Acta Mechanica Sinica

, Volume 35, Issue 1, pp 156–173 | Cite as

Band gap analysis of periodic structures based on cell experimental frequency response functions (FRFs)

  • Li-Jie Wu
  • Han-Wen SongEmail author
Research Paper


An approach is proposed to estimate the transfer function of the periodic structure, which is known as an absorber due to its repetitive cells leading to the band gap phenomenon. The band gap is a frequency range in which vibration will be inhibited. A transfer function is usually performed to gain band gap. Previous scholars regard estimation of the transfer function as a forward problem assuming known cell mass and stiffness matrices. However, the estimation of band gap for irregular or complicated cells is hardly accurate because it is difficult to model the cell exactly. Therefore, we treat the estimation as an inverse problem by employing modal identification and curve fitting. A transfer matrix is then established by parameters identified through modal analysis. Both simulations and experiments have been performed. Some interesting conclusions about the relationship between modal parameters and band gap have been achieved.


Band gap Modal analysis Parameter identification Periodic structure Transfer matrix 



This work was supported by the National Natural Science Foundation of China (Grant No. 11272235).


  1. 1.
    Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photoniccrystals. Princeton University Press, Princeton (1995)Google Scholar
  2. 2.
    Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)CrossRefGoogle Scholar
  3. 3.
    John, S.: Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486–2489 (1987)CrossRefGoogle Scholar
  4. 4.
    Wang, Y.Z., Li, F.M., Kishimoto, K.: Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress. Eur. J. Mech. 29, 182–189 (2010)CrossRefGoogle Scholar
  5. 5.
    Wang, Y.Z., Li, F.M., Kishimoto, K., et al.: Wave band gaps in three-dimensional periodic piezoelectric structures. Mech. Res. Commun. 36, 461–468 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Yan, Z.Z., Wang, Y.S.: Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Phys. Rev. B 74, 224303 (2006)CrossRefGoogle Scholar
  7. 7.
    Yan, Z.Z., Wang, Y.S.: Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method. Phys. Rev. B 78, 094306 (2008)CrossRefGoogle Scholar
  8. 8.
    Casadei, F., Rimoli, J.J., Ruzzene, M.: Multiscale finite element analysis of wave propagation in periodic solids. Finite Elem. Anal. Des. 108, 81–95 (2016)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bardi, I., Peng, G., Petersson, L.E.R.: Modeling periodic layered structures by shell elements using the finite-element method. IEEE Trans. Magn. 52, 1–4 (2016)CrossRefGoogle Scholar
  10. 10.
    Ozaki, S., Hinata, K., Senatore, C., et al.: Finite element analysis of periodic ripple formation under rigid wheels. J. Terramech. 61, 11–22 (2015)CrossRefGoogle Scholar
  11. 11.
    Zhang, Y., Shang, S., Liu, S.: A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials. Acta. Mech. Sin. 33, 368–381 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wu, Z.J., Li, F.M., Zhang, C.: Vibration band-gap properties of three-dimensional kagome lattices using the spectral element method. J. Sound Vib. 341, 162–173 (2015)CrossRefGoogle Scholar
  13. 13.
    Wu, Z.J., Li, F.M., Wang, Y.Z.: Vibration band gap behaviors of sandwich panels with corrugated cores. Comput. Struct. 129, 30–39 (2013)CrossRefGoogle Scholar
  14. 14.
    Wen, S.R., Wu, Z.J., Lu, N.L.: High-precision solution to the moving load problem using an improved spectral element method. Acta Mech. Sin. 34, 68–81 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mead, D.J.: Wave propagation in continuous periodic structures: research contributions from southampton. J. Sound Vib. 190, 495–524 (1996)CrossRefGoogle Scholar
  16. 16.
    Gupta, G.S.: Natural frequencies of periodic skin-stringer structures using a wave approach. J. Sound Vib. 16, 567–580 (1971)CrossRefGoogle Scholar
  17. 17.
    Gupta, G.S.: Dynamics of Periodically Stiffened Structures Using a Wave Approach. University of Southampton, Southampton (1970)Google Scholar
  18. 18.
    Lin, Y.K., McDaniel, T.J.: Dynamics of beam-type periodic structures. J. Eng. Ind. 91, 1133 (1969)CrossRefGoogle Scholar
  19. 19.
    Ruzzene, M., Tsopelas, P.: Control of wave propagation in sandwich plate rows with periodic honeycomb core. J. Eng. Mech. 129, 975–986 (2003)CrossRefGoogle Scholar
  20. 20.
    Richards, D., Pines, D.J.: Passive reduction of gear mesh vibration using a periodic drive shaft. J. Sound Vib. 264, 317–342 (2003)CrossRefGoogle Scholar
  21. 21.
    Wang, Y.Z., Li, F.M., Huang, W.H., et al.: The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals. J. Mech. Phys. Solids 56, 1578–1590 (2008)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, Y.Z., Li, F.M., Wang, Y.S.: Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain. Int. J. Mech. Sci. 106, 357–362 (2016)CrossRefGoogle Scholar
  23. 23.
    Wang, Y.Z., Wang, Y.S.: Active control of elastic wave propagation in nonlinear phononic crystals consisting of diatomic lattice chain. Wave Motion 78, 1–8 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Waki, Y.: On the Application of Finite Element Analysis to Wave Motion in One-Dimensional Waveguides. University of Southampton, Southampton (2007)Google Scholar
  25. 25.
    Mace, B.R., Duhamel, D., Brennan, M.J., et al.: Finite element prediction of wave motion in structural waveguides. J. Acoust. Soc. Am. 117, 2835–2843 (2005)CrossRefGoogle Scholar
  26. 26.
    Duhamel, D., Mace, B.R., Brennan, M.J.: Finite element analysis of the vibrations of waveguides and periodic structures. J. Sound Vib. 294, 205–220 (2006)CrossRefGoogle Scholar
  27. 27.
    Zhong, W.X., Williams, F.W.: On the direct solution of wave propagation for repetitive structures. J. Sound Vib. 181, 485–501 (1995)CrossRefGoogle Scholar
  28. 28.
    Zhong, W.X., Williams, F.W., Leung, A.Y.T.: Symplectic analysis for periodical electro-magnetic waveguides. J. Sound Vib. 267, 227–244 (2003)CrossRefGoogle Scholar
  29. 29.
    Hendrickx, W., Dhaene, T.: A discussion of “Rational approximation of frequency domain responses by vector fitting”. IEEE Trans. Power Syst. 21, 441–443 (2006)CrossRefGoogle Scholar
  30. 30.
    Gustavsen, B.: Relaxed vector fitting algorithm for rational approximation of frequency domain responses. In: IEEE Workshop on Signal Propagation on Interconnects, Berlin, Germany (2006)Google Scholar
  31. 31.
    Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Deliv. 14, 1052–1061 (1999)CrossRefGoogle Scholar
  32. 32.
    Jensen, J.S.: Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J. Sound Vib. 266, 1053–1078 (2003)CrossRefGoogle Scholar
  33. 33.
    Cremer, L., Heckl, M., Petersson, B.A.T.: Structure Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 3rd edn. Springer, Berlin (2005)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina

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