# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 4 Dynamic Precision Reliability Modeling of Multilink Mechanisms with Clearances  
### The Theory of Stress-Strength Interference  
In the field of mechanical equipment, once key components fail, it is highly likely to cause the entire equipment to malfunction. The reason why parts fail is often because the load they bear during actual operation exceeds the ultimate bearing capacity of the part at the corresponding time. Specifically, during the operation of the component, if its strength value is higher than the stress value it is subjected to, the component can operate normally; on the contrary, when the stress value exceeds the strength value, the part will fail. In addition, due to the randomness of various factors affecting stress and strength, both stress and strength exhibit dispersion characteristics. The relationship between stress and strength can be referred to in Figure 9. From the figure, it can be observed that when the mean value of strength is greater than the mean value of stress, situations where the strength is less than the stress may occur within the \"interference zone\".  
In accordance with the theory of stress-strength interference, reliability is the probability of strength exceeding stress, which is denoted as:  
\[R=P(S-s>0)=P(Z>0) \tag{48}\]  
where \(S\) denotes strength, \(s\) represents stress, and \(Z=S-s\) indicates the safety margin. When \(Z>0\), the mechanical system operates normally. Assuming that both stress and strength are normally distributed, and leveraging property that sum (or difference) of normally distributed variables is also normally distributed, safety margin \(Z\) likewise follows a normal distribution.  
\[f(Z)=\frac{1}{\sqrt{2\pi\sigma_{Z}}}e^{-\frac{1}{2}(\frac{Z-\mu_{Z}}{\sigma_{ Z}})^{2}} \tag{49}\]  
where \(\mu_{S}\) and \(\sigma_{S}\) represent mean and mean square deviation of strength, respectively, \(\mu_{s}\), \(\sigma_{s}\) represent mean and mean square deviation of stress, respectively, and \(\mu_{Z}=\mu_{S}-\mu_{s}\), \(\sigma_{Z}=\sqrt{\sigma_{S}{}^{2}+\sigma_{s}{}^{2}}\).  
The reliability is the probability that the safety factor \(Z>0\), can be represented as:  
\[R=P(Z>0)=\int\limits_{0}^{\infty}\frac{1}{\sqrt{2\pi\sigma_{Z}}}e^{-\frac{1}{ 2}(\frac{Z-\mu_{Z}}{\sigma_{Z}})^{2}}dZ \tag{50}\]  
Transform Equation (50) into the standard normal distribution form  
\[R=\int\limits_{0}^{\infty}f(Z)dZ=\int\limits_{z_{0}}^{\infty}\varphi(z)dz= \varphi(-z_{0}) \tag{51}\]  
where \(\varphi(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^{2}}\), \(z=\frac{Z-\mu_{Z}}{\sigma_{Z}}\), \(z_{0}=-\frac{\mu_{Z}}{\sigma_{Z}}=-\frac{\mu_{S}-\mu_{S}}{\sqrt{\sigma_{S}{} ^{2}+\sigma_{s}{}^{2}}}\).  
Figure 9: Model of stress–strength interference.  
Define the reliability indicator \(\beta_{1}\) as  
\[\beta_{1}=\frac{\mu_{Z}}{\sigma_{Z}}=\frac{\mu_{S}-\mu_{s}}{\sqrt{\sigma_{S}{}^{2 }+\sigma_{s}{}^{2}}} \tag{52}\]  
From Equations (51) and (52), the reliability is calculated as  
\[R=\int\limits_{0}^{\infty}f(Z)dZ=\Phi(\beta_{1}) \tag{53}\]  
The above equation represents the formula for calculating the structural reliability when there are strengths and stresses obeying a normal distribution.