Advertisement

Edge states and frequency response in nonlinear forced-damped model of valve spring

  • Majdi Gzal
  • O. V. GendelmanEmail author
Original paper
  • 57 Downloads

Abstract

This study explores the nonlinear dynamics of helical compression valve springs. To this end, the spring is mathematically modeled as a finite nonhomogenous one-dimensional mass–spring–damper discrete chain. Periodic displacement, which mimics the actual camshaft profile, is assumed at the upper end of the chain, while the other end is fixed. For the linear dynamics, the amplitudes of the periodic response are determined directly; they decrease toward the fixed end of the spring. Then, in order to meet more realistic conditions, the displacement of the upper mass is assumed to be nonnegative. This condition is realized by introducing an appropriate impact constraint. We assume that the impact is described by Newton impact law with restitution coefficient less than unity. For the case of one impact per period (1IPP) of excitation, exact periodic solutions are derived. The interplay between the nonhomogenous structure, multi-frequency excitation and nonlinearity leads to two qualitatively different states of the periodic responses; we refer to them as propagating states and edge states. The propagating states are characterized by weak localization, and the edge states—by strong localization at the forced edge. The stability of the system is analyzed using Floquet theory. Generic pitchfork and Neimark–Sacker bifurcations are observed. Analytical solutions conform to numerical simulations and experimental tests conducted on real valve springs.

Keywords

Valve spring Frequency response Impacts Edge states Discrete breathers Stability 

Notes

Acknowledgements

We acknowledge Nordia Springs Ltd for the help with the experimental setting used in the paper.

Funding

The authors are grateful to Israel Science Foundation (Grant No. 1696/17) and to the Neubauer Family foundation for financial support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.

