# Random Vibration Fatigue Analysis Using a Nonlinear Cumulative Damage Model
## 3 Modified Nonlinear Fatigue Damage Accumulation Model Considering the Vibration Load Effect
### Calculating the Bending Stress  
When a PSD vibration loading is applied to a component or product to cause the same damage as its dynamic environment will cause, a base input acceleration is applied. Consequently, a different response acceleration occurs, as represented in Figure 2[29; 30].  
The reason for the difference between the input and response acceleration is due to the material's natural frequency and the effect of its own mass when it is exposed to the stress of vibration (PSD). Based on the generated displacement, it is measured as  
\[Q=\frac{D_{1}}{D_{2}} \tag{15}\]  
Figure 1: Vectorial representation of the movement.  
where \(Q\) is an amplification factor, \(D_{1}\) is the base (input) displacement and \(D_{2}\) is the response displacement. Thus, the acceleration response is determined as  
\[A_{res}=\frac{2\pi^{2}F^{2}D_{2}}{G} \tag{16}\]  
where \(F\) is the frequency applied by the PSD and \(G\) is the gravity constant (\(9.81\frac{m}{g^{2}}\)). Now, based on the generated moment (M), we determine the dynamic factor that includes the effect stress caused by the vibration PSD [29]. The reaction moment for bending stress, is given by,  
\[M=F\ L \tag{17}\]  
where \(F\) corresponds to the effect of the mass of the component multiplied by the acceleration,  
\[F=m\ A \tag{18}\]  
Therefore, by considering the concept of effective mass (\(m_{e}\)), which represents the mass of an object that accelerates (vibrates) when an external force is exerted on it [29], its equation for the analysis is given by,  
\[m_{e}\approx 0.225\rho L+m \tag{19}\]  
where \(\rho\) is the density of the component's material, \(L\) is the length and m is the mass applied to the component. Thus, the bending moment \(M_{f}\) is given by,  
\[M_{f}=m_{e}\ A\ L \tag{20}\]  
Consequently, the generated bending stress \(\sigma_{f}\) is determined as  
\[\sigma_{f}=\frac{K\ M_{f}\ C}{I} \tag{21}\]  
Therefore, because Equation (21) includes Equation (20), then the scale factor of the stress induced by the stress generated by the vibration movement, here called dynamic load factor [2], is given as  
\[\sigma_{dynamic}\ =\ \left(\frac{Km_{e}\hat{L}C}{I}\right)A \tag{22}\]  
where \(K\) is the stress concentration factor in the component, \(C\) is the distance to the neutral axis, \(\hat{L}\) is the distance from the fixed point of the component to the point of application of the mass, \(A\) is the constant of gravity and \(I\) is the moment of inertia given by,  
\[I=\frac{1}{12}wt^{3} \tag{23}\]  
where \(w\) is the width of the component and \(t\) is the thickness of the material.  
Once, from Equation (22), the dynamic factor \(\sigma_{dynamic}\) is determined, and the response acceleration \(A_{res}\) is determined from Equation (16), the vibration bending stress \(\sigma_{vb}\) response to the vibration profile (base input acceleration) is determined as  
\[\sigma_{vb}=\sigma_{dynamic}*A_{res} \tag{24}\]  
Thus, finally, Equation (24) represents the bending stress that must be used in the vibration analysis to determine the expected useful life of the analyzed product that is being exposed to the environmental vibration.  
Now, it is necessary to determine the cycles of vibration load applied \(n_{i}\) at the given bending stress level \(\sigma_{vb}\).