oscillating Belt With Dry Friction
,” ASME J. Comput. Nonlinear Dyn.
, 1
(3
), pp. 212
–220
.10.1115/1.219887420.
Piiroinen
, P. T.
, and Kuznetsov
, Y. A.
, 2008
, “An Event-Driven Method to Simulate Filippov Systems With Accurate Computing of Sliding Motions
,” ACM Trans. Math. Software (TOMS)
, 34
(3
), p. 13
.10.1145/1356052.135605421.
Mehran
, K.
, Giaouris
, D.
, and Zahawi
, B.
, 2010
, “Stability Analysis and Control of Nonlinear Phenomena in Boost Converters Using Model-Based Takagi–Sugeno Fuzzy Approach
,” IEEE Trans. Circuits Syst. I
, 57
(1
), pp. 200
–212
.10.1109/TCSI.2009.201938922.
Tanaka
, K.
, Ikeda
, T.
, and Wang
, H.
, 1998
, “Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Designs
,” IEEE Trans. Fuzzy Syst.
, 6
(2
), pp. 250
–265
.10.1109/91.66902323.
Tanaka
, K.
, and Wang
, H. O.
, 2001
, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach
, John Wiley & Sons
, Newark, NJ
.24.
Ting
, C.-S.
, 2006
, “Stability Analysis and Design of Takagi–Sugeno Fuzzy Systems
,” Inf. Sci.
, 176
(19
), pp. 2817
–2845
.10.1016/j.ins.2005.06.00525.
Narimani
, M.
, and Lam
, H.-K.
, 2009
, “Relaxed LMI-Based Stability Conditions for Takagi–Sugeno Fuzzy Control Systems Using Regional-Membership-Function-Shape-Dependent Analysis Approach
,” IEEE Trans. Fuzzy Syst.
, 17
(5
), pp. 1221
–1228
.10.1109/TFUZZ.2009.202595926.
Dong
, J.
, and Yang
, G.-H.
, 2011
, “Control Synthesis of T–S Fuzzy Systems Based on a New Control Scheme
,” IEEE Trans. Fuzzy Syst.
, 19
(2
), pp. 323
–338
.10.1109/TFUZZ.2010.209647027.
Yoshitake
, Y.
, and Sueoka
, A.
, 2000
, “Forced Self-Excited Vibration With Dry Friction
,” Applied Nonlinear Dynamics and Chaos of Mechanical Systems With Discontinuities
, Vol. 28
, World Scientific Publishing
, London
, pp. 237
–257
.10.1142/9789812796301_001028.
Slotine
, J.-J. E.
, and Li
, W.
, 1991
, Applied Nonlinear Control
, Prentice-Hall
, Englewood Cliffs, NJ
.29.
Boyd
, S.
, Ghaoui
, L. E.
, Feron
, E.
, and Balakrishnan
, V.
, 1994
, Linear Matrix Inequalities in System and Control Theory
, SIAM
, Philadelphia, PA
.10.1137/1.978161197077730.
Khalil
, H. K.
, and Grizzle
, J.
, 2002
, Nonlinear Systems
, Vol. 3
, Prentice Hall
, Upper Saddle River, NJ
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