References

  1. 1.
    Wahl, A.M.: Mechanical Springs, vol. 229, 2nd edn. McGraw-Hill, New York (1963)Google Scholar
  2. 2.
    Phlips, P.J., Schamel, A.R., Meyer, J.: An efficient model for valvetrain and spring dynamics. SAE Technical Paper 890619 (1989)Google Scholar
  3. 3.
    Huber, R., et al.: An efficient spring model based on a curved beam with non-smooth contact mechanics for valve train simulations. SAE Int. J. Eng. 3(1), 28–34 (2010) CrossRefGoogle Scholar
  4. 4.
    Gzal, M., Groper, M., Gendelman, O.: Analytical, experimental and finite element analysis of elliptical cross-section helical spring with small helix angle under static load. Int. J. Mech. Sci. 130, 476–486 (2017)CrossRefGoogle Scholar
  5. 5.
    Lin, Y., Pisano, A.P.: Three-dimensional dynamic simulation of helical compression springs. Trans. ASME J. Mech. Des. 112(4), 529–537 (1990)CrossRefGoogle Scholar
  6. 6.
    Mottershead, J.E.: Finite elements for dynamical analysis of helical rods. Int. J. Mech. Sci. 22, 267–283 (1980)zbMATHCrossRefGoogle Scholar
  7. 7.
    Schamel, A., Hammacher, J., Utsch, D.: Modeling and measurement techniques for valve spring dynamics in high revving internal combustion engines. SAE Technical Paper 930615 (1993)Google Scholar
  8. 8.
    Trias, E., Mazo, J.J., Orlando, T.P.: Discrete breathers in nonlinear lattices: experimental detection in a Josephson array. Phys. Rev. Lett. 84, 741–744 (2000)CrossRefGoogle Scholar
  9. 9.
    Ustinov, A.V.: Imaging of discrete breathers. Chaos 13, 716 (2003)CrossRefGoogle Scholar
  10. 10.
    Trombettoni, A., Smerzi, A.: Discrete solitons and breathers with Dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353 (2001)CrossRefGoogle Scholar
  11. 11.
    Brazhnyi, V.A., Konotop, V.V.: Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B 18(14), 627 (2004)zbMATHCrossRefGoogle Scholar
  12. 12.
    Sato, M., Hubbard, B.E., Sievers, A.J.: Colloquium: nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys. 78(1), 137–157 (2006)CrossRefGoogle Scholar
  13. 13.
    Christodoulides, D., Joseph, R.: Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794 (1988)CrossRefGoogle Scholar
  14. 14.
    Eisenberg, H.S., Silberberg, Y., Morandotti, R., Boyd, A., Aitchison, J.S.: Dynamics of discrete solitons in optical waveguide arrays. Phys. Rev. Lett. 83, 2726–9 (1999)CrossRefGoogle Scholar
  15. 15.
    Theocharis, G., Kevrekidis, P.G., Porter, M.A., Daraio, C., Kevrekidis, Y.: Localized breathing modes in granular crystals with defects. Phys. Rev. E 80, 066601 (2009)CrossRefGoogle Scholar
  16. 16.
    Hoogeboom, G., Kevrekidis, P.G.: Breathers in periodic granular chain with multiple band gaps. Phys. Rev. E 86, 061305 (2012)CrossRefGoogle Scholar
  17. 17.
    Boechler, N., Theocharis, G., Job, S., Kevrekidis, P.G., Porter, M.A., Daraio, C.: Discrete breathers in one-dimensional diatomic granular crystals. Phys. Rev. Lett. 104(24), 5 (2010)CrossRefGoogle Scholar
  18. 18.
    Farantos, S.C.: Periodic orbits in biological molecules: phase space structures and selectivity in alanine dipeptide. J. Chem. Phys. 126, 175101 (2007)CrossRefGoogle Scholar
  19. 19.
    Gninzanlong, C.L., Ndjomatchoua, F.T., Tchawoua, C.: Discrete breathers dynamic in a model for DNA chain with a finite stacking enthalpy. Chaos 28, 043105 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Flach, S., Gorbach, A.V.: Discrete breathers: advances in theory and applications. Phys. Rep. 467(1), 1–116 (2008)zbMATHCrossRefGoogle Scholar
  21. 21.
    Gendelman, O.V., Manevitch, L.I.: Discrete breathers in vibro-impact chains: analytic solutions. Phys. Rev. E 78(2), 026609 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gendelman, O.V.: Exact solutions for discrete breathers in a forced-damped chain. Phys. Rev. E 87(6), 062911 (2013)CrossRefGoogle Scholar
  23. 23.
    Perchikov, N., Gendelman, O.V.: Dynamics and stability of a discrete breather in a harmonically excited chain with vibro-impact on-site potential. Phys. D 28, 292–293 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Shiroky, I.B., Gendelman, O.V.: Discrete breathers in an array of self-excited oscillators: exact solutions and stability. Chaos 26, 103112 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Grinberg, I., Gendelman, O.V.: Localization in finite vibroimpact chains: discrete breathers and multibreathers. Phys. Rev. E 94, 032204 (2016)CrossRefGoogle Scholar
  26. 26.
