Damage-sensitive and domain-invariant feature extraction for vehicle-vibration-based bridge health monitoring

D AMA GE-SENSITIVE AND DOMAIN-INV ARIANT FEA TURE EXTRA CTION FOR VEHICLE-VIBRA TION-B ASED BRIDGE HEAL TH MONITORING Jingxiao Liu ? Bingqing Chen † Siheng Chen ‡ Mario Ber g ´ es † J acobo Bielak † HaeY oung Noh ? ? Ci vil and En vironmental Engineering, Stanford Univ ersity † Ci vil and En vironmental Engineering, Carnegie Mellon Uni versity ‡ Mitsubishi Electric Research Laboratories (MERL) ABSTRA CT W e introduce a physics-guided signal processing approach to extract a damage-sensitive and domain-in variant (DS & DI) feature from acceleration response data of a vehicle tra v- eling ov er a bridge to assess bridge health. Motiv ated by indirect sensing methods benefits, such as low-cost and low- maintenance, vehicle-vibration-based bridge health mon- itoring has been studied to efficiently monitor bridges in real-time. Y et applying this approach is challenging because 1) physics-based features extracted manually are generally not damage-sensitiv e, and 2) features from machine learning techniques are often not applicable to different bridges. Thus, we formulate a v ehicle bridge interaction system model and find a physics-guided DS & DI feature, which can be e x- tracted using the synchrosqueezed wa velet transform repre- senting non-stationary signals as intrinsic-mode-type compo- nents. W e v alidate the effecti veness of the proposed feature with simulated e xperiments. Compared to con ventional time- and frequency-domain features, our feature provides the best damage quantification and localization results across different bridges in fiv e of six experiments. Index T erms — structural health monitoring, domain- in variant features, synchrosqueezed w av elet transform 1. INTRODUCTION Bridges are key components of transportation infrastructure, albeit one in eleven bridges in the U.S. were structurally defi- cient [1]. The high cost and time usually required to inspect aging bridges desperately calls for adv anced sensing and data analysis techniques. The use of vibration signals collected from tra velling ve- hicles to monitor structures (Figure 1) has recently become a viable alternativ e to traditional structural health monitoring approaches [2, 3, 4, 5, 6, 7]. This approach does not require intensiv e deployment and maintenance. Previous work on vehicle-vibration-based bridge health monitoring (or indirect BHM) mainly falls into two categories: modal analysis and data-driv en approaches. Modal analysis focuses on identify- ing modal parameters of a bridge, such as natural frequencies [8, 9, 10], mode shapes [11, 10] and damping [12]. Data- driv en approaches use signal processing and machine learn- ing techniques to extract informative features for diagnosing damage [7, 13, 14]. 0 0.5 1 1.5 2 Time (s) -1 0 1 Amplitude (m/s 2 ) Record 1 0 0.5 1 1.5 2 Time (s) -1 0 1 Amplitude (m/s 2 ) Record 2 Feature extraction Damage diagnosis Data acquisition system Tra n s p o rt a t i on authority Vehi cle Acc ele rati on r ecor ds Veh i cl e Acceleration records Tra ve li ng across Local damage Bridges in a transpo rtation network Fig. 1 . V ehicle-vibration-based BHM. Features extracted from vehicle vibration signals are used to diagnose damages. Howe ver , to make the indirect BHM (IBHM) approaches practical, there are three main challenges to address. First, since the vibration signals are indirect measurements of struc- ture’ s vibrations, modal parameters identified using modal analysis, e.g.,[8, 9, 11, 10, 12], are sensitive to the vehicles properties, environmental factors and noise. Second, purely data-driv en methods, e.g., [7, 13, 14, 15], can suf fer from ov erfitting to the a vailable data (i.e., a set of bridges with known damage labels) and achieve significantly worse per- formance when applied to other bridges. Third, labeled data are limited in quantity . Especially for full-scale bridges, it is expensi ve, time-consuming, and impractical to obtain vehicle vibrations with corresponding damage labels. It is also unre- alistic to damage bridges for sourcing damage labels. W e introduce a physics-guided signal-processing algo- rithm to extract an informativ e feature, which can estimate and localize damage in a bridge. T o handle the first challenge, the extracted feature should be sensitiv e to damage instead of to uncertainties. In this work, we use the synchrosqueezed wa velet transform (SWT) [16, 17] to represent the vehicle ac- celeration signal in the time-frequenc y plane and reconstruct a DS component using the in verse SWT (ISWT) within a frequency band determined by pre-identified system proper- ties. W e use the SWT and ISWT because SWT can represent the non-stationary and time-v arying vehicle acceleration as a superposition of intrinsic mode functions (IMF)-type com- ponents (our desired feature has the same type), and ISWT T o appear in Pr oc. ICASSP2020, May 04-08, 2020, Bar celona, Spain c  IEEE 2020 can reconstruct our non-stationary component within a fre- quency range without having the mode mixing effect [18]. T o addresses the second and the third challenges, a DS & DI fea- ture 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 e xtracted by a trainable model with the supervision of damage states, diagnosing damage using this feature does not suffer from ov erfitting and can across multiple bridges. W e verify the DS & DI properties of our feature by visualization and by comparing it to other time- and frequenc y-domain features for estimating stiffness reductions and locations. The main contributions of this paper are 1) W e cast the IBHM problem as a signal decomposition, thus af fording us the tools from the signal processing community; 2) W e use the synchrosqueezed wav elet transform to extract a DS & DI feature for IBHM; and 3) W e validate the DS & DI properties of it through extensi ve e xperiments. 2. PHYSICAL FOUNDA TIONS T o model the VBIS, we consider a commonly used model, a sprung mass (representing vehicle) tra veling with a con- stant speed on a simply supported beam (representing bridge). This model provides physical foundations of the IBHM prob- lem, which help us to formulate this problem as a signal- decomposition problem. V ehicle-bridge interaction system. The deri v ation of the theoretical formulation of the VBIS follo ws the same assump- tions and system geometry as presented in [7], albeit a local damage is considered in this work. Let x be the coordinate of the beam with the origin at the left support; ρ, A, µ, E I ( x ) the density , cross section area, damping coefficient and stiffness of the beam, respecti vely; δ ( x ) the Dirac delta function; g the gravity constant; v the moving speed of the vehicle; and y ( t ) the vertical displace- ment of the vehicle chassis; m v , k v , c v the weight, stiffness and damping coef ficient of the vehicle, respecti vely . The v er- tical acceleration of the v ehicle chassis, ¨ y ( t ) is our measure- ments. The equations of motion for the VBIS are ρA ∂ 2 u ( x, t ) ∂ t 2 + µ ∂ u ( x, t ) ∂ t + ∂ 2 ∂ x 2  E I ( x ) ∂ 2 u ( x, t ) ∂ x 2  = δ ( x − v t )( m v g + m v ¨ y ( t )) , (1) m v ¨ y ( t )+ c v ˙ y ( t ) + k v y ( t ) = c v ˙ u ( v t, t ) + k v u ( v t, t ) . (2) Using modal superposition, we write u ( x, t ) = P ∞ n =1 φ n ( x ) q n ( t ) , where φ n ( x ) , q n ( t ) are the n -th mode shape and mode dis- placement, respectiv ely . The stiffness of the beam with local damage is defined as E I ( x ) = ( E I 0 if x 6∈ [ x s − l s 2 , x s + l s 2 ] E I 0 (1 − R s ) if x ∈ [ x s − l s 2 , x s + l s 2 ] (3) where x s is the central location of the damage; l s and R s are the damage length and the percentage reduction of the stiffness, respectively; E I 0 is the stiffness of the undamaged beam segment. x s and R s are the parameters we want to in- fer . Note that many previous works [8, 11, 9, 7] consider the stiffness as a constant. In this paper , we solve the VBIS with the stiffness-reduction-type local damage. Damage-sensitive and domain-in variant features. W e 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 v ehicle as ¨ y ( t ) = ∞ X n =1 C 1 n sin( ω v t ) + ∞ X n =1  C 2 n φ n ( v t ) sin( ˜ ω n t ) + C 3 n ˙ φ n ( v t ) cos( ˜ ω n t ) + C 4 n ¨ φ n ( v t ) sin( ˜ ω n t )  + ∞ X n =1 C 5 n ( φ n ( v t ) ¨ φ n ( v t ) + ˙ φ n ( v t ) 2 ) , (4) where C 1 n , C 2 n , C 3 n , C 4 n , and C 5 n are constants depending on properties of the bridge and the vehicle; ˜ ω n = 2 πf n = q ˜ k n / ˜ m n , ˜ m n = R L 0 ρAφ 2 n ( x ) dx , ˜ k n = R L 0 E I ( x )( φ 00 n ( x )) 2 dx are the n -th mode’ s equiv alent resonant frequency , mass, and stiffness, respecti vely; and ω v = 2 π f v is the natural fre- quency of the vehicle. Note that when we solve Eq. (2), because m v << ˜ m n , we hav e m v g + m v ¨ y ( t ) ˜ m n ≈ m v g ˜ m n . For our VBIS, local damage has relativ ely small influence on the bridge mode shape. Thus, we can approximate C 51 ∝ ∼ ω 2 v ˜ ω 2 1 m v g ˜ k 1 ( ˜ ω 2 1 − ω 2 d 1 ) ω 2 d 2 ( ω 2 v − ω 2 d 2 ) , where ω dn = 2 π f dn = nπ v /L . W e can obtain a physical understanding of the v ehicle ac- celeration by analyzing Eq. (4). First, Eq. (4) has the follow- ing three components within three frequency bands: • P ∞ n =1 C 1 n sin( ω v t ) may encoder damage information in C 1 n . The dominant frequency of this term is ω v , which changes as vehicle properties change; • P ∞ n =1  C 2 n φ n ( v t ) sin( ˜ ω n t ) + C 3 n ˙ φ n cos( ˜ ω n t ) + C 4 n ¨ φ n ( t ) sin( ˜ ω n t )  may encode damage information in C 2 n , C 3 n , C 4 n , ˜ ω n and ω β n ( t ) . The dominant fre- quency of this term is ˜ ω n ± ω β n ( t ) , which changes as bridge properties change; • P ∞ n =1 C 5 n ( φ n ( v t ) ¨ φ n ( v t ) + ˙ φ n ( v t ) 2 ) may encode damage information in both C 5 n and ω β n ( t ) , and the multiplication of the deri vati ves of φ n ( v t ) amplifies the damage information. Also, the dominant frequency of this term is 2 ω β n ( t ) , which does not change as the bridge and vehicle properties change. For each n , the third component is thus DS & DI once we multiply it by factor 1 /C 5 n . Second, because of the local influence of the damage, φ n ( v t ) is non-stationary and its in- stantaneous frequency , which is defined as ω β n ( t ) , is time- varying and ≥ ω dn . Third, for VBIS, ω β n ( t ) is very small and generally does not overlap with the v ehicle frequency ω v and the shifting bridge frequency ˜ ω n ± ω bn ( t ) . This prop- erty of the v ehicle 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. 2 3. SIGNAL DECOMPOSITION METHOD FOR IBHM T o extract the DS & DI feature from the vehicle acceleration signal, we can cast the IBHM problem as a signal decompo- sition problem. W e decompose the non-stationary vehicle ac- celeration and extract the desired feature by reconstructing the non-stationary component within a frequency band that in- cludes 2 ω β n ( t ) . W e consider approximating the DS & DI fea- ture for the first mode that is y d ( t ) = φ 1 ( v t ) ¨ φ 1 ( v t ) + ˙ φ 1 ( v t ) 2 . Note φ 1 ( v t ) has time-varying instantaneous frequencies. Also, y d ( t ) 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 [16, 17] to represent the vehicle acceleration signal in the time-frequency plane and reconstruct the desired feature using ISWT . The recon- structed feature is used to estimate and localize damage. SWT has three adv antages in solving our problem. First, it assumes that signals are approximately harmonic locally and hav e a slo wly time-varying instantaneous frequency . This trans- form has the ability to decompose a non-stationary and time- varying signal as a superposition of IMF-type components. Second, comparing with the conv entional time-frequency methods, such as