Simulation and stability analysis of impacting systems with complete chattering

Nonlinear Dynamics

Volumn 58, Issue 1-2, 2009, Pages 85-106

  Nordmark, Arne B ^a   Piiroinen, Petri T ^b  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b NATIONAL UNIVERSITY OF IRELAND   (Ireland)

Author keywords

Bifurcations; Chattering; Discontinuity mappings; Impacting systems; Nonlinear dynamics; Numerical methods; Simulation; Stability analysis

Indexed keywords

BIFURCATIONS; CHATTERING; DISCONTINUITY MAPPINGS; IMPACTING SYSTEMS; NONLINEAR DYNAMICS; SIMULATION; STABILITY ANALYSIS;

BIFURCATION (MATHEMATICS); CONVERGENCE OF NUMERICAL METHODS; DYNAMICAL SYSTEMS; MAPPING; NUMBER THEORY;

SYSTEM STABILITY;

EID: 70349220063     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-008-9463-y     Document Type: Article

References (38)

1. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
2. Beyn, W.-J., Champneys, A.R., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., Sandstede, B.: Numerical continuation, and computation of normal forms. In: Handbook of Dynamical Systems, vol. 2, pp. 149-219. Elsevier Science, Amsterdam (2002). Chap. 4
3. A. Blakeborough 2001 An analytical approach of church bells to earthquake excitation J. Earthq. Eng. 5 69 92 10.1142/S1363246901000273
4. Brogliato, B.: Impacts in Mechanical Systems-Analysis and Modelling. Lecture Notes in Physics, vol. 551. Springer, Berlin (2000)
5. C.J. Budd F. Dux 1994 Chattering and related behaviour in impact oscillators Phil. Trans. R. Soc. Lond. A 347 365 389 0816.70018 10.1098/rsta.1994.0049
6. C.J. Budd P.T. Piiroinen 2006 Corner bifurcations in non-smoothly forced impact oscillators Physica D 220 127 145 1113.37060 10.1016/j.physd.2006.07.001 2253407 (Pubitemid 44216016)
7. F.B. Cunha D.J. Pagano U.F. Moreno 2003 Sliding bifurcations in planar variable structure systems IEEE Trans. Circuit Syst. 50 8 1129 1134 10.1109/TCSI.2003.815221 2008164
8. H. Dankowicz P.T. Piiroinen 2002 Exploiting discontinuities for stabilization of recurrent motions Dyn. Syst. 17 317 342 1063.93040 1975117
9. Dercole, F., Kuznetsov, Yu.A.: SlideCont: An Auto97 driver for sliding bifurcation analysis. Department of Mathematics, Universiteit Utrecht, The Netherlands (2002)
10. M. di Bernardo C.J. Budd A.R. Champneys 1998 Grazing, skipping and sliding: analysis of the non-smooth dynamics of the DC/DC buck converter Nonlinearity 11 858 890 1632641
11. M. di Bernardo K.H. Johansson F. Vasca 2001 Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations Int. J. Bifurc. Chaos 11 4 1121 1140 10.1142/S0218127401002584
12. M. di Bernardo P. Kowalczyk A. Nordmark 2002 Bifurcations of dynamical systems with sliding: derivation of normal-form mappings Physica D 170 175 205 1008.37029 10.1016/S0167-2789(02)00547-X 1925797 (Pubitemid 34940164)
13. M. di Bernardo C.J. Budd A.R. Champneys P. Kowalczyk A.B. Nordmark G. Olivar P.T. Piiroinen 2007 Bifurcations in non-smooth dynamical systems SIAM Rev. 50 4 629 701 10.1137/050625060
14. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise Smooth Dynamical Systems-Theory and Applications. Springer, London (2008)
15. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Yu.A., Sandstede, B., Wang, X.-J.: Auto97: Continuation and bifurcation software for ordinary differential equations (with HomCont). Computer Science, Concordia University, Montreal, Canada. ftp.cs.concordia.ca/doedel/doc/auto (1997)
