Operational modal analysis (OMA) is a powerful vibration analysis tool and widely used for structural health monitoring (SHM) of various system systems such as vehicles and civil structures. Most of the current OMA methods such as pickpicking, frequency domain decomposition, natural excitation technique, stochastic subspace identification (SSI), and so on are under the assumption of white noise excitation and system linearity. However, this assumption can be desecrated by inherent system nonlinearities and variable operating conditions, which often degrades the performance of these OMA methods in that the modal identification results show high fluctuations. To overcome this deficiency, an improved OMA method based on SSI has been proposed in this paper to make it suitable for systems with strong nonstationary vibration responses and nonlinearity. This novel method is denoted as correlation signal subsetbased SSI (CoSSSI) as it divides correlation signals from the system responses into several subsets based on their magnitudes; then, the average correlation signals with respective to each subset are taken into as the inputs of the SSI method. The performance of CoSSSI was evaluated by a simulation case and was validated through an experimental study in a further step. The results indicate that CoSSSI method is effective in handling nonstationary signals with low signal to noise ratio (SNR) to accurately identify modal parameters from a fairly complex system, which demonstrates the potential of this method to be employed for SHM.
Operational modal analysis (OMA) is, in fact, not really a new discipline. The beginning of OMA could be going back to the sixties and early seventies, which was developed along with the experimental modal analysis (EMA) [
OMA is cheaper and faster to conduct since it only measures the responses
The dynamic characteristics of the whole structural system can be obtained instead of its small parts
A linear model of structural systems under operational conditions can be obtained since the random excitations are of broadband in nature
OMA is suitable for complex and complicated structures due to the fact that the close modes can be identified through multiinput/multioutput (MIMO) modal identification algorithm
Because of these advantages, there are numerous OMA methods that have been developed in last decades. Generally, they can be classified into two categories: frequency domain (FD) and time domain (TD). The earliest FD technique is based on the power spectrum density (PSD) peakpicking algorithm. The natural frequencies are directly obtained from the choice of peaks in the PSD graph. The peakpicking technique has proved its effectiveness in the modal identification method when system’s modes are well separated [
Besides the FD techniques, the TD techniques were also developed very quickly in the last decades. For instance, the natural excitation technique (NExT) is a popular and powerful TD method, which was proposed in 1990s [
Furthermore, stochastic subspace identification (SSI) method is another widely employed TD technique for OMA. It was proposed as an extension of the subspace statespace system identification method [
Because of the effectiveness of the ACSSSI method, it was employed to identify the modal parameters related to the vehicle suspension system. However, it was found that the ACSSSI method was unable to accurately extract the target modal parameters due to the severer excitation condition. Consequently, a new method is needed to extract the modal parameters linked to the suspension parameters. In this study, the main objective of this paper is to present a novel method based on the ACSSSI method [
The rest of this paper has been divided into four sections. Section
As referred previously, most of the OMA techniques were developed under the assumption that the measured responses are stationary. However, the fieldmonitored data are usually nonstationary such as the platform under the wave impacts and the bridge with the timevarying traffic loading [
The nonstationary problem has been addressed in [
Obtain
Calculate the correlation functions of each segment for different channels. The correlation functions can be calculated as follows when the reference is
Conduct average step to all of the obtained correlation functions to calculate the corresponding average correlation functions of each channel. It can be expressed as the following equation:
The averaged correlation functions are employed as the measured responses to construct the Hankel matrix in the SSI method. The reason for the correlation signals obtained from different data sets can be averaged is based on the fact that the phase information between different records is preserved by the referencebased correlation signals. The average step will thus enhance the contents with regular or periodic components by suppressing the irregular random contents in different data records; the regular and periodic components contain the information of modal parameters [
However, the lowamplitude correlation signals are often related with the vibration modes with higher damping coefficients; in other words, they are less frequently excited. It indicates that such an averaging technique over a full set of data may lead to an inadequate identification result for these modes with high damping properties. As a result, lessexcited modes cannot be identified reliably. Moreover, the variation of the amplitude of the correlation signals may result from the inevitable effects of the system nonlinearity.
Based on the above analysis, the deficiency of the ACSSSI method is evident. Therefore, the performance of ACSSSI has to be enhanced to make it suitable for extreme nonstationary and quasinonlinear scenarios. In this paper, a novel method “correlation subsetbased SSI” (CoSSSI) was proposed which is based on the algorithm of ACSSSI. Although ACSSSI has deficiency, it has been proved that the average step conducted to the correlation signals calculated from the short segments of signal are effective in suppressing nonstationary effect, and that is why the same step is applied in the CoSSSI method. However, the correlation signals are divided into several subsets according to their amplitudes, and each subset of correlations signals are averaged rather than considering all correlation signals as a full set. Finally, the system modal parameters can be obtained by merging the similar modal parameters identified by ACSSSI with respective to each subset. Moreover, the merging step is performed according to the discrepancy of the identified frequencies and modal assurance criterion (MAC).
Furthermore, unlike ACSSSI, the main contribution of the CoSSSI is dividing the correlation signals into different subsets according to their magnitudes. This key step is fulfilled by calculating the root mean square (RMS) value of each correlation signal segment and later identifying the correlation signal segments that belong to the respective subsets based on those corresponding RMS values. Particularly, if
For clarity, the CoSSSI method is further summarized with a flow chart, shown in Figure
Flow chart of CoSSSI.
In plenty of OMA progress, stabilization diagram (SD) is a popular and efficient tool to filter out the false modes. The SD is performed to check the consistency of the modal properties by setting threshold values for the frequency, damping ratio, and the modal assurance criterion (MAC) between two adjacent orders; only the mode which satisfies all of the three thresholds will be allowed to plot a point on the SD. Moreover, the system’s true modes will produce stable points but the spurious modes will not. The tolerance can be calculated based on the following equations:
In this section, a classical 3DOF vibration system, shown in Figure
3DOF vibration system.
Theoretical modal parameters of a 3DOF system.
Mode 1  Mode 2  Mode 3  

Frequency  7.01 Hz  19.65 Hz  28.39 Hz 
Damping ratio  1.12%  3.15%  4.55% 
In the simulation case, the 3DOF system has been excited by three independent random inputs from a bandpass stationary white noise and a number of multiple random impulsive impacts. The impulsive excitations attempt to mimic the occasionally pulse inputs in real applications, such as the bump on the road. The responses
According to the theoretical resonance frequency of the third mode, the sampling frequency for the numerically solving system model was set at 500 Hz and sampling time was 60 s for each occasion. An example of the acceleration responses of the 3DOF system with measurement noise is shown in Figure
Example of the timedomain responses and corresponding PSD.
The corresponding power spectrum density (PSD) of each block has been presented under the timedomain signals. It shows that only the first and second modes at the frequencies of 7.06 Hz and 19.65 Hz are clear, whereas the third mode is not that much prominent due to the damping ratio is higher (shown in Table
In this section, to illustrate the superiority of CoSSSI, the other two methods, CovSSI and ACSSSI methods, are also employed in this simulation case. For the CovSSI, the dataset used to identify the modal parameters is of 60 seconds time duration for the 3DOF system with measurement noise; the sampling frequency is 500 Hz. Apart from that, twenty more Monte Carlo simulations were carried out to generate the sufficient signals for modal parameters identification by the ACSSSI and the CoSSSI methods. Moreover, the correlation signals were calculated in each Monte Carlo simulation and therefore, twenty correlation signal segments were obtained. For the ACSSSI, the twenty segments were averaged in single time, and then the averaged signals were employed to identify the modal parameters. However, as referred previously, the ensemble average might loss some significant signatures of the correlation signals with small amplitudes. Therefore, the twenty segments of correlation signals were divided into three subsets according to their magnitudes in the CoSSSI, and the modal parameters were identified with respective to each subset of correlation signals. The SD identified by the three methods for the 3DOF system under two SNR scenarios, SNR = 10 and 0.5, are presented in Figures
Stabilization diagram of CovSSI. (a) SD of CovSSI (SNR = 10). (b) SD of CovSSI (SNR = 0.5).
Stabilization diagram of ACSSSI. (a) SD of ACSSSI (SNR = 10). (b) SD of CovACS (SNR = 0.5).
Stabilization diagram of CoSSSI. (a1) CoSSSI (SNR = 10, J = 1/1st subset). (a2) CoSSSI (SNR = 10, J = 2/2nd subset). (a3) CoSSSI (SNR = 10, J = 3/3rd subset). (b1) CoSSSI (SNR = 0.5, J = 1/1st subset). (b2) CoSSSI (SNR = 0.5, J = 2/2nd subset). (b3) CoSSSI (SNR = 0.5, J = 3/3rd subset).
It can be seen from Figures
As referred earlier, a second threshold is set up to filter out the spurious modes. In this simulation case, 60 orders (rows) of the Hankel matrix are calculated, which can be seen from the lefty axle. The stable modes are chosen as the percentage of stable points over 50% of the SDs for CovSSI and ACSSSI; this threshold for CoSSSI is stricker which is set at 70%. An example of the second threshold result identified by CovSSI of the signal with SNR = 10 is shown in Figure
Example of selecting modes by the rate of stable points over orders (CoSSSI, SNR = 10).
Based on these two thresholds, the natural frequency and damping ratio identified by CovSSI, ACSSSI, and CoSSSI methods are listed in Tables
CovSSI results.
Mode 1  Mode 2  Mode 3  


Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Theoretical value  7.0129  Null  1.12%  Null  19.6468  Null  3.15%  Null  28.3904  Null  4.55%  Null 
SNR = 10  7.0380  0.37%  1.25%  11.61%  19.6663  0.09%  3.11%  1.27%  28.4377  0.17%  4.3%  5.49% 
SNR = 0.5 












ACSSSI results.
Mode 1  Mode 2  Mode 3  


Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Theoretical value  7.0119  Null  1.12%  Null  19.6468  Null  3.15%  Null  28.3904  Null  4.55%  Null 
SNR = 10  7.0090  0.04%  1.08%  3.57%  19.6889  0.21%  3.29%  4.44%  28.3172  0.26%  4.81%  5.71% 
SNR = 0.5  7.0093  0.04%  1.19%  6.25%  19.6797  0.17%  3.41%  8.25% 




CoSSSI results.
Mode 1  Mode 2  Mode 3  


Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Error ( 

Theoretical value  7.0119  Null  1.12%  Null  19.6468  Null  3.15%  Null  28.3904  Null  4.55%  Null  


SNR = 10  S1  7.0275  0.22%  2.17%  93.75%  19.3324  1.60%  4.45%  41.27%  28.5183  0.45%  6.53%  43.52% 
S2  7.0118  0  1.97%  75.89%  19.3377  1.57%  5.91%  87.62%  28.5781  0.66%  6.71%  47.47%  
S3  7.0405  0.41%  5.17%  361.61%  19.3322  1.6%  3.92%  24.44%  28.5076  0.41%  5.83%  28.13%  


SNR = 0.5  S1  7.0170  0.07%  3.47%  209.82%  19.5947  0.27%  5.14%  63.17%  28.4288  0.14%  6.53%  43.52% 
S2  7.0234  0.16%  2.58%  130.36%  19.5935  0.27%  3.74%  18.73%  28.5125  0.43%  3.50%  23.08%  
S3  7.0220  0.14%  4.45%  297.32%  19.5887  0.30%  3.51%  11.43%  28.4736  0.29%  5.53%  21.54% 
In order to better illustrate the identification results, the identification errors of frequency and damping are presented in bar figures, which are shown in Figures
Identified frequency errors. (a) Identified frequency errors (SNR = 10). (b) Identified frequency errors (SNR = 0.5).
Identified damping errors. (a) Identified damping errors (SNR = 10). (b) Identified damping errors (SNR = 0.5).
However, the mode shape is the most significant index for SHM whenever we adopt any modal parameters identification methods. It is wellknown that MAC values are widely employed to compare two mode shapes to see whether they are close or not. In this simulation study, all of the identified mode shapes are illustrated by the MAC values by comparing with the theoretical mode shapes, shown in Figures
MAC of CovSSI results. (a) CoVSSI (SNR = 10).
MAC of ACSSSI results. (a) ACSSSI (SNR = 10). (b) ACSSSI (SNR = 0.5).
MAC of CoSSSI results. (a1) CoSSSI (SNR = 10). (a2) CoSSSI (SNR = 10). (a3) CoSSSI (SNR = 10). (b1) CoSSSI (SNR = 0.5). (b2) CoSSSI (SNR = 0.5). (b3) CoSSSI (SNR = 0.5).
All of the three methods have been evaluated by using a 3DOF vibration system. It seems highly probable that CoSSSI is superior to other two methods, especially treating high noise signals; however, signals collected under operational conditions always contained with high noise.
The experiments carried out in this paper are to identify the vehicle suspensionrelated modal parameters by collecting the vibration signal from a car body at four corners. The experimental car is a commercial car, and its model is Vauxhall Zafira. The signals were collected during car running on a traditional UK rustic road. Moreover, four accelerometers were employed to collect the vibration signals from the vehicle body which caused by the road excitation. The four transducers are piezoelectric accelerometers which are produced by SINOCERA and the model is CAYD185. This is a widely used kind of transducer because of its wide frequency measurement range, which is from 0.5 Hz to 5000 Hz. A fourchannel data acquisition system and a laptop were adopted to collect and store the signals, respectively. The data acquisition system model is YE6231 and is also manufactured by SINOCERA with maximum sampling frequency of 96,000 Hz.
In this experiment, the accelerometers were mounted at the four corners of the car, and they were kept much close to the connection point of suspension. This is to obtain better quality vibration signals from the suspension system which are related to the road excitation. The tested car and a schematic of data acquisition system are presented in Figure
(a) Test car; (b) data acquisition equipment; (c) accelerometer; (d) schematic of test system.
The purpose of the method proposed in this paper is to identify the modal parameters of vehicle under running condition. Therefore, the data were collected when the vehicle was driven on the typical UK suburb roads with speed limits from 20 to 40 miles/hr. In order to confirm no loss of information in the modal identification process, the sampling frequency was set much higher than the requirement of Nyquist sampling theory; the sampling frequency was 4000 Hz, and each test sample was recorded with the time duration of 240 s. Moreover, the test was repeated 4 times by driving on the same road section. Although the sensors were installed close to the suspension, the collected signals still contained high noise because this is a field test and there are thousands of reasons that can introduce unwanted measurement noise. Furthermore, the vehicle was running on the real road, not on a test platform; therefore, the speed was not always constant. The changing speed will cause nonstationary vibration. In addition, the random big excitations such as the hump on the road will also result in nonstationary responses of the vehicle.
An example of the collected signals is presented in Figure
Timedomain signal and corresponding PSD. (a) Ch = FL. (b) Ch = FR. (c) Ch = RL. (d) Ch = RR.
In this section, only ACSSSI and CoSSSI methods are applied to identify the modal parameters of the car when it was in normal operation. As referred previously, the vehicle test was repeated four times with the sampled time duration of 240 s and sample rate of 4000 Hz. Firstly, the data of each test were segregated into six segments (40 s for each segment). Therefore, there are 24 (4 times × 6 segments = 24) data segments in total. Secondly, the correlation signals of each data segment were calculated. Then, for the ACSSSI method, the correlation signals were averaged in a single time; for the CoSSSI, the correlation signals were categorised into three subsets according to their amplitudes, and each subset was averaged. During the identification process of these two methods, the same threshold (
The SD identified by ACSSSI is presented in Figure
Stabilization diagram identified by ACSSSI. (a) SD of ACSSSI. (b) Selecting modes by the rate of the frequency over orders for on road vehicle modal identification.
Modal parameters identified by ACSSSI. (a) 1.64 Hz, 21%. (b) 2.133 Hz, 17%.
In the second place, the SDs identified by CoSSSI are presented in Figures
Stabilization diagram identified by CoSSSI. (a1) SD of CoSSSI (J = 1/1st subset). (a2) Selecting mode by the rate of the stable frequency over orders for on road vehicle modal identification (J = 1). (b1) SD of CoSSSI (J = 2/2nd subset). (b2) Selecting mode by the rate of the stable frequency over orders for on road vehicle modal identification (J = 2). (c1) SD of CoSSSI (J = 3/3rd subset). (c2) Selecting mode by the rate of the stable frequency over orders for on road vehicle modal identification (J = 3).
Modal parameters identified by CoSSSI. (a) J = 1(1^{st} subset). (b) J = 2(2^{nd} subset). (c) J = 3(3^{rd} subset).
What is more, roll is a significant mode in the theoretical vertical vehicle dynamic analysis. However, it is noticeable that the roll mode has not appeared in the identification results. In view of the vehicle and road design requirements, the roll mode has to be avoided for the safety. The results demonstrating no roll mode has illustrated the robustness of the proposed method in a further step. Consequently, the CoSSSI method has identified all of the vehicle suspension systemrelated modes under a high second threshold (80%), which indicates the reliability of the identified results.
An improved OMA method, denoted as CoSSSI, was proposed in this paper to accurately identify the modal parameters when the system responses are highly nonstationary and contained high noise. As the inherent nonlinearity of engineering systems often results in nonstationary vibration responses due to changes in modal properties under different operating conditions, the method then categories such responses into a number of subsets based on energy levels and implement SSI subsequently on the ensample averaged data for accurate and consistent identification. The performance of CoSSSI was evaluated by a 3DOF typical vibration system under various SNR condition. Then, an experimental study of vehicle running on the practical suburb roads was carried out to verify the performance of CoSSSI in a further step. Both simulation analysis and the experimental results provide compelling evidence that the CoSSSI method is superior to the traditional CovSSI and ACSSSI methods. In other words, the CoSSSI method can provide a more accurate and reliable modal identification results when the structure is under severe situations; the accurate results ensure the reliability of the SHM.
The experimental data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to thank the China Scholarship Council (CSC; grant no. 201608060041) and the National Natural Science Foundation of China (NSFC; grant no. 51775181) for the sponsorship of the project carried out in this study. Moreover, the authors would like to express their appreciation to Mr. Debanjan Modal for his work to improve English in this paper.