Contact reliability modeling and assessment of electrical connectors with multi-aperture slotted leaf spring contacts

Abstract

Multi-aperture electrical connectors are widely used in the aerospace system to provide high-reliability and high-quality connections. The existing reliability assessment methods only focus on the single-aperture electrical connectors, and ignored the influence of the aperture difference on the reliability of electrical connectors, which may lead to errors in the evaluation results. In order to assess the contact reliability of multi-aperture electrical connectors at storage environment, this paper build a mathematical relationship between structure parameter of contact pairs and contact force by analyzing the slotted leaf spring jack reed structure of the contact pair and using the cantilever beam model. Based on the growth law of the oxide layer on the surface of the contact pair, the degradation trajectory model of the contact pair performance was set up which incorporating the structure parameters of contact pairs, such as pin diameter, reed length, radius of curvature of the inner and outer surfaces, and the acceleration model was established by combining the Arrhenius equation. Then based on the established degradation trajectory model, the degradation failure distribution model of contact pairs was derived, and the contact reliability evaluation model of multi-aperture electrical connectors was established by using the weakest ring model. The test scheme of constant stress accelerated degradation of electrical connectors was formulated and carried out. Finally, this paper verified the correctness of the model through analyzing the test data, and obtained the reliable life of multi-aperture electrical connectors under the storage environment temperature, which realized the contact reliability assessment of multi-aperture electrical connectors.

Keywords

Multi-aperture electrical connector contact performance degradation reliability modeling accelerated degradation test reliability assessment

Introduction

As the main component of electrical signal transmission in the electronic systems, electrical connector is widely used in aerospace, national defense, military, and other fields. Contacts are the main component of electrical connector to realize electrical signal transmission, and the contact reliability directly determines the life of electrical connector.

At present, the electrical connectors used for different systems can be divided into containing a single aperture and multi-aperture class according to the different sizes of contact pairs, and also can be divided into slotted leaf spring, wire spring hole, and twist pin type according to the different structural forms of contact pairs. Among them, the electrical connectors with multi-aperture slotted leaf spring contact pairs have the advantages of diverse and stable signal transmission, and the application range of this type of electrical connector will be more extensive with the increasing of diversity and complexity of signal transmission requirements.

In China and other countries, previous studies of contact reliability modeling and assessment of electrical connectors were all focused on single-aperture electrical connectors, and ignored the effects of aperture difference of contact pairs on contact reliability of electrical connectors. Chen et al.¹ deduced the life distribution model of aerospace electrical connectors by using the extreme value distribution theory, and obtained the reliability index of Y11X series aerospace electrical connectors under the comprehensive action of ambient temperature and vibration stress through the comprehensive stress accelerated life test. In order to shorten the test time, Chen et al.² also established a statistical model of degradation failure, and proposed the theory and method of accelerated life test of aerospace electrical connectors based on performance degradation by studying the statistical analysis method of degradation data. Pan et al.³ established a contact performance degradation model for electrical connectors based on the winner process under random vibration stress, and established the relationship between contact life and vibration magnitude through carrying out stepped stress accelerated degradation test, which realized the rapid contact reliability assessment of the electrical connector under the vibration environment. Xu et al.⁴ established a contact performance degradation model and reliability assessment model for electrical connectors based on the mechanism of contact performance degradation under fretting wear conditions. Through fretting cycle test, they obtained the variation law of contact reliability with fretting cycle under different fretting amplitude. Ling et al.⁵ established a fretting wear life estimation model based on Archard’s equation and achieved reliability assessment of electrical connectors fretting wear conditions. Zhong et al.⁶ solved the reliability modeling problem of electrical connectors under the temperature-insertion stress condition, and laid a foundation for the further research on the reliability assessment and the accelerated degradation testing plan for electrical connectors in the storage environment with task profiles. Ren et al.⁷ summarized some problems of physical failure model and reliability assessment of circular electrical connectors, and proposed a new method combining accelerated degradation test and physical failure modeling. Wang et al.⁸ proposed a systematic method for the reliability assessment of connector and verified the effectiveness of the method by applying it to leaf spring connectors, and verified the effectiveness of the method by applying it on blade-spring connector. However, a large number of test data showed that the degradation rate of different aperture contact pairs has difference in the same test environment. If the influence of the aperture difference between contacts on the reliability of electrical connectors is ignored, the results of reliability evaluation would produce errors.

