Structural and Multidisciplinary Optimization

, Volume 42, Issue 2, pp 293–304 | Cite as

Effect of approximation fidelity on vibration-based elastic constants identification

  • Christian GoguEmail author
  • Raphael Haftka
  • Rodolphe Le Riche
  • Jerome Molimard
Research Paper


Some applications such as identification or Monte Carlo based uncertainty quantification often require simple analytical formulas that are fast to evaluate. Approximate closed-form solutions for the natural frequencies of free orthotropic plates have been developed and have a wide range of applicability, but, as we show in this article, they lack accuracy for vibration based material properties identification. This article first demonstrates that a very accurate response surface approximation can be constructed by using dimensional analysis. Second, the article investigates how the accuracy of the approximation used propagates to the accuracy of the elastic constants identified from vibration experiments. For a least squares identification approach, the approximate analytical solution led to physically implausible properties, while the high-fidelity response surface approximation obtained reasonable estimates. With a Bayesian identification approach, the lower-fidelity analytical approximation led to reasonable results, but with much lower accuracy than the higher-fidelity approximation. The results also indicate that standard least squares approaches for identifying elastic constants from vibration tests may be ill-conditioned, because they are highly sensitive to the accuracy of the vibration frequencies calculation.


Identification Bayesian identification Response surface approximations Dimensionality Reduction Plate vibration 



This work was supported in part by the NASA grant NNX08AB40A. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Aeronautics and Space Administration.


  1. Blevins RD (1979) Formulas for natural frequency and mode shape. Van Nostrand Reinhold, New YorkGoogle Scholar
  2. Buckingham E (1914) On physically similar systems: illustrations of the use of dimensional equations. Phys Rev 4:345–376CrossRefGoogle Scholar
  3. Dickinson SM (1978) The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh’s method. J Sound Vib 61:1–8zbMATHCrossRefGoogle Scholar
  4. Gogu C, Haftka RT, Le Riche R, Molimard J, Vautrin A, Sankar BV (2008) Comparison between the basic least squares and the Bayesian approach for elastic constants identification. J Phys: Conf Ser 135:012045CrossRefGoogle Scholar
  5. Gogu C, Haftka RT, Bapanapalli S, Sankar BV (2009a) Dimensionality reduction approach for response surface approximations: application to thermal design. AIAA J 47(7):1700–1708CrossRefGoogle Scholar
  6. Gogu C, Haftka RT, Le Riche R, Molimard J, Vautrin A, Sankar BV (2009b) Bayesian statistical identification of orthotropic elastic constants accounting for measurement and modeling errors. In: 11th AIAA non-deterministic approaches conference, AIAA paper 2009-2258, Palm Springs, CAGoogle Scholar
  7. Gürdal Z, Haftka RT, Hajela P (1998) Design and optimization of laminated composite materials. Wiley Interscience, New YorkGoogle Scholar
  8. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New YorkzbMATHGoogle Scholar
  9. Mottershead JE, Friswell MI (1993) Model updating in structural dynamics: a survey. J Sound Vib 167:347–375zbMATHCrossRefGoogle Scholar
  10. Myers RH, Montgomery DC (2002) Response surface methodology: process and product optimization using designed experiments, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  11. Pedersen P, Frederiksen PS (1992) Identification of orthotropic material moduli by a combined experimental/numerical approach. Measurement 10:113–118CrossRefGoogle Scholar
  12. Vaschy A (1892) Sur les lois de similitude en physique. Ann Télégr 19:25–28Google Scholar
  13. Viana FAC, Goel T (2009) Surrogates toolbox v1.1 user’s guide.
  14. Waller MD (1939) Vibrations of free square plates: part I. Normal vibrating modes. Proc Phys Soc 51:831–844CrossRefGoogle Scholar
  15. Waller MD (1949) Vibrations of free rectangular plates. Proc Phys Soc B 62(5):277–285CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Christian Gogu
    • 1
    • 2
    Email author
  • Raphael Haftka
    • 2
  • Rodolphe Le Riche
    • 1
  • Jerome Molimard
    • 1
  1. 1.Centre Science des Matériaux et des StructuresEcole des Mines de Saint EtienneSaint Etienne Cedex 2France
  2. 2.Mechanical and Aerospace Engineering DepartmentUniversity of FloridaGainesvilleUSA

Personalised recommendations