PSI - Issue 78

A. Romanazzi et al. / Procedia Structural Integrity 78 (2026) 777–784

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1. Introduction Vibration based damage detection is based on tracking the changes in dynamic properties of a system, such as damping, natural frequencies and mode shapes. Different methods have been proposed in the literature to identify these changes, such as neural network approaches by Shi and Chang (2012), wavelet analysis by Yi et al. (2006), Fourier transform based methods by Ditommaso et al. (2012), response function by Hsu and Loh (2013), and time series analysis by Romanazzi et al. (2023). Recently, further approaches based on statistical and data-driven methods have been developed to identify the modal parameters of a dynamic system from output-only acceleration data, which model the structural dynamic behaviour as time-invariant linear system resorting to polynomial data fitting, complex pole relations and linear regressions. Among the others, Auto-Regressive (AR) and Auto-Regressive Moving-Average (ARMA) models are commonly used. In general, such signal processing techniques assume that signals are stationary, and the system under consideration is linear and time-invariant, as in Chopra (1995), Ibrahim and Mikulcik (1973), and Juang and Pappa (1985). However, complex excitations such as earthquake ground-motions are non-stationary signals and might induce structural damage, which leads to non-linear and time-variant systems, as in Entezami and Shariatmadar (2019), and Gong et al. (2010). Consequently, common vibration-based techniques cannot be applied to such cases, as the results would represent an average of the spectral amplitude over the duration of the signals, losing information on the local spectrum and thus the time varying properties. With this regard, different approaches have been developed to analyse time-variant dynamic responses as by Gabor (1946), Cohen (1989), and Young (1993), and among the others, time-frequency methods have recently gained increasing application in structural monitoring, as Wavelet Transform (WT), Empirical Mode Decomposition (EMD), Hilbert-Huang Transform (HHT), Short Time Fourier Transform (STFT), by Perez-Ramirez et al. (2016). In the framework of data-driven approaches, an extension to the AR models is the Auto-Regressive with eXogenous variables model (ARX), in which the input time series (the eXogenous variables) is considered to fit the polynomial model on the output time series, providing robustness to the identification algorithm. Subsequently, the frequency response of the ARX fitted model can be analysed to identify the dynamic properties of a system. The advantage of ARX approach lies in the robustness and reliability of the solution which includes itself already possible noise and the non-stationarity of the exogenous variable, despite the source of the external forcing as in Lennart (1999). In general, for Single-Input Single-Output (SISO) records, the relationship between input ( ) and output ( ) at time t of a system can be expressed as a linear difference equation as: ( ) = − 1 ( − 1) − ⋯ − ( − )+ 1 ( − 1) + ⋯ + ( − ) (1) Where the parameters and are defined as the number of observations to be considered for the autoregressive polynomial (AR-order) and exogenous polynomial (X-order) of the model, respectively. In addition, the time delay order is added being the number of input instants that occur before the input affects the output. In equation (1), ( − 1) … ( − ) is the previous outputs on which the current output depends, and ( − 1) … ( − − +1) is the previous and delayed inputs on which the current output depends. Leading to a compact form: ( ) = ( ) (2) With: = [ 1 … 1 … ] (3) ( ) = [− ( − 1) … − ( − ) ( − 1) … ( − − +1)] (4)