2 Sensitivity Evaluation of Subspace-Based Damage Detection Method to Different Types of Damage 13 singular value decomposition in order to compute the left null space S. Defining Hfor the possibly damaged state of the system, the left null space matrixSin the reference state is characterized bySTHD0([2–4]). Therefore the residual vector —n can be written as —n D pnvec STH (2.4) inwhich, n represents the number of samples measured for computing H. This residual can now be used in order to check if any change is made in the model due to damage. The residual vector —n is asymptotically Gaussian with zero mean in reference state; significant changes in its mean value indicates the structure is moved from its reference state. In order to check this change from the residual vector mean, the 2 test can be performed as following [4–7]. 2 D—T n† 1— n (2.5) Herein, †represents the covariance matrix of the residual in the reference state, and can be shown as †DE —n— T n (2.6) It is worth mentioning that the covariance of the input noise "k is assumed to not change between the reference state and the possibly damaged state. Details of the covariance computation are found in [3]. By monitoring the value of 2 and comparing it to a threshold value, the state of the damage of the system can be estimated. This threshold can be simply evaluated using several data sets measured from the structure in its reference state. Subsequently, some other data sets measured from the reference state are used to check the threshold. Then the 2 value is computed for the possibly damaged structure. If the computed 2 value is higher than the threshold it can be inferred that the structure may be damaged. In other words, damage that provokes a change in the statistics of the measured data leads to an increase in the 2 value. 2.3 Damage and Data Simulation The ideal test data that can be used to evaluate a damage detection technique is to damage a real structure progressively and measure its response continuously [7]. Having a clear understanding of the conditions of the structure before damaging plays a critical role in the results. Furthermore, in addition to the cost of the procedure, it is not practical to damage a structure and restore it to its undamaged condition for the next test, especially when different elements of the structure are needed to be damaged separately and to various extents. Simulating the damage in a structure and subsequently generating data that represents the ambient vibration test data can be a useful approach to evaluate damage detection techniques. This data can be an acceptable benchmark to evaluate the functionality of these techniques by allowing control on the test conditions, e.g. structural properties and damage effects. In order to investigate the effect of noise on these techniques, a predefined amount of white noise can be superposed to the data. For simplicity, in this paper there is no additional noise imposed on the results and its effect on the damage detection technique investigated in future study. In order to evaluate the functionality of the subspace-based damage detection technique, the ambient vibration test data can be simulated for different damage types and amounts. In order to simulate this data, a finite element model of the structure is created and then calibrated to the real structure. It should be mentioned that calibration of the structure does not have a straight effect on the damage detection technique. In other words, the damage detection technique should be able to detect the damage in any structural model including the uncalibrated one as long as the base of comparison is identical. However in this study, calibration to a real structure is performed to obtain a realistic model and simulate the damage in it. The damage in different elements of the model is simulated by reducing the dimensions of one or some short elements in the intended location of damage. The amount of the damage is defined in terms of ratio of this reduction. Several points of the structure are excited using white noise excitation in all three directions. Different excitations are imposed on the structure in order to excite the structure as randomly as possible. This excitation can be done by acceleration or load forces in different points of the structure. Subsequently, the simulated data can be obtained by measuring acceleration time histories of the nodes typically measured and instrumented in a bridge. The simulated data can then be analyzed in order to compute the natural frequencies and their corresponding mode shapes. These can be used to check which mode shapes can be captured by the simulated white noise excitation. Based on
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