Bivariate degradation modeling and reliability analysis based on a shared frailty factor with truncated normal distribution

For a complex product, a single performance characteristic(PC) often fails to fully reflect its degradation process, making it essential to consider the joint degradation of multiple PCs. In this paper, we propose a bivariate degradation model based on a shared frailty factor with the truncated normal distribution, using Wiener processes to characterize the marginal distributions of the PCs. The assumption of the truncated normal distribution aligns better with the physical background where the degradation rates of PCs are non-negative during actual degradation processes. Furthermore, a method for inferring unknown parameters is developed by employing the expectation maximization algorithm. Under this modeling assumption, it became possible to obtain an analytical expression for the product's lifetime distribution on the basis of the concept of the first hitting time. Therefore, in this paper, we further extend the normal distribution integral lemmas to the case of the truncated normal distribution, and provide analytical expression for the cumulative distribution function of the product lifetime. Finally, the rational effectiveness of the proposed model and methods is validated through a numerical simulation example and a case study on wheel wear.