This study is concerned with the diagnosis of discrepancies in a steel truss bridge by identifying dynamic properties from the vibration response signals of the bridges. The vibration response signals collected at bridges under three different vehicular speeds of 10 km/hr, 20 km/hr, and 30 km/hr are analyzed using statistical features such as kurtosis, magnitude of peaktopeak, root mean square, crest factor as well as impulse factor in time domain, and Stockwell transform in the timefrequency domain. The considered statistical features except for kurtosis show uncertain behavior. The Stockwell transform showed lowresolution outcomes when the presence of noise in the recorded vibration responses. The elimination of noise and extraction of meaningful dynamic properties from the vibration responses is done by applying a new method which comes from the fusion of Hilbert transform with Spectral kurtosis and bandpass filtering. The outcomes obtained from Hilbert transform processed residual signals which are further filtered using bandpass filter show more robustness and accuracy in characterizing bridge modal frequencies from the noisy vibration responses. The proposed method produces a highresolution frequency response which can unveil the joint discrepancy in the bridge structure.
The damage detection using vibrationbased signal processing techniques in bridges has been widely applied in recent decades [
Various bridge health monitoring measures were taken in past to detect these structural deficiencies as early as possible and provide suitable countermeasures associated with the particular problem such that there can be an enhancement in the service span of the bridge [
The timedomain analysis includes the use of statistical and stochastic features in performing timehistory signals analysis [
The loosening of joint connections in a bridge produces transient variation in vibration responses due to alterations in resonant modal properties such as natural frequencies etc. of the bridge [
The present study clarifies the limitations of statistical (time domain), and Stransform (timefrequency domain) techniques in eliminating noise involved in the recorded vibration response signals from the steel truss bridge, and demonstrates the proposed technique of analyzing recorded vibration response signals at different vehicular speeds through combined Hilbert transform, Spectral kurtosis and bandpass filtering technique is efficient in extracting modal frequencies of the steel truss bridge. The obtained modal frequencies are used to determine the location of discrepancies in the steel truss bridge. The structurally deficient nodes showed either missing or low amplitude of modal frequencies due to loss of stiffness in the joining members at a particular location.
The throughtype steel truss bridge considered for the study is a simply supported bridge having a span of 40 m with a single lane carriageway. It is supported by the roller at one end and hinged at another end. There is an equal spacing of 4 m interval among the vertical members along the span of the bridge as shown in
A summary of the definitions, process, and application of each technique used in the recorded vibration signal analysis are presented in this section. During the analysis, it has been found that the behavior of the nodes is random and the response hence is shown for all nodes under different signal processing techniques.
The
For any vibration signal y(i) in the time domain, with i = 1, 2, …, n, where n represents the different data points present in the signal. The
where the maximum peak values (
The crest factor feature showed marginal variation at different nodes of the bridge as shown in
The kurtosis feature showed a significant irregular rise in amplitude at 6, 7, and 8 downstream nodes for both 10 km/hr and 30 km/hr vehicular speeds, while the amplitude of these nodes decreased suddenly with respect to other nodes for 20 km/hr vehicular speed. This indicates the possibility of having structural flexibility deficiencies at these locations. The nodes in the upstream truss exhibit absolute ascending order variation in amplitudes except for marginal variation in nodes 14 and 16 as shown in
The impulse factor values on downstream truss showed an absolute increasing trend for nodes 3, 4, 6, 7, and 10 while on upstream truss no definite pattern is observed as shown in
The idea behind calculating some popular statistical features is to investigate raw signals in the time domain and identify nodes showing irregular behavior. Except for kurtosis other features were unable to give any significant variation in the behavior of nodes of the bridge. The staggering difference between the upstream and downstream truss nodes response is due to varying stiffness in members of the trusses. From the graphs, it is observable that maximum significant variation occurs in nodes 6, 7, and 8 and the kurtosis feature shows a reasonable change hence for further analysis kurtosis is considered. The downstream nodes 6, 7, and 8 showed peculiar behavior in comparison to all other nodes. However, magnitude of PeaktoPeak also indicates a peculiar pattern at nodes 6, 7, and 8 but kurtosis is comparatively more distinct. However, on the upstream truss, the kurtosis results at all the nodes follow a similar pattern whereas the magnitude of PeaktoPeak feature show randomness and do not yield any useful information.
