PSI - Issue 38

L. Heindel et al. / Procedia Structural Integrity 38 (2022) 159–167 L. Heindel et al. / Structural Integrity Procedia 00 (2021) 000–000

160

2

that reduce the cost of measuring setups and increase their flexibility of application. Virtual sensing (VS) aims to estimate unmeasured physical quantities based on existing sensor data. It exploits redun dancies in measuring systems, which can be used to omit superfluous sensors or to facilitate sensing in regions which are di ffi cult to instrument. This is usually achieved by either model-based or data-driven approaches. Applications of model-based VS can be found in structural monitoring, were finite element models and modal analysis were success fully combined to estimate stress histories from strain measurements in Hjelm et al. (2005) or to predict the full-field response of a system from a limited number of measurements in Kullaa (2019). In a separate study by Ching and Beck (2007), system responses are predicted under uncertain excitation using a Kalman Filter. Data-driven approaches to VS are parameterized using measurement data and are therefore applicable in situations where information about the structure and the properties of a system are not available. Frequency Response Function (FRF) based methods, de scribed in Natke (1988), perfectly capture linear relationships between measurement signals and are therefore used for test rig descriptions, see Cryer et al. (1976); Hay and Roberts (2007). Machine Learning algorithms like the non linear autoregressive exogenous model (NARX) featured in Rouss et al. (2009) can be used to estimate responses of non-linear systems. Since these approaches rely on data exclusively, they are also able to generate predictions of system responses from system input data, which is referred to as Forward Prediction (FP) in the context of fatigue test commissioning. This contribution provides a hybrid framework for VS and FP, which combines FRF models with Long Short-Term Memory (LSTM) networks, introduced in Hochreiter and Schmidhuber (1997); Gers et al. (1999), in order to accu rately predict non-linear responses of dynamic systems with multiple input and output channels. LSTM networks are a type of recurrent artificial neural networks, which learn to approximate complex sequential relationships from datasets and have been applied to numerous highly non-linear problems like speech recognition or translation. A combination of both approaches can be achieved by either training the LSTM network to predict the error of the FRF model for a given set of inputs, or by incorporating the FRF model predictions into the training dataset of the LSTM as a baseline solution. The paper is structured as follows: In section 2, basics on FRF models as well as LSTM networks will be established and their application in hybrid modeling approaches for VS are described. The following section 3 features experi mental data from a servo hydraulic fatigue test bench, which is used to demonstrate the proposed methods in both a VS and an FP context. Finally, a discussion of results and an outlook are given in section 4.

2. Model setup

In many practical applications, incomplete information or time constraints complicate the use of a physical model for system approximation. As a result, generally applicable algorithms like FRF models are used to establish relation ships between physical quantities of interest.

2.1. Frequency response function model

The measurement data in a dynamic system can be divided into input data x ( t ) and output data y ( t ), which each consist of discrete temporal sequences of measurements x k ( t ) and y l ( t ), corresponding to specific data channels indexed by k and l . In such a system, the frequency response function matrix H (j ω ) provides the relationship (1) between the Fourier transformed input and output channels X j ω = F x ( t ) and Y j ω = F y ( t ) . In order to estimate the individual elements H kl (j ω n ) = , (2) Y (j ω ) = H (j ω ) X (j ω )

¯ S kl (j ω n ) ¯ S kk (j ω n )

the e ff ective Power Spectral Densities (PSD) ¯ S kk and ¯ S kl can be computed using the Fast Fourier Transform (FFT). The data used in the parameterization of FRF models is usually generated by exciting the input channels with noise signals