# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 2 Common Frequency Domain Methods
The Dirlik method [29; 30] is an empirical closed formula that includes an exponential distribution function and two Rayleigh distribution functions. The two distribution functions are linearly combined to simulate the probability density distribution of rain flow amplitude. The specific calculation formula is shown in Equation (20):  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]5  
where:  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]6  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]7  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]8  
\[D^{NB}=v_{0}\int_{0}^{\infty}\frac{p(S)}{f(S)}\mathrm{d}S \tag{8}\]9  
It is proved by experiments that Dirlik method is very accurate. However, the main problem of this method lies in that it is an empirical formula completely obtained through experimental simulation without any complete theoretical explanation. In addition, the Dirlik method does not take into account the influence of average stress, so this method is difficult to apply to non-Gaussian random processes with non-zero mean value and large error. 7. Zhao-Baker method (ZB). The Zhao-Baker method [31] describes the probability density of the rain flow amplitude through a linear combination of a Rayleigh distribution and a Weibull distribution function. The expression is shown in Equation (21):  
\[D^{WL}=\rho_{WL}D^{NB} \tag{9}\]0  
The weighting coefficient \(\eta\) in Equation (21) is:  
\[D^{WL}=\rho_{WL}D^{NB} \tag{9}\]1The other two parameters \(A^{\prime}\), \(B^{\prime}\)are expressed by the following formula:  
\[D^{WL}=\rho_{WL}D^{NB} \tag{9}\]2  
\[B^{\prime}=\left\{ \begin{array}{ll}1.1&\alpha_{2}<0.9\\ 1.1+9(\alpha_{2}-0.9)&\alpha_{2}\geq 0.9\end{array} \right.\]  
The Zhao-Baker method is especially suitable for some specific power spectra. 8. Single moment method (SM). In the single moment method, the structural fatigue damage per unit time can be calculated is shown in Equations (22) and (23):  
\[D^{SM}=\zeta_{SM}D^{NB} \tag{22}\]  
\[\zeta_{SM}=\frac{\left(m_{\alpha^{\prime}}/m_{0}\right)^{\alpha/2}}{v_{0}} \tag{23}\]  
The SM method improves the Rayleigh approximation to some extent, and is suitable for some wideband random processes. 9. Nakagami method (NK). The Nakagami method [34] believes that the rain flow amplitude probability density distribution follows the Nakagami distribution, the expression is shown in Equation (22):  
\[p(s)=\frac{2v^{\
u}}{\Gamma(\
u)\Omega^{\
u}}s^{2\
u-1}\exp(-\frac{\
u}{ \Omega}s^{2}) \tag{24}\]  
where:  
\[\Omega=P_{1}+P_{2}\sigma+P_{3}\sigma^{2}+P_{4}\sigma^{3}+P_{5}\sigma^{4}+P_{ 6}\sigma^{5}+\frac{P_{7}}{\varepsilon}+\frac{P_{8}}{\varepsilon^{2}}+\frac{P _{9}}{\varepsilon^{3}}+\frac{P_{10}}{\varepsilon^{4}}\]  
\[\
u=Q_{1}+Q_{2}\ln(\sigma)+\frac{Q_{3}}{\varepsilon}+\frac{Q_{4}}{\varepsilon ^{2}}+\frac{Q_{5}}{\varepsilon^{3}}+\frac{Q_{6}}{\varepsilon^{4}}+\frac{Q_{7 }}{\varepsilon^{5}}.\]  
**3. Analysis of the Change of Spectrum Parameter and Its Influencing Factors**  
In order to obtain the applicable range of different spectrum width parameters for the different frequency-domain methods, in addition to the change of the spectrum width parameter, the influence of different spectral types and different material parameters must also be considered. Therefore, through the numerical simulation of different simulation spectra and three different materials, the accuracy and applicability of the above 10 methods (including TB method divided into TB1 and TB2) were compared.