## 1 Introduction
The PM used in repairable systems is usually called the proportional intensity model (PIM). For example, [PERSON] et al. [26] combined the advantages of the kernel method and PIM, developing a kernelized PIM (KPIM). [PERSON] et al. [27] extended the PIM into a two-effect model which considers operation and maintenance. Both of these two kinds of models are widely used in reliability studies that consider the impact of WCs. However, in electromechanical systems, WCs are often dynamic in actual operations. Since the AM directly reflects the WCs' effects over time, reliability under dynamic WCs can be handled by utilizing the equivalent time method [28], which converts the operation time under other WCs to standard WCs. Based on this idea, [PERSON] et al. [29] obtained the failure rate of production equipment under dynamic WCs. On this basis, we discuss the FI of repairable systems under dynamic WCs. Although the linear assumption of the AM provides an effective channel to obtain reliability under dynamic WCs, the class of semiparametric models will introduce extra parameters which lead to increasing computational difficulties when performing parameter estimation by using the AM [30]. By comparison, the PM is a non-parametric model, which can ignore the form of the baseline function when estimating parameters. This reduces the computational difficulty caused by the introduction of extra parameters. However, the PM cannot accurately describe the reliability under dynamic WCs due to the lack of past effects [28]. It should be noted that the FI of repairable electromechanical systems can be described by the Weibull process. The literature [31] reports that, when the baseline of the PM takes the form of the Weibull or the exponential function, the AM and PM can be mathematically converted into each other. Through this transformation, the lack of attention given to the past cumulative effects in PIM can be addressed. This paper first explains the failure probability from the perspective of damage, which constructs the relationships over time, failure probability, and the effects of WCs by converting the probability function over time to cumulative damage. The original probability function is also regarded as a composite function related to time. Then, the correlations among FIs in different WCs are described by using the equal damage model, which is the equivalent time model interpreted from the perspective of damage. Subsequently, the PIM is used to obtain the effects of WCs on the FI. The corresponding factor coefficients in the WCs are first calculated without knowing the time distribution. Since the failure process of electromechanical systems can be described by the Weibull process, the linear assumption can be used in the PIM to eliminate the problem of past cumulative effects not being considered. Finally, the FIs in dynamic WCs are determined by the improved PIM. The rest of this paper is organized as follows: Section 2 discusses the relationship between cumulative damage and failure probability. The proposed PIM, which considers the past effects, is given in Section 3. In Section 4, case studies prove the effectiveness of the method proposed in Section 3. Finally, conclusions are given in Section 5.