The paper takes a type of electrical connector as the research object. On the basis of failure mechanism analysis, a mathematical model between structural parameters of contacts and contact reliability of electrical connectors was established by combining the structural characteristics of contacts. Then through constant stress accelerated degradation test, the contact reliability model was verified, and the reliability assessment of the multi-aperture electrical connector was realized.

Contact failure mechanism of electrical connector

Contact failure mechanism analysis

A type of electrical connector is composed of contact parts, insulation parts, seals, shell, and locking separation mechanism. Among them, the contact parts are composed of three kinds of contact pair with Φ1.1, Φ2.2, and Φ3.5 mm aperture specifications, which adopt slotted leaf spring structure.

In peacetime, the electrical connector will be stored for a long time with missiles and other weapons. In the storage environment, the main failure mode of electrical connectors is contact failure, and the growth of oxide film on the surface of contact parts is the main reason for contact failure.⁹ The basal material of the contact part is copper, and the surface is gold-plated, but the gold-plated layer is very thin because of the economic cost limited. Therefore, the surface of the coating will inevitably have defects such as pores, which will lead to the $O_{2}$ and $H_{2} O$ in the air contacting with the base material through the coating, then the oxidation reaction generates between the substrate material and the coating, thus generating the $C u_{2} O$ film.

4 Cu + O_{2} \to 2 C u_{2} O

Temperature can significantly affect the growth rate of the oxide film, because of the increase of temperature can increase the internal energy of the $C u^{+}$ , which will accelerate the diffusion of $C u^{+}$ to the surface. With the continuous increase of the film thickness, the contact resistance is also increasing, and the contact failure of the electrical connector is caused finally.

Based on the results of contact failure mechanism analysis, the contact reliability model of electrical connectors is established, the roadmap of the modeling is shown in Figure 1.

Figure 1.

The roadmap for modeling contact reliability of the electrical connector.

Influence of aperture structure parameters of contact pair on contact failure mechanism

At present, many scholars have studied the contact force model,^10–14 while these models are mainly based on dynamic contact forces generated by impact and collision of moving objects. The contact pressure between contact pairs belongs to the static contact force. The slotted leaf spring contact pair consists of a pin and a jack. The pin is rigid part of cylindrical and the jack is slotted to form several reeds, then the reed is closed evenly to the center through the shell nosing device, the contact pair provides contact pressure through the closed reed to ensure reliable contact of the contact parts. The diffusion of ions is carried out in most cases by means of the vacancy mechanism with the minimum activation energy required for diffusion. According to the diffusion mechanism of ions, the growth of oxide film is not only related to temperature, but also affected by contact pressure.¹⁵ The contact pressure will affect the energy required to overcome the barrier for ion migration in the oxide film. The higher the contact pressure is, the greater the energy required to overcome the barrier for ion migration will be, thus reducing the ion mobility and ultimately slowing down the growth rate of the film layer. The contact pressure is determined by the structure of the contact parts, and the structure diagram of the contact pair is shown in Figure 2.

Figure 2.

The structure diagram of contact pair: (a) the assembly drawing contact pair and (b) the axial and radial section drawing of contact pair.

where $L$ is the length of reed, $R_{0}$ is the radius of pin, $R_{1}$ is the radius of curvature of the outer surface of the reed, $R_{2}$ is the radius of curvature of the inner surface of the reed, $f$ is the size of shell nosing.

The reed is simplified into a cantilever structure, as shown in Figure 3.

Figure 3.

The cantilever structure of the reed.

The expression of the contact pressure $F$ can be obtained through the mechanical model of the cantilever as follows:

F = \frac{3 E I_{z} δ}{L^{3}}

(1)

where $E$ is the elastic modulus of the base material, $δ$ is the deflection of the reed, $I_{z}$ is the moment of inertia of the transverse section of the reed.

Reed deflection $δ$ can be calculated by dimension chain:

δ = f - R_{2} + R_{0}

(2)

The transverse section diagram of reed is shown in Figure 4.

Figure 4.

The transverse section diagram of reed.