16. M.H. Fredriksson D. Borglund A.B. Nordmark 1999 Experiments on the onset of impacting motion using a pipe conveying fluid Nonlinear Dyn. 19 261 271 0965.70003 10.1023/A:1008322725617
17. Glocker, C.: Set-Valued Force Laws. Lecture Notes in Applied Mechanics, vol. 1. Springer, Berlin (2001)
18. C.K. Halse R.E. Wilson M. di Bernardo M.E. Homer 2007 Coexisting solutions and bifurcations in mechanical oscillators with backlash J. Sound Vib. 305 854 885 10.1016/j.jsv.2007.05.010 2322714 (Pubitemid 47069465)
19. Jerrelind, J., Stensson, A.: Braille printer dynamics. In: ASME Design Engineering Technical Conferences, DETC99/VIB-8032 (1999)
20. P. Kowalczyk M. di Bernardo A.R. Champneys S.J. Hogan M. Homer Yu.A. Kuznetsov A.B. Nordmark P.T. Piiroinen 2006 Two-parameter non-smooth bifurcations of limit cycle: classification and open problems Int. J. Bifurc. Chaos 16 3 601 629 1141.37341 10.1142/S0218127406015015 2228836 (Pubitemid 43823092)
21. Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Berlin (2004)
22. A.B. Nordmark 1991 Non-periodic motion caused by grazing incidence in impact oscillators J. Sound Vib. 2 279 297 10.1016/0022-460X(91)90592-8
23. A.B. Nordmark 1997 Universal limit mapping in grazing bifurcations Phys. Rev. E 55 266 270 10.1103/PhysRevE.55.266
24. A.B. Nordmark 2001 Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators Nonlinearity 14 1517 1542 1001.70020 10.1088/0951-7715/14/6/306 1867090
25. A.B. Nordmark 2002 Discontinuity mappings for vector fields with higher-order continuity Dyn. Syst. 17 359 376 1032.37009 1975119
26. Nordmark, A.B.: On chattering bifurcations in 1 d.o.f. impact oscillator models. Manuscript in preparation
27. A.B. Nordmark P. Kowalczyk 2006 A codimension-two scenario of sliding solutions in grazing-sliding bifurcations Nonlinearity 19 1 26 1102.37034 10.1088/0951-7715/19/1/001 2191616 (Pubitemid 41806748)
28. Osirio, G., di Bernardo, M., Santini, S.: Chattering and complex behavior of a can-follower system. In: ENOC-2005, Fifth Euromech Nonlinear Dynamics Conference (2005)
29. F. Peterka 1974 Part 1: Theoretical analysis of n-multiple (1/n)-impact solutions CSAV Acta Tech. 19 462 473
30. F. Peterka 1974 Results of analogue computer modelling of the motion. Part 2 CSAV Acta Tech. 19 569 580 0479.70022
31. Pfeiffer, F., Glocker, Ch.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)
32. P.T. Piiroinen H.J. Dankowicz 2005 Low-cost control of repetitive gait in passive bipedal walkers Int. J. Bifurc. Chaos 15 6 1959 1973 1092.70521 10.1142/S0218127405013083 2158330 (Pubitemid 41204858)
33. L.F. Shampine M.W. Reichelt 1997 The MATLAB ODE suite SIAM J. Sci. Comput. 18 1 22 0868.65040 10.1137/S1064827594276424 1433374
34. S.W. Shaw 1986 On the dynamic response of a system with dry friction J. Sound Vib. 108 2 305 325 10.1016/S0022-460X(86)80058-X
35. S.W. Shaw P.J. Holmes 1983 Periodically forced linear oscillator with impacts: chaos and long-periodic motions Phys. Rev. Lett. 51 623 626 10.1103/PhysRevLett.51.623 714300
36. Stronge, W.J.: Impact Mechanics. Cambridge University Press, Cambridge (2000)
37. D.J. Wagg G. Karpodinis S.R. Bishop 1999 An experimental study of the impulse response of a vibro-impacting cantilever beam J. Sound Vib. 228 242 264 10.1006/jsvi.1999.2318
38. J. Zhang K.H. Johansson J. Lygeros S. Sastry 2001 Zeno hybrid system Int. J. Robust Nonlinear Control 11 5 435 451 0977.93047 10.1002/rnc.592 1822022 (Pubitemid 32371922)