# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 1 Introduction  
Modal identification involves the determination of structural modal parameters, such as natural frequency, damping ratios, and mode shapes, using structural input and measured response data. It is widely used in structural performance evaluation, structural vibration control, structural fault diagnosis, and health monitoring [1, 2, 3]. Modal identification is typically categorized into Experimental Modal Analysis (EMA) and Operational Modal Analysis (OMA), depending on whether the system inputs are known or unknown during the analysis process. OMA takes the environmental excitations (wind, traffic, micro-vibrations, etc.) as the system inputs and models them as a uniform broadband random noise. Since the OMA methods do not require manual intervention, they can be continuously applied throughout the life cycle of the structure, offering significant economic and practical advantages [4, 5]. However, the OMA methods also exhibit certain limitations, particularly when the environmental excitation is of low intensity. In such cases, the measured response exhibits a low amplitude and a reduced signal-to-noise ratio, which makes it challenging to accurately determine the damping ratio [6, 7]. In addition, for some mechanical structures, the environmental excitations under operating conditions contain harmonic excitation, which can significantly affect the identificationresults when the harmonic excitation amplitude is high [8]. EMA uses excitation and response signals for the modal parameter identification, and one of its main advantages is that the loading equipment (impulse hammer, shaking table, etc.) can achieve different types of excitation. It can obtain measurement signals with a high signal-to-noise ratio through controllable experimental conditions, achieving the accurate identification of high-performance equipment modal parameters [9]. Due to limited excitation energy, EMA is more suitable for the modal parameter identification of small- and medium-sized structures; in addition, EMA does not easily simulate the actual working conditions of the structure. The main idea of EMA frequency domain methods is to obtain the modal parameters by fitting the frequency response function (FRF) of the measured data and the theoretical model, e.g., the weighted least-squares [10], rational fraction polynomial (RFP) [11], and polyreference least-squares complex frequency domain methods (PolyMAX) [12]. When the frequency resolution is low and spectral leakage exists, the identification accuracy of the frequency domain method is not satisfactory. Therefore, scholars have simultaneously carried out research on the time domain identification methods of modal parameters, with the widely used methods including the least-squares complex exponential (LSCE) [13], the polyreference complex exponential (PRCE) [14], the Eigensystem realization algorithm (ERA) [15], and so on. The EMA methods described above only produce point estimates of the modal parameters and do not provide information about their uncertainties. However, many uncertainties need to be taken into account in the process of modal identification, including the limitation of the number of test samples, the errors caused by incomplete modeling of physical phenomena, the measurement noise, etc. These uncertainties lead to statistical uncertainties in the structural dynamic parameters and are of great importance to further develop the EMA method, so that it can take these uncertainties into account. In statistical inference theory, there are two ways to quantify the identification uncertainty of EMA methods: the \"frequency\" approach and the Bayesian approach. In \"frequency theory\", identification uncertainty is defined as the variance of a random variable estimated in repeated experiments. [PERSON] et al.