Isolated resonances and nonlinear damping

  • Giuseppe Habib
  • Giuseppe I. Cirillo
  • Gaetan Kerschen
Original Paper
  • 39 Downloads

Abstract

We analyze isolated resonance curves (IRCs) in single-degree-of-freedom systems possessing nonlinear damping. Through the combination of singularity theory and the averaging method, the onset and merging of IRCs, which coincide to isola and simple bifurcation singularities, respectively, can be analytically predicted. Numerical simulations confirm the accuracy of the analytical developments. Another important finding of this paper is that we unveil a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs. Considering a damping force possessing several minima and maxima confirms the general validity of the analytical result. It also evidences a very complex scenario for which different IRCs are created, co-exist and then merge together to form a super IRC which eventually merges with the main resonance peak.

Keywords

Isolated resonance curves Isola Singularity theory Nonlinear damping 

References

  1. 1.
    Abramson, H.N.: Response curves for a system with softening restoring force. J. Appl. Mech. 22(3), 434–435 (1955)MathSciNetGoogle Scholar
  2. 2.
    Bouc, R.: Influence du cycle d’hystérésis sur la résonance non linéaire d’un circuit série. Colloq. Inter. du CNRS 148, 483–489 (1964)Google Scholar
  3. 3.
    Hayashi, C.: The influence of hysteresis on nonlinear resonance. J. Frankl. Inst. 281(5), 379–386 (1966)CrossRefGoogle Scholar
  4. 4.
    Hagedorn, P.: Parametric resonance in certain nonlinear systems, In: Periodic Orbits, Stability and Resonances, pp. 482–492. Springer (1970)Google Scholar
  5. 5.
    Iwan, W.D.: Steady-state dynamic response of a limited slip system. J. Appl. Mech. 35(2), 322–326 (1968)CrossRefGoogle Scholar
  6. 6.
    Furuike, D.M.: Dynamic response of hysteretic systems with application to a system containing limited slip. California Inst. Technology (1971)Google Scholar
  7. 7.
    Iwan, W.D., Furuike, D.M.: The transient and steady-state response of a hereditary system. Int. J. Non-Linear Mech. 8(4), 395–406 (1973)CrossRefMATHGoogle Scholar
  8. 8.
    Koenigsberg, W., Dunn, J.: Jump resonant frequency islands in nonlinear feedback control systems. IEEE Trans. Autom. Control 20(2), 208–217 (1975)CrossRefGoogle Scholar
  9. 9.
    Hirai, K., Sawai, N.: Jump phenomena and frequency islands in nonlinear feedback systems (in Japanese), In: Working Group for Nonlinear Probl., Inst. Electron. Commun., pp. 39–48 (1977)Google Scholar
  10. 10.
    Hirai, K., Sawai, N.: A general criterion for jump resonance of nonlinear control systems. IEEE Trans. Autom. Control 23(5), 896–901 (1978)CrossRefGoogle Scholar
  11. 11.
    Fukuma, A., Matsubara, M.: Jump resonance in nonlinear feedback systems-part I: approximate analysis by the describing-function method. IEEE Trans. Autom. Control 23(5), 891–896 (1978)CrossRefGoogle Scholar
  12. 12.
    Capecchi, D., Vestroni, F.: Periodic response of a class of hysteretic oscillators. Int. J. Non-Linear Mech. 25(2–3), 309–317 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Doole, S., Hogan, S.: A piece wise linear suspension bridge model: nonlinear dynamics and orbit continuation. Dyn. Stab. Syst. 11(1), 19–47 (1996)CrossRefMATHGoogle Scholar
  14. 14.
    Duan, C., Rook, T.E., Singh, R.: Sub-harmonic resonance in a nearly pre-loaded mechanical oscillator. Nonlinear Dyn. 50(3), 639–650 (2007)CrossRefMATHGoogle Scholar
  15. 15.
