Keywords: frame model, structural damage, inverse analysis, stiffness, filtering algorithms, observation data, observation noise, natural frequency, health monitoring system.

Most of architectural building structures will accumulate damage during their service life. In order to assure the structural safety and human comfort, it is necessary to monitor the state of structural system at regular intervals. Against the forward problem in the dynamic analysis relating to earthquake-resistant structural design, structural damage identification analysis has been categorized as the inverse problem [1].

The inverse problem in general must be analyzed under the consideration of stochastic properties of mathematical model and structural system because the observation data are usually measured in the presence of noise. Filtering algorithms have been well known as one of the inverse analysis method that is able to consider the stochastic properties [2]. In this study, the identification analysis to detect the damage of frame structural model was performed as the framework of inverse problem in computational mechanics. Natural frequencies corresponding to each eigen-mode characterized to the change of structural stiffness are effectively used as observation data. The frame model and outline to measure the observation data and observation error used in our inverse analysis is shown in Figure 1.

As the inverse analysis method three kinds of filtering algorithm based on the Wiener filter, the projection filter and the parametric projection filter are employed in filter equation as follows [3]:

• Filter equation

  (67)

  
  

• Wiener filter

  		(68)
  		(69)

  
  

• Projection filter

  (70)

  
  

• Parametric projection filter

  (71)

  
  

where is the filter gain given by the Wiener filter, projection filter and parametric projection filter, is observation error covariant matrix, is sensitivity matrix, is state vector and is the observations given as natural frequencies, respectively. It is noteworthy that the projection filter and parametric projection filter do not depend on the covariant matrix in contrast with the Wiener filter, and the parametric projection filter is including the parameter to be regularized the filtering process.

The notable characteristics of each filtering algorithm in applying the inverse problem are made clear through several numerical calculations.

1

M. Tanaka, G.S. Dulikravich (Edited), Inverse Problems in Engineering Mechanics, Elsevier, 2001.

2

Catlin, E.D., Estimation, Control, and the Discrete Kalman Filter, Springer-Verlag, Berlin, 1989.

3

R. Endo, Y. Kawakami, T. Imai, N. Tosaka, Identification Analysis of Structural Damage on Unit-Linked Offshore Floating Models, Int. J. of Offshore and Polar Engineering, Vol.9, No.3, September, 1999.

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