# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
###### Abstract
[PERSON] further studied the wear damage modeling of clearance hinges and the joint failure of motion mechanism in an aerospace vehicle [13]. [PERSON] studied the motion principle of an internal combustion engine, analyzing and predicting the wear process [14]. However, an ASME Fellow, Professor [PERSON], stated that repeated impacts might cause the plastic deformation of metal parts, which is more likely to occur than wear failure [15]. The same is true of metal revolute joints, and the performance of the motion mechanism may be influenced. [PERSON] studied the fracture property of a gear system based on contact and energy release rate. The worn regulation and deformation of the gear were both investigated [16]. Competing risk (failure) models are aimed at analyzing the reliability or failure probability for systems under multi-competing risk modes. The theory of competing risks was developed by demographers and biostatisticians [17]. [PERSON] built a distribution-based reliability model for systems under extreme shocks and natural degradation by the Brown-Proschan model and non-homogeneous Poisson process. Based on this model, the survival probabilities with time can be obtained [18]. [PERSON] established a reliability model for a competing failure system with self-recovery features, based on which a semiconductor laser is studied as an example [19]. The competing risk models are also applied in accelerated tests. [PERSON] proposed a new estimating procedure for competing risk data with a missing cause of failure [20]. [PERSON] developed a testing methodology based on the reliability target allocation for reliability demonstration under competing failure modes at accelerated conditions. Data from previous research were used to illustrate the methodology [21]. For systems under competing risk modes of external shocks (hard failure) and degradation process (soft failure), [PERSON] conducted a series of research [2; 3; 4; 6; 17; 18; 19]. In reference [2], the reliability and maintenance policy for systems were studied by a fundamental competing risk model. The changing degradation rate and shifting failure thresholds were considered in reference [3] and reference [4] respectively, on the basis of the model in reference [2]. Further, in reference [5], the changing degradation rate and shifting failure thresholds were combined in one model. In reference [22; 23] the shocks were separated according to their effects on different components, which led to more definite reliability results. Reference [24] studied maintenance policies for repairable systems under competing failure modes by different shock models. There are still some insufficiencies in previous models for systems under competing risk modes of external random shocks and degradation processes, especially for mechanical systems under wear and tribology. First, external shocks will cause increments in the degradation process, while the increments are regarded as the same distribution as in the previous paper. However, a shock with a larger size may lead to a larger increment, while a shock with a smaller size may lead to a smaller one. It is inappropriate to use the same distribution for all the increments. Secondly, many previous papers only obtained the system reliability, but the _competing failure probability_ is ignored. The competing failure probability means the probability that one risk mode fails while the other risk mode does not fail. The risk mode with a larger competing failure probability is more dangerous than the other risk mode. The value of competing failure probabilities may change even for an identical system reliability value. Moreover, the competing failure probability for soft failure (soft failure occurs when there is no hard failing) should approach 1, which implies the fact that the system will certainly fail due to wastage when the working time is long enough. However, the results calculated according to previous models do not conform to this fact. Thus, the modified probability value is demonstrated.