# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 1 Introduction  
Mechanical components and structures subjected to vibration are affected by dynamic loads, which induce fatigue damage due to the cycling loading application [1, 2]. The generated fatigue damage is primarily related to vibration history loading, geometry, and material properties [3]. The generated fatigue damage directly determines the component's reliability, replacement policy, and warranty costs. Thus, the reliability index characteristic has an important role that is determined mathematically and is used to describe the fatigue damage behavior of mechanical components [4]. That description can be performed by using a probabilistic approach [5]. Then, for mechanical components or systems, according to their application and data complexity, an accurate method must be selected to determine an effective level of service life reliability [6]. Now, since data from fatigue damage are affected by significant scatter, and damage is provoked as a response to random forces, gathering fatigue damage data is generally a difficult activity [7]. One of the more commonly applied methods to work with fatigue damage is the Miner's rule [8, 9]. To consider the random nature of the generated damage, here, we use a probabilistic time-dependent approach [7]. In the fatigue damage accumulation models, the principal factors involved are load sequence, type of load, overloads, plasticization, and type of material. Consequently, for the damage accumulation analysis, we require a probabilistic concept, or a physical quantity related to the probability of occurrence [10]. Thus, the measurement of damage helps us to calculate the probabilities of failure. Since the accumulation of random vibration fatigue damage entails increasing deterioration, an increasing hazard function is required, and the mostrecommendable cumulative distribution function (cdf) used to estimate the fatigue damage is the Weibull cdf [11; 12; 13]. In a vibration profile, each row has its own stationary frequency and amplitude which allow for the performance of a vibration analysis with a statistical approach. Since it is possible to estimate the stress from the acceleration responses, here, the Weibull distribution is used. Thus, based on the increasing behavior of the cumulated damage, in this paper, the increasing random damage behavior is used in the Weibull cdf to determine the failure percentile that the observed cumulated damage represents in the used vibration profile. Then, once the damage percentiles are determined based on their corresponding cycles of the S-N curve, the Weibull scale parameter is determined. Similarly, the Weibull shape parameter is determined directly from the cumulated response stress of the used profile (see Equations (9) and (10) and Section 3.2). Thus, in Section 2.3, a probabilistic methodology to characterize the fatigue damage induced by random vibration is developed by using the Weibull distribution, which uses a relation of the scale and shape parameters with the mechanical vibration fatigue damage. The methodology includes probabilistic estimations based on the Weibull scale and shape parameters that are governed by a new way of analyzing the fatigue damage accumulation presented in Section 2.1. Then, the vibration analysis is based on the fatigue damage, which is determined directly from the principal stresses \(\sigma_{1}\) and \(\sigma_{2}\); therefore, the methodology is efficient because it uses the damage as the platform to project and represent its random behavior and consequently the component's fatigue life. With the purpose to assess it, the methodology is applied in Section 3 to a probabilistic failure analysis of a panel support made of cold drawn steel AISI 1025. The mechanical component is submitted to a random vibration loading profile of 10-55 Hertz with an amplitude of 1.5 mm peak to peak during a 2 h period per block. The testing was performed by using an electrodynamic vibration system in which the vibration profile loading was applied 29 times.