## 3 Fatigue Life Calculation in Frequency Domain  
### Fatigue Life Calculation  
The random vibration stress studied in this paper is a stationary non-Gaussian random process, represented by \(X(t)\). In the frequency domain, the PSD of the random stress \(X(t)\) is denoted as \(G_{XX}\left(f\right)\), which is a one-sided spectrum. The \(i\)-th order spectral moment \(m_{i}\) of \(G_{XX}\left(f\right)\) is defined as  
\[m_{i}=\int_{0}^{+\infty}f^{i}G_{XX}(f)\text{d}f \tag{8}\]  
where the unit of frequency \(f\) is Hz. The variance in the stochastic process \(X(t)\) can be represented as \(\sigma_{X}^{2}=m_{0}\). The spectral width parameter is denoted as  
\[\alpha_{i}=\frac{m_{i}}{\sqrt{m_{0}m_{2i}}} \tag{9}\]  
The peak frequency \(v_{p}\) is defined as  
\[v_{p}=\sqrt{\frac{m_{4}}{m_{2}}} \tag{10}\]  
The \(S\)-\(N\) curve of a material is  
\[N\cdot S^{k}=Cwhere \(k\) and \(C\) are material parameters, \(S\) is the stress amplitude, and \(N\) is the corresponding number of cycles to failure or life. According to the Miner cumulative damage rule, fatigue damage can be defined as  
\[D=\sum_{i=1}^{q}\frac{n_{i}}{N(S_{i})} \tag{12}\]  
where \(n_{i}\) is the number of rainflow cycles in the \(i\)-th block of stress amplitude \(S_{i}\), \(q\) is the total number of stress blocks, and \(N(S_{i})\) is the life corresponding to stress amplitude \(S_{i}\), which can be calculated from the \(S\)-\(N\) curve in Equation (11). Assuming that the total number of peak occurrences in the stochastic process \(X(t)\) is equal to the total number of cycles \(\sum_{i=1}^{q}n_{i}\), then the probability of stress amplitude \(S=S_{i}\) is given by  
\[f_{i}=\frac{n_{i}}{\sum_{i=1}^{q}n_{i}} \tag{13}\]  
Therefore, the PDF of the stress amplitude \(S\) is \(f_{i}\). Then, the fatigue damage in Equation (12) can be written as  
\[D=\sum_{i=1}^{q}\frac{n_{i}}{N(S_{i})}=\sum_{i=1}^{q}\frac{f_{i}\sum_{i=1}^{q} n_{i}}{N(S_{i})} \tag{14}\]  
Combining the expression of the \(S\)-\(N\) curve in Equation (11), the fatigue damage can be written as  
\[D=\sum_{i=1}^{q}\frac{f_{i}\sum_{i=1}^{q}n_{i}}{C/S^{k}}=C^{-1}\sum_{i=1}^{q} n_{i}\sum_{i=1}^{q}f_{i}S^{k} \tag{15}\]  
The number of peak \(\sum_{i=1}^{q}n_{i}\) occurrences in the time unit corresponds to the peak frequency \(v_{p}\). If the PDF of the stress amplitude \(S\) is represented with a continuous function \(p(S)\), then the fatigue damage in the time unit time can be written as  
\[D_{\rm g}=v_{p}C^{-1}\int_{0}^{+\infty}S^{k}p(S){\rm d}S \tag{16}\]  
where the function \(p(S)\) represents the PDF of stress amplitude in the rainflow cycles. Determining \(p(S)\) is crucial for frequency domain random fatigue analysis, because Equation (15) is applicable to both Gaussian and non-Gaussian situations. But so far, there is no analytical expression for \(p(S)\) under non-Gaussian conditions. So, this article is dedicated to studying the PDF of stationary non-Gaussian stress amplitudes.  
For the computation of non-Gaussian fatigue using the correction method, the damage \(D_{\rm g}\) caused by Gaussian fatigue under the same PSD can be multiplied by a correction factor \(\lambda_{\rm ng}\) to obtain damage caused by non-Gaussian fatigue, which is represented as  
\[D_{\rm ng}=\lambda_{\rm ng}D_{\rm g} \tag{17}\]  
The non-Gaussian correction factor [12] is defined as  
\[\lambda_{\rm ng}=e^{\frac{n^{3/2}}{n}(\frac{(S_{\rm g}-3)}{5}-\frac{S^{2}}{4})} \tag{18}\]  
where \(K_{u}\) and \(S_{k}\), respectively, represent the kurtosis and skewness of the signal, while \(m=-1/k\) denotes the slope of the \(S\)-\(N\) curve as described in Equation (11)