The Determination of Dispersion Curves from Measurements by the Matrix Pencil Method

  • T. SchusterEmail author
  • F. Schöpfer
Part of the Research Topics in Aerospace book series (RTA)


In this chapter, we present an application of the Matrix Pencil Method (MPM) to compute dispersion curves from laser vibrometer measurement data. Assuming that only finitely many elementary wave modes contribute to the measured out-of-plane component of the displacement field, this method estimates the number of essential wave modes and simultaneously extracts the frequencies of these modes by computing rank reducing numbers of two Hankel matrices that are built by the measurement data. Since the frequencies depend on the wave vector, it is possible to deduce the dispersion diagrams from them. The performance of the method is demonstrated by means of experimental data.


  1. 1.
    Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, AmsterdamzbMATHGoogle Scholar
  2. 2.
    Ahmad A (2011) Numerical simulation of lamb waves in plates using a semi-analytical finite element method. VDI Fortschritt-Berichte 20(437)Google Scholar
  3. 3.
    Alleyne D, Cawley P (1990) A 2-dimensional Fourier transform method for the quantitative measurement of Lamb modes. In: Ultrasonics Symposium, 1990 Proceedings, IEEE 1990, vol 2. pp 1143–1146Google Scholar
  4. 4.
    von Ende S, Lammering R (2007) Investigation on piezoelectrically induced Lamb wave generation and propagation. Smart Mater Struct 16:1802–1809CrossRefGoogle Scholar
  5. 5.
    Galan JM, Abascal R (2002) Numerical simulation of Lamb wave scattering in semi-infinite plates. Int J Numer Methods Eng 53:1145–1173CrossRefGoogle Scholar
  6. 6.
    Giurgiutiu V (2008) Structural health monitoring with piezoelectric wafer active sensors. Academic Press, CambridgeGoogle Scholar
  7. 7.
    Grondel S, Assaad J, Delebarre C, Blanquet P, Moulin E (1999) The propagation of Lamb waves in multilayered plates: phase-velocity measurement. Meas Sci Technol 10:348–353CrossRefGoogle Scholar
  8. 8.
    Harris FJ (1978) On the use of windows for harmonic analysis with the discrete Fourier transform. Proc IEEE 66:51–83CrossRefGoogle Scholar
  9. 9.
    Hua Y, Sarkar T (1990) Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans Acoust Speech Signal Process 38(5):814–824MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kausel E (1986) Wave propagation in anisotropic layered media. Int J Numer Methods Eng 23:1567–1578CrossRefzbMATHGoogle Scholar
  11. 11.
    Lowe MJS (1995) Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans Ultrason Ferroelectr Freq Control 42(4):525–542CrossRefGoogle Scholar
  12. 12.
    Prada C, Balogun O, Murray TW (2005) Laser-based ultrasonic generation and detection of zero-group velocity Lamb waves in thin plates. Appl Phys Lett 87:1–3CrossRefGoogle Scholar
  13. 13.
    Prosser WH, Seale MD, Smith BT (1999) Time-frequency analysis of the dispersion of lamb modes. J Acoust Soc Am 105(5):2669–2676CrossRefGoogle Scholar
  14. 14.
    Rose J (1999) Ultrasonic waves in solid media. Cambridge University Press, CambridgeGoogle Scholar
  15. 15.
    Sarkar TK, Pereira O (1995) Using the matrix pencil method to estimate the parameters of a sum of complex exponentials. IEEE Antennas Propag Mag 37(1):48–55CrossRefGoogle Scholar
  16. 16.
    Schöpfer F, Binder F, Wöstehoff A, Schuster T (2010) A mathematical analysis of the Strip Element Method for the computation of dispersion curves of guided waves in anisotropic layered media. Math Probl Eng 22:311–329MathSciNetzbMATHGoogle Scholar
  17. 17.
    Schöpfer F, Binder F, Wöstehoff A, Schuster T, Ende S, Föll S, Lammering R (2013) Accurate determination of dispersion curves of guided waves in plates by applying the matrix pencil method to laser vibrometer measurements. CEAS Aeronaut J 4:61–68CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsSaarland UniversitySaarbrückenGermany
  2. 2.Department of MathematicsCarl von Ossietzky University OldenburgOldenburgGermany

Personalised recommendations