The Uncertainty of ambient vibration test results in stochastic subspace methods

In stochastic subspace methods, the most important factor influencing the dynamic specifications is the dimensions of the Hankel matrix include the number of rows and columns. Using small matrix dimensions is unlikely to identify existing poles, and selecting very large dimensions not only increases the likelihood of virtual and bias poles but also increases computational costs. In this study, it is intended that the optimal dimensions of the Hankel matrix in the balanced stochastic subspace method be calculated in such a way that in addition to covering the existing poles, it also has a minimum computational cost. For this purpose, the condition number of the Hankel Matrix and Energy Indicator is used in two steps. The steps are as follows: First, calculate the optimal order of each cycle, and then use the optimal order to draw the condition number of the system matrix for different dimensions and calculate the desired dimension from its convergence. To verify the accuracy of the proposed method, the ambient vibration test of the Namin Entrance Bridge has been used. This bridge is located at the entrance of Namin city, 25 km from the center of Ardabil province, Iran, which includes two spans of 27.10m with a concrete deck. The deck of the bridge is located on beams with I sections, which are 2.5m away from each other, and the whole set of beams and deck is located on a system of foundations and piles with a diameter of 120cm. This bridge being the only entrance to the city and is exposed to various traffic loads, it was necessary to monitor the dynamic characteristics of the bridge as modal frequencies and damping ratios to evaluate the performance and ensure the health of the bridge structure. According to the numerical analysis and the length of the data (12000), the minimum order and the maximum number of cycles are 22 and 55, respectively. By diverging the curvature of the energy indicator graph, the optimal order is determined in the initial 5-12% of the singular values of cycles. For example, the maximum order of the 6th cycles was obtained, 28-62. Also, from the convergence of the maximum condition number of cycles from the 8th   cycle, the optimal dimension was selected 352. In a general summary, it can be said that the use of the energy indicator concept in finding the effective order of the stability diagram has a significant effect on reducing the uncertainty of the extracted results. So that from the three identified stable poles, two poles have been extracted in the effective-order area. Also, using the concept of conditional number to find the optimal dimension of the system was effective, so that by drawing a stability diagram for the 15th cycle, it was found that the identified modal characteristics were not significantly different from the results of the optimal cycle (8th). Finally, the extracted modal properties have an acceptable agreement with the numerical model and frequency domain decomposition method (FDD). The modal frequencies of both methods (FDDand B-SSI) have a good correlation but the damping ratios were very different. In frequency domain methods the damping ratios being very sensitive to the quality of data collection, one can expect that the results of the subspace method are closer to reality.