Structural Identification and Damage Detection from Noisy Modal Data Firdaus E. Udwadia 1 Abstract: In this paper we present a simple, yet powerful, method for the identification of stiffness matrices of structural and mechanical systems from information about some of their measured natural frequencies and corresponding mode shapes of vibration. The method is computationally efficient and is shown to perform remarkably well in the presence of measurement errors in the mode shapes of vibration. It is applied to the identification of the stiffness distribution along the height of a simple vibrating structure. An example illustrating the method’s ability to detect structural damage that could be highly localized in a building structure is also given. The efficiency and accuracy with which the method yields estimates of the system’s stiffness from noisy modal measurement data makes it useful for rapid, on-line damage detection of structures. DOI: 10.1061/ASCE0893-1321200518:3179 CE Database subject headings: Damage assessment; Identification; Structural safety; Measurement; Noise; Model verification; Modal analysis. Introduction Modal testing of structures is an extensive field in civil, aero- space, and mechanical engineering. It is generally used to understand/predict the dynamic behavior of a structure when sub- jected to low amplitude vibrations. Often modal information is also used to identify/estimate the structural parameters of a sys- tem, under the assumption that it has classical normal modes of vibration Caughey and O’Kelley 1963. Such identification leads to improved mathematical models that can be used in either pre- dicting and/or controlling structural response to dynamic excita- tions. Several different approaches to the parameter identification problem have appeared in the literature Baruch and Itzhak 1978; Udwadia and Ghodsi 1984; Kabe 1985; Wei 1989; Kalaba and Udwadia 1993; Mottershead and Friswell 1993; Kenigsbuch and Halevi 1997; Udwadia and Proskurowski 1998; Koh et al. 2000. One approach is the so-called model updating method. Here a suitable analytical model of a structural system is developed using the equations of motion, and its numerical representation is ob- tained. Validation of the numerical model through modal testing is then sought. Such tests usually provide some of the frequencies of vibration usually the lower frequencies and the corresponding mode shapes. When these frequencies and mode shapes obtained from modal testing are compared with those obtained from the numerical model, they generally do not agree with one another. Discrepancies between the results from experimental testing and theoretical modeling arise due to a variety of reasons: simplifica- tions used in developing the analytical model, uncertainties in the structural description like those in material properties and bound- ary conditions, and experimental errors during modal testing. The problem of updating a numerical model so that it is as much as possible in conformity with experimental modal test data is re- ferred to as the updating problem, and over the years it has re- ceived considerable attention. In this paper we investigate a direct approach to structural identification through the use of modal test data. No a priori es- timates are used. It should be pointed out that such experimental test data is seldom “complete,” i.e., all the mode shapes of vibra- tion and the corresponding natural frequencies are seldom avail- able, for there is a practical limit to the range of frequencies that a structural or mechanical system can be tested for. Hence the idea is to obtain suitable models through the use of incomplete information, i.e., information on only a limited number of mode shapes and frequencies of vibration. We shall illustrate our method assuming that normal classical modes exist and that the damping factors are small, as is the common occurrence in struc- tural and mechanical systems. System Model Consider a structural system modeled by the linear differential equation Mx ¨ + C ˆ x ˙ + K ˆ x =0 1 where x = n by 1 vector, and M = n by n symmetric positive- definite mass matrix, K ˆ = symmetric stiffness matrix, and C ˆ =damping matrix. We shall assume that the elements of the mass matrix, M, are sufficiently well known, and that the system is classically damped. We could then rewrite Eq. 1 as y ¨ + Cy ˙ + Ky =0 2 where K = M -1/2 K ˆ M -1/2 and C = M -1/2 C ˆ M -1/2 . 1 Professor of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, Univ. of Southern California, Los Angeles, CA 90089- 1453. E-mail: fudwadia@usc.edu Note. Discussion open until December 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and pos- sible publication on April 25, 2003; approved on June 4, 2004. This paper is part of the Journal of Aerospace Engineering, Vol. 18, No. 3, July 1, 2005. ©ASCE, ISSN 0893-1321/2005/3-179–187/$25.00. JOURNAL OF AEROSPACE ENGINEERING © ASCE / JULY 2005 / 179