# Vibration Fatigue Life Reliability Cable Trough Assessment by Using Weibull Distribution
## 1 Introduction  
In the industry, accelerated vibration tests are used to qualify mechanical products against vibration loads that induce dynamic loads to the product and lead to fatigue damage, specifically due to resonance and natural frequencies [1, 2]. Thus, mechanical elements and systems must be designed and validated to support vibration environments. Mechanical components might fail due to yielding, ultimate limit, buckling, and bending [3], where fatigue is commonly the leading failure mode, and it is mainly generated by random vibration. The nature of random vibrations is complex, and it can be represented with a Fourier analysis, where the random motion is presented as a series of sines and cosines waves that are cycling at their own frequency and amplitude [4]. All the series of frequencies occurring at the same time in the mechanical component induce structural resonances, and it is analyzed by using the power spectral density (PSD) [5], which represents the energy in the time signal at different frequencies and load signal [6]. Since random vibration induces variable amplitude loadings, from which, it is difficult to determine or predict the fatigue life [7], then the measuring unit from the PSD is the root mean square (rms). However, because the PSD represents energy (acceleration), then in the vibration analysis is also required to know the equivalence of the acceleration response commonly in gravities (g), to stress units (Psi). For that purpose, we use the dynamic factor [8]. Fortunately, because the random vibration is a random function of time, then it can be analyzed by using a probability density function. Consequently, a vibration stress model is required to translate the stress and stress ranges to the statistical model derived from the stress approach [9]. Within the literature, there are models to quantify fatigue damage and perform fatigue life prediction, such as simple models, mechanical damage models, statistical models, andmodels that combine mechanical and probabilistic considerations. Here, we employ a methodology to combine random mechanical vibrations and probabilistic considerations by using the Weibull distribution. The efficiency of the Weibull distribution is based on its principle of consistency [10]. Consequently, we are performing the analysis by considering that, when a force is applied to the area of the component, the stress is transmitted to the entire body of the material and the probability of onset of rupture is the same at any point in the body of the material. In the analysis, the shape parameter \(\beta\) is a key value that represents the fatigue induced by the vibration load and, it is determined by the maximal \(\sigma_{1}\) and the minimal \(\sigma_{2}\) principal applied stresses values. Additionally, the scale parameter \(\eta\) is dependent on the \(\beta\) value and on the required product's, reliability \(R(t)\). Additionally, due to random vibration generating variant stress, the steps to determine the reliability \(R(t)\) for variable stress are given in Section 4.4.  
The structure of the article is as follows. Section 2 includes the generalities of the fatigue life and random vibration. In Section 3 the proposed Weibull random vibration method is formulated. In Section 4, the numerical application is given. Finally, in Section 5 the conclusions are stated.