# Research on the Mechanism and Dynamic Characteristics of Intermitt Failure of Electrical Connectors
## 2 Analysis of the Electrical Connector Intermittent Failure Mechanism
### Analysis of the Mechanism of Electrical Connector Intermittent Failure Induced by Vibration  
In practice, the vibration is one of the primary factors that could cause the relative movement of the contact pairs. To investigate the electrical connector intermittent failure induced by vibration, the dynamic model of electrical connector under vibration is constructed, as shown in Figure 5.  
According to Figure 5, the differential equation of the dynamic model is obtained as follows:  
\[m_{3}\ddot{z_{c}}+2c_{4}\dot{z_{c}}+2k_{4}z_{c}=F(t)-(m_{3}-m_{4})\ddot{z_{4}}-m _{1}\ddot{z_{1}}-m_{2}\ddot{z_{2}}-2k_{2}(z_{2}-z_{4}) \tag{5}\]  
The expression of \(z_{c}\) is a sine or cosine wave when the vibration equation \(F(t)\) follows a sine or cosine wave. The amplitude of the relative displacement \(z_{c}\) is maximized if the electrical connector is forced to vibrate at the resonance frequency. Let \(z_{c}(t)=Z_{c}\mathrm{sin}2\pi\mathrm{i}\theta\), where \(Z_{c}\) denotes the amplitude of \(z_{c}\).  
By analyzing the mechanism of intermittent failures induced by vibration is shown in Figure 6, and the contact resistance curve of a contact spot under sinusoidal vibration can be calculated.  
In Figure 6, the shaded region of the larger circle refers to the plowing region, denoted by 1. The outer region refers to the oxidized region, denoted by 2. The smaller circle represents the \"a-spot \". The conductivity of the contact spot depends on the distribution of oxidized film on the rigid plate and the relative position of the asperity and the rigid plate.  
Assuming that the displacement direction of the asperity remains consistent with the arrowhead, the curve of the contact resistance \(R_{c}\) could be plotted in Figure 6. Initially, the asperity comes into contact with the rigid plate in region 2, forming a \"metal-oxide\" contact spot that is nonconductive. As the asperity slides to region 1, the \"metal-oxide\" contact spot turns to an \"entire-metallic\" contact, which is conductive. If the asperity continues to reciprocate with the rigid plate, then the contact resistance value fluctuates. Once the level of vibration stress is high enough, intermittent failure may occur, and, the higher the vibration stress is, the higher the probability is for electrical connector intermittent failure.  
Figure 5: Dynamic model of an electrical connector. (**a**) Physical model, (**b**) Simplified model.  
The dynamic behavior of electrical connector intermittent failure in a reciprocating cycle is analyzed as follow. Assuming that the asperity is completely within region 1 at time 0 and, considering the randomness of direction of plowing, the relative position between asperity and rigid plate is random. Let the distance from the center of the asperity to the left border of region 1 be \(n\) and the distance to the right border of region 1 be \(m,n>m\), as shown in Figure 7.  
According to the different amplitudes of reciprocating motion \(Z_{\rm c}\), a simple discussion of the dynamic behaviors of intermittent failure during a reciprocating cycle is given below.  
Case 1: When \(Z_{\rm c}<m\), the asperity is always within region 1 during a reciprocating cycle, and the contact resistance is stabilized at a lower value.  
Case 2: When \(m<Z_{\rm c}<n\), the asperity slides to region 2 from the right boundary of the region 1; the intermittent failure could be observed only once.  
Case 3: When \(n<Z_{\rm c}\), the asperity slides to region 2 from both the left and right boundaries of region 1; the intermittent failure could be observed twice.  
Figure 6: Schematic diagram of intermittent failure induced by vibration.  
Figure 7: The relative position between an asperity and the rigid plate.  
Figure 8 shows that the duration and numbers of intermittent failure shows different behaviors under different conditions. To investigate the evolution of intermittent failure dynamic behaviors, the dynamic model of intermittent failure should be studied.