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Thus temporal consistency of the identified properties in the form of the modal participation (or contribution) factors is highly dependent on the choice of reference nodes for calculation of the cross-correlation function since the reference node might have relatively less degree to which some of the modes participate in the response than others. DAMAGE DETECTION METHOD USING PARAMETER SUBSET SELECTION METHOD Most of the model based damage detection algorithms are subject to the issue of ill-posedness due to a large number of damage parameters defined in finite element models and the limited number of global model parameters. To overcome such a limitation, a new parameter subset selection method based on dynamic residual force vector is employed [Yun et al. 2008a, Yun et al. 2008b]. Major advantage of the method is that it can locate multiple damage locations with reasonable accuracy. In this section, brief conceptual procedures will be revisited. Formulation using Dynamic Residual Force Vector Herein it is assumed that structural damage can be modeled as a reduction of Young’s modulus in each finite element. Theoretical rationale is that the structural damage will change the dynamic residual force vector (R) leaving a residual vector (b) defined as a difference of the dynamic residual force vectors between damaged and undamaged structures. The residual vector is expressed as i di ui di di di di ; b R R R Kφ Mφ = − = − λ (10) where λdi and φdi are the ith measured eigenvalues and mode shapes of the damaged structure, respectively; K and M are the stiffness and mass matrices of the structure, respectively. Because the dynamic residual force vector for an undamaged structure, Rui will be approximately equal to the null vector, the residual values of the b vector originate from Rdi. When neglecting the higher order terms of Taylor series, the linearized equation can be rewritten as = Sθ b (11) where b is the difference in the residual force vectors between the measured and predicted damaged states; S is the sensitivity matrix consisting of sensitivities of the residual force vectors with respect to damage parameters. Taking the derivative of the residual force vector with respect to the jth damage parameter, sensitivities of the residual force vector can be expressed as i i i ii i ij j j j j∂ ∂ ∂λ ∂ ∂= + − − λ∂θ ∂θ ∂θ ∂θ ∂θR φ φ K φ K Mφ M (12) It is assumed that the mass matrix does not change due to damage. Thus, the sensitivity of the eigenvalues and mode shapes are calculated as in Reference [Fox et al. 1968]. Therefore, the sensitivity matrix S in terms of the residual force vectors is (neq m)x(p) ij for i 1,2,...,m and j 1,2, ...,pR S R × ∂= ∈ = = ∂θ (13) where ‘neq’, ‘m’ and ‘p’ are, respectively, the number of active DOFs, the number of measured modes, and the number of damage parameters. Then the problem is to find the subset of parameters that minimizes the residuals within these equations. Because the number of rows of the S matrix is greater than the number of columns, the problem is ill-posed in an over-determined sense. Therefore, the parameter subset selection method is described for purposes of regularization in the following section. Damage Localization by Parameter Subset Selection Method In the identification of damage parameters, sub-optimal problems are often sequentially formulated using the forward selection approach [Lallement et al. 1990]. Detail procedures can be referred to References [Friswell et al. 1997, Yun et al. 2008a, Yun et al. 2008c, Song et al. 2007]. In each sub-optimal problem, one damage parameter is selected out of the remaining damage parameters. Overall, the main task of forward selection is to select a column vector in the S matrix that best represents the residual vector b. This procedure seeks the basis vector aj that is closest to the damage residual vector b. If the first basis vector aj1 and its corresponding damage parameter θj1 are