The moment of inertia of the transverse section of the reed $I_{z}$ can be expressed as¹⁶:

I_{z} = \frac{2 α_{3} + \sin 2 α_{3}}{8} (R_{1}^{4} - R_{2}^{4}) - \frac{4 (R_{1}^{3} - R_{2}^{3}) \sin^{2} α_{3}}{9 α_{3} (R_{1}^{2} - R_{2}^{2})}

(3)

where $α_{3} = \frac{π}{N} - \arcsin (\frac{w}{R_{1}})$ , $α_{3}$ is half of the central angle of the outer circle, $N$ is the number of reeds, $w$ is the slotted half width.

As shown in the expression of the contact pressure, when the materials of the contact pair are the same, the size of the contact pressure is mainly determined by the size parameters of the contact pair such as the length of the reed, the amount of the orifice, the radius of curvature of the outer surface, the radius of curvature of the inner surface. Meanwhile, the radius of the pin, and the size of the contact pressure will affect the growth rate of the oxide film. Therefore, the contact pressure between different aperture contact pairs is different, which leads to different failure conditions of different aperture contact pairs.

Degenerate trajectory model for contact pairs

Mathematical model of contact resistance of slotted leaf spring contact pair

At the microscopic level, the real conductive region of the contact pair is composed of raised contact spots on the contact surface. The contact resistance of the contact pair consists of three parts: the bulk resistance of the material, the contraction resistance of the contact spot, and the film resistance of the contact spot. Ignoring the difference in the randomness of the morphological structure of each contact spot,¹⁷ it is considered that the jack reed has the same structure and is completely centering when inserted. The contact resistance of the contact pair with three slotted reeds can be expressed as:

r = r_{v} + \frac{r_{s, i}}{N} + \frac{r_{m, i}}{N}

(4)

where $r_{v}$ is the bulk resistance of the contact pair, $r_{s, i}$ is the contraction resistance of a single reed, $r_{m, i}$ is the film resistance of a single reed, $N$ is the number of reeds.

According to the failure mechanism, the contact failure of electrical connectors in storage environment is caused by the increase of film resistance, while the bulk resistance and contraction resistance are mainly determined by the material, structure, and process parameters. When the contact parts leave the factory, these parameters have been determined and can be regarded as fixed values, denoted as $r_{0}$ , and the contact resistance can be further expressed as:

r = r_{0} + \frac{r_{m, i}}{N}

(5)

Mathematical model of film resistance

Barkan and Tuohy¹⁸ established the calculation formula of film resistance through the study of contact resistance:

r_{m} = \frac{2 σ_{m}}{F / H}

(6)

where $σ_{m}$ is the tunnel resistivity of the film, $H$ is the surface hardness of the coating material.

According to Dietrich and Fisher’s theoretical analysis and experimental derivation of tunneling effect,⁹ the relationship between the tunnel resistivity of the film and the average thickness of the film is as follows:

σ_{m} = k_{1} {\bar{χ}}^{c}

(7)

where $k_{1}$ and $c$ is the constant.

With the continuous increase of the film thickness, the growth rate of film layer will gradually slow down. Therefore, the relationship between film thickness $χ$ and time $t$ can be expressed as¹⁹:

χ = k_{2} t^{1 / 2}

(8)

where $k_{2}$ is the reaction rate constant, which depends only on temperature.

Here, it is considered that the pore shape of the contact surface is approximately circular, and the oxide film layer will become more uniform with time. If the porosity of the contact zone is represented by $ρ_{m}$ and the contact area is represented by $A_{a}$ , the relationship between the average thickness of the oxide film $\bar{χ}$ and time $t$ can be expressed as:

\bar{χ} = \frac{ρ_{m} A_{a} Δ V}{A_{a}} = ρ_{m} π {\bar{a}}^{2} k_{2} t^{1 / 2}

(9)

where $Δ V$ is the oxide volume generated by a single pore, $\bar{a}$ is the average radius of a single pore.

