# Vibration Fatigue Life Reliability Cable Trough Assessment by Using Weibull Distribution
## 2 General Random Vibration Background  
The principal objective of the reliability testing of a mechanical component or system is that it will meet its field operation conditions [11], and laboratory testing has an important role to evaluate product reliability by performing accelerated life testing. In particular, a mechanical fatigue stress test is performed based on the Basquin equation given as [11].  
\[N_{fi}=A*\sigma_{ai}^{-m} \tag{1}\]  
where \(N_{fi}\) is the total number of cycles that the component can sustain at a given stress amplitude level \(\sigma_{ai}\) and the constant material parameters \(A\) and \(m\) represent the intercept and the slope of the S-N curve, respectively. Consequently, the mechanical fatigue life estimation can be performed principally by two main groups, crack nucleation and stress/strain material element [12]. In this work, stress analysis is selected. The induced stress is generated by random vibrations. Whereas the behavior of a vibrating system, as is the structural component subjected to an oscillatory excitation force, is led by Equation (2), which represents the equation of motion, which relates the internal forces of the system with the external forces of excitement [13].  
\[m\ddot{x}+c\ddot{x}+kx=F_{0} \tag{2}\]  
In Equation (2), \(m\), \(c\), and \(k\) are the mass, damping, and stiffness of the system, that represent its inertial, dissipative, and elastic properties, respectively. \(F_{0}\) is the excitation force applied to the mass. Thereby in mechanical vibrations exist two main types of vibration testing, the deterministic/swept sine and the random one. However, random vibration testing represents an application most realistic, due to the variance of the level of frequencies and amplitude ranges [14]. In random vibration, the fatigue life analysis is according to the time series approach, and frequency PSD load inputs [15]. Thus, in the estimation of fatigue life for a random vibration environment, an analysis of the amplitude distribution and frequency spectrum is usually required [16]. In addition to that, the mechanical component that will be exposed due to its function to random vibration loads must have a defined fatigue or reliability requirement. Thus, the design team establishes a reliability objective based on the fatigue life. Since random vibration is a random function, there is a certain probability that the movement value will be within a certain range of values, which is why it is described in statistical terms. In fatigue life models, the factors involved are load sequence, type of load, overloads, plasticization, and type of material, among others. Additionally, according to [9], for its analysis, we require a probabilistic concept or a physical quantity that is related to the probability of occurrence. Therefore, the measurement of stress, provoked by random vibration lets us determine the probabilities of failure. Generally, the Weibull distribution has been used to perform the probabilistic analysis [17]. Next, the proposed random vibration stress method and its steps are presented.