1 Introduction
Bridges are key components of transportation infrastructure, albeit one in eleven bridges in the U.S. were structurally deficient [asce]. The high cost and time usually required to inspect aging bridges desperately calls for advanced sensing and data analysis techniques.
The use of vibration signals collected from travelling vehicles to monitor structures (Figure 1) has recently become a viable alternative to traditional structural health monitoring approaches [lederman2017track, lederman2017sparse, lederman2017data, liu2019dynamic, liu2019detecting, liu2019damage]. This approach does not require intensive deployment and maintenance. Previous work on vehiclevibrationbased bridge health monitoring (or indirect BHM) mainly falls into two categories: modal analysis and datadriven approaches. Modal analysis focuses on identifying modal parameters of a bridge, such as natural frequencies [yang2005vehicle, yang2015extraction, matarazzo2018scalable], mode shapes [yang2014constructing, matarazzo2018scalable] and damping [yang2019bridge]. Datadriven approaches use signal processing and machine learning techniques to extract informative features for diagnosing damage [liu2019damage, lederman2014damage, malekjafarian2019machine].
However, to make the indirect BHM (IBHM) approaches practical, there are three main challenges to address. First, since the vibration signals are indirect measurements of structure’s vibrations, modal parameters identified using modal analysis, e.g.,[yang2005vehicle, yang2015extraction, yang2014constructing, matarazzo2018scalable, yang2019bridge], are sensitive to the vehicle’s properties, environmental factors and noise. Second, purely datadriven methods, e.g., [liu2019damage, lederman2014damage, malekjafarian2019machine, liu2020diagnosis], can suffer from overfitting to the available data (i.e., a set of bridges with known damage labels) and achieve significantly worse performance when applied to other bridges. Third, labeled data are limited in quantity. Especially for fullscale bridges, it is expensive, timeconsuming, and impractical to obtain vehicle vibrations with corresponding damage labels. It is also unrealistic to damage bridges for sourcing damage labels.
We introduce a physicsguided signalprocessing algorithm to extract an informative feature, which can estimate and localize damage in a bridge. To handle the first challenge, the extracted feature should be sensitive to damage instead of to uncertainties. In this work, we use the synchrosqueezed wavelet transform (SWT)
[daubechies2009synchrosqueezed, daubechies2011synchrosqueezed] to represent the vehicle acceleration signal in the timefrequency plane and reconstruct a DS component using the inverse SWT (ISWT) within a frequency band determined by preidentified system properties. We use the SWT and ISWT because SWT can represent the nonstationary and timevarying vehicle acceleration as a superposition of intrinsic mode functions (IMF)type components (our desired feature has the same type), and ISWT can reconstruct our nonstationary component within a frequency range without having the mode mixing effect [mandic2013empirical]. To addresses the second and the third challenges, a DS & DI feature is obtained by multiplying the reconstructed component by a DI factor. It is obtained from the solution of a vehicle bridge interaction system (VBIS). Because this feature is DI and not extracted by a trainable model with the supervision of damage states, diagnosing damage using this feature does not suffer from overfitting and can across multiple bridges. We verify the DS & DI properties of our feature by visualization and by comparing it to other time and frequencydomain features for estimating stiffness reductions and locations.The main contributions of this paper are 1) We cast the IBHM problem as a signal decomposition, thus affording us the tools from the signal processing community; 2) We use the synchrosqueezed wavelet transform to extract a DS & DI feature for IBHM; and 3) We validate the DS & DI properties of it through extensive experiments.
2 Physical foundations
To model the VBIS, we consider a commonly used model, a sprung mass (representing vehicle) traveling with a constant speed on a simply supported beam (representing bridge). This model provides physical foundations of the IBHM problem, which help us to formulate this problem as a signaldecomposition problem.
Vehiclebridge interaction system. The derivation of the theoretical formulation of the VBIS follows the same assumptions and system geometry as presented in [liu2019damage], albeit a local damage is considered in this work.
Let be the coordinate of the beam with the origin at the left support; the density, cross section area, damping coefficient and stiffness of the beam, respectively; the Dirac delta function; the gravity constant; the moving speed of the vehicle; and the vertical displacement of the vehicle chassis; the weight, stiffness and damping coefficient of the vehicle, respectively. The vertical acceleration of the vehicle chassis, is our measurements. The equations of motion for the VBIS are
(1)  
(2) 
Using modal superposition, we write , where are the th mode shape and mode displacement, respectively.
