# Modeling of Distorted Degradation Data Based on Oil Analysis
## 4 Case Studies  
In this section, we validate the practical applicability of our proposed degradation model by investigating wear degradation data in the power-shift steering transmission (PSST) of tracked vehicles. PSST is characterized by its high loading capacity. The wear in PSST is complex, occurring by the plastic displacement of surface and near-surface material and by the detachment of particles. The metal particles of various shapes and sizes that are formed by the wear are distributed in the oil as contaminants. Our experiment is an on-road test rather than a laboratory test. To be specific, the tracked vehicle runs on the road and the oil is sampled at discrete time points. Then, the spectrometric oil analysis technique is used to measure concentrations of metal particles within oil samples. In consideration of oil consumption, at some sampling points, fresh oil is added to the PSST after sampling. The Cu (copper) element represents the overall wear level. Therefore, the observations for Cu in the PSST of Vehicle 1 are shown in Figure 1 and listed in Table 6. We can see that the concentrations of Cu are non-monotone although the actual wear process is monotone. This is due to the fact that the addition of fresh oil reduces the concentration of Cu in the PSST. Consequently, it is necessary to correct the observed data and thus recover the actual degradation curve. Unfortunately, as shown in Table 6, the data obtained only include  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Sampling Time [h]** & **Observed Concentration [ppm]** \\ \hline
1 & 0 & 27.7000 \\
2 & 13.5 & 40.1500 \\
3 & 26 & 49.6500 \\
4 & 41 & 57.9000 \\
5 & 50 & 47.3000 \\
6 & 63.5 & 43.9500 \\
7 & 76 & 42.9500 \\  
the sampling time and the observed concentration, without the oil top-up time and the oil top-up quantity. Thus, we model these observed data based on **Case III** defined in Section 2 and then, respectively, use the parametric and non-parametric approaches to estimate the parameters of the model. In the parametric approach, the parametric representation of the physical degradation process is critical. The fresh oil is added after running for some time; therefore, the first few observed concentrations are not affected by the fresh oil and can represent the actual physical degradation. According to the first few observed concentrations in Figure 1, we can assume that the running time and actual physical degradation are non-linearly expressed as follows:  
\[h(t)=a+bt^{c},\]  
where \(t\) is the time and \(a,b,c\) are unknown parameters. In the non-parametric approach, as introduced in Section 2, we use a natural cubic smoothing spline to fit the actual physical degradation curve. Figure 5 shows the results for the parametric and non-parametric approaches, including the observed data, the detected oil top-up points, the actual degradation (the corrected curve), and the bootstrap confidence intervals. We can see that the number of oil top-ups is relatively large. Frequent oil top-ups may be due to high-intensity work or a high leakage rate. Moreover, we notice that there is a big gap between the observed degradation and the corrected one. The \"uncorrected\" (observed) data are non-monotone and relative while the \"corrected\" (actual) data are monotone and absolute. This implies that the corrected curve reflects the actual physical degradation process better than the observed data. Therefore, it is meaningful to use our methodology to correct the observed data. Based on actual experience, the failure threshold of Cu is set to 160 ppm. Figure 5 also shows the FHT for failure occurrence based on parametric and non-parametric approaches. Furthermore, we can see that the actual wear degradation has not reached the failure threshold yet, and thus Vehicle 1 can still work normally for a while. The observed degradation data in the PSST of Vehicle 2 are shown in Table 7. Figure 6 shows the results obtained, respectively, by the parametric approach and the non-parametric approach.