selected, Gram-Schmidt orthogonalization is generally performed on the remaining column vectors to ensure a well-conditioned sub-matrix of S. This procedure will be continued ranking damage parameters with most likely damaged element selected first. Efroymson suggested a stepwise regression algorithm to be used to decide whether a new parameter should be included in the subset [Efroymson 1960]. However, after all the elements are ranked by the subset selection, a subset of damaged elements can be finally selected based on the variation of the F-to-enter ratio [Yun et al. 2008c]. If the F-to-enter ratio drops down to extremely small number, it indicates cut-off step in the ranked order. DAMAGE QUANTIFICATION In this study, Steady-State Genetic Algorithm (SSGA) has been applied to update the finite element model within optimization framework. The modal flexibility matrix is employed as the objective function. With its particular sensitivity to structural damage, the modal flexibility methodology is numerically advantageous to use over the flexibility matrix because it does not require the inverse of the modal matrix. Therefore, the computation time is significantly less demanding. NUMERICAL CASE STUDIES In this example, the proposed SHM procedure from system identification to damage detection has been verified with numerical case studies considering environmental effects such as damping level and sensor noise and error. Example Truss Structure and Numerical Modeling In this example, a 14 bay planar truss is selected to demonstrate the performance of the proposed SHM methodology for the purpose of damage diagnosis. The truss is modeled using 53 truss elements with 28 nodes as shown in FIGURE 1. The members are steel bars with a tube cross section having an inner diameter of 3.1mm and an outer diameter of 17.0 mm. The physical properties are: the elastic modulus of the material=1.999x1011 N/m2; and the mass density=7,827 kg/m3. The resulting numerical model has 53 DOFs. In this numerical study, sensors are assumed to be placed at each node. 271 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 17 18 19 20 21 22 23 24 25 2628 29 30 31 32 33 34 35 36 37 38 39 40 53 52 51 50 49 48 47 46 45 44 43 42 4112 3 4 5 6 7 8 9 10 11 12 13 141516 17 18 19 20 21 22 23 24 25 26 27 2814x0.4 m0.4 m17.0 mm 3.1 mm271 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 17 18 19 20 21 22 23 24 25 2628 29 30 31 32 33 34 35 36 37 38 39 40 53 52 51 50 49 48 47 46 45 44 43 42 4112 3 4 5 6 7 8 9 10 11 12 13 141516 17 18 19 20 21 22 23 24 25 26 27 2814x0.4 m0.4 m17.0 mm 3.1 mm FIGURE 1 FOURTEEN-BAY PLANAR TRUSS STRUCTURE 2D FINITE ELEMENT MODEL The truss structure is modeled as a linear time-invariant model in state-space form. A classical damping matrix of the structure is formed using a modal damping ratio applied to all the modes. For numerical simulations under ambient excitation, a built-in MATLAB function (lsim.m) is used. Modal Identification by NExT/ERA Techniques under Environmental Changes In this example, modal properties (e.g. natural frequency, mode shapes, damping ratios and modal participation factors) are identified using NExT/ERA method under environmental changes. The natural frequencies of the numerical model are in a range of 31.93 Hz to 2,979.77 Hz. By using the sampling frequency 2,028 Hz, the ambient vibration measurements are purposely set to have the highest identifiable natural frequency less than 1024 Hz by which total 19 modes are identifiable. Figure 2(a) shows a CPSD function between the response of a reference DOF 6 and the response of a DOF 34. For the calculation of the CPSD, 4,096 points are used for each window and a total of 360 windows are averaged. 