The stiffness of the beam with local damage is defined as
(3) 
where is the central location of the damage; and are the damage length and the percentage reduction of the stiffness, respectively; is the stiffness of the undamaged beam segment. and are the parameters we want to infer. Note that many previous works [yang2005vehicle, yang2014constructing, yang2015extraction, liu2019damage] consider the stiffness as a constant. In this paper, we solve the VBIS with the stiffnessreductiontype local damage.
Damagesensitive and domaininvariant features. We now solve Eq. (2), look for a DS & DI feature and formulate the IBHM problem as a signal decomposition problem. By omitting the damping of the beam and the vehicle, we write the acceleration of the moving vehicle as
(4)  
where , , , , and are constants depending on properties of the bridge and the vehicle; , , are the th mode’s equivalent resonant frequency, mass, and stiffness, respectively; and is the natural frequency of the vehicle. Note that when we solve Eq. (2), because , we have .
For our VBIS, local damage has relatively small influence on the bridge mode shape. Thus, we can approximate , where .
We can obtain a physical understanding of the vehicle acceleration by analyzing Eq. (4). First, Eq. (4) has the following three components within three frequency bands:

may encoder damage information in . The dominant frequency of this term is , which changes as vehicle properties change;

may encode damage information in , , , and . The dominant frequency of this term is , which changes as bridge properties change;

may encode damage information in both and , and the multiplication of the derivatives of amplifies the damage information. Also, the dominant frequency of this term is , which does not change as the bridge and vehicle properties change.
For each , the third component is thus DS & DI once we multiply it by factor . Second, because of the local influence of the damage, is nonstationary and its instantaneous frequency, which is defined as , is timevarying and . Third, for VBIS, is very small and generally does not overlap with the vehicle frequency and the shifting bridge frequency . This property of the vehicle acceleration signal also indicates that the third component is a good candidate for our desired feature because it can be extracted from the original signal.
3 Signal decomposition method for IBHM
To extract the DS & DI feature from the vehicle acceleration signal, we can cast the IBHM problem as a signal decomposition problem. We decompose the nonstationary vehicle acceleration and extract the desired feature by reconstructing the nonstationary component within a frequency band that includes . We consider approximating the DS & DI feature for the first mode that is Note has timevarying instantaneous frequencies. Also, is the sum of harmonic functions, hyperbolic functions and multiplication of harmonic and hyperbolic functions, so that the desired feature can be expressed as a superposition of IMFs.
Our proposed method uses SWT [daubechies2009synchrosqueezed, daubechies2011synchrosqueezed]
to represent the vehicle acceleration signal in the timefrequency plane and reconstruct the desired feature using ISWT. The reconstructed feature is used to estimate and localize damage. SWT has three advantages in solving our problem. First, it assumes that signals are approximately harmonic locally and have a slowly timevarying instantaneous frequency. This transform has the ability to decompose a nonstationary and timevarying signal as a superposition of IMFtype components. Second, comparing with the conventional timefrequency methods, such as shorttime Fourier transform (STFT) and continuous wavelet transform (CWT), this empirical model decomposition (EMD)like approach can further sharpen the timefrequency representation and enhance frequency localization
[daubechies2011synchrosqueezed]. Third, to localize the damage, we need to reconstruct the timedomain signal in the damagerelated frequency band as the vehicle moves. ISWT can directly reconstruct a component within a selected frequency band and avoid the modemixing effect encountered by the EMD method [mandic2013empirical]. The SWT has three steps:1) Calculate the wavelet coefficients of the signal
(5) 
where is the scale, is the time offset, and is the complex conjugate of wavelet. We use analytic Morlet wavelet;
2) Estimate the instantaneous frequencies for the signal
(6) 
3) Reallocate the energy of CWT coefficients to enhance frequency localization. At discrete scales , the SWT of is only calculated at the centers of the frequency range , where . The SWT is
(7) 
where . Component of the original signal in band is estimated by ISWT:
(8) 
where returns the real part of the function; represents the Fourier transform of the complex conjugate of the wavelet. Further details about the implementation of SWT and ISWT can be found in [daubechies2009synchrosqueezed, daubechies2011synchrosqueezed]. Our feature extraction algorithm is shown in Algorithm 1.
4 Experimental results
Finite element simulations and the dataset. Finite element models (FEMs) of the VBIS are employed to create a dataset. We have five bridges (Bridge 1, 2, 3, 4 & 5) simulated as EulerBernoulli beams of lengths , , , and , respectively; natural frequency of , , , and (for ), respectively; and they all have the same undamaged stiffness, . We use the same properties (with reference to [yang2015extraction]) for the simulated vehicle: , , and .
