Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

Nonlinear Analysis: Real World Applications

Volumn 8, Issue 4, 2007, Pages 1272-1292

  Luo, G W ^a   Zhang, Y L ^a   Chu, Y D ^a   Zhang, J G ^a  

^a LANZHOU JIAOTONG UNIVERSITY   (China)

Author keywords

Clearance; Codimension two bifurcation; Periodic motion; Vibration

Indexed keywords

CHAOS THEORY; MOTION ESTIMATION; THEOREM PROVING; VIBRATION ANALYSIS;

CODIMENSION TWO BIFURCATION; PERIODIC MOTION; PITCHFORK BIFURCATION; SYMMETRICAL RIGID STOPS;

HOPF BIFURCATION;

EID: 34047249923     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2006.07.003     Document Type: Article

References (25)

1. Aidanpää J.O., and Gupta B.R. Periodic and chaotic behavior of a threshold-limited two-degree-of-freedom system. J. Sound Vib. 165 2 (1993) 305-327
2. Balachandran B. Dynamics of an elastic structure excited by harmonic and harmonic impactor motions. J. Vib. Control 9 3-4 (2003) 265-279
3. Budd C., Dux F., and Cliffe A. The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators. J. Sound Vib. 184 3 (1995) 475-502
4. Carr J. Applications of centre manifold theory. Applied Mathematical Sciences vol. 35 (1981), Springer, Berlin-Heidelberg, New York
5. de Souza S.L.T., and Caldas I.L. Calculation of Lyapunov exponents in systems with impacts. Chaos, Solitons Fractals 19 3 (2004) 569-579
6. Feigin M.I. The increasingly complex structure of the bifurcation tree of a piecewise-smooth system. J. Appl. Math. Mech. 59 6 (1995) 853-863
7. Feng Q., and He H. Modeling of the mean Poincare map on a class of random impact oscillators. Euro. J. Mech. A/Solids 22 2 (2003) 267-281
8. Huang Z.L., Liu Z.H., and Zhu W.Q. Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations. J. Sound Vib. 275 1-2 (2004) 223-240
9. Ivanov A.P. Stabilization of an impact oscillator near grazing incidence owing to resonance. J. Sound Vib. 162 3 (1993) 562-565
10. Kanso E., and Papadopoulos P. Pseudo-rigid ball impact on an oscillating rigid foundation. Int. J. Non-Linear Mech. 39 7 (2004) 1129-1145
11. Knudsen J., and Massih A.R. Dynamic stability of weakly damped oscillators with elastic impacts and wear. J. Sound Vib. 263 1 (2003) 175-204
12. Kuznetsov Y.A. Elements of Applied Bifurcation Theory (1998), Springer, New York
13. Leine R.I., Van Campen D.H., and Glocker Ch. Nonlinear dynamics and modeling of various wooden toys with impact and friction. JVC/J. Vib. Control 9 1-2 (2003) 25-78
14. Luo A.C.J. An unsymmetrical motion in a horizontal impact oscillator. J. Vib. Acoust. Trans. ASME 124 3 (2002) 420-426
15. Luo G.W. Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops. Phys. Lett. A 323 (2004) 210-217
16. Luo G.W., and Xie J.H. Bifurcations and chaos in a system with impacts. Physica D 148 (2001) 183-200
17. Luo G.W., and Xie J.H. Hopf bifurcations and chaos of a two-degree-of- freedom vibro-impact system in two strong resonance cases. Int. J. Nonlinear Mech. 37 1 (2002) 19-34
18. Mikhlin Y.V. Direct and inverse problems encountered in vibro-impact oscillations of a discrete system. J. Sound Vib. 216 2 (1998) 227-250
19. Murphy K.D., and Morrison T.M. Grazing instabilities and post-bifurcation behavior in an impacting string. J. Acoust. Soc. Am. 111 2 (2002) 884-892
20. Nordmark A.B. Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145 2 (1991) 279-297
21. Peterka P., and Vacik J. Transition to chaotic motion in mechanical systems with impacts. J. Sound Vib. 154 1 (1992) 95-115
22. Wagg D.J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator. Chaos, Solitons Fractals 22 3 (2004) 541-548
23. Yu K., and Luo A.C.J. The periodic impact responses and stability of a human body in a vehicle traveling on rough terrain. J. Sound Vib. 272 1-2 (2004) 267-286
24. G.W. Luo, J.N. Yu, J.G. Zhang, Periodic-impact motions and bifurcations of a dual component system. J. Nonlinear Anal.: Real World Appl. 7(4) (2006) 813-828.
25. G.W. Luo, Hopf-flip bifurcations of vibratory systems with impacts. J. Nonlinear Anal.: Real World Appl. 7 (5) (2006) 1029-1041.