    Duan, C., Singh, R.: Isolated sub-harmonic resonance branch in the frequency response of an oscillator with slight asymmetry in the clearance. J. Sound Vib. 314(1), 12–18 (2008)CrossRefGoogle Scholar
  16. 16.
    Elmegård, M., Krauskopf, B., Osinga, H., Starke, J., Thomsen, J.J.: Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment. Nonlinear Dyn. 77(3), 951–966 (2014)CrossRefGoogle Scholar
  17. 17.
    Bureau, E., Schilder, F., Elmegård, M., Santos, I.F., Thomsen, J.J., Starke, J.: Experimental bifurcation analysis of an impact oscillator-determining stability. J. Sound Vib. 333(21), 5464–5474 (2014)CrossRefGoogle Scholar
  18. 18.
    Lee, S., Howell, S., Raman, A., Reifenberger, R.: Nonlinear dynamic perspectives on dynamic force microscopy. Ultramicroscopy 97(1), 185–198 (2003)CrossRefGoogle Scholar
  19. 19.
    Misra, S., Dankowicz, H., Paul, M.R.: Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy. Phys. D: Nonlinear Phenom. 239(1), 33–43 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (1995)CrossRefMATHGoogle Scholar
  21. 21.
    Perret-Liaudet, J., Rigaud, E.: Superharmonic resonance of order 2 for an impacting hertzian contact oscillator: theory and experiments. J. Comput. Nonlinear Dyn. 2(2), 190–196 (2007)CrossRefGoogle Scholar
  22. 22.
    Rega, G.: Nonlinear vibrations of suspended cables-part II: deterministic phenomena. Appl. Mech. Rev. 57(6), 479–514 (2004)CrossRefGoogle Scholar
  23. 23.
    Lenci, S., Ruzziconi, L.: Nonlinear phenomena in the single-mode dynamics of a cable-supported beam. Int. J. Bifurc. Chaos 19(03), 923–945 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    DiBerardino, L.A., Dankowicz, H.: Accounting for nonlinearities in open-loop protocols for symmetry fault compensation. J. Comput. Nonlinear Dyn. 9(2), 021002 (2014)CrossRefGoogle Scholar
  25. 25.
    Arroyo, S.I., Zanette, D.H.: Duffing revisited: phase-shift control and internal resonance in self-sustained oscillators. Eur. Phys. J. B 89(1), 1–8 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mangussi, F., Zanette, D.H.: Internal resonance in a vibrating beam: a zoo of nonlinear resonance peaks. PLoS ONE 11(9), e0162365 (2016)CrossRefGoogle Scholar
  27. 27.
    Takács, D., Stépán, G., Hogan, S.J.: Isolated large amplitude periodic motions of towed rigid wheels. Nonlinear Dyn. 52(1), 27–34 (2008)CrossRefMATHGoogle Scholar
  28. 28.
    Luongo, A., Zulli, D.: Parametric, external and self-excitation of a tower under turbulent wind flow. J. Sound Vib. 330(13), 3057–3069 (2011)CrossRefGoogle Scholar
  29. 29.
    Zulli, D., Luongo, A.: Bifurcation and stability of a two-tower system under wind-induced parametric, external and self-excitation. J. Sound Vib. 331(2), 365–383 (2012)CrossRefGoogle Scholar
  30. 30.
    Dimitriadis, G.: Introduction to Nonlinear Aeroelasticity. Wiley, Hoboken (2017)CrossRefGoogle Scholar
  31. 31.
    Van Heerden, C.: Autothermic processes. Ind. Eng. Chem. 45(6), 1242–1247 (1953)CrossRefGoogle Scholar
  32. 32.
    Hlaváček, V., Kubíček, M., Jelinek, J.: Modeling of chemical reactors-XVIII stability and oscillatory behaviour of the CSTR. Chem. Eng. Sci. 25(9), 1441–1461 (1970)CrossRefGoogle Scholar
  33. 33.
    Uppal, A., Ray, W., Poore, A.: The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. Chem. Eng. Sci. 31(3), 205–214 (1976)CrossRefGoogle Scholar
  34. 34.
    Razón, L.F., Schmitz, R.A.: Multiplicities and instabilities in chemically reacting systems-a review. Chem. Eng. Sci. 42(5), 1005–1047 (1987)CrossRefGoogle Scholar
  35. 35.
