Research on improved modal parameter identification method using Hilbert-Huang transform

In the wind tunnel test, the full-bridge aeroelastic model needs to simulate both the shape and dynamic characteristics of the real bridge, and modal parameters are key dynamic parameters. Therefore, it is essential to identify the modal parameters of the model accurately. To get accurate modal parameters of the long-span bridge model in the wind tunnel test, the applications of the Hilbert-Huang transform for modal parameter identification were analyzed in this paper. Then a band-pass filter is designed to filter the original signal so that the intrinsic mode function obtained by empirical mode decomposition can satisfy the single-component signal requirement and eliminate the mode mixing effect effectively. Meanwhile, the endpoint data extension method based on SVM (Support Vector Machine) was presented to restrain the end effects of empirical mode decomposition. Finally, taking the Oujiang Bridge as the engineering background, the improved algorithm was applied to modal parameter identification of the bridge under ambient excitation. The modal parameters such as modal frequency and damping ratio were obtained. The reliability of the improved method was verified by comparing the identified modal parameters with the results of the finite element method, and it turns out that the improved method can reduce the frequency identification error of vertical bend, lateral bend, and torsion to 1.01%, 4.07%, and 1.68%. The results indicated that the improved method based on the Hilbert-Huang transform can accurately identify the main modal parameters of the structure and can be better applied to identify the modal parameters of long-span bridge structures.


