PSI - Issue 38

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

165

L. Heindel et al. / Structural Integrity Procedia 00 (2021) 000–000

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which generalizes the fatigue damage accumulation resulting from a signal s in multiple spacial directions ψ . Here, the signal channels are denoted by s x , s y and s z , while ψ x , ψ y and ψ z are the coordinates of the unit direction vector ψ . The Multi-Rain ratio damage MR y ∗ , y = (10) compares the Multi-Rain damage of prediction and true response, equaling one if both signals are identical. Unfortunately, the Multi-Rain ratio is prone to error compensation, which can occur if large amplitudes are under- and small amplitudes are overestimated. A good prediction is therefore only achieved when both the RMS error is low and the Multi-Rain ratio is close to one. The experimental dataset can be used to demonstrate the hybrid modeling approach for both VS and FP problems. In both cases, the hybrid models hybrid 1 and hybrid 2 introduced in subsection 2.3 are compared to the pure FRF model, as well as a pure LSTM network. Since the process of LSTM network training involves a large number of hyper-parameters which have to be evaluated on a large dataset, a global optimization in this parameter space is not feasible. Instead, the best models in each category were chosen after a series of parameter studies on an high perfor mance computing cluster at the Center for Information Services and HPC (ZIH) at TU Dresden. The FRF model was parameterized using only measurement data of regions with mostly constant sti ff ness, displayed in green in Figure 3, and is identical in both the pure FRF and hybrid methods. The aim of the VS experiment is to predict the force responses of the test rig from its displacement measurements. The results of this study are depicted in the first row of Figure 4. It is apparent that both hybrid combinations significantly outperform the pure FRF and LSTM models in terms of RMS. All LSTM based models achieve very low errors for the fatigue serviceloads with an o ff set, compared to the linear FRF prediction. Similar results can be found regarding the Multi-Rain ratio, which is closest to one for the hybrid 2 model across all scaling and o ff set variations. In the FP study, the displacement and force responses of the system are estimated from the drive signal force channels, which control the actuation of the test rig. The results are displayed separately for displacement and force predictions in rows 2 and 3 of Figure 4, although they were produced by the same LSTM networks. Here, the displacement predictions clearly demonstrate the strengths of hybrid modeling. While the RMS error of the pure LSTM network is high compared to the FRF prediction for all rescaled serviceloads, the LSTM performs much better in the o ff set examples. The hybrid models provide an overall better performance, since they match the FRF model in the rescaled examples but improve upon its shortcomings in highly non-linear cases. The Multi-Rain ratio shows that the pure FRF and LSTM models respectively under- or overestimate the fatigue damage, while the hybrid predictions remain much closer to the ideal result of one. While the force channels are generally predicted with an overall higher error compared to the displacements, the best results in terms of both RMS and Multi-Rain metrics are again achieved using the second hybrid modeling approach. Significant di ff erences in prediction quality exist between VS and FP, as well as between displacement and force prediction during FP. The main di ff erence between virtual sensing and forward prediction consists in which system components are relevant for the signal prediction. In the case of force prediction from displacements during VS, only dynamic parameters like the dynamic sti ff ness and dampening properties of the hydro-mounts have to be learned. In contrast, the dynamic behavior of the entire test setup must be predicted during FP. This is further complicated by a linearly controlled active system which exhibits nonlinear dynamic behavior, instead of the nonlinear dynamic behav ior of a passive hydro mount. The most likely explanation for the fact that the displacement channels are nevertheless well predicted in the FP case is the high sti ff ness in the outer regions of the static characteristics. It result in almost constant displacements at high force amplitudes, which are much easier to predict than the force channels themselves. The best performing models in both VS and FP studies are of the type hybrid 2 and feature a subsequence length L of 256, an overlap factor o of 0.5 as well as a single LSTM block containing 39 memory cells. In the VS case, the highest prediction quality was achieved after 501 epochs of training with a learning rate of λ = 0 . 0002, while the FP model was trained for 800 epochs with λ = 0 . 0001. max d ψ y ∗ max d ψ y 3.3. Evaluation