4.1. Mahalanobis distance outlier detection method Mahalanobis distance is a distance measure used for multivariate statistics defined as y y S y y k k kD T 1 , (4) where yk is a multi-dimensional feature vector for which the normalized distance from the mean is being calculated, y and S are a mean vector and a covariance matrix based on all acquired vectors defining the nominal condition, respectively. The low distance value Dk can be obtained if yk is similar to the set of feature vectors that defines the nominal condition. The outlier detection procedure is then summarized as follows. (1) Acquire experimental responses (experimental data set) from the target structure. (2) Calculate a feature vector from each measured set. (3) Derive a Mahalanobis distance each experimental data set k, given as 1 E E E E E E T y y S y y k k kD . (5) Notice that mean feature vector Ey and covariance matrix SE are calculated from all acquired experimental data. A set of Dk E is then the experimental baseline distribution of Mahalanobis distance. (4) Create a feature vector from a numerical output, and calculate a Mahalanobis distance DN using the mean vector and the covariance matrix from the experimental data set; Ey and SE, 1 N E E N N E T y y S y y D . (6) (5) Compare DN with the experimental baseline distribution DE. If the feature vector form numerical output were similar to those from the experimental data set, DN would show a lower value. The validity of a numerical model was then expected to be investigated statistically by comparing DN with the experimental baseline distribution. 4.2. Damping parameters validation using linear system responses 4.2.1. Experimental baseline distribution and numerical run set An experimental baseline distribution was created by acquiring twenty acceleration data sets (Exp. data #1~#20) under the same linear condition (w/o the bumper-column mechanism) of the test-bed structure. The input force was the band-limited random excitation with frequency range of 20-200Hz. Accelerations from each floor was acquired; the length of data was 16384 points with sampling frequency of 640Hz. A numerical run set consisted of 200 calculation outputs was created using variable damping parameter sets; variable 2nd and 4th modal damping ratios. 200 parameter sets were sampled from the ranges of the two damping ratios indicated in Table 3 using Latin hypercube sampling method. 200 time-response accelerations (Run #1~#200) were then calculated by using each parameter set. Notice that the input force in all calculations was the same time-history that was measured in one of the experimental data set acquisitions; Exp. data #10. Table 3. Parameter ranges for Latin hypercube sampling Minimum Maximum 2nd mode damping ratio ]2 0.01 0.08 4th mode damping ratio ]4 0.001 0.02 159
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