58 K. Demirlioglu et al. of vehicle. In addition to these, Yang et al. [14] revealed that the contact-point (CP) response does not contain vehicle frequencies that hamper identifying the bridge frequencies. This observation was verified by numerical simulations [14] and later by conducting an in-situ test [15]. Yang et al. [16] indicated that using the CP response enables extracting more bridge frequencies and constructing mode shapes more accurately than the ones obtained from the vehicle response. Also, some researchers have developed a different concept for mode shape identification [17–20]. In their approach, the acceleration signal measured on the vehicle is separated into a number of equal segments along the bridge deck to form a data set at each segment that carries information on the mode shapes of that segment. Then, the mode shapes of the bridge can be obtained by applying an Operational Modal Analysis (OMA) to each data record. These studies made substantial contributions to the mode shape identification of bridges using indirect measurements. However, there is still a lack of studies that objectively evaluate the accuracy of these methodologies under varying conditions. This article aims at assessing the efficacy of two of the most commonly employed methods that apply (i) variational mode decomposition together with the amplitude histories of the Hilbert transform [16] and (ii) the reference-based Stochastic Subspace Identification (SSI) [19] in extracting bridge mode shapes. These methods will be referred to as Method 1 [16] and Method 2 [19], respectively, for brevity. They could be considered representatives of two different approaches in drive-by monitoring. In order to investigate their performance in estimating the bridge mode shapes, this chapter uses the application of these two distinct methods on the same single-span bridge that has similar properties to the ones used in [9, 16, 19]. For a proper comparison, the same quarter-car is employed in the numerical simulations of both methods. However, the CP response is adopted for Method 1 as proposed by [16], while the vehicle response is utilized for Method 2 [19]. Finally, their sensitivity to different parameters is evaluated using a parametric study that considers the effects of vehicle speed, sampling rate, and road roughness profile on constructing the mode shapes of the bridge. 6.2 Numerical Simulations A simply supported bridge with a total span length of 25 m is modeled using ABAQUS software and Euler-Bernoulli beam elements as shown in Fig. 6.1. The bridge is discretized into 50 equal-length elements of 0.5 m. The elasticity modulus of the bridge is assumed to be E = 30 GPa, the moment of inertia of its cross-section is I = 0.2 0 m4 , and its mass per lengt h ρ =2000 kg/m. The first three vertical modal frequencies of the bridge are computed as 4.35, 17.23, and 38.06 Hz, respectively, through Eigen-value analysis. The vehicle moving over the bridge is simulated with a quarter-car model, i.e., using a lumped-mass spring model as shown in Fig. 6.1. The vehicle’s characteristic mass, stiffness coefficient, and damping coefficient are adopted as m v =200 0kg , k v =550,000 N/m, andc v = 2000 N.s/m, respectively. The frequency of the vehicle is 2.63 Hz. The acceleration responses of the vehicle and the contact point (CP) are obtained during the crossing of the vehicle over the bridge with a velocity of 5 m/s using a sampling rate of 1000 Hz (Fig. 6.2a, b). The measured responses are processed using the Fast Fourier transform (FFT) to obtain the frequency spectra; see Fig. 6.2c, d. Figure 6.2c indicates that the spectrum obtained from the vehicle response consists of four main frequencies: the vehicle frequency f v =2.63 Hz; the driving frequency f d =0.2 Hz, which is computed as the ratio of the vehicle velocity to the length of the bridge; and the two frequencies for each modal frequency of the bridge which are shifted (f b ± fd/2) due to the Doppler’s effect [21]. As a result, the first, second, and third frequencies of the bridge are identified as 4.40, 17.20, and 37.80 Hz, respectively, by taking the average of their corresponding shifted frequencies [21]. Moreover, Fig. 6.2c also demonstrates that the bridge frequencies become less visible in the FFT spectrum for the higher modes [16]. However, using the CP’s response completely eliminates the vehicle frequency from the FFT spectrum, as demonstrated in Fig. 6.2d. It significantly enhances the amplitudes of the second and third modal frequencies of the bridge and make them more detectable in the FFT spectrum, as seen in Fig. 6.2d. Fig . 6. 1 The single-span bridge
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