Hanchao Yuan1a* Boyu Ning1b
1JiangSu University, School of Physics and Electronic Engineering, Zhenjiang, 212013, China
aE-mail:2222126007@stmail.ujs.edu.cn
bE-mail：1390637502@qq.com
ABSTRACT: Modal parameters of structures, including frequency, mode shapes, and damping ratios, are inherent characteristics essential for understanding structural dynamics, detecting structural damage, and conducting safety assessments. Recently, deep learning method has enjoyed a broad application in the identification of structural modal parameters to facilitate automated detection. However, uncertainties due to measurement errors can make the modal parameter identification much less accurate. In response, a machine-learning method is proposed for the structural modal parameter identification based on interval analysis. This method transforms modal parameter identification into a deep learning problem, with deterministic acceleration data as input and interval-based modal data as the output. Finally, numerical simulation examples are used for validation. A model bridge was employed to further validate the algorithm presented in this paper. According to these results, the algorithm is capable of accurately predicting the structural modal parameters, demonstrating potential for practical engineering applications.
Keywords: modal identification, interval analysis, uncertainty analysis, deep learning, artificial neural networks
1. Introduction
Structural modal parameters, including frequency, mode shapes, and damping ratios, are inherent characteristics reflecting the response of structures to external forces and characterizing the unique structure behavior. These parameters are crucial for understanding the dynamic structure properties, detecting structural damage, and conducting safety assessments [1]. Hence, accurately identifying structural modal parameters is significantly important. Together with finite element analysis, modal parameter identification forms one of the two main pillars of structural dynamics. The finite element method addresses the “forward problem” in structural dynamics, while modal parameter identification deals with the “inverse problem.” For linear stationary systems, the vibration differential equation can be decoupled from physical coordinates to modal coordinates through modal analysis and thus enable the identification of modal parameters. Modal parameter identification is a primary solution to engineering vibration problems and has important applications in mechanical engineering, aerospace, and civil engineering and other relevant fields.
Traditional methods for modal parameter identification take into account the input-output relationship of structure and primarily encompass three types categorized by identification domain: frequency domain, time domain, and time-frequency domain. Frequency domain methods, which are developed from classical spectral theory, assume the input signal conforms to white noise characteristics, making the power spectral density (PSD) function specific to the response signal approximate the frequency response function. Power spectra-based methods encompass the peak picking method[2], frequency domain decomposition (FDD) method[3], and least squares (LS) complex frequency domain method[4]. Modern spectral analysis has also led to the generation of time-domain methods with regard to the modal parameter identification. These methods follow two approaches: one involves response functions or matrices to solve for modal parameters through eigenvalue decomposition, such as time series methods and data-driven stochastic subspace methods, while the other approach preprocesses the system’s output response data to obtain approximate free response or impulse response signals, which are then used in classical time domain identification methods (eigensystem realization algorithm[5] and covariance-driven stochastic subspace method, etc.)[6]. The first two methods assume white noise or stationary non-white noise excitation for modal parameter identification. However, in practical situations, it is often found that real operating environment excitation and structural response deviate from these assumptions, displaying characteristics of non-white noise and non-stationary variations. To address these issues, researchers have combined time domain and frequency domain identification processes into new analytical approaches, with wavelet transform[7]and Hilbert-Huang Transform (HHT)[8] being typical time-frequency domain modal parameter identification algorithms.
Currently, machine learning[10] is a highly effective method across various disciplines. With a powerful fitting effect, machine learning is capable of correctly finding the global minimum regardless of the state of the target function (non-convex and non-smooth). Han et al. [9] adopted deep learning for the solution of high-dimensional partial differential equations, which was proved to be highly accurate and cost-benefit. Li et al. [11]applied deep learning methods for solving control problems related to partial differential equations by transforming the deep residual network into an equation control problem. In the study by Chen et al. [12], a new deep neural network model was introduced, taking the network output as a black-box solver for partial differential equations. In the study by Sun et al. [13], fast magnetic resonance imaging was conducted relying on the alternating direction method of multipliers (ADMM) network, and the network showed improved computation speed and reconstruction efficiency. Carleo et al. [14] employed artificial neural networks for the solution of quantum mechanics problems, which could effectively and accurately describe the prototype interaction spin models in both one and two dimensions.
Besides, machine learning has also enjoyed a wide application in various fields related to civil engineering[15], namely lost-data recovery [16], damage detection [17], and structure-aware 3D reconstruction [18]. Machine learning was employed in the study of Tang et al. for the data anomaly detection [19]and compressive-sensing data reconstruction [20]. Bao et al. [21] proposed the machine learning paradigm for SHM, and Demarie et al. [22] developed the same method to deal with automatic long-term SHM, despite the requirement for engineering judgement. Mao et al. [23] applied principal components analysis and clustering techniques for the acquisition structural modal parameters in the premise of giving priority to the determination of the model order. The machine learning method in the study of Bao et al. [24] was designed for the modal identification, despite the obvious impact exerted by mode independence.
Currently, most deep learning-based methods specific to modal parameter identification are deterministic. However, due to environmental testing noise and inherent uncertainties in inverse problems, uncertainty analysis methods align more closely with the nature of these problems. Therefore, this paper introduces interval analysis to expand traditional deep learning-based modal parameter identification into an uncertainty analysis approach. First, Stochastic Subspace Identification (SSI) and FDD are adopted for identifying the structure’s modal parameters. Interval analysis is then employed to convert the modal data into interval variables. Next, an artificial neural network (ANN) is constructed, where the network input comprises the structure’s acceleration response dataset, and the output consists of the interval modal variables. Finally, a finite element model (FEM) of a simply supported beam is used to test the constructed neural network model. Numerical examples demonstrate the high computational accuracy of the method proposed here
The rest of the paper falls into 4 sections. Section 2 briefly reviews the FDD and SSI techniques, Section 3 details the proposed method, and Section 4 explains numerical shear frame building structures. Section 5 is the conclusion.
2. Modal Analysis Methods
2.1 Frequency domain decomposition
FDD is a widely used modal analysis method operating in the frequency domain, which can serve for effectively identifying structure’s modal parameters under operational conditions where the input forces are unknown or uncontrollable. The FDD method combines spectral analysis principles and utilizes the PSD matrices of the output signals.
Defining the unknown input excitation force of the system [25] as the signal x(t) and the measured output response signal as y(t), we use the equation below to interpret the relationship between the system’s output PSD matrix Gyy(jw), the corresponding frequency response function matrix H(jw), and the autocorrelation matrix Gxx(jw) of the input signal as follows
The system’s frequency response function matrix can be expressed by its poles and residues:
Where n represents the order of the mode, λk is the kth order pole, * denotes the conjugate, and Rkis is the residue of H(jw).
Assuming the input signal is a white noise, the input power spectrum becomes a constant matrix. The PSD equation can then be expressed by poles and residues:
Where Ak represents the k-th order residue matrix specific to the output PSD, and H denotes the complex conjugate transpose of the matrix.
It is allowed to express the residue Ak by modal shape vectors:
where and
denote the modal shape and participation vector, respectively.
To ensure the performance of SCD on the PSD matrix, the resulting estimate of the PSD can be written as:
The FDD method converts a multi-degree-of-freedom system (MDOF) into single-degree-of-freedom (SDOF) systems using SVD. The singular value spectrum refers to a superposition of several SDOF spectral densities, with each DOF corresponding to a fundamental natural frequency. Given the formation of a continuous curve by the singular value spectrum, a strong correlation should be constructed between the natural frequency and corresponding neighboring regions, which can be classified as the same mode.
By extracting the singular values that are located near the natural frequency and meet the correlation conditions, with their corresponding positions maintained within the window, we can obtain the auto-spectral density function specific to the SDOF system. The function will have the same length as the window, with the rest set to zero (zero-padding).
2.2 SSI
This section mainly introduces the covariance-based SSI method. The initial step in the SSI process involves the calculation of the Hankel matrix, which can be obtained as follows:
Where yk represents the output data at a certain moment arranged in columns, with k=0,1 …, 2i +N- 2. The number of columns of yk is l, also the number of output channels, and N indicates the number of columns in the Hankel matrix. Yp is the past matrix, nd Yf are is future matrix. Yp comprises the first i rows of the Hankel matrix. Yf can take two forms, different in the first block of rows being delayed by one and two sampling times, respectively, compared to the last block of rows of Yp, with both having i blocks of rows. The total data length is s, where s =2i+N-1. Theoretically, N should approach infinity; however, based on empirical evidence, N should be selected to satisfy N > 20i. Thus, the selection of i should satisfy:
The modal parameters of the system are ultimately obtained through calculations between the covariances of the output vectors. Therefore, the covariance matrix of the output data shall be computed. All covariances form a Toeplitz matrix:
The covariance between the state vector and the output vector is established. In determining the system’s mode shapes, the required output matrix (C) comprises the first (l) rows of the observation matrix:
Equation below explains the SVD of a Toeplitz matrix:
In above equation, U and V are orthogonal matrices. S is a singular value matrix, a diagonal matrix with elements arranged in descending order. The basis for determining the system order using eigenvalues lies in the number of nonzero values in S:
Then:
The process of solving for the state matrix A is relatively simple. Although the number of nonzero singular values in S theoretically determines the system order, in practical applications, the high-order singular values of S are usually not zero. In such cases, it is allowed to identify the system order considering the jump in singular values. However, when the jump in singular values is not significant, the accurate determination of the system order can be hardly achieved. In this case, the stability graph method can assist in determining the system order.
3.ANN-based interval model prediction model
3.1 Interval analysis
Interval analysis is a non-probabilistic method for handling uncertainty problems. It is particularly suitable for quantifying and analyzing uncertainties when there is insufficient information to describe the probabilistic model of uncertain parameters. The method primarily engages in representing uncertain parameters as intervals, with the minimum value as the lower bound and maximum value[26] as the upper bound of the interval. By analyzing the upper and lower bounds of these uncertain parameters, an interval estimation result can be obtained, indicating the possible ranges of values for the parameters.
The uncertain parameter X belongs to the scope of interval variable:
Next, equation (16) explains the midpoint (or mean value) Xm and radius (or deviation amplitude) ∆X of X:
Where, ∆X serves as a quantitative indicator of the magnitude of parameter uncertainty.
We can then define the four arithmetic operations between two uncertain parameters:
This means
3.2 ANN model
To automatically predict the structural modal parameters, a multilayer ANN model is used. It comprises two hidden layers, each with 20 neurons. The Levenberg-Marquardt back-propagation algorithm serves for the optimization of the weights, while tangent sigmoid transfer functions act as the activation functions for all neurons. The schematic diagram of the network is shown in Error: Reference source not found.
To predict the structural modal parameters, a FEM must be trained to obtain sufficient data. In this study, structure acceleration response is used as the input to the ANN, and the output consists of the structural modal parameters. Unlike traditional modal parameter identification methods, the output modal parameters are represented as interval-valued rather than deterministic modal values.
To achieve this, the modal data of the training set is identified using the SSI and FDD methods described in section 2. However, the identification results for the same acceleration data cannot be completely consistent due to algorithms and uncertainties. Therefore, interval analysis is used to preprocess the data, with frequency as an example.
To meet the modal requirements under environmental excitation, acceleration time history data from the operational phase of the structure is employed. For data characterization, the acceleration data is converted into transfer ratio functions:
Training is critical for a neural network to learn modal identification capabilities. For effective training, the sample data approximate preprocessing and encoding, such as denoising, normalization, and label encoding, to improve the model’s generalization performance. Additionally, it is essential to prepare enough sample data that covers various possible structural characteristic scenarios, equipping the model with strong identification capabilities. The representation of input and output data for the ANN is as follows:
The input consists of the transfer ratio function amplitudes from various sensors of the structure, where m represents the number of transfer ratio functions. The target output consists of interval-valued modal data, with m sets in total.
Fig. 2 shows the analysis process summarized as follows:
1) Establish a finite element numerical model of the measured structure. Alter the physical parameters of the structure (mass, stiffness, and damping) to obtain the structure acceleration response under environmental excitation.
2) Apply the SSI and FDD methods to the acquisition of the modal data regarding the structure. Apply interval analysis methods to obtain the corresponding interval modal data and divide the data into training set and test set.
3) Train the ANN using the training and test sets obtained in Step 2.
4) Input the actual engineering acceleration data to be tested into the trained ANN to obtain the interval modal data of the structure.
| Fig. 1. ANN architecture |
| Fig. 2. The procedure of the proposed approach |
4.Numerical example
Consider a 10-story shear-type frame structure with the following parameters: for each floor, the mass M is m1=m2=…m10=5×105Kg, and the inter-story stiffness K is k1=k2=…k10=1.5×108N/m. The Rayleigh damping coefficient is defined as C=0.0193M+9.711×10-4K. The environmental excitation is modeled as Gaussian white noise. In the simulation, the time step is dt = 0.01 s, with the simulation duration being 500 s. The Runge-Kutta method serves for the solution of the dynamic equation. Figure 3 illustrates the typical acceleration response time history of one of the floors alongside the ground excitation time history. It can be observed that the input excitation and the response are stationary.
Table 1. Model frequencies (Hz) and damping ratios
| Order | 1st | 2nd | 3rd | 4th | 5th | 6th |
| Frequency(Hz) | 0.41 | 1.22 | 2.01 | 2.75 | 3.43 | 4.04 |
| Damping Ratio (%) | 0.50 | 0.50 | 0.69 | 0.89 | 1.09 | 1.27 |
Fig. 4 shows the theoretical mode shapes for the 1st four modes of the structure. Table 1 lists the theoretical frequencies and damping ratios for the 1st six modes. To train the ANN model, 100 numerical simulations were conducted, with 80 samples as the training set and 20 as the validation set. The acceleration response transfer function served as the input to the ANN, while the interval-valued modal data as the output.
According to the interval modal results predicted by the ANN (Figs.7-9), the mean interval modal results align closely with the theoretical solutions. However, due to the inclusion of uncertainty factors, the identification results are represented as intervals, where the lower and upper bounds target the minimum and maximum values of the structural modal parameters, respectively. Additionally, the interval radius is larger for higher-order modes, indicating greater uncertainty in the identification of these modes.
For ascertaining the way noise intensity impacted the uncertainty of identification results, Gaussian white noise with varying intensities was added to the test response data:
Fig. 10 illustrates the variation of the interval radius of the structural modal frequencies under different noise intensities. It can be observed that as noise intensity increases, the interval radius increases linearly. Furthermore, the interval radius for higher-order modes is larger than that for the lower-order modes.
5.Experimental example
Accurate identification of structural modal parameters is crucial for structural health monitoring, fault diagnosis, and dynamic performance evaluation in modern engineering. Traditional methods often rely on precise mathematical models and high-quality measurement data. However, real-world engineering problems frequently involve uncertainties and noise, limiting the applicability of conventional approaches. To address these challenges, this chapter presents an innovative method combining interval analysis and deep learning for the identification of structural modal parameters, validated through experimental studies, as shown in Fig. 11 and Fig. 12.
Interval analysis offers a means to handle uncertainties by representing measurements as intervals rather than single values, better reflecting the inherent uncertainty in data. This experiment utilizes interval analysis to establish preliminary estimation ranges for structural modal parameters, providing training data for the deep learning model. Select an appropriate deep learning architecture, such as Convolutional Neural Networks (CNN), tailored to the characteristics of modal parameter identification tasks. Design the network structure and determine key parameters, including the number of layers, nodes, and activation functions. Train the model using interval data generated from interval analysis to ensure its capability to handle data uncertainties.
Select a representative set of structural models as test subjects, such as simply supported beams or cantilever beams. Utilize high-precision sensors and data acquisition systems to record the vibrational responses of the structures under various excitation conditions. Collect vibration signals from the structures, and process them through filtering and denoising to obtain clear response signals. Convert the processed signals into interval data using interval analysis methods. Input the preprocessed data into the deep learning model for training, optimizing model parameters and training strategies to enhance identification accuracy. Evaluate the model’s performance using cross-validation to ensure its generalization capabilities.
Table 1 Identified modal parameters
| 1st | 2nd | 3rd | |
| Exact | 0.56 | 2.45 | 3.11 |
| Obtained | 0.54 | 2.41 | 3.01 |
The acceleration responses of the model bridge under different excitation conditions were analyzed in detail. Under environmental excitation, the maximum acceleration response at the mid-span of the bridge deck was approximately 0.15g, mainly concentrated in the frequency range of 1-10 Hz, as shown in Figure 13. By performing a Fast Fourier Transform (FFT) on the acceleration time-history signal, the obtained spectrum clearly displayed the peaks corresponding to the identified vibration modes. Additionally, short-term high-frequency responses caused by vehicle loads were observed, with the maximum amplitude reaching up to 0.3g. These acceleration response characteristics not only verified the previous vibration mode identification results but also provided important basis for the dynamic performance evaluation and fatigue life prediction of the bridge.
To evaluate the performance of artificial neural networks (ANN) in predicting the dynamic response of bridges, a three-layer feedforward neural network model was constructed. 70% of the data was used for training, 15% for validation, and the remaining 15% for testing, as shown in Figure 14. The prediction results indicated that the RMSE of the ANN model on the test set was 0.05g, and the relative error was within 5%. By comparing the time-history curves of the predicted and measured values, it was found that the ANN model could accurately capture the main characteristics of bridge vibration, but there was a slight deviation in the prediction of some peak responses. This suggests that the ANN model has good potential in predicting the dynamic response of bridges, but there is still room for further optimization.
In this study, the operational modal analysis method was adopted to identify the vibration modes of the model bridge. By using the ANN algorithm and the structural response data collected from the acceleration sensors under environmental excitation, the first five vibration modes of the bridge were successfully identified. The identification results show that the fundamental frequency of the bridge is 2.35 Hz, corresponding to the first bending mode. The second and third vibration modes are 4.78 Hz and 7.12 Hz respectively, presenting typical high-order bending characteristics, as shown in Figure 15. These identification results lay the foundation for subsequent structural health monitoring and dynamic characteristic analysis.
| Fig. 14 Mode shape identification results |
6.Conclusion
To account for the uncertainties in structural dynamic response testing, a deep learning-based method was proposed to deal with the interval identification with regard to structural modal parameters. This approach leverages modal independence and sparse structural response data in the time-frequency domain to achieve the systematic identification of structural modal parameters. We conducted numerical simulations and experiments to validate the algorithm presented in this paper. conclusions were primarily drawn:
1)The method effectively identifies modal parameters under environmental excitation. The mean value of the interval represents the structural modal parameters, while the standard deviation of the interval indicates the intensity of uncertainty factors.
2) The input uncertainty factors significantly affect the accuracy of the neural network’s identification. Simulations revealed that the stronger the environmental noise, the greater the standard deviation of the identification results.
3) Increases in the amount of data enhance the stability of the identification results, thereby reducing the likelihood of random errors in the predictions.
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