short-time Fourier transform (STFT) and continuous wa velet transform (CWT), this empirical model decomposition (EMD)-like approach can further sharpen the time-frequency representation and enhance frequency lo- calization [17]. Third, to localize the damage, we need to reconstruct the time-domain signal in the damage-related frequency band as the vehicle mo ves. ISWT can directly reconstruct a component within a selected frequency band and av oid the mode-mixing effect encountered by the EMD method [18]. The SWT has three steps: 1) Calculate the wa velet coef ficients of the signal W y ( a, b ) = 1 √ a Z ∞ −∞ ¨ y ( t )Φ ∗  t − b a  dt, (5) where a is the scale, b is the time offset, and Φ ∗ ( t ) is the com- plex conjugate of w avelet. W e use analytic Morlet wav elet; 2) Estimate the instantaneous frequencies for the signal ω y ( a, b ) = − j ∂ b [ W y ( a, b )] W y ( a, b ) , for W y ( a, b ) 6 = 0; (6) 3) Reallocate the ener gy of CWT coefficients to enhance frequency localization. At discrete scales a k , the SWT of ¨ y ( t ) is only calculated at the centers ω c of the frequency range ω c ± ∆ ω c , where ∆ ω c = ω c − ω c − 1 . The SWT is T y ( ω c , b ) = 1 ∆ ω c X a k : | ω y ( a k ,b ) − ω c |≤ ∆ ω c / 2 W y ( a k , b ) a − 3 / 2 k ∆ a k , (7) where ∆ a k = a k − a k − 1 . Component of the original signal in band [ ω l β 1 , ω u β 1 ] is estimated by ISWT : ˆ ¨ y ( t ) = R h 1 / 2 Z ∞ 0 ˆ Φ ∗ ( ξ ) dξ ξ X ω c ∈ [ ω l β 1 ,ω u β 1 ] T y ( ω c , t ) i , (8) where R [ · ] returns the real part of the function; ˆ Φ ∗ ( ξ ) repre- sents the Fourier transform of the complex conjugate of the wa velet. Further details about the implementation of SWT and ISWT can be found in [16, 17]. Our feature extraction algorithm is shown in Algorithm 1. Algorithm 1 DS & DI feature extraction for IBHM Require: Initialize known parameters: m v , k v , v , L , ω dn 1: Input : v ertical acceleration of the moving vehicle: ¨ y ( t ) 2: Estimate system properties, including ˜ ω 1 and ˜ k 1 , using system identification methods, e.g. [10, 19]; 3: Compute the SWT , T y ( ω c , b ) , for vehicle acceleration; 4: Choose a frequency band, [ ω l β 1 , ω u β 1 ] , for extracting the desired feature based on the identified ω d 1 ; 5: Calculate ISWT , ˆ ¨ y ( t ) , within the selected frequenc y band in order to approximately reconstructing the third compo- nent of Eq. 4; 6: Output : the DS & DI feature y d ( t ) = ˆ ¨ y ( t ) /C 51 4. EXPERIMENT AL RESUL TS Finite element simulations and the dataset. Finite ele- ment models (FEMs) of the VBIS are employed to create a dataset. W e hav e fi ve bridges (Bridge 1, 2, 3, 4 & 5) simulated as Euler-Bernoulli beams of lengths 25 m , 25 m , 25 m , 20 m and 30 m , respectively; natural frequenc y of 2 n 2 H z , 2 . 5 n 2 H z , 3 n 2 H z , 2 . 5 n 2 H z and 2 . 5 n 2 H z (for n = 1 · · · 10 ), respectively; and they all have the same un- damaged stiffness, E I 0 = 14 . 54 GP a × m 4 . W e use the same properties (with reference to [9]) for the simulated vehicle: m v = 100 k g , f v = 6 . 5 H z , and v = 3 m/s . Damage proxy is introduced by reducing the stiffness of one element at different locations. W e use the same beam element with the length of 0 . 6 m for all bridges. The stiffness reduction ( R s ) ranges from 70% to 30% with an interval of 10% . For each stiffness reduction lev el, the simulation is run at sev en different damage locations ( x s is ev ery 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 distrib ution with zero mean and 0 . 