# Random Vibration Fatigue Analysis Using a Nonlinear Cumulative Damage Model
###### Abstract  
The paper's content allowed us to determine the fatigue life of a component that is being subjected to a random vibration environment. Its estimation is performed in the frequency domain with loading frequencies being closer to the system's natural frequency. From loads' amplitude and their interaction effect, we derive a nonlinear damage model to cumulate the generated fatigue damage. The exponent value of 0.4 from the Manson-Halford curve damage model was replaced by a vibration bending stress relation that considers the effect and interaction of loads. The analysis is performed from a progressive accelerated vibration spectrum to predict the fatigue life estimation. From this accelerated scenario, the accelerated coefficients and cumulated damage are both determined. The proposed nonlinear model is based on the following facts: (1) vibration and bending stress \(\sigma_{\mathrm{pb}}\) values are obtained from the response acceleration of power spectral density (PSD) applied and (2) the model can be applied to any mechanical component analysis where the corresponding acceleration responses \(A_{res}\) and the dynamic load factor \(\sigma_{dynamic}\) values are known. The steps to determine the expected fatigue damage accumulation \(D\) by using the curve damage are given. fatigue damage; random vibration; resonant frequency; acceleration response; non-linear accumulative model +
Footnote †: journal: Applied Sciences
also the deterministic conditions of the environmental vibration. Additionally, there is the [PERSON]'s method [6], which is based on empirical closed form expression of the probability density function (PDF) of the rainflow amplitude, that, in the analysis, employs the stress amplitudes and coefficients functions of the power spectral density PSD of the stress. Various other frequency fatigue damage models have been developed and reviewed [7], however, at the phase of determining the damage accumulation, most of them use the linear damage rule. The linear damage accumulation rule, also known as Miner's rule, is widely used in fatigue life prediction analysis [8]. It is given by  
\[D=\sum\frac{n_{i}}{N_{1}}=1 \tag{1}\]  
where \(D\) is the fatigue damage, \(n_{i}\) represents the number of the cycles of applied load at a given stress level \(\sigma_{i}\) and \(N_{i}\) are the number of cycles to failure at \(\sigma_{i}\). However, this model does not consider changes in the effect of stress and does not consider the load sequence of application of stress in the component [9]. In practice, the fatigue damage induced in the mechanical components can be analyzed based on stress, strain and crack growth rate [10]. Moreover, according to [11; 12], the effect of the stress is an important parameter that address the fatigue lifetime of the component. In this research, the analysis is performed based on the stress approach by using the random vibration PSD function of the stress response. Because the vibration stress is generated by the variable amplitudes of the cyclical loads, then it is not recommended to estimate the vibration fatigue lifespan by a linear analysis that is independent on the generated response stress level and without considering their interactions [13]. Consequently, since fatigue damage is a complex process that involves many factors [14], then a nonlinear damage analysis is recommendable [15; 16]. For these reasons, in this paper, we present a modification of the damage curve model [17] that includes in its assessment of the damage accumulation the interaction effect load and the sequence load of the random vibration applied. Since the inputs for this nonlinear proposed model are the response acceleration values that provide the bending stresses by using a dynamic factor and by replacing the constant exponent value of 0.4 on the damage curve with a response stress relation, the model's efficiency is that it can be applied to any mechanical element to estimate and perform an analysis of the fatigue damage accumulation induced by a PSD of random vibration.