    Doedel, E.: The computer-aided bifurcation analysis of predator-prey models. J. Math. Biol. 20(1), 1–14 (1984)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Pavlou, S., Kevrekidis, I.: Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies. Math. Biosci. 108(1), 1–55 (1992)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Starosvetsky, Y., Gendelman, O.: Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning. J. Sound Vib. 315(3), 746–765 (2008)CrossRefGoogle Scholar
  38. 38.
    Starosvetsky, Y., Gendelman, O.: Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system. J. Sound Vib. 312(1), 234–256 (2008)CrossRefGoogle Scholar
  39. 39.
    Gourc, E., Michon, G., Seguy, S., Berlioz, A.: Experimental investigation and design optimization of targeted energy transfer under periodic forcing. J. Vib. Acoust. 136(2), 021021 (2014)CrossRefGoogle Scholar
  40. 40.
    Starosvetsky, Y., Gendelman, O.: Vibration absorption in systems with a nonlinear energy sink: nonlinear damping. J. Sound Vib. 324(3), 916–939 (2009)CrossRefGoogle Scholar
  41. 41.
    Habib, G., Detroux, T., Viguié, R., Kerschen, G.: Nonlinear generalization of Den Hartog’s equal-peak method. Mech. Syst. Signal Process. 52, 17–28 (2015)CrossRefGoogle Scholar
  42. 42.
    Habib, G., Kerschen, G.: A principle of similarity for nonlinear vibration absorbers. Phys. D: Nonlinear Phenom. 332, 1–8 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Detroux, T., Habib, G., Masset, L., Kerschen, G.: Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber. Mech. Syst. Signal Process. 60, 799–809 (2015)CrossRefGoogle Scholar
  44. 44.
    Alexander, N.A., Schilder, F.: Exploring the performance of a nonlinear tuned mass damper. J. Sound Vib. 319(1), 445–462 (2009)CrossRefGoogle Scholar
  45. 45.
    Cirillo, G., Habib, G., Kerschen, G., Sepulchre, R.: Analysis and design of nonlinear resonances via singularity theory. J. Sound Vib. 392, 295–306 (2017)CrossRefGoogle Scholar
  46. 46.
    Gatti, G., Brennan, M.J., Kovacic, I.: On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system. Phys. D: Nonlinear Phenom. 239(10), 591–599 (2010)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Gatti, G., Kovacic, I., Brennan, M.J.: On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator. J. Sound Vib. 329(10), 1823–1835 (2010)CrossRefGoogle Scholar
  48. 48.
    Gatti, G., Brennan, M.J.: On the effects of system parameters on the response of a harmonically excited system consisting of weakly coupled nonlinear and linear oscillators. J. Sound Vib. 330(18), 4538–4550 (2011)CrossRefGoogle Scholar
  49. 49.
    Gatti, G.: Uncovering inner detached resonance curves in coupled oscillators with nonlinearity. J. Sound Vib. 372, 239–254 (2016)CrossRefGoogle Scholar
  50. 50.
    Gatti, G., Brennan, M.J.: Inner detached frequency response curves: an experimental study. J. Sound Vib. 396, 246–254 (2017)CrossRefGoogle Scholar
  51. 51.
    Detroux, T., Renson, L., Masset, L., Kerschen, G.: The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Comput. Methods Appl. Mech. Eng. 296, 18–38 (2015)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Kuether, R.J., Renson, L., Detroux, T., Grappasonni, C., Kerschen, G., Allen, M.S.: Nonlinear normal modes, modal interactions and isolated resonance curves. J. Sound Vib. 351, 299–310 (2015)CrossRefGoogle Scholar
  53. 53.
    Hill, T., Neild, S., Cammarano, A.: An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures. J. Sound Vib. 379, 150–165 (2016)CrossRefGoogle Scholar
  54. 54.
