Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Lamb-Wave Based Structural Health Monitoring in Polymer Composites

Part of the book series: Research Topics in Aerospace ((RTA))

  • 1436 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    With kind permission from Springer Science+Business Media: Fig. 1 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  2. 2.

    With kind permission from Springer Science+Business Media: Fig. 2 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  3. 3.

    With kind permission from Springer Science+Business Media: Fig. 3 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  4. 4.

    With kind permission from Springer Science+Business Media: Fig. 4 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  5. 5.

    With kind permission from Springer Science+Business Media: Fig. 5 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  6. 6.

    With kind permission from Springer Science+Business Media: Fig. 6 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

  7. 7.

    With kind permission from Springer Science+Business Media: Fig. 7 from F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll and R. Lammering, Accurate determination of dispersion curves of guided waves in plates by applying the Matrix Pencil Method to laser vibrometer measurement data, CEAS Aeronautical Journal, Article ID 10.1007/s13272-012-0055-7, 2013.

References

  1. Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, Amsterdam

    MATH  Google Scholar 

  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. 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 1143–1146

    Google Scholar 

  4. von Ende S, Lammering R (2007) Investigation on piezoelectrically induced Lamb wave generation and propagation. Smart Mater Struct 16:1802–1809

    Article  Google Scholar 

  5. Galan JM, Abascal R (2002) Numerical simulation of Lamb wave scattering in semi-infinite plates. Int J Numer Methods Eng 53:1145–1173

    Article  Google Scholar 

  6. Giurgiutiu V (2008) Structural health monitoring with piezoelectric wafer active sensors. Academic Press, Cambridge

    Google Scholar 

  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 10:348–353

    Article  Google Scholar 

  8. Harris FJ (1978) On the use of windows for harmonic analysis with the discrete Fourier transform. Proc IEEE 66:51–83

    Article  Google Scholar 

  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):814–824

    Article  MathSciNet  MATH  Google Scholar 

  10. Kausel E (1986) Wave propagation in anisotropic layered media. Int J Numer Methods Eng 23:1567–1578

    Article  MATH  Google Scholar 

  11. Lowe MJS (1995) Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans Ultrason Ferroelectr Freq Control 42(4):525–542

    Article  Google Scholar 

  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 87:1–3

    Article  Google Scholar 

  13. Prosser WH, Seale MD, Smith BT (1999) Time-frequency analysis of the dispersion of lamb modes. J Acoust Soc Am 105(5):2669–2676

    Article  Google Scholar 

  14. Rose J (1999) Ultrasonic waves in solid media. Cambridge University Press, Cambridge

    Google Scholar 

  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):48–55

    Article  Google Scholar 

  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 22:311–329

    MathSciNet  MATH  Google Scholar 

  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 4:61–68

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saarland University, PO Box 151150, 66041, Saarbrücken, Germany

    T. Schuster

  2. Department of Mathematics, Carl von Ossietzky University Oldenburg, 26111, Oldenburg, Germany

    F. Schöpfer

Authors
  1. T. Schuster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Schöpfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Schuster .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg, Hamburg, Germany

    Rolf Lammering

  2. Institute of Mechanics, Otto von Guericke University Magdeburg, Magdeburg, Germany

    Ulrich Gabbert

  3. Institute of Adaptronics and Function Integration, Braunschweig University of Technology, Braunschweig, Germany

    Michael Sinapius

  4. Faculty of Mathematics, Saarland University, Saarbrücken, Germany

    Thomas Schuster

  5. DLR - German Aerospace Center, Braunschweig, Germany

    Peter Wierach

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Schuster, T., Schöpfer, F. (2018). The Determination of Dispersion Curves from Measurements by the Matrix Pencil Method. In: Lammering, R., Gabbert, U., Sinapius, M., Schuster, T., Wierach, P. (eds) Lamb-Wave Based Structural Health Monitoring in Polymer Composites. Research Topics in Aerospace. Springer, Cham. https://doi.org/10.1007/978-3-319-49715-0_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-49715-0_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-49714-3

  • Online ISBN: 978-3-319-49715-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation