Bridge modal identification based on successive variational mode decomposition using a moving test vehicle

Abstract

Bridge modal identification using an instrumented test vehicle as a moving sensor is promising but challenging. A key factor is to extract bridge dynamic components from vehicle responses measured when the bridge is operating. A new method based on an advanced adaptive signal decomposition technique, the successive variational mode decomposition (SVMD), has been developed to estimate the bridge modal parameters from the dynamic responses of a passing test vehicle. When bridge-related dynamic components are extracted from the decomposition, the natural excitation technique and/or random-decrement technique based fitting methods are used to estimate the modal frequencies and damping ratios of the bridge. Effects of measurement noise, moving speed and vehicle properties on the decomposition are investigated numerically. The superiority of SVMD in the decomposition is verified by comparing to another adaptive decomposition technique, the singular spectrum decomposition. The results of the proposed method confirm that the bridge modal frequencies can be identified from bridge related components with high accuracy, while damping ratio is more sensitive to the random operational load. Finally, the feasibility of the proposed method for bridge monitoring using a moving test vehicle is further verified by an in-situ experimental test on a cable-stayed bridge. The components related to the bridge dynamic responses are successfully extracted from vehicle responses.

Keywords

bridge modal identification moving test vehicle adaptive signal decomposition successive variational mode decomposition singular spectrum decomposition

Introduction

Bridge modal identification using an instrumented test vehicle has drawn great attention due to its significant potential for quick scan of bridge structural health conditions (Locke et al., 2020; Yang et al., 2020). The sensing vehicle measuring bridge dynamic responses under operational conditions is more convenient and cost-effective compared to the conventional fixed sensing networks mounted on the structure. Modal parameters identified from vehicle responses can be used to assess the health conditions of bridge (Mei et al., 2019). However, the extraction of useful dynamic information from vehicle responses for bridge health monitoring is not trivial (Hester and González, 2017; Tan et al., 2019). The vertical dynamic response of vehicle passing over a bridge is a multi-component signal mainly including the bridge dynamic component, vehicle dynamic component and driving component (Yang et al., 2004). When bridge surface roughness is considered, the vehicle related dynamic component is amplified and the bridge related dynamic information becomes less visible.

To enhance the drive-by bridge modal identification, signal processing techniques have been used to reduce the effects of road surface roughness and to extract bridge related dynamic information. Wavelet transform based methods have been used to separate the bridge dynamic information from the vehicle dynamic response (Jian et al., 2020; Li et al., 2021). However, the selection for the optimal wavelet parameters is an arbitrary process that may cause uncertainties (Tan et al., 2020). Empirical mode decomposition (EMD) as a data-adaptive technique has been used to decompose vehicle response (Yang and Chang, 2009; Yang and Lee, 2018) into a set of intrinsic mode functions (IMFs). The recovered IMFs by repeated siftings process in EMD made bridge frequencies more visible in the first few IMFs. The IMFs extracted from vehicle response were used as damage indicators of bridge structure (Kildashti et al., 2020; O’Brien et al., 2017). EEMD method introduced by Wu and Huang (2009) to address the mode mixing problem of the EMD was used to identify bridge modal frequencies from vehicle response (Zhu and Malekjafarian, 2019). The results showed that EEMD method provided better performance on the decomposition of vehicle responses compared to EMD. Singular spectrum analysis (SSA) (Elsner and Tsonis, 1996) is another powerful technique for time series decomposition and eigenvalue identification that has been applied in the field of drive-by bridge modal identification. Yang et al. (2013) applied SSA method to identify the bridge frequencies from the test vehicle response. A combination of SSA with band-pass filter can filter out the vehicle-related dynamic components to improve the scan of bridge modal parameters. Li et al. (2019a) proposed a drive-by blind modal identification method (SSA-BSS) by combining SSA and second-order blind identification. However, SSA method requires manual selection of the embedding length. A new adaptive method, singular spectrum decomposition (SSD), for decomposing time series into narrow-banded components was proposed in (Bonizzi et al., 2014). The method is originated from SSA with an automated choice of fundamental parameters. Singular spectrum decomposition method has been shown to retrieve different components concealed in the data accurately to many fields. However, to the best knowledge of the authors, SSD has not been used to analyze vehicle responses for drive-by bridge inspection.

Moreover, Dragomiretskiy and Zosso (2014) proposed the variational mode decomposition (VMD), a noniterative and adaptive signal processing method. Due to its solid mathematical theoretical foundation compared with EMD, VMD-based methods have been used in different areas, such as the analysis of seismic signal (Li et al., 2018), underwater acoustic signal (Li et al., 2019b), structural system identification (Ni et al., 2018) and load data of mechanical systems (Fu et al., 2020). Tian and Zhang (2020) utilized VMD to decompose vehicle-induced bridge responses into IMFs to extract dynamic properties of the VBI coupled system. Yang et al. (2021) used VMD to extract the mono-components from contact-point responses of a VBI model to identify the frequencies and damping ratios of the bridge. The results demonstrated that VMD performed more efficiently and elegantly than EMD/EEMD in extracting the mono-component responses. Despite of its extensive application, the performance of VMD is greatly affected by the manually preset mode number and mode frequency bandwidth control parameter (Zhang et al., 2018). Nazari and Sakhaei (2020) proposed a novel successive variational mode decomposition (SVMD) method which extracts the components successively from signals. It is more effective for the signal decomposition than VMD when the number of components is unknown.

From the above literature review, it is found that the operational bridge modal identification from vehicle responses is challenging. Therefore, a method based on SVMD is developed to estimate the bridge modal parameters using a passing test vehicle. It can successively decompose vehicle responses into meaningful mono-components without requiring the number of components contained in the vehicle responses. This study investigates its performance in the decomposition of vehicle responses by comparing with that of SSD. The bridge related components are extracted to identify frequencies and damping ratios by incorporating natural excitation technique (NExT) and/or random-decrement technique (RDT) based modal identification algorithms, respectively.

The rest of the paper is organized as follows. Vehicle-bridge interaction model considering operational load presents the vehicle bridge interaction model that describes the dynamics of bridge structure under operational moving load and the response measurement of moving test vehicle. Bridge modal identification based on adaptive signal decomposition from a test vehicle briefly introduces the SVMD for the decomposition of vehicle responses. The NExT and RDT to be used for damping ratio identification are also described. Extensive numerical study is conducted in Numerical study to demonstrate the decomposition results of the adaptive techniques. Feasibility of the incorporated damping ratio identification methods is investigated. Finally, the vehicle response measured from an in-situ vehicle-bridge interaction test is used to further verify the effectiveness of the decomposition methods which is followed by conclusions.

Vehicle-bridge interaction model considering operational load

For the implementation of vibration-based bridge health monitoring, sufficient external load is usually required to excite the bridge structure to a certain extent (Makki Alamdari et al., 2021). For a bridge subjected to a medium to large volume of random operational traffic, the spatio-temporal load pattern can be modeled as a random white noise with sufficient accuracy (Sadeghi Eshkevari et al., 2020). Therefore, the VBI model considered for bridge modal identification is shown in Figure 1. The moving load P in the figure represents the operational load and a widely used single-degree-of-freedom quarter car is utilized as the instrumented test vehicle. The acceleration response measured from the sensing vehicle is used for bridge modal identification. The operational load enters the bridge ahead of the sensing vehicle with a moving speed $v_{1}$ and the speed of the moving vehicle is $v_{2}$ .

Figure 1.

The model of drive-by bridge inspection in operational condition.

Bridge model under operational load

The simply supported bridge with a length L can be modelled with finite element model. When the moving operational load $P (t)$ is incorporated into the bridge system, the motion of equation of bridge can be written as

M_{b} {\overset{\cdot\cdot}{d}}_{b} (t) + C_{b} {\overset{\cdot}{d}}_{b} (t) + K_{b} d_{b} (t) = H_{r} (t) P (t)

(1)

where $M_{b}$ , $C_{b}$ , and $K_{b}$ are the mass, damping, and stiffness matrices of the bridge, respectively; ${\overset{\cdot\cdot}{d}}_{b}$ , ${\overset{\cdot}{d}}_{b}$ , and $d_{b}$ are the acceleration, velocity, and displacement responses at the element nodes of the bridge, respectively; $H_{r} (t) P (t)$ is the equivalent nodal load vector for the finite element analysis. $H_{r} (t)$ is based on the Hermitian cubic interpolation function for the calculation of equivalent nodal force from moving load $P (t)$ . The entries of $H_{r} (t)$ are zeros except at the degrees-of-freedom (DOFs) corresponding to the nodal displacements of the beam element on which the load is acting, with

H_{r} = {0 \dots 0 \dots H_{1} \dots 0}^{T} \in R^{NN \times 1}

(2)

where NN is the total number of DOFs for the bridge model; the shape function $H_{1}$ evaluates the moving load on the j-th beam element and its components are given as

H_{1} = {\begin{matrix} \begin{matrix} 1 - 3 {(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})}^{2} + 2 {(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})}^{3} \\ ({\bar{x}}_{1} (t) - (j - 1) l_{e}) {(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}} - 1)}^{2} \end{matrix} \\ \begin{matrix} 3 {(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})}^{2} - 2 {(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})}^{3} \\ ({\bar{x}}_{1} (t) - (j - 1) l_{e}) ({(\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})}^{2} - (\frac{{\bar{x}}_{1} (t) - (j - 1) l_{e}}{l_{e}})) \end{matrix} \end{matrix}}^{T}

(3)

with $(j - 1) l_{e} \leq {\bar{x}}_{1} (t) \leq j l_{e}$ and $l_{e}$ is the length of finite element for the moving load on the j-th finite element. In this study, the vehicle is used as a moving sensor to measure the bridge response subjected to the operational load P(t). The vehicle mass is very small compared to that of the bridge. It is reasonable to neglect the dynamic effects of vehicle on the bridge in this scenario (Kong et al., 2014). Therefore, the vehicle load to the bridge is not considered in equation (1).

Dynamics of the test vehicle

The vehicle parameters are: $m_{v}$ the mass of vehicle, $k_{s}$ and $c_{s}$ the stiffness and damping of suspension spring and damper, respectively. The equation of motion of vehicle can be expressed as

m_{v} {\overset{\cdot\cdot}{y}}_{v} (t) + c_{v} {\overset{\cdot}{y}}_{v} (t) + k_{v} y_{v} (t) = F_{cp} (t)

(4)

where $y_{v}$ is the displacement response of vehicle; $F_{cp} (t) = k_{v} d_{cp} (t) + c_{v} {\overset{\cdot}{d}}_{cp} (t)$ ; $d_{cp} (t) = w ({\hat{x}}_{1} (t), t) + r ({\hat{x}}_{1} (t))$ is the displacement input to the sensing vehicle at location ${\hat{x}}_{1} (t)$ and ${\overset{\cdot}{d}}_{cp} (t)$ is its time derivative. The operational load moves over the bridge with a speed $v_{1}$ and it enters the bridge with a distance $L_{ap}$ ahead of the sensing vehicle. The moving operational load is used to excite the bridge for vibration-based structural modal parameter identification. Based on equation (1), the operational load is transformed into equivalent nodal forces at the nodes of the bridge beam elements. It can be viewed as applying the random load along the bridge span to consider its spatio-temporal characteristics (Sadeghi Eshkevari et al., 2020). The response of the test vehicle is then analyzed for the bridge modal identification. Under these conditions, the speed of operational load has little effect on the analysis in this study. Therefore, the moving speed of the operational load is simply set as $v_{1} = L / L (L + L_{ap}) v_{2} (L + L_{ap}) v_{2}$ to simulate the random excitation to the bridge. Newmark-beta method is used to solve the equations (1) and (3) to obtain the dynamic responses of the bridge structure and sensing vehicle.

Model of road surface roughness

A widely used random roughness surface simulated based on ISO-8606 (1995) is considered. The random roughness in time domain can be given as follows (Henchi et al., 1998)

r (x) = \sum_{i = 1}^{N_{f}} \sqrt{4 S_{d} (f_{i}) Δ f} \cos (2 π f_{i} x + θ_{i})

(5)

where $S_{d} (f)$ is the displacement power spectral density of road surface roughness; $f_{i} = i Δ f$ is the spatial frequency(cycles/m); $Δ f = 1 / 1 N_{f} Δ N_{f} Δ$ , and $Δ$ is the distance interval between successive ordinates of the surface profile; $N_{f}$ is the number of data points; $θ_{i}$ is a set of independent random phase angle uniformly distributed between 0 and $2 π$ . The degree of road roughness is determined by the $S_{d} (f_{0})$ value, where $f_{0} (= 0.1 cycles / = 0.1 cycles m m)$ is the reference spatial frequency. Class A road roughness defined using specified $S_{d} (f_{0})$ value in ISO specification is considered. The roughness pattern in time domain is given in Figure 2.

Figure 2.

Road roughness profile.

Bridge modal identification based on adaptive signal decomposition from a test vehicle

SVMD

For the multi-component vehicle response with N data points, ${\overset{\cdot\cdot}{y}}_{v} (t)$ , its mono-components can be separated using SVMD iteratively. Unlike the VMD that finds IMFs simultaneously, the SVMD is performed to decompose signal into IMFs one after the other by successively applying variational mode extraction (Nazari and Sakhaei, 2018). This procedure is continued until all modes are extracted. Assume that the signal ${\overset{\cdot\cdot}{y}}_{v} (t)$ is decomposed into two parts: the k-th mode $u_{k} (t)$ and the residual signal $y_{r} (t)$ as follows

{\overset{\cdot\cdot}{y}}_{v} (t) = u_{k} (t) + y_{r} (t)

(6)

where $y_{r} (t)$ contains the sum of previously obtained modes and the un-processed part of signal $y_{u} (t)$ as

y_{r} (t) = \sum_{i = 1}^{k - 1} u_{i} (t) + y_{u} (t)

(7)

The decomposition method is based on the following four criteria

(1) Each mode should be compact around its center frequency. The k-th mode minimizes the following criterion:

J_{1} = ‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖_{2}^{2}

(8)

where $j = \sqrt{- 1}$ , $\partial_{t}$ denotes the derivative with respect to time, * the convolution operation, $Δ$ the L²-norm and $ω_{L}$ is the center frequency of the k-th mode.

(2) The energy of the residual signal $y_{r} (t)$ should be minimized at frequencies where $u_{k} (t)$ has effective components. This constraint is realized by using a proper filter ${\hat{β}}_{k} (ω)$ with frequency response of

{\hat{β}}_{k} (ω) = \frac{1}{α {(ω - ω_{k})}^{2}}

(9)

To get minimized spectral overlap between $y_{r} (t)$ and $u_{k} (t)$ , the energy of filtered $y_{r} (t)$ by ${\hat{β}}_{k} (ω)$ should be minimized. The following criterion needs to be realized

J_{2} = ‖ β_{k} (t) * y_{r} (t) ‖_{2}^{2}

(10)

where $β_{k} (t)$ is the impulse response of the filter ${\hat{β}}_{k} (ω)$ .

(3) Besides the minimization of criteria $J_{1}$ and $J_{2}$ , $u_{k} (t)$ should have less energy at frequencies around the center frequencies of the previously obtained modes. This constraint can be satisfied using proper filters with the frequency responses as

{\hat{β}}_{i} (ω) = \frac{1}{α {(ω - ω_{i})}^{2}}; i = 1, 2, \dots, k - 1

(11)

This added criterion is represented as follows

J_{3} = \sum_{i = 1}^{k - 1} ‖ β_{i} (t) * y_{r} (t) ‖_{2}^{2}

(12)

where $β_{i} (t)$ is the impulse response of the filter ${\hat{β}}_{i} (ω)$ .

(4) The last constrain is to guarantee the reconstruction of ${\overset{\cdot\cdot}{y}}_{v} (t)$ from k modes and the un-processed part of the signal:

{\overset{\cdot\cdot}{y}}_{v} (t) = u_{k} (t) + y_{u} (t) + \sum_{i = 1}^{k - 1} u_{i} (t)

(13)

When k-1 modes are known, the problem of extracting the k-th mode can be expressed as a constrained minimization problem, in which a combination of $J_{1}$ , $J_{2}$ and $J_{3}$ is minimized subject to the constrain of

\begin{matrix} min {α J_{1} + J_{2} + J_{3}} subject to : u_{k} (t) + y_{r} (t) = {\overset{\cdot\cdot}{y}}_{v} (t) \\ u_{L} . ω_{L} . y_{r} \end{matrix}

(14)

where $α$ is a parameter for balancing $J_{1}$ , $J_{2}$ and $J_{3}$ , which can be solved through Lagrangian multiplier method. The weighting parameter $α$ is one of the most important parameters of SVMD. A simple heuristic method to change $α$ in each iteration is used to avoid the problems related to low or high value of $α$ . The algorithm of SVMD with varying $α$ is presented in (Nazari and Sakhaei, 2020).

Bridge modal identification using a moving test vehicle

In this study, it is assumed that the properties of sensing vehicle are known, and vehicle’s frequencies are not coincided with those of bridge. Therefore, the bridge related dynamic mono-components can be extracted from vehicle response for the identification of bridge modal frequency and damping ratio (Yang et al., 2021). When the mono-components related to the bridge dynamic modes are extracted, the frequency can be easily identified and the damping ratio for each mode can be estimated by the least-squares fitting an exponential decay to the envelop of the impulse response function (IRF) of the system. Since the IRF is not directly available, the NExT and/or RDT are applied to extract the impulse response function from the mono-component.

The NExT and RDT can be used to estimate the impulse response function by calculating the auto-correlation function of the measured response of a structure under random or ambient excitation. The auto-correlation function is closely related to the free-decay response of the structure (James et al., 1993). For NExT, the correlation function can be estimated using direct procedure with time domain data or via calculating spectral density functions. The calculation of correlation function requires a preset time lag T. While the principle of RDT is to estimate random decrement signatures by averaging time segments of the responses. These segments are selected under certain triggering conditions. For the application of RDT, two key parameters, i.e. the trigger threshold $ϕ$ and time lag T to determine the segment number, need to be preset. The most used triggering conditions are level-crossing, positive point, local extrema, and zero-up crossing. In this study, the level-crossing triggering condition is utilized with the recommendation of $ϕ = \sqrt{2} σ$ , where $σ$ is the standard deviation of the signal. The outcome of the fitting procedure depends on the number of cycles considered within the IRF. It is better to have a sufficient length of the time lag T to cover the low frequencies of the structure (Kordestani et al., 2018). The envelop of IRF can be obtained using the Hilbert transform. Therefore, the flowchart of the proposed drive-by bridge modal identification is shown in Figure 3.

Figure 3.

Flow chart of the bridge modal identification using moving test vehicle.

Numerical study

Numerical study is conducted to analyze the effectiveness of the method for extracting mono-components from vehicle responses and drive-by bridge modal identification. The properties of the bridge are: length $L = 35 m$ , density $ρ = 5000 kg / 5000 kg m m$ , and flexural rigidity $EI = 2.178 e 10 N m^{2}$ . The damping ratio is set as 0.01 and the theoretical values of the first three bridge modal frequencies are 2.68, 10.71 and 24.09 Hz, respectively. The properties of the sensing vehicle are: body mass m_v = 466.5 kg, suspension stiffness k_s = 9.00 × 10⁵ N/m, suspension damping c_s = 0.14 × 10³ N s/m and its fundamental frequency f_v is 6.99 Hz. The mass ratio between vehicle and bridge is 0.27% that is small enough to assume that the vehicle does not change the dynamic response of the bridge. The operational load to simulate the traffic on the bridge is a randomly generated load as shown in Figure 4(a) and the sampling frequency is 200 Hz. The approach length before the vehicle arriving at the bridge is 35m. When the moving speed of the vehicle is 2 m/s, the simulated dynamic response of the vehicle and the power spectral density (PSD) by Fourier transform of the response considering Class A roughness is presented in Figure 4(b). In the PSD of the response, frequencies f_b1 and f_b2 are the first two modal frequencies of the bridge; f_v is the vehicle dynamic frequency.

Figure 4.

The operational load and vehicle response. (a) The operational load, (b) Vehicle response and PSD of response.

Drive-by bridge modal identification using SVMD and SSD

The performance of SVMD for decomposing the vehicle responses is investigated by comparing to that of SSD. The noise polluted measurement of the vehicle response is simulated as y_noisy = y_true+noise%× SD(y_true)×WGN, where y_true is the calculated vehicle response; $noise %$ is the noise level in percentage; $SD (y_{true})$ is the standard deviation of $y_{true}$ and WGN is the standard Gaussian white noise. Noise level of 5% is considered, unless otherwise mentioned. Singular spectrum decomposition and SVMD are used to decompose the vehicle response respectively. The separated components in time and frequency domains are shown in Figures 5(a) and (b), respectively. Three components are obtained from the vehicle response. The first and third components are related to the first and second dynamic modes of bridge, respectively. The second component is related to the component of vehicle dynamic response. It can be seen that the components can be separated clearly. The results by SSD and SVMD are similar for the extraction of first bridge dynamic mode and vehicle dynamic mode. Moreover, the SVMD can extract the second bridge dynamic component slightly better than SSD as shown in Figure 5(b).

Figure 5.

The decomposed components by singular spectrum decomposition and successive variational mode decomposition. (a) In time domain, (b) In frequency domain.

Parametric study

Different measurement noise

The effects of the measurement noise on the decomposition performance of the methods are studied by considering different noise levels to be added into the simulated vehicle acceleration. Three different noise levels are considered, i.e. 0%, 5%, and 15%. The decomposition on the noise polluted measurements is performed using SSD and SVMD respectively. The obtained components and their spectra considering different noise levels are shown in Figure 6(a) and (b), respectively. In the figure, three components related to the first bridge mode, the vehicle mode and the second bridge mode can be clearly identified, respectively. The results confirm the robustness of two techniques to the measurement noise. In the rest part of the numerical study, 5% measurement noise is used in the simulation.

Figure 6.

The decomposed components considering different measurement noise. (a) Using singular spectrum decomposition, (b) Using successive variational mode decomposition.

Effect of the vehicle speed

In the previous study, the vehicle speed is set as 2 m/s. To study the effects of vehicle speed on the decomposition, a higher vehicle speed is considered, i.e. 6 m/s. SSD and SVMD are used to decompose the vehicle response and the decomposed components are shown in Figure 7 along with their spectra. The results show that only the vehicle related dynamic component is clearly extracted. The components related to the bridge are heavily contaminated due to the effects of surface roughness and higher vehicle speed. When a high vehicle speed is used, the dynamic response of test vehicle will be enhanced. The vehicle-related dynamic component in the response become more prominent that makes the bridge-related dynamic component less visible. Besides, a lower vehicle speed can lead to a longer measurement time of the response that is beneficial to the identification of higher vibration mode of the bridge and a higher accuracy of the modal parameters. Therefore, it is confirmed that a low speed of sensing vehicle is beneficial to the drive-by bridge health monitoring. Moreover, it can be seen that SVMD outperforms SSD in extracting purer modes.

Figure 7.

The decomposed components in time and frequency domains when vehicle speed is 6 m/s. (a) In time domain, (b) In frequency domain.

Extraction of close modes between the vehicle and bridge

In above studies the frequency ratio between the vehicle and bridge frequency is 6.99/2.68 = 2.61. To further study the performance of the adaptive methods, a close frequency case between the vehicle and bridge is discussed. The stiffness of the vehicle suspension is set as 1/4 of the original value. Therefore, the fundamental frequency of the vehicle becomes 3.49 Hz and the frequency ratio between the vehicle and bridge is 1.30. The vehicle response is analyzed using those two methods and the results are shown in Figure 8. Figure 8(a) shows the components and their spectra using SSD and the decomposed components by SVMD are shown in Figure 8(b). In Figure 8(a), the first component is dominated by the first bridge mode, the second component includes both the vehicle and the first bridge modes and there is a clear peak related to the second bridge mode in the third component. The results show that the vehicle and the first bridge modes cannot be separated successfully by SSD. In Figure 8(b), three components are separated successfully and the first, second and third components are related to the first bridge, the vehicle and the second bridge modes respectively. The results show that SVMD can identify both the vehicle and bridge modes for the close mode case.

Figure 8.

The decomposed components in time and frequency domains. (a) Using singular spectrum decomposition, (b) Using successive variational mode decomposition.

Bridge modal parameter identification using multiple passes

The effectiveness of adaptive signal decomposition using SVMD has been discussed in Drive-by bridge modal identification using SVMD and SSD and Parametric study. This section is to study the bridge modal identification using extracted dynamic components. The dynamic modes decomposed by SVMD are used to estimate the bridge frequencies and damping ratios. To evaluate the accuracy of the proposed drive-by bridge modal identification method, the Monte Carlo method with 50 simulations is used to generate the vehicle response dataset to simulate multiple passes of the sensing vehicle considering random operational load. Each of these responses is analyzed by SVMD, and the components related to the first two dynamic modes of bridge are used for the identification of frequency and damping ratio. Another two different damping ratio values of bridge, i.e. 0.02 and 0.03 are also considered in simulating vehicle responses. The mean values and the standard deviation (std) of the identified frequencies for 50 passes are presented in Table 1. It can be seen that the mean values are very close to the theoretical values and the errors are all less than 1.5%. The results confirm that the bridge modal frequencies can be identified with high accuracy using the developed method.

Table 1.

Identified frequency considering different damping ratios.

Identified frequency (Hz)
Damping ratio	0.01		0.02		0.03
	Mean	std	Mean	std	Mean	std
First mode	2.674	0.0561	2.667	0.0419	2.658	0.085
Second mode	10.589	0.0842	10.559	0.1012	10.549	0.182

The curve-fitting modal identification methods based on NExT and RDT are used to estimate the damping ratios from bridge dynamic components. As mentioned in Bridge modal identification using a moving test vehicle, the time lag T need to be preset. As such, five time lags, $T = ([12, 13, 14, 15, 16] / [12, 13, 14, 15, 16] f_{bi} f_{bi})$ are used to identify the damping ratios of the bridge, where $f_{bi}$ is the bridge frequency. Table 2 presents the mean value and standard deviation of the identified damping ratios using the decomposition technique and two damping ratio identification methods with different time lags. It can be seen that the mean value of the damping ratios by multiple passes are close to the true values when damping ratio is 0.01 and 0.02, respectively, and the errors are within 10%. When the damping ratio 0.03 is considered, the maximum error of the damping ratio for the second mode is about 15%. The results show that the proposed method could identify the damping ratio with an acceptable accuracy.

Table 2.

Identified damping ratio considering random operational load.

	Damping ratio (%)											True value		Error (%)
T	12/f_bi		13/f_bi		14/f_bi		15/f_bi		16/f_bi		Mean	Mode No	Value (%)
T	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Mode No	Value (%)
RDT	1.03	0.48	1.07	0.49	1.09	0.49	1.11	0.49	1.11	0.48	1.08	1st mode	1.00	8.0
NExT	1.02	0.46	1.04	0.47	1.05	0.48	1.07	0.48	1.08	0.48	1.05	1st mode		5.0
RDT	0.92	0.51	0.94	0.52	0.95	0.52	0.96	0.52	0.97	0.52	0.95	2nd mode		5.0
NExT	0.94	0.49	0.96	0.50	0.98	0.51	0.99	0.52	1.00	0.53	0.97	2nd mode		3.0
RDT	2.07	1.00	2.01	0.90	1.94	0.82	1.86	0.76	1.79	0.70	1.93	1st mode	2.00	2.3
NExT	2.10	0.97	2.00	0.92	1.91	0.87	1.83	0.84	1.77	0.80	1.92	1st mode		2.7
RDT	1.87	0.67	1.85	0.67	1.82	0.67	1.78	0.67	1.74	0.67	1.81	2nd mode		9.5
NExT	1.90	0.61	1.91	0.62	1.90	0.62	1.89	0.62	1.86	0.61	1.89	2nd mode		5.5
RDT	3.07	1.79	3.03	1.78	2.98	1.77	2.92	1.73	2.84	1.69	2.97	1st mode	3.00	1.0
NExT	3.17	1.86	3.15	1.86	3.14	1.87	3.12	1.87	3.10	1.87	3.14	1st mode		4.7
RDT	2.61	0.86	2.58	0.86	2.54	0.86	2.50	0.85	2.45	0.85	2.54	2nd mode		15.0
NExT	2.86	0.92	2.82	0.92	2.77	0.92	2.73	0.92	2.68	0.92	2.77	2nd mode		7.7

For multiple tests using the vehicle, each of the identified first modal frequency and damping ratio of the bridge are shown in Figure 9. The result shows that both the frequency and damping ratio vary around the true values and the mean value from multiple tests can reduce the uncertainty due to the random operational load and measurement noise. The result also confirms that the identification of damping ratio involves more uncertainty and inaccuracy.

Figure 9.

The identified modal parameters from multiple tests.

Experimental study on a cable-stayed bridge

In-situ vehicle-bridge interaction test is conducted to further verify the proposed method. Figure 10 shows the bridge and test vehicle for the experimental investigation. The bridge is a single lane cabled-stayed bridge with a span 46m and a width 6m crossing a busy highway. A long-term monitoring system has been installed on the bridge including a dense array of accelerometers under the bridge deck. The dynamic monitoring system continuously records the vibration response of the bridge and produces a file with an acceleration time series every 10 min at a sampling rate of 600 Hz. A total number of 66 10-mimute-file were elaborately selected from 22 days of monitoring data which were analyzed using the covariance-driven stochastic subspace identification method for modal parameter identification (Sun et al., 2017). The mean value of the natural frequency for each mode from the datasets is used as reference baseline and the first five modal frequencies of bridge are presented in Table 3. It was found that not all the modes were extracted from every signal dataset measured from bridge. The last column of Table 3 is the percentage of identification for each particular mode among 66 datasets. The percentage of identification for the third and fourth modes among the datasets is relatively small which means that these two modes are not dominated and less likely to be identified.

Figure 10.

Bridge and vehicle used for experimental test.

Table 3.

Identified bridge frequencies using sensors on the bridge (Sun et al., 2017).

Mode no	Frequency (Hz)	Vibration mode	Percentage of the identified modes (%)
1	2.032	First vertical bending mode	100
2	3.548	A mixture of the second vertical bending and torsion	85
3	3.649	Second vertical bending mode	81
4	5.584	Third vertical bending mode	67
5	6.136	A mixture of torsion and bending modes	82

For the bridge modal identification using a test vehicle, a Hyundai Tucson 2006 vehicle is used. A wireless accelerometer (manufactured by BeanAir) is installed on the top surface of the dashboard of the vehicle. The vehicle responses are measured when it stops on the ground and on the bridge deck with its engine idling, respectively. Figure 11 shows the measured responses and their spectra. The results show that there are two dominated peaks at 17.5 Hz and 23.3 Hz in the spectrum and they are the vehicle engine-related frequencies as it is idling. In the results when the vehicle parks on the bridge deck, the peak at about 2.0 Hz is also visible and that is related to the first bridge vibration mode. The vehicle is driven multiple times on the ground with different speeds at 10, 20 and 30 km/h, respectively. The dynamic responses measured from the wireless sensor are used for spectrum analysis with Fourier transform. After analyzing all the responses, the first three vibration frequencies of the vehicle body when it is moving are about 1.1, 1.5–1.8 and 2.2–2.7 Hz, respectively.

Figure 11.

Response measurements when vehicle stops on the road and bridge.

The dynamic response measured from the wireless sensor when the vehicle passes the bridge at a speed 10 km/h is shown in Figure 12. Successive variational mode decomposition and SSD are used to decompose the vehicle response. Figure 13(a) and (b) show the decomposed components and their spectra, respectively. In Figure 13(a), five components are extracted using SVMD. The frequencies of first two components are 1.05 Hz and 1.56 Hz and they are related to the vehicle dynamic responses. Other three components are around 2.05 Hz, 3.56 Hz and 6.23 Hz. Compared with the results using sensors on the bridge, these three components are corresponding to the first, second and fifth dynamic modes of the bridge respectively. The results show that the SVMD can successfully extract the bridge related dynamic components from vehicle response. Figure 13(b) shows three components extracted by SSD. The results show that the bridge related components are not extracted successfully. This further confirms the numerical results that the SVMD can decompose the mono-components from the vehicle responses when the frequencies of the vehicle and bridge are close.

Figure 12.

Response measured in the vehicle and the response spectrum.

Figure 13.

The decomposed components using successive variational mode decomposition and singular spectrum decomposition. (a) Using successive variational mode decomposition, (b) Using singular spectrum decomposition.

Conclusions

This study investigates the operational bridge modal identification based on the adaptive decomposition of vehicle responses using SVMD. The performance of SVMD is compared to that of SSD. Results of the parametric analysis demonstrate that the SVMD can extract the mono-components from vehicle responses effectively. The investigation confirms that the SVMD performs better than the SSD, especially when the frequencies of the components in the vehicle response are close. The NExT and RDT based modal identifications are incorporated to analyze the bridge related dynamic components to estimate the modal frequencies and damping ratios. The bridge modal parameters are identified accurately by computing the mean value of multiple tests when the damping ratio is 0.01 and 0.02, respectively. The damping ratio identification is more sensitive to the operational load than the frequency identification and the multiple tests can improve the accuracy of damping ratio identification when the bridge is subjected to random operational load. The contact-point response of the bridge usually calculated from vehicle response contains more dynamic information related to the bridge. The proposed SVMD based method has potential to improve the bridge modal identification using the contact-point response due to its effectiveness and accuracy in signal decomposition, that deserves further study in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported in part by research funding of the National Natural Science Foundation of China (52108288, 52078461, U1709207, 51878433), Key R&D program of Zhejiang (2019C03098) and Zhejiang Provincial Postdoctoral Science Foundation (ZJ2020024). The financial aid is gratefully acknowledged.

ORCID iD

Jiantao Li

Xinqun Zhu

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