Damage proxy is introduced by reducing the stiffness of one element at different locations. We use the same beam element with the length of for all bridges. The stiffness reduction () ranges from to with an interval of . For each stiffness reduction level, the simulation is run at seven different damage locations (
is every eighth of the span), and for each damage scenario, the simulation is run ten times. For each simulation run, random forces, which follows a Gaussian distribution with zero mean and
variance, are applied on each node of the FEMs. This added disturbing force is a process noise that propagates through time and varies in space. In total, we generate (bridges) (reduction levels) (damage locations) (undamaged bridge)] (trials) vehicle acceleration records.Visualization of the proposed feature. We verify the DS & DI properties of our feature by visualization. The data for visualization is created by FEMs without adding the disturbing force. In our experiments, we use the proposed method to extract the desired feature within . Figures 2 (a) and (b) show vehicle accelerations and our proposed feature with different damage locations, respectively. Figures 2 (c) and (d) show the signals and the feature with different stiffness reductions. By visualization, we can easily localize and compare the damage simulated by reducing stiffness. To verify if the proposed feature is DI, in Figure 3 we visualize the ISWT of accelerations collected from the simulated vehicle traveling on Bridges 1 to 5 with different simulated damage before and after multiplying by . We can observe that, after the multiplication, the features for different bridges having the same damage match each other, which indicates that our feature is DI. Though the results also contain boundary effects of the transform that require further investigation, it is easy to see that at the normalized locations of damage the proposed feature exhibit high sensitivity.
Damage localization and quantification. To further verify if our proposed feature is DS & DI, we use a multitask learning model proposed in [liu2019damage] to estimate and localize stiffness reductions in a supervised and a semisupervised fashion. The model’s input includes raw accelerations (raw data); bandpass filtered signals within , , and (Bandpass 1, 2&3); inverse CWT within the above three bands (ICWT 1, 2 & 3); the first three IMFs calculated using EMD (IMF 1, 2 & 3); ISWT within the three bands (Ours, ISWT 2 & 3); and spectrograms calculated using STFT, CWT and SWT.
Feature  Supervised  Different  Different  

DLE  SRE  DLE  SRE  DLE  SRE  
Raw data  0.28  0.15  0.70  0.51  0.57  0.55 
Bandpass 1  0.32  0.26  0.56  0.44  0.48  0.49 
Bandpass 2  0.26  0.16  0.39  0.42  0.67  0.69 
Bandpass 3  0.38  0.31  0.46  0.41  0.64  0.91 
STFT  0.59  0.15  0.71  0.34  0.47  0.35 
ICWT 1  0.18  0.08  0.37  0.27  0.38  0.28 
ICWT 2  0.24  0.22  0.40  0.35  0.52  0.37 
ICWT 3  0.26  0.17  0.56  0.51  0.37  0.34 
CWT  0.43  0.52  0.48  0.61  0.63  0.31 
IMF 1  0.28  0.21  0.36  0.34  0.33  0.36 
IMF 2  0.27  0.45  0.40  0.56  0.82  0.84 
IMF 3  0.35  0.28  0.52  0.43  0.44  0.28 
Ours  0.17  0.08  0.36  0.27  0.38  0.27 
ISWT 2  0.30  0.26  0.39  0.39  0.55  0.53 
ISWT 3  0.28  0.18  0.39  0.33  0.58  0.30 
SWT  0.56  0.38  0.62  0.45  0.43  0.65 
1. DLE means damage location estimation.
2. SRE means stiffnens reduction estimation
The supervised task examines how sensitive each input is to damage. The second and third columns in Table 1 present this task’s results (30% for testing) in terms of root mean squared error (RMSE). Using the proposed feature, we obtain the best stiffness reduction estimation and localization results.
Table 1 also shows results for the semisupervised regressions, where we have two subtasks: test if the feature is DI across bridges with different lengths (Different ) and with different natural frequencies (Different ). As shown in columns 4 to 7 of the table, using our feature, we obtain the best stiffness reduction estimations for the two subtasks, and the best damage localization results for bridges having different lengths and the same frequency. For damage localization in the second subtask, IMF 1 provides the best result.
5 Conclusion
We introduce a physicsguided signal decomposition method to extract a DS & DI feature from vehicle accelerations for IBHM. The SWT is used to represent the data in the timefrequency plane, and the desired feature is reconstructed using ISWT within a damagerelated frequency band. We verify and evaluate the DS & DI properties of the extracted feature for IBHM using the simulated data generated from FEMs. Among the six experiments we conducted, five of them exhibited the best damage quantification and localization results across different bridges using our proposed feature.
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