# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 2 Common Frequency Domain Methods
Since the number of mean positive crossing and the number of peak occurrence in unit time of the narrow-band random process are equal, the fatigue damage formula of the structure in unit time in this case is transformed into:  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]  
In Equation (8), f(S) is the N value in the S-N curve. The Rayleigh distribution has higher applicability to the narrow-band random process, but it has poor applicability to the wide-band random process, because the probability of occurrence of stress cycles in the low-stress region in the wide-band random process will be higher than that of the narrow-band process. At this time, if the narrow-band method is used to estimate the fatigue damage and fatigue life, it is likely to overestimate and obtain a very conservative life value, which will have a greater impact on engineering applications. Wirsching-Light method (WL). In the Wirsching-Light method, the fatigue damage per unit time of the structure is shown in Equation (9):  
\[D^{WL}=\rho_{WL}D^{NB} \tag{9}\]  
where \(\rho_{\text{WL}}\) is related to the spectral width parameter \(\varepsilon\) and the slope \(\alpha\) of the S-N curve, the specific expression is shown in Equation (10):  
\[\rho_{WL}=A(\alpha)+[1-B(\alpha)](1-\varepsilon)^{B(\alpha)} \tag{10}\]  
The parameters \(A\) and \(B\) are both related to the slope value \(\alpha\). The specific formula is:  
\[A(\alpha)=0.926-0.033\alpha\]  
\[B(\alpha)=1.587\alpha-2.323\]  
3. \(\alpha\)0.75 method (AL). In the \(\alpha_{0.75}\) method, the structural fatigue damage per unit time can be calculated is shown in Equation (11):  
\[D^{AL}=\alpha_{0.75}^{2}D^{NB} \tag{11}\]  
4. [PERSON] and [PERSON] method (OC). [PERSON] and [PERSON] [32] studied the Rayleigh distribution under different spectral width parameters and proposed the following new correction parameter, as shown in Equations (12) and (13):  
\[D^{OC}=\zeta_{k^{\prime}}D^{NB} \tag{12}\]  
\[\zeta_{k^{\prime}}=\frac{1}{\alpha_{2}}\Bigg{(}\sqrt{\frac{m_{2}m_{k^{\prime} }}{m_{0}m_{k^{\prime}+2}}}\Bigg{)}^{k} \tag{13}\]  
where k' = 2.0/k. The Wirsching-Light method, \(\alpha_{0.75}\) method, and Ortiz and Chen method all proposed to modify the Bendat narrow-band fatigue life estimation with a broadband coefficient, so as to obtain the fatigue life of wideband stochastic process. However, these methods ignore the probability distribution information of stress amplitude and only calculates the fatigue life, which are only applicable to some specific cases. 5. Tovo-Benasciutti method (TB). Through extensive research, [PERSON] [28] found that the rain damage of the Gaussian process is always bounded, D\({}_{\text{RFC}}\) is between the upper and lower limits of Rayleigh fatigue damage D\({}_{\text{NB}}\) and the damage D\({}_{\text{RC}}\) of the variable range mean counting method, as shown in Equation (14):  
\[D_{RC}\leq D_{RFC}\leq D_{NB} \tag{14}\]  
Therefore, the rain damage value can be calculated through a certain linear combination, and the weight coefficient b is selected to obtain the structural damage, as shown in Equations (15) and (16):  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]0  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]1Therefore:  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]2  
In Equation (17), for the value of parameter b, there are currently two forms are shown in Equations (18) and (19):  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]3  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]4  
According to the different values of parameter b, the TB method is divided into TB1 and TB2 methods. In the wideband random process, the narrow-band fractional step method and its improvement method can get conservative results, while the T-B method can get more accurate results. 6. Dirlik method (DI).