Introduction
Among the research methods of bridge wind engineering, wind tunnel test technology is the most widely used, relatively mature, and dominant technique.In the wind tunnel test, the model needs to simulate the dynamic characteristics of the actual structure to satisfy the test requirement.Therefore, it is essential to identify the modal parameters of the model accurately. 1,2owever, the existing modal parameter identification methods for full-bridge aeroelastic models have significant limitations and cannot effectively identify the higher-order modal parameters of the models.Based on this, this paper presents an in-depth study of the Hilbert-Huang transform algorithm for modal parameter identification under environmental excitation and proposes an improved algorithm.
[5] In engineering, the most common methods of modal parameter identification are the traditional methods and the methods under ambient excitation. 6The traditional method generates free vibrations by using artificial excitation.By measuring the displacement response data at each measurement point, the modal parameters of the structure can be identified. 2The method performs well for low-order modes and has adequate accuracy, but the high-order mode response is always mixed with other orders of mode response, which lead to low-quality results or failure of the method.To address this problem, Zhenshan et al. 7 analyzed the modality of the full model of the Runyang suspension bridge by using varied time-based transfer function analysis methods, which can improve the accuracy of structural modal test analysis under impulse excitation.By applying an initial displacement to the girder as artificial excitation, Fuyou et al. 8 extracted the modal parameters of the aeroelastic model of the Sutong Cable-Stayed Bridge in the bridge formation state and the maximum single cantilever construction state.
According to the different processing domains, the method under ambient excitation can be divided into frequency domain method, time domain method, and time-frequency domain method.The frequency domain method uses the frequency response function obtained from the input and output to identify the modal parameters of the structure.By performing a sinusoidal excitation force scan and using the resonance characteristics of the vibration system, the frequency domain method can excite the main mode of the system at each order, thus determining the modal parameters such as the frequencies, damping ratios, and vibration shapes.The time domain method directly uses the time history curve of the system response to identify the modal parameters of the structure, and its raw data is the time history of the system response (such as free response and impulse response).The combination of these two methods is the time-frequency domain method, which is a hot topic in modal parameter identification.Figure 1 shows the common modal parameter identification methods under ambient excitation.
Unlike the traditional method, the method under ambient excitation takes the environmental excitations acting on the structure as the system's input.As the system is driven by environmental excitations, no artificial excitations need to be imposed on the structure.Therefore, the method can complete mode identification only according to the output data of the system. 9,10Thus, the method can complete the modal parameter identification without affecting the normal operation of the structure.Besides, the method eliminates the need for expensive and sizable artificial excitation equipment and avoids the possible damage to the structure caused by artificial excitation.Based on the improved Continuous Wavelet Transform, the modal parameters of the Oujiang Bridge under ambient excitation were identified by Zhang et al. 11 Guo et al. 12 proposed a method using reconstructed displacements and Stochastic Subspace Identification (SSI) to identify the modal parameters of structures.Based on the enhanced Hilbert Vibration Decomposition (HVD), Li and Cao 13 decomposed the initial signal data into different single-component dynamic responses and calculated the structural modal parameters by the empirical envelope method.Xiaoning et al. 14 proposed a method for automatically identifying each order of the system based on frequency stability and mode stability.][17] For modal parameter identification of structures, various signal decomposition techniques were used, including Empirical Mode Decomposition (EMD), Wavelet Transform (WT), 18 and Stochastic Subspace Identification (SSI). 17Mao et al. 19,20 extracted the structural modal parameters of Sutong Cable-Stayed Bridge based on field monitoring data.Based on the Vold-Kalman order tracking filter, Feng et al. 21proposed a novel scheme for Vold-Kalman filter bandwidth selection to guarantee the consistency and accuracy of the condition monitoring process of offshore wind turbines.The time-domain statistic indicator (sample entropy) was used for the diagnostic of planetary gear faults. 22The signal decomposition techniques can also be combined with machine learning and deep learning algorithms.The features of the image dataset consisting of apple, grape, and tomato plants can be extracted by a twodimensional discrete wavelet transform (2D-DWT).Then the CNN classifier can classify diseases of apple, grape, and tomato plants in real-time. 23By combining the over, and optimizing the intrinsic mode function (IMF) estimated outputs with a grey wolf optimizer (GWO), Altan et al. 24 proposed a new hybrid model for wind speed forecasting.
Due to the ability to decompose the nonlinear and nonstationary signal, the EMD is an adaptive signal decomposition method. 25,26However, the original EMD has some drawbacks that need to be addressed, like the mode mixing problem, which means that multiple signals of different scales appear in the same intrinsic mode function (IMF) or some signals of similar scales reside in different IMFs.Huang et al. 25 considered that the mode mixing problem of EMD is due to the presence of intermittent signals and proposed the intermittency test.To overwhelm the scale separation problem, Wu and Huang 26 introduced a new method called Ensemble EMD (EEMD), where white noise is added to the initial signal data many times. 27,28There is no doubt that the EEMD can significantly improve the stability of EMD algorithm.However, the method also contaminates the IMFs with white noise.In order to improve the efficiency of EEMD algorithm, the Complimentary EEMD (CEEMD) was introduced. 27ilbert-Huang Transform (HHT) algorithm is a time-frequency domain analysis method for nonlinear and nonstationary signals. 25,29It mainly consists of Empirical Mode Decomposition (EMD) and Hilbert transform. 29First, the EMD divides the original signal into a series of orthogonal oscillatory components called Intrinsic Mode Functions (IMFs).Then Hilbert transformation is performed on each component of the IMF series to obtain the Hilbert spectrum, which provides a measure of the contribution of the total amplitude for each frequency value of each IMF.Compared with the wavelet transform (WT), HHT does not rely on priori basis functions.It is an adaptive analysis method that can be self-adjusting according to the local characteristics of the signal. 30Therefore, the HHT has an exclusive advantage in the tract of nonlinear and nonstationary signals. 29The HHT time-frequency analysis technique is becoming an important research direction and a hot spot in various fields of society.Lan et al. 31 used Wavelet and improved Hilbert-Huang transform method to study the spectrum distribution and energy of turbine pressure pulsation.Trung et al. 32 proposed the HHT method based on the improved ensemble empirical mode decomposition (iEEMD) algorithm to identify the instantaneous natural frequency degradation due to damages of the fixed offshore structure under the random wave excitation.Arslan and Karhan 33 classified the healthy and diseased phonocardiogram (PCG) signals based on the HHT method and machine learning algorithms.Yin et al. 34 used the HHT method to analyze the electromagnetic radiation (EMR) waveform characteristics of the coal failure process.
Although the HHT method has evolved significantly and applied to various fields of society since its introduction, the EMD still has some dilemmas related to its analytical formulation. 30Therefore, this paper proposed a new method to improve the modal parameter identification of HHT.The core of the whole identification process of HHT is to extract IMF through EMD.Whether the ideal IMF can be obtained will directly affect the accuracy of modal identification results.In general, under noise interference, the mode mixing effect of IMF components obtained by EMD decomposition is serious, and there are multiple time scales in most IMF components.
In this study, a band-pass filter is designed to filter the original signal so that the IMF obtained by EMD can satisfy the single-component signal requirement and eliminate the mode mixing effect effectively.Meanwhile, the end effect of EMD is also severe.In order to solve the problem, the endpoint data extension method based on SVM was presented in this paper.Finally, the numerical simulation case of the Oujiang Bridge was utilized as an engineering background to verify the improved algorithm.The proposed method can be applied to better identify the modal parameters of long-span bridge structures.

Hilbert-Huang transform identifies modal parameters
For a linear system with a single degree of freedom, the displacement response under impact load is as follows: where v 0 , v d , and j are the undamped circular frequency, damped circular frequency, and damping ratio of the system, respectively.The Hilbert transform of the signal x t ð Þ is x(t), then its analytic signal z t ð Þ can be expressed as: where amplitude A t ð Þ and phase u t ð Þ can be expressed as follows: Take the logarithm of the amplitude in equation ( 3), and take the derivative of time to obtain the following form: The phase derivative of equation ( 3) can be obtained in the following form: According to the relationship between the undamped natural frequency, damped natural frequency, and damping ratio, the undamped frequency f 0 and damping ratio of the system j can be obtained as follows: The general process of HHT to identify the modal parameters of the system under ambient excitation is shown in Figure 2. First, EMD was applied to the response data of the system, and a set of IMF components representing different order modal responses of the structure were obtained.Since these components contain the free vibration response of the system and the forced vibration response under random excitation, the Random Decrement Method or Natural Excitation Technique can be applied to the IMF component to obtain the free attenuation response signal of each mode of the system.Next, the same procedure as that used to identify the modal parameters of a single-degree linear system under impact load is applied to process the free attenuation response signal.Then the frequencies and damping ratios of each order of the system are obtained.Since the logarithmic curve of amplitude A t ð Þ and the curve of phase u t ð Þ obtained in the actual process are not ideal straight lines, the slope k 1 and k 2 can be obtained by the least-squares linear fitting.

Elimination of mode mixing effect based on bandpass filter
According to the time-scale characteristics of the data to be processed as the decomposition standard, EMD decomposes the original signal into a series of IMF components with a time scale from small to large.However, when there is strong noise interference or large discontinuity in the signal, the components of different time scales will be decomposed into the same IMF, or components of the same time scale will be decomposed into different IMF components, thus resulting in the mode mixing effect.
To illustrate the mode mixing effect, a signal x t ð Þ composed of two components and white noise is simulated numerically: where f 1 = 0.44 Hz, f 2 = 0.68 Hz, x n is the White Gaussian noise with a variance of 1.The signal sampling frequency was 20 Hz and the sampling time was 50 s.Figure 3 shows the time history of the first-order signal component, second-order signal component, white noise, and the composite signal of three signals.
After the above signal carried on the empirical mode decomposition to get eight IMF components, as shown in Figure 4, the phenomenon can be observed that the mode mixing effect of the IMF component obtained by empirical mode decomposition is serious under noise interference.And multiple time scales appear in most IMF components.For example, the low-frequency component with a large time scale, which is inconsistent with the main features of the IMF3 component, appears in the IMF3 component.To solve the above problems, a band-pass filter is used to filter the original signal, so that the IMF obtained after empirical mode decomposition can meet the single-component signal requirement.The specific steps are as follows: (1) By Fourier spectrum analysis of the original signal x t ð Þ, the first n-order natural frequencies are estimated f 1 , f 2 , Á Á Á , f n f g .
(2) For the i-order frequency f i , a band-pass filter with a passband range of ½f p1 , f p2 is designed to obtain a new narrowband signal x i (t), where f p1 \ f i \ f p2 .(3) Perform empirical mode decomposition on the narrowband signal x i t ð Þ .Generally speaking, the first-order IMF component is very close to the i-order modal response component of the structure. 354) Repeat step (2) and step (3) n times to obtain the first n-order modal response components of the original signal.
Fourier spectrum analysis was performed on the signal shown in equation ( 8), and its spectrum was obtained as shown in Figure 5 with the solid black line.Thus, the passband range of the band-pass filter is determined to be (0.3, 0.     be seen from the figure that the time scale in each IMF component is consistent.Thus, the new method has almost eliminated the mode mixing effect and has a good decomposition effect.

