13 Mass Scaling of Mode Shapes Based on the Effect of Traffic on Bridges: A Numerical Study 101 Table 13.2 FEM model natural frequencies Mode no. 1 2 3 Natural frequency !0—(rad/s) 12.8 51.4 115.8 Time (s) 0 100 50 150 200 250 300 0 0.01 0.02 0.03 0.04 Normalized M1 (t) Fig. 13.4 Time history of traffic assigned to node 1 under specific traffic conditions: D1.85, D0.9, v D15 (km/h) Table 13.3 Traffic characterestics Front headway Case no. Velocity—v (km/h) 1 1.97 0.9 15.1 2 1.94 0.94 15.9 3 1.69 0.96 21.2 4 1.55 0.87 23.2 m.t/ D 2 66 66 66 66 64 m11 CM1.t/ m1;2n m22 : : : : : : m2i 1;2i 1 CMi.t/ : : : m2n;1 m2n;2n 3 77 77 77 77 75 (13.22) where n is number of nodes. It should be noted that since the traffic excitation is stationary and time period is long enough, there are no concerns regarding the use of time-variant mass matrix in the modal analysis techniques. In order to evaluate the validity of the proposed method for estimating mass changes, four near congestion real traffic scenarios for front headway of passenger cars are acquired from [10] and are tested using the method. The details of the traffic conditions are presented in Table 13.3. Therefore, in the interest of imposing artificial traffic excitation, lognormal distribution parameters presented in Table 13.3 are considered and headway times between vehicles are generated using the lognormal distribution from Eq. (13.10). Response of the structure is then determined by Newmark method. Shears forces and moments which are determined using Eqs. (13.13, 13.14) are used as input for the system with the addition of Gaussian white noise. The sensors are assumed to be located on all of the nodes, and acceleration of each node is recorded during the time period for each simulation, and then low-pass filtered to have the signal in the frequency range of the desired mode shapes. Fast Fourier Transform (FFT) method is used for filtering to avoid phase and amplitude errors [14]. Correlation function matrix of the acceleration response is then constructed using Eq. (13.1) and is then used in the NExT-ERA process for output-only modal identification of the structure. The mass change matrix is created using Eq. (13.21) for each traffic condition and is subsequently used in the scaling factor equations. Equation (13.6) is used to obtain scale factors and the modal parameters extracted from the free flow condition and the values extracted from the mentioned congested traffic conditions are used as inputs for this equation. A hundred simulations are performed for each traffic setup and the calculated scale factors are normalized to the exact scale factors obtained from FEM model which are shown for the first three modes of vibration in Figs. 13.5, 13.6, 13.7 and 13.8: The mean value and standard deviation regarding each case for the first three modes are calculated. Mean values of normalized scale factors and Coefficients of Variations (CV), along with average frequency shifts are presented in Table 13.4. The results show that values of CVs are lower than 5% for all cases of traffic situations and frequency shift is 2% for each case identically. However, there is an inclination in mean values which indicate that mass changes produced by Eq. (13.21) are slightly overestimated.
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