Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments

Physica D: Nonlinear Phenomena

Volumn 238, Issue 18, 2009, Pages 1868-1896

  Lee, Young S ^a   Nucera, Francesco ^b   Vakakis, Alexander F ^c   McFarland, D Michael ^d   Bergman, Lawrence A ^d  

^a Horeshoe Cir Jett Hall   (United States)

^b UNIVERSITY MEDITERRANEA OF REGGIO CALABRIA   (Italy)

^c UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^d United States of America   (United States)

Author keywords

Passive and broadband vibration absorber; Targeted energy transfer (TET); Vibro impact nonlinear energy sink (VI NES)

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); DYNAMICS; ENERGY TRANSFER; OSCILLATORS (ELECTRONIC); OSCILLATORS (MECHANICAL);

BROADBAND VIBRATION; ESSENTIAL NONLINEARITY; HAMILTONIAN SYSTEMS; NONLINEAR ENERGY SINK; QUASI-PERIODIC ORBITS; SINGLE DEGREE OF FREEDOMS; TARGETED ENERGY TRANSFERS; ZERO INITIAL CONDITIONS;

HAMILTONIANS;

EID: 68549134930     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2009.06.013     Document Type: Article

References (72)

1. Gendelman O., Manevitch L., Vakakis A., and M'Closkey R. Energy pumping in coupled mechanical oscillators, Part I: Dynamics of the underlying hamiltonian systems. Trans. ASME, J. Appl. Mech. 68 (2001) 34-41
2. Vakakis A., and Gendelman O. Energy pumping in coupled mechanical oscillators, Part II: Resonance capture. Trans. ASME, J. Appl. Mech. 68 (2001) 42-48
3. Lee Y., Vakakis A., Bergman L., McFarland D.M., Kerschen G., Nucera F., Tsakirtzis S., and Panagopoulos P. Passive nonlinear targeted energy transfer (TET) and its applications to vibration absorption: A review. Proc. Inst. Mech. Eng. Part K, J. Multi-Body Dynam. 222 2 (2008) 77-134
4. Vakakis A., Gendelman O., Bergman L., McFarland D.M., Kerschen G., and Lee Y. Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems: I and II (2008), Springer-Verlag, Berlin, New York
5. Nucera F., Vakakis A., McFarland D.M., Bergman L., and Kerschen G. Targeted energy transfers in vibro-impact oscillators for seismic mitigation. Nonlinear Dynam. 50 3 (2007) 651-677
6. Nucera F., Iacono F.L., McFarland D.M., Bergman L., and Vakakis A. Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results. J. Sound Vib. 313 1-2 (2008) 57-76
7. F. Nucera, D.M. McFarland, L. Bergman, A. Vakakis, Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Computational results, J. Sound Vib. (in press)
8. Georgiadis F., Vakakis A., McFarland D.M., and Bergman L. Shock isolation through passive energy pumping caused by nonsmooth nonlinearities. Internat. J. Bifur. Chaos 15 6 (2005) 1989-2001
9. F. Georgiades, Nonlinear localization and targeted energy transfer phenomena in vibrating systems with smooth and non-smooth stiffness nonlinearities, Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 2006
10. Karayannis I., Vakakis A., and Georgiades F. Vibro-impact attachments as shock absorbers. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 222 (2008) 1899-1908
11. Shaw S., and Rand R. The transition to chaos in a simple mechanical system. Internat. J. Non-Linear Mech. 24 (1989) 41-56
12. Babitsky V. Theory of Vibro-impact Systems (1998), Springer Verlag, Berlin, New York
13. Brogliato B. Non-smooth Mechanics (1999), Springer Verlag, Berlin, New York
14. Kryzhevich S., and Pliss V. Chaotic modes of oscillation of a vibro-impact system. J. Appl. Math. Mech. (PMM) 69 (2005) 13-26
15. Thota P., Zhao X., and Dankowicz H. Co-dimension-two grazing bifurcations in single-degree-of-freedom impact oscillators. J. Comput. Nonlinear Dynam. 1 (2006) 328-335
16. Thota P., and Dankowicz H. Continuous and discontinuous grazing bifurcations in impacting oscillators. Physica D 214 (2006) 187-197
17. Masri S., and Caughey T. On the stability of the impact damper. Trans. ASME, J. Appl. Mech. 33 3 (1966) 586-592
18. Masri S., and Ibrahim A. Response of the impact damper to nonstationary random excitation. J. Acoust. Soc. Am. 53 1 (1973) 200-211
