94 P. Singh and A. Sadhu ∼ Hdl,n =Hdl,nω 2 b,n−1S 2 n−1 (10.9) ∼ Hdr,n =Hdr,nω 2 b,n+1S 2 n+1 (10.10) ∼ Hv,n =Hv,nω 2 v (10.11) ∼ Hbl,n =Hbl,nω 2 b,n(1−Sn) 2 (10.12) ∼ Hbr,n =Hbr,nω 2 b,n(1−Sn) 2 (10.13) The acceleration response of the vehicle traveling over the bridge can be generated using Eq. 10.3. Five terms involved in vehicle response in Eq. 10.3 can be labeled into three groups as driving frequencies, including (n−1)πv L and (n+1)πv L ; vehicle frequency ωv; and bridge frequencies, including ωbn − nπv L and ωbn + nπv L . n indicates the index of the vibration modes of the bridge. It may be observed that the bridge frequencies ωbn are shifted by an equal amount to the vehicle speed ±nπv L . The importance of bridge frequency terms ωbn− nπv L andωbn+ nπv L is crucial, and it will be verified whether these terms are practically visible in the vehicle response. 10.2.2 WPT Wavelet packet transform (WPT) is a powerful time-frequency decomposition technique that can be used to decompose a particular signal into its low- and high-frequency components. This method results in both approximated and detailed coefficients [17]. Recently, [18] used a scaled bridge model for the damage detection in the shear connectors. The damage was characterized using changes in wavelet packet energy. Over the recent years, WPT has also been explored for structural damage identification [19–21]. The decomposition process using WPT is a recursive filter-decimation operation. After j levels of decomposition, a signal f (t) can be represented as [18]: f(t) = 2j i =1 fi j (t) (10.14) A linear combination of wavelet packet functions ψi j,k (t) can be used to represent the wavelet packet component signal fi j (t) [18]: fi j (t) = ∞ k =−∞ ci j,k(t)ψ i j,k(t) (10.15) where a wavelet packet is a function with three indices, i, j, and k, which correspond to modulation, scale, and translation, respectively. Furthermore, wavelet packet coefficients, ci j,k (t), can be calculated as: ci j,k(t) = ∞ −∞ f(t)ψi j,k(t)dt (10.16) In another study, wavelet packet energy spectrum was used to create a real-time and early damage alarming system for an operational metro subway line. [22] utilized the signal denoising and wavelet packet energy spectrum to estimate and construct the damage indicators, namely, energy ratio deviation and energy ratio variance. Computational issues and accurate sensor placement were the shortcomings of this study. The application of WPT in the detection of micro-damage initiation
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