# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 2 Common Frequency Domain Methods  
In frequency-domain method, autocorrelation function is generally used to describe the time-varying characteristic of the stationary random process. The power spectral density describes the stationary process in the frequency domain and can express the energy distribution of the random process [24]. Theoretically, the autocorrelation function and power spectral density conform to the Wiener-Sinchin theorem, as shown in Equation (1):  
\[\begin{array}{c}S_{x}(\omega)=\int_{-\infty}^{\infty}R_{x}(\tau)e^{-j\omega \tau}d\tau\\ R_{x}(\tau)=\frac{1}{2\pi}\int_  
by combining with the fatigue damage theory. The random process can be divided into narrowband approximation and wideband correction according to the bandwidth characteristics. Since the amplitude distribution method is simpler, intuitive, and convenient for application, it has been widely used in the engineering field. The frequency domain methods adopted in this paper are all based on the amplitude distribution method.  
For the stationary Gaussian random process, there are many methods to estimate the damage of the structure under random load. For narrow-band stationary Gaussian stochastic processes, [PERSON] proposed to use Rayleigh distribution to describe the probability density distribution of rain flow amplitude, and it has been confirmed [25; 26]. For the broadband stable Gaussian random process, due to the large difference in the distribution of rain flow amplitude, there is no unified fatigue life prediction model for academic analysis, but many scholars have proposed many different frequency domain empirical formulas. In 1980, after studying the power spectral density of different shapes, [PERSON] [2] proposed a correction factor to correct the fatigue damage calculated by the narrow-band method. [PERSON] and [PERSON] [27] proposed a new spectral width parameter \(\alpha_{0.75}\) through a large number of numerical simulations, and used the correction factor to improve the narrow-band method, so this method is also called the \(\alpha_{0.75}\) method. The [PERSON] [28] method divides the broadband stochastic process into two parts, high frequency and low frequency, and treats them as narrow-band stochastic processes. Then the rain-fatigue damage of the original stochastic process is equivalent to these two narrow-band spectral fatigue damage superposition. [PERSON] [29; 30], [PERSON] and [PERSON] [31], etc., proposed a different stress amplitude probability density function model by combining known probability density distribution functions. [PERSON] and [PERSON] [32] proposed new correction parameters by studying the Rayleigh distribution under different spectral widths. [PERSON] and [PERSON] [33] obtained a single-moment method suitable for estimating the random fatigue life of the bimodal spectrum through a large number of numerical simulations and rain flow amplitude analysis. The Nakagami method [34] believes that the Nakagami distribution model can not only accurately simulate the shape of the rain flow amplitude probability density distribution model, but also can make a good approximation of its tail compared to other distribution models.  
1. Narrowband method (NB).  
For the narrow-band random process, [PERSON] [25] proposed to describe the probability density distribution of the rain-flow amplitude with the Rayleigh distribution. He assumed that each cycle of the rain-flow amplitude of the narrow-band random process was completely symmetric, that is, the equivalent peak-valley cycle. The specific functional expression is shown in Equation (7):  
\[p(S)=\frac{S}{\sigma^{2}}\exp\left(-\frac{S^{2}}{2\sigma^{2}}\right) \tag{7}\]  
where \(\sigma^{2}\) is the mean stress square value of the random process, and S is the stress amplitude.