19. Filippov A. Differential Equations with Discontinuous Righthand Sides (1988), Kluwer Academic Publishers
20. Pfeiffer F., and Glocker C. Contacts in multibody systems. J. Appl. Math. Mech. 64 5 (2000) 773-782
21. Blazejczyk-Okolewska B. Analysis of an impact damper of vibration. Chaos Solitons Fractals 12 (2001) 1983-1988
22. Leine R., and Nijmeijer H. Dynamics and Bifurcations in Non-Smooth Mechanical Systems (2004), Springer-Verlag, Berlin, New York
23. Peterka F., and Blazejczyk-Okolewska B. Some aspects of the dynamical behavior of the impact damper. J. Vib. Control 11 (2005) 459-479
24. Sun J.Q., and Luo A. Bifurcation and Chaos in Complex Systems (2006), Elsevier
25. Shaw S., and Holmes P. A periodically forced piecewise linear oscillator. J. Sound Vib. 90 1 (1983) 129-155
26. Shaw J., and Shaw S. The onset of chaos in a two-dof impacting system. Trans. ASME, J. Appl. Mech. 56 (1989) 168-174
27. Shaw S. The dynamics of a harmonically excited system having rigid amplitude constraints. I: Subharmonic motions and local bifurcations. II: Chaotic motions and global bifurcations. Trans. ASME, J. Appl. Mech. 52 2 (1985) 453-464
28. Gorelyshev I., and Neishtadt A. On the adiabatic perturbation theory for subsystems with impacts. J. Appl. Math. Mech. 70 (2006) 4-17
29. Mikhlin Y., Vakakis A., and Salenger G. Direct and inverse problems encountered in vibro-impact oscillations of a discrete system. J. Sound Vib. 216 (1998) 227-250
30. Leine R., Campen D.V., and van de Vrande B. Bifurcations in nonlinear discontinuous systems. Nonlinear Dynam. 23 (2000) 105-164
31. Czolczynski K., and Kapitaniak T. On the existence of a stable periodic solution of two impacting oscillators with damping. Internat. J. Bifur. Chaos 14 11 (2004) 3931-3947
32. Wen G., Xie J., and Xu D. Onset of degenerate hopf bifurcation of a vibro-impact oscillator. Trans. ASME, J. Appl. Mech. 71 (2004) 579-581
33. Dupac M., and Marghitu D. Nonlinear dynamics of a flexible mechanism with impact. J. Sound Vib. 289 (2006) 952-966
34. Leine R. Bifurcations of equilibria in non-smooth continuous systems. Physica D 223 (2006) 121-137
35. Lin W., Ni Q., and Huang Y. Bifurcations and chaos in a forced cantilever system with impacts. J. Sound Vib. 296 (2006) 1068-1078
36. Halse C., Wilson R., di Bernardo M., and Homer M. Coexisting solutions and bifurcations in mechanical oscillators with backlash. J. Sound Vib. 305 (2007) 854-885
37. Luo G., Zhang Y., Chu Y., and Zhang J. Co-dimension-two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops. Nonlinear Anal. 8 (2007) 1272-1292
38. Zhuravlev V. A method of analyzing vibro-impact systems using special functions. Izv. Akad. Nauk. SSSR, MTT 2 (1976) 30-34
39. Zhuravlev V. Investigation of some vibro-impact systems by the method of non-smooth transformations. Izv. Akad. Nauk. SSSR, MTT 6 (1977) 24-28
40. Azeez M., Vakakis A., and Manevitch L. Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees-of-freedom. J. Appl. Math. Mech. (PMM) 63 4 (1999) 527-530
41. Pilipchuk V. The calculation of strongly nonlinear systems close to vibration-impact systems. J. Appl. Math. Mech. (PMM) 49 (1985) 572-578
42. Pinnington R. Energy dissipation prediction in a line of colliding oscillators. J. Sound Vib. 268 (2003) 361-384
43. Salapaka S., Dahleh M., and Mezic I. On the dynamics of a harmonic oscillator undergoing impacts with a vibrating platform. Nonlinear Dynam. 24 (2001) 333-358
44. Quinn D. The dynamics of two parametrically excited pendula with impacts. Internat. J. Bifur. Chaos 15 6 (2005) 1975-1988
45. Li K., and Darby A. Experiments on the effect of an impact damper on a MDOF system. J. Vib. Control 12 5 (2006) 445-464
46. Murphy K., and Morrison T. Grazing instabilities and post-bifurcation behavior in an impacting string. J. Acoust. Soc. Am. 111 2 (2002) 884-892
47. Hu B., and Schiehlen W. Multi-time scale simulation for impact systems: From wave propagation to rigid-body motion. Arch. Appl. Mech. (Ingenieur Archiv) 72 11 (2003) 885-898
