# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 4 Comparative Analysis
### Random Fatigue Life Estimation  
For the three materials listed in Table 1 and the 44 simulated spectrum curves above, the 10 frequency-domain methods and rain flow counting methods are used to calculate different random fatigue life estimation results \(T_{\text{XX}}\). The life cycle \(T_{\text{RF}}\) obtained by rain-flow counting method is used as the exact solution, in order to observe the error of each frequency domain method and the exact solution more intuitively, the relative error value RE is taken for analysis. The specific calculation formula is shown in Equation (29):  
\[RE=\frac{T_{RF}-T_{XX}}{T_{RF}} \tag{29}\]  
Performing statistical analysis on the data in the above table and calculating according to Equation (27) obtains the relative error value when calculating random fatigue life by different frequency-domain methods shown in Figures 6-8. Figure 6: Comparison of relative errors in estimating fatigue life by frequency-domain method (steel): (**a**) Single peak spectrum; (**b**) bimodal spectrum; (**c**) multi-peak spectrum; (**d**) band-limited white noise spectrum. Figure 7: Cont. Analyzing Figures 6-8, we can find that each frequency-domain method has different applicability to different spectrum width parameters and different materials, but it has similar applicability to different spectrum types. Therefore, it can be considered that the accuracy of these 10 frequency-domain methods for calculating random fatigue life is only related to the spectral width parameters and materials, and not to the spectral type. The influence of the material on it is mainly reflected by the  
Figure 8: Comparison of relative errors in estimating fatigue life by frequency-domain method (spring steel): (**a**) Single peak spectrum; (**b**) bimodal spectrum; (**c**) multi-peak spectrum; (**d**) band-limited white noise spectrum. Figure 7: Comparison of relative errors in estimating fatigue life by frequency-domain method (aluminum alloy): (**a**) Single peak spectrum; (**b**) bimodal spectrum; (**c**) multi-peak spectrum; (**d**) band-limited white noise spectrum. influence of the slope \(k\) of the S-N curve of the material. When the material is fixed, that is, the \(m\) value is fixed, the relative error value of each frequency domain method changes regularly with the change of the spectral width parameter. In addition to the NB method that has been proven to calculate the narrow-band random process, the remaining nine frequency-domain methods are commonly used for the broadband stationary Gaussian random process, so they all show the rule that the errors keep decreasing with the increase of the spectral width parameter, even lower than zero. When the spectrum width parameter is fixed, the applicabilities of each frequency domain method to different slope \(k\) values are also different. In general, the increase of \(k\) value will cause the calculation error of various frequency-domain methods to increase accordingly, which is very unfavorable for the random fatigue life analysis of materials with a large slope. The specific analysis of the change law of each frequency domain method separately can get the applications of these 10 commonly used frequency domain methods:  
1. NB method: For the narrow-band stationary Gaussian random process, the distribution of the rain flow amplitude can generally be approximated by the Rayleigh distribution, which has been theoretically confirmed. Through analysis, the NB method is only suitable for narrow-band random processes, but it has poor applicability to the material slope \(k\) value. 2. WL method: It is a modification of the NB method. When the spectrum width parameter is small, the accuracy of the calculation result is higher than other frequency-domain methods, and the adaptability to the slope \(k\) value of the S-N curve is better; when the parameter is large, its accuracy is smaller than those of the frequency domain method applicable to the broadband process. The specific \(\varepsilon\) application range is roughly 0.1-0.4. 3.