## 2 Proposing a New Failure Threshold
### Proposing a New Criterion for the FT  
REBs are usually the weak and failure-prone elements of rotary machines. Various mechanisms may lead to the failure of these elements [26]. If an REB is designed accurately, adequately lubricated and installed, continuously aligned during operation and is kept away from moisture, corrosive materials and excessive loading, the only mechanism that can lead to the failure of the REB is rolling contact fatigue (RCF) [27]. The failure caused by RCF can include soft failure and hard failure stages. In the soft failure stage, degradation occurs gradually over a long period of time and the REB does not entirely lose its efficiency. However, in the hard failure stage, the REB undergoes sudden impacts that can impose catastrophic damage on the equipment and the machine completely loses its function [4]. When the hard failure stage begins, fast growth is observed in the vibration level trend (Figure 1). Guidelines highly recommended terminating the operation of a machine when its vibration level passes a specific limit. Manufacturers and some standard codes [3] suggest values for this limit based on the type and properties of the machines. This limit is employed to define a high-risk zone for the equipment. Moreover, the transformation of degradation from soft failure to hard failure corresponds to the changing vibrations from the stable stage to the unstable stage. The beginning the unstable stage can be interpreted as entering the high-risk zone. Therefore, if one can predict this transition point (from soft failure to hard failure), it can be used as the FT and define the end of life for the machine. Furthermore, instead of considering a static FT, predicting a probability distribution of the FT can enhance the reliability estimation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Mathematical Description** & **Parameter Range** \\ \hline Gaussian & \(\int_{-\infty}^{\Phi^{-1}(u)}\int_{-\infty}^{\Phi^{-1}(v)}\frac{1}{2\pi\sqrt{1 -\theta^{2}}}\text{exp}(\frac{2\text{day}-\text{x}^{2}-\text{y}^{2}}{2(1- \theta^{2})})\) & \(\theta\,\in\,[-1,\,1]\) \\ [PERSON] & \(\text{max}\,\left(\text{u}^{-\text{0}}+\text{v}^{-\text{0}}-1\,,\,0\right)^{ \frac{\pi}{2}}\) & \(\theta\,\in\,[-1,\,\infty)\backslash 0\) \\ [PERSON] & \(\text{exp}\{-\{\left(-\ln(u)\right)^{\theta}+\left(-\ln(v)\right)^{\theta} \left[\frac{1}{\theta}\right]\}}\) & \(\theta\,\in\,[-1,\infty)\) \\ [PERSON] & \(\frac{-1}{\Phi}\ln[1+\frac{\text{exp}\left(-\theta u\right)-1\text{exp}\left( -\theta v\right)-1}{\text{exp}\left(-\theta\right)-1}]\) & \(\theta\,\in\,R\backslash 0\) \\ [PERSON] & \(1\,-\,\left[(1\,-\,\text{u})^{\theta}+(1-v)^{\theta}\,\,-\,(1-u)^{\theta}(1\, -\,\text{v})^{\theta}\right]^{\frac{1}{\theta}}\) & \(\theta\,\in\,[1,\,\infty)\) \\ [PERSON]- & \(\text{u}+\text{v}\,-\,1+(1\,-\,\text{u})(1-\,\text{v})\text{exp}[-\theta\ln(1 \,-\,\text{u})\ln(1-\,\text{v})]\) & \(\theta\,\in\,[0,\,1]\) \\ [PERSON] & \(\frac{-1}{\Phi}\log[1+\frac{\text{exp}\left(-\theta u\right)-1\text{exp}\left( -\theta v\right)-1}{\text{exp}\left(-\theta\right)-1}]\) & \(\theta\,\in\,(0,\,\infty)\) \\ [PERSON] & \(\text{uv. exp}\,\,\left\{-\left[\left(-\ln(u)\right)^{-\theta}+\left(-\ln(v) \right)^{-\theta}\right]^{\frac{1}{\theta}}\right\}\) & \(\theta\,\in\,(0,\,\infty)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical description of copulas [25]. Figure 1: Soft failure and hard failure stages in a run-to-failure test [9]. There is no accurate and unanimous definition for the physical point of failure for a machine. Therefore, the recommendations of experts in the relevant field can be helpful in considering a limit for the high-risk region and decreasing failure possibility. In this paper, it is proposed that the high fluctuation occurrence in the vibration trend can be used as the FT in order to prevent machine operation in the high-risk zone. High fluctuation is assumed here to be an amount of 2 dB or \(\pm\)25% variation of the vibration feature.