The Fourier transform has the limitation of simultaneously visualizing frequency and time and it is also not suitable for nonstationary signals [
Stockwell transform involves combined elements of both Short Time Fourier transform (STFT) and wavelet transform (WT) that gives outcome in the timefrequency spectrum [
The ShortTime Fourier Transform (
where, f = frequency and r, t = time variables
The uncertainty principle states that the product of timebandwidth cannot be reduced without limits. The Gaussian window (
where,
‘
The new base function obtained is shown as
where
The mathematical expression of Stransform is shown as
The inverse Stransform can be expressed as
where
The plots of Stransform have been shown for downstream nodes 2, 6, 7, and 8 at different vehicular speeds in
For downstream nodes, 1^{st}, 2^{nd} and 3^{rd} modal frequencies at vehicular speeds of (a) 10 km/hr are in the range of 2.81 Hz to 16.75 Hz, 7.25 Hz to 11.33 Hz, and 16.25 Hz to 16.99 Hz, respectively (b) 20 km/hr are in the range of 4.67 Hz to 6.57 Hz, 8.28 Hz to 10.66 Hz and 14.75 Hz to 16.66 Hz, respectively (c) 30 km/hr are in the range of 3.49 Hz to 4.70 Hz, 7.98 Hz to 12.57 Hz and 16.56 Hz to 16.75 Hz, respectively.
The detailed values of all the frequencies obtained for downstream nodes around 1^{st} (4.56 Hz), 2^{nd} (10.44 Hz), and 3^{rd} (16.66 Hz) analytical modal frequencies from Stockwell transform at all vehicular speeds are shown in
(a) 10 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
2  4.75  16.25  
3  4.75, 6.25, 
7.25, 10.50  
4  –  8.66, 

5  5.70  10.57  16.66 
6  4.70  –  – 
7  3.50  –  – 
8  –  –  
9  –  10.66  16.66 
10  5.10  10.66  16.66 
(b) 20 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
2  5.98  10.33  16.44 
3  16.66  
4  5.33  10.33  
5  4.77  8.28  16.28 
6  4.70  9.89  – 
7  4.80, 6.28  10.33  – 
8  4.67  –  – 
9  –  10.55  
10  16.25  
(c) 30 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
2  10.75  16.66  
3  4.67  10.75, 11.86  
4  –  12.25  16.75 
5  –  8.28, 10.57  16.66 
6  11.46  –  
7  4.75  8.28, 10.33  – 
8  4.70  –  – 
9  –  16.85  
10  4.65  10.44 
Note: All frequencies provided in the table are in Hz. *The bold values in the table indicate the minimum and maximum frequency values. ‘’ Not present
The mean of frequencies obtained around 1^{st}, 2^{nd,} and 3^{rd} modes for 10 km/hr vehicular speed are 4.92 Hz, 9.74 Hz, and 16.58 Hz, respectively. For 20 km/hr vehicular speed are 5.31 Hz, 10.15 Hz, and 16.17 Hz, respectively. For 30 km/hr vehicular speed are 4.49 Hz, 10.46 Hz, and 16.71 Hz, respectively.
The plots of Stransform have been shown for upstream nodes 13, 15, 16, and 18 at different vehicular speeds in
For upstream nodes, 1^{st}, 2^{nd} and 3^{rd} modal frequencies at vehicular speeds of (a) 10 km/hr are in the range of 4.66 Hz to 6.98 Hz, 7.98 Hz to 12.04 Hz, and 14.33 Hz to 16.86 Hz, respectively (b) 20 km/hr are in the range of 4.75 Hz to 5.75 Hz, 7.50 Hz to 12.55 Hz and 13.25 Hz to 17.75 Hz, respectively (c) 30 km/hr are in the range of 5.33 Hz to 7.98 Hz, 8.34 Hz to 12.25 Hz and 14.55 Hz to 17.75 Hz, respectively.
The values of all the frequencies obtained for upstream nodes around 1^{st} (4.56 Hz), 2^{nd} (10.44 Hz), and 3^{rd} (16.66 Hz) analytical modal frequencies from Stockwell transform at all vehicular speeds are shown in
(a) 10 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
13  –  8.15, 11.82  16.66 
14  10.66  14.66, 15.66, 

15  6.33  11.98, 10.33  
16  –  9.33, 10.33, 
15.22, 16.66 
17  5.10  10.85  16.57 
18  –  9.10, 10.57  16.57 
19  –  16.50  
20  –  10.01, 11.76  16.25 
21  –  9.75  16.50 
(b) 20 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
13  
14  –  10.33  
15  6.20  11.25  16.20 
16  10.20  16.66  
17  4.75  9.66, 10.25  16.56 
18  –  10.85  15.71 
19  –  10.85  16.78 
20  –  10.57, 11.43  15.42, 16.25 
21  –  8.25, 9.49, 10.50  14.50, 16.75 
(c) 30 km/hr  
Node No.  1^{st} Mode  2^{nd} Mode  3^{rd} Mode 
13  –  14.66, 15.66, 16.56  
14  –  10.75, 
14.25, 16.75, 
15  5.75  10.75  16.25 
16  7.66, 8.99, 9.66, 10.66  14.33  
17  –  10.33  15.55 
18  5.57  10.57  16.28 
19  –  
20  –  9.37, 10.33  16.87 
21  –  8.39, 10.33, 12.20  14.80, 16.74 
Note: All frequencies provided in the table are in Hz. *The bold values in the table indicate the minimum and maximum frequency values. ‘’ Not present or marginally present.
The mean of frequencies obtained around 1^{st}, 2^{nd,} and 3^{rd} modes for 10 km/hr vehicular speed are 5.77 Hz, 10.34 Hz, and 16.06 Hz, respectively. For 20 km/hr vehicular speed are 5.36 Hz, 10.30 Hz, and 15.99 Hz, respectively. For 30 km/hr vehicular speed are 5.55 Hz, 10.16 Hz, and 15.79 Hz, respectively.
From the Stockwell transform plots it is concluded that at different vehicular speeds the frequencies of downstream and upstream nodes are not exactly same. The resolution of output plots showing the frequencies present in the vibration response signals got improved with the application of Stockwell transform with respect to time features respectively. However, the presence of noise in the vibration signals generated unwanted frequencies in the vicinity of desired modal frequencies. The nodes 6, 7, and 8 in downstream truss showed undesired behavior with the presence of higher intensity frequencies only around 1^{st} mode at all the vehicular speeds.
To ensure that there exists partial flexibility in the joints at these locations further denoising of the vibration signals is required. Various studies are showing a broader range of areas of application of Hilbert transform on noisy vibration signals for obtaining better visualization of hidden frequencies [
where,
Hilbert transform is a frequency independent time domain involution that maps real timedomain value into another value. It is also called 90° phase shifter and does not affect the nonstationary characteristics of a modulating signal. This can be obtained mathematically by the following
where,
where,
From the time domain analysis performed earlier it is known that the kurtosis feature shows better performance than others, hence the signal obtained after Hilbert transform having the highest kurtosis value is selected to obtain a spectral response. The spectral kurtosis provides the highest kurtosis value, frequency present along with its duration in the signal. Further, the bandpass filter is used to refine the final outcome vibration signal. The modal frequencies are obtained which are further utilized to identify the bridge nodes having discrepancies, respectively.
After the analysis of popular statistical features, and improved method in timefrequency, i.e., Stransform. The identification of flexible nodes requires more confidence. From the performed analysis it has been concluded that timedomain analysis only gives the identification of response of speed variation. The timefrequency analysis gives more information regarding the condition of nodes from frequencies obtained with poor resolution due to the presence of noise in vibration signals measured from the bridge. However, by using previously established techniques, due to the presence of noise some nodes not able to give a clear understanding of the node which may mislead the interpretation. Thus, a methodology is proposed that is efficiently able to remove the noise and successfully provide a clear view of the present fundamental modal frequencies of the bridge. The methodology adopted here is shown in the flowchart in
Step 1: Initially power spectrum analysis is performed and then Hilbert transform is computed.
Step 2: The Hilbert envelope spectrum is calculated from the Hilbert envelope signal to identify the modal frequencies. The
The visualization of the contaminated signal in the time domain along with the computation of kurtosis of the signal was done.
Step 3: Kurtosis is a statistical measure that defines how heavily the tails of distribution differ from the tails of a normal distribution as shown in
The typical spectrogram of spectral kurtosis is shown in
Step 4: The Spectral Kurtosis is applied to the specified window length obtained from kurtogram plots. The concept of
where
“−2” removes the complexity of the signal
The complex envelope (
where
The complex envelope
The refined spectral kurtosis plots for typical nodes 2 and 6 at different vehicular speeds of 10 km/hr, 20 km/hr, and 30 km/hr are shown in
It is concluded that all downstream and upstream nodes showed a high range of frequencies except for nodes 6, 7, and 8, respectively. This indicates that the Spectral Kurtosis values give a clearcut indication of the consolidation in frequencies in the lower range for deficient nodes.
Step 5 BandPass filter is applied to the spectral kurtosis outcome signals for retrieving a more enhanced amplitude modulated signal. The low frequency and high frequency cut offs used in band pass filter for effective filtration of the noise from spectral kurtosis signal are 0 Hz to 35 Hz, respectively.
The spectrum responses of downstream nodes from the proposed methodology at different vehicular speeds respectively are shown in
For the downstream nodes at 10 km/hr vehicular speed the 1^{st} modal frequency is obtained for all the nodes except for nodes 7 and 9, respectively. The nodes 5, 6, and 8 showed a low amplitude of 1^{st} modal frequency. The 2^{nd} modal frequency is obtained with low amplitude for all the nodes except for nodes 7 and 9. The 3^{rd} modal frequency at all the nodes has a significant amplitude except for nodes 6 and 7. At 20 km/hr vehicular speed the 1^{st} modal frequency is obtained at all the nodes except for nodes 7 and 9 respectively. The nodes 2, 3, and 4 showed higher amplitude for 1^{st} modal frequency relatively to the other nodes. The 2^{nd} modal frequency is obtained at all the nodes. The nodes 6, 7, 9, and 10 showed higher amplitude relatively to other nodes. The 3^{rd} modal frequency is obtained with significant amplitude at all the nodes except at nodes 6, 7, and 8, respectively. At 30 km/hr vehicular speed the 1^{st} modal frequency is obtained at all the nodes. The nodes 5, 6, 7, and 8 showed a low amplitude for 1^{st} modal frequency relatively to the other nodes. The 2^{nd} modal frequency is obtained at all the nodes. The nodes 6, 7, 8, and 9 showed a low amplitude for 2^{nd} modal frequency relatively to the other nodes. The 3^{rd} modal frequency is present at all the nodes with high amplitude except for nodes 6, 7, and 8, respectively. From the obtained modal frequencies of the downstream nodes for different vehicular speeds, it is concluded that the irregular behavior at nodes 6, 7, and 8 shows the presence of deficiencies at these locations of the steel truss bridge.
The spectrum responses of bridge upstream nodes with the proposed methodology at different vehicular speeds as shown in
For the upstream nodes at 10 km/hr vehicular speed the 1^{st} modal frequency is obtained at all the nodes. The nodes 13, 14, 20, and 21 showed higher amplitude relatively to the other nodes. The 2^{nd} modal frequency is present at all the nodes. All the nodes showed a low amplitude for 2^{nd} modal frequency except nodes 19 and 21. The 3^{rd} modal frequency is present at all the nodes. The nodes 15, 16, 17, 18, and 21 showed higher amplitude relatively to the other nodes. At 20 km/hr vehicular speed the 1^{st} modal frequency is present at all the nodes. The nodes 16, 17, and 19 showed a low amplitude for 1^{st} modal frequency relatively to the other nodes. The 2^{nd} modal frequency is present at all the nodes except node 15 respectively. The nodes 17, 18, 19, and 21 showed a high amplitude for 2^{nd} modal frequency relatively to the other nodes. At 30 km/hr the 1^{st} modal frequency is present at all the nodes. The nodes 18, 19, 20, and 21 showed a high amplitude relatively to the other nodes. The 2^{nd} modal frequency is present at all the nodes. The nodes 15, 16, 18, and 19 showed higher amplitude relatively to the other nodes. The 3^{rd} modal frequency is present at all the nodes. The nodes 14, 17, and 19 showed low amplitude relatively to the other nodes.
It is observed in the present study that the adopted method provides better denoised outcome of frequencies present in the recorded vibration response signals from the bridge than statistical techniques, and Stransform. The improved resolution of frequency component makes it easy to identify the deficient node through significant variation in modal frequencies. The adopted method of Hilbert transform in combination with spectral kurtosis and bandpass filter is a generalized method which can be applied on any stationary and nonstationary signals collected from any structure (e.g., fault detection in induction motor, turbine, etc.).
In the present study, the novelty is that the proposed methodology uses combined Hilbert transform, spectral kurtosis, bandpass filter and kurtogram for the selection of window length for a highresolution frequency response which is utilized to unveil the irregularity in the steel truss bridge structure using various speeds of vehicle. The limitation of study is that only first three modal frequencies are obtained from the experimentally measured vibration response data for all the nodes of the bridge. The higher nodes are not able to be extracted accurately and they showed uncertain behaviour with low amplitude. It is observed from the analytical model analysis that the modal load participation factor ratios for first three modes are 96.26%, 93.35%, 91.58%, respectively. The first three modes have shown dominant behaviour to analyze the dynamic behaviour of the bridge structure. The deviation of experimentally obtained modal frequencies w.r.t analytical frequencies shows discrepancies in behaviour of bridge structure. The nodes with increased flexibility and reduced stiffness were identified with the indication of low or missing modal frequency at all three vehicular speeds. The structural deficiency in joining members at the particular nodes can cause peculiarity in nodes behavior. The healthy nodes exhibit modal frequencies in vicinity of analytically obtained modal frequencies of bridge.
Statistical analysis features, i.e., Crest factor, kurtosis, Impulse Factor, and PeaktoPeak feature lack confidence for determining the particular behavior as the pattern for a large number of nodes is not obtained with clarity. The kurtosis gives the best pattern among all the statistical features as the speed of the vehicle increases. The kurtosis feature showed a significant irregular rise in amplitude at nodes having structural flexibility deficiencies.
In the timefrequency domain, the Stransform showed better resolution of modal frequencies plots due to its scalable window and crossterm issue elimination. The unwanted frequencies are obtained in the Stransform plots and require further denoising of the signal to eliminate the noise from vibration signals. The mode value of the obtained frequencies is observed to be the closest to analytical frequency for both downstream and upstream nodes at all vehicular speeds.
The proposed methodology utilized Hilbert transform, spectral kurtosis, and bandpass filter in combination to extract the hidden dynamic modal information with high resolution. The methodology is performed to obtain the enhanced amplitude modulated signal as compared to the outcomes of statistical features, and Stransform methods. The elimination of the noise is significantly observed with the application of Hilbert envelope analysis and bandpass filtering. The variation in obtained distinct modal frequencies is used to obtain the deficient nodes present in the steel truss bridge.
The authors would like to thank the Himachal Pradesh Public Works Department, Government of Himachal Pradesh, India for allowing the National Institute of Technology, Hamirpur to conduct the experiment on the steel truss bridge in the state. The authors also thank Dr. Suresh Kumar Walia for providing necessary experimental data for further signal processing.
Angular frequency of
root mean square
Complex envelope
the scaling factor used to vary gaussian window width
The envelope of
Fourier transform of signal
Spectral kurtosis
Gaussian window
time averaging operator
Hilbert transform
translation parameters
maximum peak values
window length
Phase angle