Restraining the end effects of EMD based on SVM
In the process of EMD, the maximum and minimum points of the signal need to be determined by cubic spline curve fitting.However, the endpoints at both ends of the signal are not necessarily extreme points, resulting in a large distortion of the upper and lower envelope obtained by fitting at the endpoints.In addition, in the process of extracting IMF components, it is necessary to continuously fit the upper and lower envelope, which leads to the gradual inward development of distortion.Finally, the IMF components are seriously distorted near the endpoints.The end effect is particularly severe in short sample signals, and it is highly likely to lead to ''contamination'' of the whole IMF component.In this paper, the endpoint data extension approach based on Support Vector Machine (SVM) is used to restrain the end effects of EMD.Meanwhile, the prediction and extension of the original signal are required in the HHT algorithm, and then the IMF components of each order are obtained through the previous steps.
SVM is a machine learning method for small sample data, which can solve the problems of classification recognition, regression, and prediction. 36,37The goal of SVM is to maximize the margin between different classes, which gives the classification of SVM higher confidence and generalization ability. 38The task of the SVM is to find an optimal hyperplane that can correctly separate the positive and negative training data.Therefore, if linear separation is not possible in a given feature space, the data are mapped into a higher-dimensional space where linear separation might become feasible. 39or the binary classification problem, assume that there are N training points x i and each training point has an indicator y i .The training set is Where x i is the input variable and its components are called features or attributes.y i 2 f1, À 1g is the output indicator.Suppose the linear equation of hyperplane is v T x + b = 0.Where v = (v 1 , v 2 , . . ., v n ) is the normal vector, which determines the direction of the hyperplane; and b is the displacement term, which determines the distance between the hyperplane and the source point.It is clear that the hyperplane can be determined by the normal vector v and the displacement b. 38 When the sample set is linear-separable, the optimization problem can be expressed as follow: When the sample set is nonlinear-separable, we need to relax the constraints appropriately.The slack variable j (j i 50, i = 1, 2, . . ., N) and the penalty factor C are added to equation (9), where the penalty factor C is artificially set and it represents the degree of tolerance for outlier points.Thus, the optimization problem becomes the following form: To solve nonlinear-separable problems, the common approach is to find the kernel function, which is found in mapping a low-dimensional sample set to a highdimensional space and can make the inner product result of the sample set in the low-dimensional space consistent with the high-dimensional space.The commonly used kernel functions include the Gaussian kernel function, polynomial kernel function, sigmoid kernel function, and so on.By introducing the kernel function k(x i , x j ) and Lagrange multiplier a i (a i 50, i = 1, 2, . . ., N), equation ( 10) can be changed into the following form: The basic process of SVM prediction extension is as follows: the original data set is used to establish a learning sample in a specific format, and the sample is trained by the SVM algorithm to obtain a prediction model between input and output.Finally, the unknown output is predicted as accurately as possible via this model to extend the data.
For the original dataset fx 1 , x 2 , Á Á Á , x n g, it is divided into l training samples of length n2l + 1, where the last value of each sample is used as the output value of training.In other words, the SVM gets the last data by training the first n2l data.The corresponding prediction model will be obtained after the training of l samples.Then the latter n-l data of the original data set, which is fx l + 1 , x l + 2 , Á Á Á , x l + n g, is used as the input data of the first prediction sample.The first prediction value x n + 1 will be obtained by the prediction model, which forms the second prediction sample fx l + 2 , x l + 3 , Á Á Á , x n + 1 g.And x n + 2 is predicted in the same way.And so on, we finally get m extended data after m times of prediction.The specific process is shown in Figure 7.
Taking the signal in Figure 3 as an example, the data were divided into 400 training samples and there are 500 data points were predicted at each end of the signal for the extension based on the sample training by the SVM method.The signal obtained after the extension is shown in Figure 8(a).Compared with the noiseless signal in Figure 8(b), it can be seen that the prediction model restores the main component features of the original data and restrains the interference of noise components commendably.The result indicates that the prediction model based on SVM performs well in the extension of the signal.
The same band-pass filter as introduced in the previous section is used to filter the above-extended signals and two narrowband signals are obtained.Then, EMD is performed on the two narrowband signals respectively to extract the first IMF component.In Figure 9, the solid black line is the IMF component, and the dashed red line is the original signal component.Figure 9(a) shows the comparison between the IMF components of the unextended signal and the original signal components while Figure 9(b) shows the result of the extended signal.By comparing the two figures, it can be clearly seen that the IMF components of the extended signal basically coincide with the original signal component near the endpoints, and the end effect is significantly restrained.However, the IMF component of the unextended signal has obvious distortion at the endpoint and the end effect is severe.Thus, we can draw the conclusion that the end effects of EMD can be restrained significantly by using the prediction model based on SVM to extend the signal.

Calculation of dynamic characteristics
Taking Oujiang Cable-Stayed Bridge as the engineering background.The span arrangement of the bridge and the distribution of the bridge tower is shown in Figure 10.The main girder of the bridge is a prestressed concrete box girder, of which the width and height in the central section are 13 and 4 m.The key modal vibration frequencies of the bridge and the corresponding vibration modes description are listed in Table 1. Figure 11 shows the diagram of the main vibration modes of the Oujiang Bridge.

Bridge response under environmental excitations
Based on the self-developed Wind-Train-Bridge coupling calculation and analysis software system BANSYS, 40 the acceleration response of the Oujiang Bridge under the combined action of the train and the fluctuating wind loads was calculated.The wind speed was set to 30 m/s in the analysis of the Wind-Train-Bridge coupling vibration of Oujiang Bridge, where C62 freight cargo was selected as the train model with the speed of 72 km/h, and the track irregularity power spectra were based on the measured data for Zhengzhou-Wuhan Line.There were seven observation points arranged along the bridge so as to monitor the acceleration responses of the girder.The arrangement of the observation points on the girder is shown in Figure 12.
Based on the BANSYS software, this paper analyzes the acceleration response of the bridge under fluctuating wind load after the train leaves the bridge.The data sampling frequency of each observation point was 100 Hz and the sampling time of each signal was 50 s.Figure 13 shows the acceleration response time history of the mid-span observation point (point 4).Due to the limitations of space, the acceleration response time history of the remaining observation points is not drawn one by one in this paper.

Modal parameter identification
MATLAB was used to realize the programming of the improved HHT algorithm as introduced above and identify the modal parameters of the vertical, transverse, and torsion of the Oujiang Bridge.The sampling frequency was 100 Hz and the sampling time was 50 s.In order to identify the vertical bending modal parameters, the SVM is used to predict and extend 1000 data points at both ends of the original signal to eliminate the end effect of EMD.Then, the distribution of each order frequency of each signal was initially estimated by Fourier spectrum analysis to determine the passband range of the bandpass filter.The curve shown in Figure 14 is the extended data of observation point 4, in which the blue line in the middle is the original signal and the red line at both ends is the predicted extension signal.The spectrum of observation point 4 is shown in Figure 15, where it can be noticed that the frequency of the vertical acceleration signal is mainly distributed within 0:3, 0:5 ½ , 0:6, 0:8 ½ , 1,1:2 ½ f g Hz.Based on this, the band-pass filter can be designed.The IMF components of the filtered signal obtained by EMD are shown in Figure 16.
The free-response signal of each order IMF component was obtained by Natural Excitation Technology, and then the logarithmic curve of amplitude and the curve of phase were obtained by Hilbert transformation.Then, the least-squares linear fitting was performed on the curve to obtain the corresponding curve slope.
According to equations ( 6) and ( 7), the modal parameters of each order of each observation point were calculated, and the final modal parameters of the system are the average values of the modal parameters of each observation point.Figure 17 shows the free response signal extracted from the first-order IMF component of observation point 4, the logarithmic curve of amplitude and its fitting curve, and the curve of phase and its fitting curve.Finally, the modal parameters of Oujiang Bridge obtained by the improved HHT algorithm are shown in Table 2, including the frequency and damping ratio of the vertical bend, transverse bend, and torsion modal.And it was compared with the results of the finite element method to verify the reliability of the new method proposed in this paper.It can be noticed that the modal parameters of each order identified by the two methods match well and the frequency identification error of vertical bend, lateral bend, and torsion can be reduced to 1.01%, 4.07%, and 1.68%, which indicated that the improved HHT algorithm can accurately identify the main modal parameters of the structure and can better separate each mode by the band-pass filter.

Conclusion
In this paper, an improved modal parameter identification method using HHT was proposed.The modal parameter identification of a long-span bridge under the combined action of the train and the fluctuating wind loads was studied.The main conclusions are as follows: (1) In the modal parameter identification of the HHT, the original signal can be filtered by using a bandpass filter, which can make the IMF obtained by EMD satisfy the requirement of a single component, and can eliminate the mode mixing effect effectively.
(2) The support vector machine (SVM) can be used to predict and extend the endpoint data, which is also    required in the HHT algorithm.Thus, not only will the original signal characteristics be preserved, but the interference of noise components can also be commendably restrained.The end effect of EMD can be suppressed effectively.(3) In the improved HHT algorithm, the frequency identification error of vertical bend, lateral bend, and torsion can be reduced to 1.01%, 4.07%, and 1.68%.Thus, the method can enhance the accuracy of identification to a certain extent and can be better applied to identifying the modal parameters of long-span bridge structures under environmental excitation.
It is noteworthy that the band-pass filter in the present study is used to intervene in the empirical modal decomposition.The parameters of the band-pass filter may not be applicable to some bridges.For general applicability, adaptive empirical modal decomposition methods shall be investigated in the future study.

Figure 1 .
Figure 1.The modal parameter identification methods under ambient excitation.

5 )
Hz and (0.6, 0.8) Hz.The spectra of the two narrowband signals obtained after filtering are shown in blue and red dotted lines in Figure5.It can be seen from the figure that the signal components outside the passband are basically eliminated after filtering and the signal components in the passband are completely retained.The amplitude of the signal obtained after filtering almost does not decay.Therefore, we can get the conclusion that the passband range of the bandpass filter is set reasonably and the filtering performs well.EMD was performed on the above two narrowband signals respectively.The first IMF component of the two signals was extracted as shown in Figure6.It can

Figure 3 .
Figure 3.Time history of simulation signal.

Figure 4 .
Figure 4. Intrinsic mode function of simulation signal.

Figure 5 .
Figure 5. Spectrum before and after filtering of simulation signal.

Figure 6 .
Figure 6.Intrinsic mode function obtained after filtering.

Figure 9 .
Figure 9.Comparison of IMF components and original components: (a) unextended signal and (b) extended signal.

Figure 13 .
Figure 13.Acceleration time history of observation point 4 (unit: m/s 2 ): (a) vertical acceleration response of the bridge, (b) lateral acceleration response of the bridge, and (c) torsional acceleration response of the bridge.

Figure 14 .
Figure 14.Extended vertical acceleration signal of observation point 4.

Figure 16 .
Figure 16.IMF Components of the Vertical Acceleration Signal of observation point 4.

Figure 17 .
Figure 17.Free response signal and fitting curves of the first IMF component of observation point 4.

Table 1 .
Main frequency and mode shape.

Table 2 .
Results of modal parameter identification.