    Shaw, A., Hill, T., Neild, S., Friswell, M.: Periodic responses of a structure with 3:1 internal resonance. Mech. Syst. Signal Process. 81, 19–34 (2016)CrossRefGoogle Scholar
  55. 55.
    Hill, T., Cammarano, A., Neild, S., Barton, D.: Identifying the significance of nonlinear normal modes. In: Proceedings of Royal Society Part A, vol. 473, pp. 20160789. The Royal Society (2017)Google Scholar
  56. 56.
    Noël, J.-P., Detroux, T., Masset, L., Kerschen, G., Virgin, L.: Isolated response curves in a base-excited, two-degree-of-freedom, nonlinear system. In: ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V006T10A043–V006T10A043. American Society of Mechanical Engineers (2015)Google Scholar
  57. 57.
    Detroux, T., Noël, J.-P., Kerschen, G., Virgin, L.N.: Experimental study of isolated response curves in a two-degree-of-freedom nonlinear system. In: Nonlinear Dynamics, vol. 1, pp. 229–235. Springer (2016)Google Scholar
  58. 58.
    Gourdon, E., Alexander, N.A., Taylor, C.A., Lamarque, C.-H., Pernot, S.: Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J. Sound Vib. 300(3), 522–551 (2007)CrossRefGoogle Scholar
  59. 59.
    Spence, A., Jepson, A.D.: The numerical calculation of cusps, bifurcation points and isola formation points in two parameter problems, In: Numerical Methods for the Bifurcation Problems, pp. 502–514. Springer (1984)Google Scholar
  60. 60.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, New York (1985)CrossRefMATHGoogle Scholar
  61. 61.
    Troger, H., Steindl, A.: Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Applied Scientists. Springer-Verlag, Wien, New York (1991)CrossRefMATHGoogle Scholar
  62. 62.
    Drazin, P.G.: Nonlinear Systems, vol. 10. Cambridge University Press, Cambridge (1992)CrossRefMATHGoogle Scholar
  63. 63.
    Janovskỳ, V., Plecháč, P.: Computer-aided analysis of imperfect bifurcation diagrams, I. Simple bifurcation point and isola formation centre. SIAM J. Num. Anal. 29(2), 498–512 (1992)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Seydel, R.: Practical Bifurcation and Stability Analysis. Springer, New York (2009)MATHGoogle Scholar
  65. 65.
    Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (2007)MATHGoogle Scholar
  66. 66.
    Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.-C.: Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009)CrossRefGoogle Scholar
  67. 67.
    Holmes, P., Rand, D.: Bifurcations of the forced van der Pol oscillator. Q. Appl. Math. 35(4), 495–509 (1978)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Parlitz, U., Lauterborn, W.: Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys. Rev. A 36(3), 1428 (1987)CrossRefGoogle Scholar
  69. 69.
    Mettin, R., Parlitz, U., Lauterborn, W.: Bifurcation structure of the driven van der Pol oscillator. Int. J. Bifurc. Chaos 3(06), 1529–1555 (1993)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Guckenheimer, J., Hoffman, K., Weckesser, W.: The forced van der Pol equation I: The slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst. 2(1), 1–35 (2003)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Hill, T., Cammarano, A., Neild, S., Wagg, D.: An analytical method for the optimisation of weakly nonlinear systems, In: Proceedings of EURODYN 2014, pp. 1981–1988. Sheffield (2014)Google Scholar

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Authors and Affiliations

  1. 1.Department of Applied MechanicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK
  3. 3.Department of Aerospace and Mechanical EngineeringUniversity of LiegeLiègeBelgium

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