###### Abstract  
A bi-Gamma distribution model is proposed to determine the probability density function (PDF) of broadband non-Gaussian random stress rainflow ranges during vibration fatigue. A series of stress Power Spectral Densities (PSD) are provided, and the corresponding Gaussian random stress time histories are generated using the inverse Fourier transform and time-domain randomization methods. These Gaussian random stress time histories are then transformed into non-Gaussian random stress time histories. The probability density values of the stress ranges are obtained using the rainflow counting method, and then the bi-Gamma distribution PDF model is fitted to these values to determine the model's parameters. The PSD parameters and the kurtosis, along with their corresponding model parameters, constitute the neural network input-output dataset. The neural network model established after training can directly provide the parameter values of the bi-Gamma model based on the input PSD parameters and kurtosis, thereby obtaining the PDF of the stress rainflow ranges. The predictive capability of the neural network model is verified and the effects of non-Gaussian random stress with different kurtosis on the structural fatigue life are compared for the same stress PSD. And all life predicted results were within the second scatter band. non-Gaussian random stress; random vibration fatigue; life prediction; neural network; probability density function; kurtosis +
Footnote †: journal: Applied  
1
Footnote †: journal: Applied  
1
Footnote 1: institutetext: Applied Mathematics and Statistics, University of Toronto, Toronto, Ontario,  
and non-stationary conditions. Many scholars have begun to explore methods for analyzing and predicting the fatigue life of a non-Gaussian case. [PERSON] et al. conducted fatigue tests on 10 HNAP steel under broadband non-Gaussian uniaxial loading and compared various fatigue prediction models. They found that both the time-domain and frequency-domain strain energy density methods provided the best and most consistent life predictions [10]. [PERSON] et al. proposed a nonlinear transformation method to convert Gaussian processes into non-Gaussian processes while maintaining the signal RMS constant under specified kurtosis conditions [11]. [PERSON] et al. analyzed the non-Gaussian and non-stationary response characteristics of structures and studied the bimodal fatigue issues under broadband excitation [5; 6]. They discovered that, when the magnitude of the load PSD remains unchanged, simply altering the kurtosis of the load does not significantly affect the structure's fatigue life. However, when the load is non-stationary, the response exhibits significant non-Gaussian characteristics, significantly reducing the structure's fatigue life. In other words, stationary non-Gaussian excitation does not cause the structural response to exhibit significant non-Gaussian characteristics, whereas non-stationary, non-Gaussian excitation leads to significant non-Gaussian responses in the structure, resulting in a noticeable decrease in fatigue life. It is generally accepted that time-domain methods can yield accurate results for both Gaussian and non-Gaussian fatigue analyses. However, due to computational convenience, frequency-domain methods are also widely used, especially in the early stages of structural design. [PERSON] et al. hypothesized that non-Gaussian damage can be estimated by multiplying the Gaussian damage under the same stress PSD by a correction factor, which is a function of the kurtosis and skewness of the non-Gaussian stress, as well as the slope of the \(S\)-\(N\) curve. This correction method involves calculating Gaussian fatigue damage using frequency-domain methods, and then introducing a correction factor to obtain non-Gaussian fatigue damage. The correction parameter is related to the mean, variance, kurtosis, and skewness of the non-Gaussian stress, as well as the slope of the \(S\)-\(N\) curve [12].