Time Series Forecasting for Structures Subjected to Nonstationary Inputs

In this paper, a method for real-time forecasting of the dynamics of structures experiencing nonstationary inputs is described. This is presented as time series predictions across different timescales. The target applications include hypersonic vehicles, space launch systems, real-time prognostics, and monitoring of high-rate and energetic systems. This work presents numerical analysis and experimental results for the real-time implementation of a Fast Fourier Transform (FFT)-based approach for time series forecasting. For this preliminary study, a testbench structure that consists of a cantilever beam subjected to nonstationary inputs is used to generate experimental data. First, the data is de-trended, then the time series data is transferred into the frequency domain, and measures for frequency, amplitude, and phase are obtained. Thereafter, select frequency components are collected, transformed back to the time domain, recombined, and then the trend in the data is restored. Finally, the recombined signals are propagated into the future to the selected prediction horizon. This preliminary time series forecasting work is done offline using pre-recorded experimental data, and the FFT-based approach is implemented in a rolling window configuration. Here learning windows of 0.1, 0.5, and 1 s are considered with different computation times simulated. Results demonstrate that the proposed FFT-based approach can maintain a constant prediction horizon at 1 s with sufficient accuracy for the considered system. The performance of the system is quantified using a variety of metrics. Computational speed and prediction accuracy as a function of training time and learning window lengths are examined in this work. The algorithm configuration with the shortest learning window (0.1 s) is shown to converge faster following the nonstationary when compared to algorithm configuration with longer learning windows. INTRODUCTION Structures experiencing high-rate dynamics are subjected to 1) large uncertainties in external loads; 2) high levels of nonstationarities and heavy disturbances, and 3) the generation of unmodeled dynamics from changes in system configuration [1]. The development of a real-time monitoring and prediction methodology that observes the current state of a structure and 1 Copyright © 2021 by ASME predicts its future state will enable active structures that can respond to high-rate dynamics in real-time [2]. If an effective framework can be developed, control commands can be initiated to prevent further harm or complete system failure [3]. One key challenge in the development of a real-time monitoring and prediction methodology is its ability to operate through nonstationarities. A nonstationary event is one in which the statistical representation of the signal changes. Stationarities can be classified as weak stationarity, covariance stationarity, or second-order stationarity [4]. If a shift in time does not induce a difference in the distribution form, a time series has stationarity, and therefore, the distribution properties (e.g., mean, variance, and covariance) are constant over time. There are a variety of cases in which a time series does not stay stationary. If these distribution properties are mishandled, the time series would display nonstationary attributes, which is a significant test for a few fields. The nonstationary time arrangement incorporates time trends, arbitrary strolls (additionally called unit roots), and seasonality. Time trends in a signal can also be thought of as low frequency components with periods longer than the considered data set. Thus, a few methodologies are created to break down the nonstationary attributes. These methodologies can be characterized into two primary sorts: time strategies (e.g., Auto-Correlation Analysis strategy, Regression technique, Seasonal Auto-Regressive Incorporated Moving Average, Break for Additive Trend and Season) and Spectro-Temporal strategies [5]. Spectro-Temporal techniques consider the portrayal of frequency varieties [6]. Time series forecasting of high-rate dynamics is difficult. Specifically, a time series forecasting technique must be robust enough to operate with noisy sensor data. Time series forecasting is performed by studying patterns in a variable (or the relationships between variables), building a model, and using this knowledge to build a model. The model is then used to extrapolate the variable into the future. This demonstrating approach is especially valuable when little information is accessible on the information-producing operation or when there is no agreeable illustrative model that relates the expectation variable to other illustrative factors. Much effort has been committed to the improvement and development of time series forecasting models [7]. The investigation of the time series can be separated into two tasks. The initial task is to acquire the structure and basic knowledge (i.e. dynamics) of the observed information. The subsequent task is to fit a model that will be used to make predictions. Observing past information can be utilized for the examination of the dynamics of a structure under nonstationary inputs as well as prediction of its future dynamics. A standard methodology in dissecting time series is to decompose the monitored variable into the three segments, trend, nonstationary, and residual [8]. For the most part, time series examination can be isolated into univariate and multivariate examinations. Univariate time series includes a period arrangement containing a solitary perception recorded consecutively over time. Multivariate time arrangement is utilized when several time series factors are included, and their connections are to be considered [9]. Common techniques for time series prediction incorporate the sliding window, smoothing, and autoregressive expectation techniques, which are broadly applied in a forecast of high rate dynamics system states, financial turn of events, environmental change, and energy interest. The sliding window technique is similar to the single dramatic smoothing strategy while the smoothing and autoregressive techniques are similar to the two-fold dramatic smoothing technique and the triple outstanding smoothing strategy, respectively [10]. The main aim of this work is to investigate the real-time implementation of time series forecasting over a nonstationary event. In this work, a change in loading is introduced into a cantilever beam structure to generate a nonstationarity event. This is intended to represent a structure subjected to a high-rate dynamic event (e.g. impact) that changes the state (i.e. damage) of the structure. This work presents a numerical analysis for the realtime implementation of a Fast Fourier Transform (FFT)-based approach for time series forecasting. For this preliminary study, a testbench structure that consists of a cantilever beam subjected to nonstationary inputs is used to generate experimental data. For online time series forecasting, the FFT-based approach is implemented in a rolling window configuration. The main contribution of this paper is a investigation into how the FFT-based approach responds before, during, and directly following a nonstationary event, while considering different learning window lengths and assumed computation times. The performance of the system is quantified using a variety of metrics that investigate the quality of the prediction. The FFT-based approach with the shortest learning time achieves the best performance. Here 0.1 s, 0.5 s, and 1 s learning window length have been considered with different computation times simulated. The effect of learning window length in different states is described with MAE (mean absolute time) and transient state. The mean absolute error (MAE) can be used to classify errors that are uniformly distributed [11]. The computation time is an approximation of the actual computation time needed for the FFT, signal extraction, and IFFT. These values are reasonable approximations for actual hardware. The effect of computational time is also analyzed in different states and described with mean error and transient time. The algorithm configuration with the shortest learning window that exceeds the lower bound set by the Nyquist sampling theorem (0.1 s) is shown to converge faster following the nonstationary when compared to algorithm configurations with longer learning windows. The shortest computational time (0.01 s) is also impactful for smaller mean errors in different states and for obtaining the shortest transient time. 2 Copyright © 2021 by ASME FIGURE 1. Experimental setup of a cantilever beam with key components and data acquisition setup. FIGURE 2. Mode shapes and frequencies for the cantilever beam setup showing: (a) mode shape 1; (b) mode shape 2, and; (c) mode shape 3. EXPERIMENTAL SETUP The experimental setup is shown in Figure 1. For the purpose of the experiment, a steel cantilever beam structure of 759 x 50.66 x 5.14 mm is used and a single Integral Electronics Piezoelectric (IEPE) accelerometer (model J352C33 manufactured by PCB Piezotronics) is mounted close to the edge of the beam structure.The location of the accelerometer is 0.46 m from the fixed point of the cantilever beam as shown in Figure 1.This accelerometer has a frequency range of 0.5 Hz to 9k Hz with a sensitivity of 100 mV/g. The sensor data is digitized using a 24bit NI-9234 IEPE signal conditioner manufactured by National Instruments. To ensure that the accelerometer was not placed at a node of the beam, the mode shapes and natural frequencies for the first three modes of the cantilever were computed via Euler’s formula [12] and are shown in Figure 2. The node of the system for the second mode is at 0.594 m while the nodes of the system for the third mode are at 0.380 m and 0.659 m. Therefore, the location of the accelerometer at 0.46 m does not lie directly at any node. The beam is excited by an electromagn