0 100 200 300 400 500 600 700 800 900 1000-220-200-180-160-140-120-100-80-60Frequency (Hz)Cross-Power-Spectral Density (dB)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.10-0.050.000.050.10-0.10-0.050.000.050.10-0.10Time (sec)Impulse Free Response in Acceleration (m/sec2) (a) (b) FIGURE 2 (A) A TYPICAL CROSS-POWER-SPECTRAL DENSITY FUNCTION BETWEEN THE RESPONSE OF A REFERENCE DOF 6 AND THE RESPONSE OF DOF 34 AND (B) ITS CORRESPONDING RECONSTRUCTED IMPULSE FREE RESPONSE THROUGH NEXT TECHNIQUE Assumed Modal Damping Ratio (%) Mode No. Accuracy and Consistency Indicators 0.1 0.5 0.9 1.3 1.7 2.1 2.5 2.9 3.3 3.7 100 x |ftrue-fid|/ ftrue 0.007 0.117 0.018 0.063 0.166 0.366 0.358 0.042 0.163 0.117 100 x |ζtrue-ζid|/ζtrue 1.065 11.91 7.262 77.40 15.26 3.419 2.719 37.33 19.23 4.217 MAC(φtrue, φid) 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.996 0.992 CMI 99.89 95.40 47.37 11.71 0.000 0.005 0.000 0.183 0.000 0.000 Mode 4 CMI_O 99.95 99.38 97.58 98.16 93.29 87.86 84.28 64.50 72.40 61.34 100 x |ftrue-fid|/ ftrue 0.0185 0.023 0.155 0.010 0.189 0.088 0.082 0.114 0.117 0.257 100 x |ζtrue-ζid|/ζtrue 3.097 9.180 4.711 1.387 3.556 8.437 6.772 0.832 3.153 0.004 MAC(φtrue, φid) 0.999 0.994 0.987 0.964 0.910 0.127 0.120 0.544 0.405 0.285 CMI 38.65 8.519 5.600 0.682 0.106 0.855 0.605 0.007 0.010 0.003 Mode 5 CMI_O 99.38 91.32 76.39 13.82 3.817 29.58 31.71 0.332 0.689 0.208 TABLE 1 EFFECTS OF MODAL DAMPING RATIO ON THE IDENTIFICATION RESULTS AND ASSESSMENT OF CONSISTENCY INDICATORS The ERA tests with different levels of the structural damping have been performed for the verification. Performances of the ERA analyses have been compared with gradually increased modal damping ratio from 0.1% to 4% with 0.1% increment. For the current analysis, the size of the Hankel matrix is fixed to have 1060 x 530. TABLE 1 summarizes the accuracy of the identified results for 4th and 5th mode and their corresponding consistency indicators. For the 4th mode, all the CMI_O values are greater than 60.0 % which is considered to be consistent with the accuracy obtained. However, CMI values are not properly representing the current accuracy of the identified results for damping ratio greater than 1.3%. As mentioned in Section 3, the inconsistent trend of CMI values is due to the choice of the reference node necessary for the output-only identification technique, NExT used in this paper. According to numerous testing of the proposed accuracy indicator for distinguishing physical modes from computationally spurious modes, it has been observed that modes with CMI_O values greater than approximately 85% are identified with high confidence and modes with CMI_O values ranging from 85% to 10% display moderate to large uncertainty. According to the current study, it is also recommended that environmental changes such as amplitudes of the ambient vibration responses should be considered when NExT/ERA techniques are employed for modal identifications. Damage Detection by Parameter Subset Selection Method and Steady-State Genetic Algorithm In this section, the proposed damage detection method has been verified following the modal identification by NExT/ERA. 0.1% modal damping ration is assumed. Two damage scenarios are assumed herein. The first damage scenario assumes a severe multiple damage case in which the damaged elements (element number 13, 20, and 33) are evenly distributed along the span. The second damage scenario assumes a challenging case in which two elements are lightly damaged (10% loss of section: element number 7, and 18). In the parameter subset selection method, threshold values for the F-to-enter statistic are determined by plotting the F-to-enter ratios during iterative steps of subset selection as shown in FIGURE 3 since they are dependent on the given problems. In current damage localization process, the sensor noise has not been considered. The parameter subset selection method is shown to be capable of selecting all of the assumed damage locations. 0 10 20 30 40 50-202468101214Selection Order of Damage ParameterF-to-enter Ratio (Fa)Element Number3 E4 D20 C13 B33 AElement Number3 E4 D20 C13 B33 AABCDE0 10 20 30 40 50-202468Selection Order of Damage ParameterF-to-enter Ratio (Fa)ABCElement Number26 C8 B18 AElement Number26 C8 B18 A (a) Damage Scenario 1