1 N 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 5 (bridges) × 5 (reduction le vels) × [7 (damage locations) +1 (undamaged bridge)] × 10 (trials) = 2000 vehicle acceleration records. V isualization of the proposed feature. W e 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 e xperiments, we use the proposed method to extract the desired feature within [ f d 1 , 1 H z ] . Figures 2 (a) and (b) show vehicle accelerations and our pro- posed feature with different damage locations, respecti vely . Figures 2 (c) and (d) show the signals and the feature with different stif fness reductions. By visualization, we can eas- 3 ily localize and compare the damage simulated by reducing stiffness. T o verify if the proposed feature is DI, in Figure 3 we visualize the ISWT of accelerations collected from the simulated vehicle trav eling on Bridges 1 to 5 with different simulated damage before and after multiplying by 1 /C 51 . W e can observe that, after the multiplication, the features for different bridges ha ving 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 in vestigation, it is easy to see that at the normalized locations of damage the proposed feature exhibit high sensiti vity . 0 2 4 6 8 10 Time (s) No damage x s =1/4L x s =2/4L x s =3/4L (a) V ehicle accelerations 0 2 4 6 8 10 Time (s) No damage x s =1/4L x s =2/4L x s =3/4L (c) Our proposed feature 0 2 4 6 8 10 Time (s) R s =0.25 R s =0.5 R s =0.75 Damaged No damage (c) V ehicle accelerations 0 2 4 6 8 10 Time (s) -5 0 5 Normalized Feature 10 5 No damage R s =0.25 R s =0.5 R s =0.75 (d) Our proposed feature Fig. 2 . Ra w signals and our proposed feature for vehicle ac- celerations with dif ferent damage locations (a, and b) and dif- ferent stiffness reduction le vels (c and d). The blue marks in- dicate the damages. W e can visually localize and compare the stiffness reductions using our proposed feature, which verifies that our feature is DS. (a) ISWT in [ f d 1 , 1 H z ] (b) Our proposed feature Fig. 3 . ISWT within [ f d 1 , 1 H z ] of vehicle accelerations tra v- eling on Bridges 1, 2, 3, 4 & 5 with different damage (a) before and (b) after multiplying by 1 /C 51 . Our features for bridges with different properties and the same damage match each other around damage locations. Howev er , the features before applying the proposed multiplication do not match each other . This visualization verifies that our feature is DI. Damage localization and quantification. T o further ver- ify if our proposed feature is DS & DI, we use a multi-task learning model proposed in [7] to estimate and localize stiff- ness reductions in a supervised and a semi-supervised fash- ion. The model’ s input includes raw accelerations (raw data); band-pass filtered signals within [ f d 1 , 1 H z ] , f n ± 0 . 5 H z , and f v ± 0 . 5 H z (Bandpass 1, 2&3); in verse CWT within the abov e three bands (ICWT 1, 2 & 3); the first three IMFs cal- culated using EMD (IMF 1, 2 & 3); ISWT within the three bands (Ours, ISWT 2 & 3); and spectrograms calculated us- ing STFT , CWT and SWT . T able 1 . Damage localization and quantification results in terms of RMSE. Lower error means better performance. Feature Supervised Different L Different ˜ ω 1 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 sensiti ve each input is to damage. The second and third columns in T able 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 stif fness reduction estimation and localization results. T able 1 also shows results for the semi-supervised regres- sions, where we have two sub-tasks: test if the feature is DI across bridges with dif ferent lengths (Dif ferent L ) and with different natural frequencies (Different ˜ ω 1 ). As shown in columns 4 to 7 of the table, using our feature, we obtain the best stiffness reduction estimations for the two sub-tasks, and the best damage localization results for bridges having different lengths and the same frequency . For damage local- ization in the second sub-task, IMF 1 provides the best result. 5. CONCLUSION W e introduce a physics-guided signal decomposition method to extract a DS & DI feature from vehicle accelerations for IBHM. The SWT is used to represent the data in the time- frequency plane, and the desired feature is reconstructed us- ing ISWT within a damage-related frequency band. 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