New modal identification method under the non-stationary Gaussian ambient excitation

  • Xiu-li Du (杜秀丽)Email author
  • Feng-quan Wang (汪凤泉)


Based on the multivariate continuous time autoregressive (CAR) model, this paper presents a new time-domain modal identification method of linear time-invariant system driven by the uniformly modulated Gaussian random excitation. The method can identify the physical parameters of the system from the response data. First, the structural dynamic equation is transformed into a continuous time autoregressive model (CAR) of order 3. Second, based on the assumption that the uniformly modulated function is approximately equal to a constant matrix in a very short period of time and on the property of the strong solution of the stochastic differential equation, the uniformly modulated function is identified piecewise. Two special situations are discussed. Finally, by virtue of the Girsanov theorem, we introduce a likelihood function, which is just a conditional density function. Maximizing the likelihood function gives the exact maximum likelihood estimators of model parameters. Numerical results show that the method has high precision and the computation is efficient.

Key words

modal identification uniformly modulated function continuous time autoregressive model Brownian motion exact maximum likelihood estimator 

Chinese Library Classification

O324 O211.63 TU311.3 

2000 Mathematics Subject Classification

60H10 70J10 70L05 


  1. [1]
    Conforto, S. and D’Alessio, T. Spectral analysis for non-stationary signals from mechanical measurements: a parametric approach. Mechanical Systems and Signal Processing 13(3), 395–411 (1999)CrossRefGoogle Scholar
  2. [2]
    Zhang, Z. Y., Hua, H. X., Xu, X. Z., and Huang, Z. Modal parameter identification through Gabor expansion of response signals. Journal of Sound and Vibration 266(5), 943–955 (2003)CrossRefGoogle Scholar
  3. [3]
    Bonato, P., Ceravolo, R., De Stefano, A., and Molinari, F. Use of cross time-frequency estimators for structural identification in non-stationary conditions and under unknown excitation. Journal of Sound and Vibration 237(5), 775–791 (2000)CrossRefGoogle Scholar
  4. [4]
    Lardies, J., Ta, M. N., and Berthillier, M. Modal parameter estimation based on the wavelet transform of output data. Archive of Applied Mechanics 73(9–10), 718–733 (2004)zbMATHGoogle Scholar
  5. [5]
    Yang, J. N., Lei, Y., Pan, S. W., and Huang, N. System identification of linear structures based on Hilbert-Huang spectral analysis, part 1: normal modes. Earthquake Engineering and Structural Dynamics 32(9), 1443–1467 (2003)CrossRefGoogle Scholar
  6. [6]
    Tse, P., Yang, W. X., and Tam, H. Y. Machine fault diagnosis through an effective exact wavelet analysis. Journal of Sound and Vibration 277(4–5), 1005–1024 (2004)CrossRefGoogle Scholar
  7. [7]
    Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., and Shi, H. H. The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society of London A454, 903–951 (1998)Google Scholar
  8. [8]
    Peng, Z. K., Tse, P. W., and Chu, F. L. An improved Hilbert-Huang transform and its application in vibration signal analysis. Journal of Sound and Vibration 286(1–2), 187–205 (2005)CrossRefGoogle Scholar
  9. [9]
    Li, J. and Chen, J. Study on identification of structural dynamic parameters with unknown input information (in Chinese). Chinese Journal of Computational Mechanics 16(1), 32–40 (1999)zbMATHGoogle Scholar
  10. [10]
    Chen, J. Y., Wang, J. Y., and Lin, G. A sructural parameter identification method without input information (in Chinese). Engineering Mechanics 23(1), 6–10 (2006)Google Scholar
  11. [11]
    Poulimenos, A. G. and Fassois, S. D. Non-stationary mechanical vibration modelling and analysis via functional series TARMA models. Proceedings of the 13th IFAC Symposium on System Identification, Rotterdam, The Netherlands, 965–970 (2003)Google Scholar
  12. [12]
    Poulimenos, A. G. and Fassois, S. D. Parametric time-domain methods for non-stationary random vibration modelling and analysis: a critical survey and comparison. Mechanical Systems and Signal Processing 20(4), 763–816 (2006)CrossRefGoogle Scholar
  13. [13]
    Brockwell, P., Davis, R. A., and Yu, Y. Continuous-time Gaussian autoregression. Statistica Sinica 17(1), 63–80 (2007)zbMATHGoogle Scholar
  14. [14]
    Karatzas, L. and Shreve, S. E. Brownian Motion and Stochastic Calculus, Springer, New York (2000)Google Scholar
  15. [15]
    Harris, C. M. and Crede, C. E. Shock and Vibration Handbook, McGraw-Hill Book Company, New York (1976)Google Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2009

Authors and Affiliations

  • Xiu-li Du (杜秀丽)
    • 1
    Email author
  • Feng-quan Wang (汪凤泉)
    • 2
  1. 1.College of Mathematical SciencesNanjing Normal UniversityNanjingP. R. China
  2. 2.College of Civil EngineeringSoutheast UniversityNanjingP. R. China

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