    Grinberg, I., Gendelman, O.V.: Localization in finite asymmetric vibro-impact chains. arXiv:1701.03055 (2017)
  27. 27.
    Grinberg, I., Gendelman, O.V.: Localization in coupled finite vibro-impact chains: discrete breathers and multibreathers. arXiv:1705.06248 (2017)
  28. 28.
    Theocharis, G., Boechler, N., Kevrekidis, P.G., Job, S., Porter, M.A., Daraio, C.: Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys. Rev. E 82(5), 056604 (2010)CrossRefGoogle Scholar
  29. 29.
    Hasan, M., Cho, S., Remick, K., Vakakis, A.F., McFarland, D.M., Kriven, W.M.: Experimental study of nonlinear acoustic bands and propagating breathers in ordered granular media embedded in matrix. Granul. Matter 17, 49–72 (2015)CrossRefGoogle Scholar
  30. 30.
    Hasan, M.A., Starosvetsky, Y., Vakakis, A.F., Manevitch, L.I.: Nonlinear targeted energy transfer andmacroscopic analog of the quantum Landau–Zener effect in coupled granular chains. Physica D 252, 46–58 (2013a)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Starosvetsky, Y., Hasan, M.A., Vakakis, A.F., Manevitch, L.I.: Strongly nonlinear beat phenomena and energy exchanges in weakly coupled granular chains on elastic foundations. SIAM J. Appl. Math. 72(1), 337–361 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    MacKay, R.S., Sepulchre, J.A.: Stability of discrete breathers. Physica D 119, 148–162 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Martínez, P.J., Meister, M., Floria, L.M., Falo, F.: Dissipative discrete breathers: periodic, quasiperiodic, chaotic, and mobile. Chaos 13, 610–623 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Marín, J.L., Falo, F., Martínez, P.J., Floría, L.M.: Discrete breathers in dissipative lattices. Phys. Rev. E 63, 066603 (2001)CrossRefGoogle Scholar
  35. 35.
    Palmero, F., Han, J., English, L.Q., Alexander, T.J., Kevrekidis, P.G.: Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: theory, computation and experiment. Phys. Lett. A 380, 402–407 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)CrossRefGoogle Scholar
  37. 37.
    König, M., Wiedmann, S., Brüne, C., Roth, A., Buhmann, H., Molenkamp, L.W., Qi, X.-L., Zhang, S.-C.: Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007)CrossRefGoogle Scholar
  38. 38.
    Garanovich, I.L., Sukhorukov, A.A., Kivshar, Y.S.: Defect-free surface states in modulated photonic lattices. Phys. Rev. Lett. 100, 203904 (2008)CrossRefGoogle Scholar
  39. 39.
    Savin, A.V., Kivshar, Y.S.: Vibrational Tamm states at the edges of graphene nanoribbons. Phys. Rev. B 81, 165418 (2010)CrossRefGoogle Scholar
  40. 40.
    Shockley, W.: On the surface states associated with a periodic potential. Phys. Rev. 56, 317 (1939)zbMATHCrossRefGoogle Scholar
  41. 41.
    Maradudin, A.A., Stegeman, G.: Surface acoustic waves. In: Kress, W., De Wette, F.W. (eds.) Surface Phonons, pp. 5–35. Springer, Berlin (1991) CrossRefGoogle Scholar
  42. 42.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1995)zbMATHCrossRefGoogle Scholar
  43. 43.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)zbMATHCrossRefGoogle Scholar
  44. 44.
    Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Methods Eng. 11, 1117–1136 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Doi, Y., Nakatani, A.: Structure and stability of discrete breather in zigzag and armchair carbon nanotubes. Lett. Mater. 6(1), 49–53 (2016)CrossRefGoogle Scholar
  46. 46.
    Zen, G., Miiftii, S.: Stability of an axially accelerating string subjected to frictional forces. J. Sound Vib. 289, 551–576 (2006)CrossRefGoogle Scholar
  47. 47.
    Fredriksson, M.H., Nordmark, A.B.: On normal form calculations in impact oscillators. Proc. R. Soc. Lond. Ser. A 456, 315–329 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems. Theory and Applications. Springer, New-York (2008)zbMATHGoogle Scholar
  49. 49.
    Pereira, D.A., Vasconcellos, R., Hajj, M.R., Marques, F.D.: Effects of combined hardening and free-play nonlinearities on the response of a typical aeroelastic section. Aerosp. Sci. Technol. 50, 44–54 (2016)CrossRefGoogle Scholar
  50. 50.
    Maki, A., Virgin, L.N., Umeda, N., Ueta, T., Miino, Y., Sakai, M., Kawakami, H.: On the loss of stability of periodic oscillations and its relevance to ship capsize. J. Mar. Sci. Technol. 24, 846–854 (2019)CrossRefGoogle Scholar
  51. 51.
    Shampine, L., Reichelt, M.: The matlab ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations