# A Fault-Signal-Based Generalizing Remaining Useful Life Prognostics Method for Wheel Hub Bearings
## 3 Methodology  
### Fault Signal Model  
The signal \(x(t)\) is formed by superimposing various fault signals \(e_{i}(t)\), environmental noise signals \(n_{j}(t)\), and system noise signals \(s_{k}(t)\), which is denoted by Equation (13).  
\[x(t)=\underset{i=1}{\overset{I}{\sum}}e_{i}(t)+\underset{j=1}{\overset{I}{ \sum}}n_{j}(t)+\underset{k=1}{\overset{K}{\sum}}s_{k}(t) \tag{13}\]Here, \(\sum\limits_{i=1}^{I}e_{i}(t)\) is a fault signal set, \(\sum\limits_{j=1}^{I}n_{j}(t)\) is an environmental noise signal set, and considering that in actual operating conditions, the intensity of the various noise signals generated by the environment is very large, far stronger even than various fault signals, so \(\sum\limits_{i=1}^{I}e_{i}(t)<<\sum\limits_{j=1}^{J}n_{j}(t)\). Further, \(\sum\limits_{k=1}^{K}s_{k}(t)\) is a system signal set. The system signal set has relative stability. Similarly, the intensity of the various noise signals generated by the environment is stronger than the signals generated by the system itself, so \(\sum\limits_{j=1}^{J}n_{j}(t)>\sum\limits_{k=1}^{K}s_{k}(t)\). However, a few intensities of fault signals and system signals are stronger than noise signals. The fault signals are mainly generated by the structure of wheel hub bearing interaction with the environment, reflecting both structural resonance and forced response. The noise signals are mainly generated by the environment, including the signals of the forced response. The system signals are mainly generated by the structure of the wheel hub bearing itself, including the signals of structural resonance. The fatigue characteristics are implicit in the fault signals.  
To make the RUL prognostics method more effective, it is important to give prominence to the fatigue characteristics in the fault signal. We give the following principles in this study.  
(a) When the external environment is the same, the severity of single fault damage is the largest for the first time, leading the severity to decrease sequentially. Therefore, Equation (13) is rewritten as Equation (14) by adding the severity factor \(e^{-\alpha_{i}t}\).  
\[x(t)=\sum\limits_{i=1}^{I}e_{i}(t)\cdot e^{-\alpha_{i}t}+\sum\limits_{j=1}^{J} n_{j}(t)+\sum\limits_{k=1}^{K}s_{k}(t) \tag{14}\]  
Here, \(\alpha_{i}\) is the attenuation factor of fault signal \(e_{i}(t)\).  
(b) The cumulative fault damage severity increases gradually when the external environment is the same. Equation (14) is rewritten as Equation (15) by accumulating \(e_{i}(\tau)\cdot e^{-\alpha_{i}\tau}\).  
\[x(t)=\sum\limits_{i=1}^{I}\sum\limits_{\tau=l_{0}}^{t}e_{i}(\tau)e^{-\alpha_{i }\tau}+\sum\limits_{j=1}^{J}n_{j}(t)+\sum\limits_{k=1}^{K}s_{k}(t) \tag{15}\]  
(c) The external environment always changes, so that the severity of the cumulative fault damage cannot be correctly performed in the signal \(x(t)\). By the secondary accumulation of the fault signal item and the environmental noise signal item, the change of the external environment can be suppressed and the severity of the cumulative fault damage can be correctly represented. As such, Equation (14) is rewritten as Equation (16).  
\[x(t)=\sum\limits_{\tau=l_{0}}^{t}(\sum\limits_{i=1}^{I}\sum\limits_{\varsigma =\tau_{0}}^{\tau}e_{i}(\tau)e^{-\alpha_{i}\varsigma}+\sum\limits_{j=1}^{J}n_ {j}(\tau))+\sum\limits_{k=1}^{K}s_{k}(t) \tag{16}\]  
The fault signal model of Equation (16) can better express the fatigue characteristics, and the RUL can be dynamically prognosticated in real time with these signal series.