# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 4 Comparative Analysis
### Rainflow Amplitude Distribution  
Since the approximate amplitude probability distribution method can intuitively describe the distribution of rain flow amplitude, it is of great theoretical significance to calculate the random fatigue life by the approximate amplitude probability distribution method. This section first studies the applicability of the above four rain flow amplitude probability density distribution functions to different spectral width parameters. Time domain simulation was performed on 44 sets of simulated spectral curves. The rain flow amplitude histogram and the cumulative probability density curve of rain flow amplitude are obtained by the rain flow counting method. By comparing it with four probability density distribution function curves of rain flow amplitude, we can see its applicability to rain flow amplitude. The probability density curves of rain flow amplitude and cumulative probability density curves of the simulated spectrum are shown in Appendix A. In order to observe the errors of the four frequency-domain methods and the rain flow counting method more intuitively, the root mean square error value is calculated for the cumulative probability density function. The specific calculation is shown in Equation (28):  
\[MSE=\sqrt{\frac{\sum\limits_{n=1}^{n}\left(C_{RF}-C_{XX}\right)^{2}}{n}} \tag{28}\]  
The variation of the root-mean-square error of the four frequency domain methods with the spectral width parameter is shown in Figure 5.  
Figure 5: _Cont._As can be seen from Figure 5 and Appendix A, the four different life prediction models have the same application conditions and changing rules under different spectrum width parameters, regardless of the form of the simulated spectrum. Since the stress RMS values of these 44 simulated spectrum curves are the same, and the variable is only the spectrum width parameter, the fitting conditions of different life prediction models to the rain flow amplitude can be obtained. The specific analysis results are:  
1. Histogram of rain flow amplitude: When the root mean square stress is the same, when the spectrum width parameter gradually increases, the stochastic process gradually transits from narrowband to wideband, the distribution of the rain flow amplitude gradually concentrates in the low amplitude area, and the proportion of the high-stress area gradually decreases. That is, the probability of occurrence of low-stress amplitude in a broadband random process is higher than that in a narrow-band random process. 2. Rayleigh distribution: When the spectral width parameter \(\varepsilon\) is small, that is, when approaching a narrow-band random process, the rain flow amplitude distribution is more in line with the Rayleigh distribution, and the fitting effect gradually becomes worse when \(\varepsilon\) gradually becomes larger. From different cumulative probability density curves, it can be seen that as \(\varepsilon\) becomes larger,  
Figure 5: Comparison of RMS errors in CDF by four frequency-domain methods: (**a**) Single peak spectrum; (**b**) bimodal spectrum; (**c**) multi-peak spectrum; (**d**) band-limited white noise spectrum. the fitting error in the low-stress region becomes larger, and the probability density value obtained by the Rayleigh distribution will gradually become smaller than the probability density value of the rain flow amplitude. This reduced stress range is also getting larger and larger, which causes a large error when fitting the broadband random process by the Rayleigh distribution. Therefore, it will make the life results too conservative, resulting in the waste of materials. 3. Dirlik model: When the spectrum width parameter \(\varepsilon\) is small, the Dirlik distribution is closer to the Rayleigh distribution. When \(\varepsilon\) gradually becomes larger, the Dirlik distribution fits the rain flow amplitude histogram better and better, and the shape is closer. The low-stress region can achieve good coincidence, high accuracy, and strong versatility.