48. Sampaio R., and Soize C. On measures of nonlinearity effects for uncertain dynamical systems - Applications to a vibro-impact system. J. Sound Vib. 303 (2007) 659-674
49. Azeez M., and Vakakis A. Proper orthogonal decomposition (POD) of a class of vibro-impact oscillations. J. Sound Vib. 240 5 (2001) 859-889
50. Vedenova Y., Manevitch L., and Pilipchuk V. The normal vibrations of a string with concentrated masses on nonlinearly elastic supports. Prik. Mat. Mech. 49 2 (1985) 203-211
51. Gendelman O.V. Modeling of inelastic impacts with the help of smooth-functions. Chaos Solitons Fractals 28 2 (2006) 522-526
52. Meimukhin D., and Gendelman O. Response regimes of integrable damped strongly nonlinear oscillator under impact periodic forcing. Chaos Solitons Fractals 32 2 (2007) 405-414
53. Nayeri R., Masri S., and Caffrey J. Studies of the performance of multi-unit impact dampers under stochastic excitation. J. Vib. Acoust. 129 (2007) 239-251
54. Namachchivaya S., and Park J. Stochastic dynamics of impact oscillators. Trans. ASME, J. Appl. Mech. 72 (2005) 862-870
55. Wagg D. A note on coefficient of restitution models including the effects of impact induced vibration. J. Sound Vib. 300 (2007) 1071-1078
56. Shaw S., and Pierre C. The dynamic response of tuned impact absorbers for rotating flexible structures. J. Comput. Nonlinear Dynam. 1 (2006) 13-24
57. Lee Y., Kerschen G., Vakakis A., Panagopoulos P., Bergman L., and McFarland D.M. Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204 1-2 (2005) 41-69
58. Kerschen G., Lee Y., Vakakis A., McFarland D.M., and Bergman L. Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM J. Appl. Math. 66 2 (2006) 648-679
59. Pilipchuk V., Vakakis A., and Azeez M. Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT). Physica D 100 (1997) 145-164
60. A. Vakakis, Analysis and identification of linear and nonlinear normal modes in vibrating systems, Ph.D. Thesis, California Institute of Technology, 1991
61. Roberts S., and Shipman J. Two-Point Boundary Value Problems: Shooting Methods (1972), American Elsevier Publishing Company, Inc., New York
62. Ha S.N. A nonlinear shooting method for two-point boundary value problems. Comput. Math. Appl. 42 10-11 (2001) 1411-1420
63. Attili B.S., and Syam M.I. Efficient shooting method for solving two point boundary value problems. Chaos Solitons Fractals 35 5 (2008) 895-903
64. Manevitch L., Mikhlin Y., and Pilipchuk V. The Method of Normal Modes for Essentially Nonlinear Systems (1989), Nauka, Moscow
65. Pilipchuk V. Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformation. J. Sound Vib. 192 (1996) 43-64
66. Vakakis A., Manevitch L., Mikhlin Y., Pilipchuk V., and Zevin A. Normal Modes and Localization in Nonlinear Systems (1996), John Wiley & Sons, Inc.
67. Salenger G., and Vakakis A. Discreteness effects in the forced dynamics of a string on a periodic array of non-linear supports. Internat. J. Non-Linear Mech. 33 (1998) 659-673
68. Kerschen G., Vakakis A., Lee Y., Mcfarland D.M., Kowtko J., and Bergman L. Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1:1 resonance manifold and transient bridging orbits. Nonlinear Dynam. 42 3 (2005) 283-303
69. Kerschen G., Gendelman O., Vakakis A.F., Bergman L.A., and McFarland D.M. Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 13 5 (2008) 959-978
70. Huang N., Shen Z., Long S., Wu M., Shih H., Zheng Q., Yen N.-C., Tung C., and Liu H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. London Ser. A. Math. Phys. Sci. 454 (1998) 903-995
71. Kerschen G., Vakakis A., Lee Y., McFarland D.M., and Bergman L. Toward a fundamental understanding of the Hilbert-Huang transform in nonlinear structural dynamics. J. Vib. Control 14 1-2 (2008) 77-105
72. G. Rilling, P. Flandrin, P. Gono̧alvès, On empirical mode decomposition and its algorithms, in: IEEE-Eurasip Workshop on Nonlinear Signal and Image Processing, Grado, Italy, June 2003