The expression of film resistance can be obtained from equation (1) and equations (6)–(9):

r_{m} = \frac{2 k_{1} H L^{3} {(ρ_{m} π {\bar{a}}^{2} k_{2})}^{c} t^{c / 2}}{3 E I_{z} δ}

(10)

By substituting equation (10) into equation (5), the performance degradation trajectory model of the contact pair can be obtained:

r = r_{0} + \frac{2 k_{1} H L^{3} {(ρ_{m} π {\bar{a}}^{2} k_{2})}^{c} t^{c / 2}}{3 NE I_{z} δ}

(11)

Equation (11) contains various parameters such as the length of the reed $L$ , the number of the reeds $N$ , the deflection of the reed $δ$ , the moment of inertia of the transverse section of the reed $I_{z}$ . For ease of application, set $β = \frac{c}{2}$ and $α = \frac{2 k_{1} H L^{3} {(ρ_{m} π {\bar{a}}^{2} k_{2})}^{c}}{3 NE I_{z} δ}$ , then equation (11) can be expressed as:

r = r_{0} + α t^{β}

(12)

where $r_{0}$ is the initial value of contact resistance, $β$ is constant, $α$ is the degradation rate of contact resistance.

Acceleration model

According to the theory of chemical reactions, the higher the temperature, the faster the reaction. The relationship between reaction rate constant and temperature can be expressed by Arrhenius equation¹⁹:

k_{2} = Λ e^{- Δ E / k (T + 273.15)}

(13)

where $Λ$ is the frequency factor, $Δ E$ is the activation energy, $k$ is the Boltzmann constant, $T$ is the temperature.

By substituting equation (13) into the expression of the degradation rate $α$ , the accelerated degradation equation of the contact resistance can be obtained:

α = \frac{2 k_{1} H L^{3} {(Λ ρ_{m} π {\bar{a}}^{2})}^{c}}{3 NE I_{z} δ} e^{- c Δ E / k (T + 273.15)}

(14)

Taking the logarithm of both sides of equation (14) can be expressed as:

\begin{matrix} \ln α = \ln (\frac{2 k_{1} H L^{3} {(Λ ρ_{m} π {\bar{a}}^{2})}^{c}}{3 NE I_{z} δ} e^{- c Δ E / k (T + 273.15)}) \\ = \ln L^{3} - \ln I_{z} - \ln δ - \ln E + \ln H - \ln N \\ + c \ln (Λ ρ_{m} π {\bar{a}}^{2}) + \ln (2 k_{1} / 3) - c Δ E / k (T + 273.15) \end{matrix}

(15)

It can be seen from equation (15) that $\ln α$ is the synthesis of many random variables. According to the central limit theorem, it can be considered that $\ln α$ obey the normal distribution.⁹ Therefore, the degradation rate $α$ of the contact resistance obey the lognormal distribution, as $α ~ LN (μ_{α}, σ^{2})$ , where $μ_{α}$ is the logarithmic mean of the degenerate distribution, $σ$ is the logarithmic standard deviation of the degenerate distribution. $μ_{α}$ can be expressed as:

\begin{matrix} μ_{α} = E (\ln α) = E (\ln L^{3}) - E (\ln I_{z}) \\ - E (\ln δ) - E (\ln E) + a + bx \end{matrix}

(16)

where $L$ , $I_{z}$ , $δ$ _, and $N$ are the structural size parameters of the contact pair, $a = E (\ln H) - E (\ln E) + cE [\ln (Λ ρ_{m} π {\bar{a}}^{2})] + \ln (2 k_{1} / 3)$ , $b = - c Δ E / 1000 k$ , $x = \frac{1000}{273.15 + T}$ , where $a$ and $b$ are unknown parameters to be estimated.

Equation (16) indicates that the degradation rate $α$ is affected by the size parameters of contact pairs such as the length $L$ , deflection $δ$ , moment of inertia of the transverse section $I_{z}$ , and number $N$ of the reeds, The deflection $δ$ and the moment of inertia of the transverse section $I_{z}$ also include the aperture structure parameters of contact pairs such as pin radius $R_{0}$ , radius of curvature of the outer surface $R_{1}$ , and radius of curvature of the inner surface $R_{2}$ of the reed. Therefore, the degradation rate of contact resistance of contact pairs with different aperture specifications is different, which leads to the difference of contact reliability of contact pairs with different aperture specifications.

Contact reliability model for multi-aperture electrical connectors

Degenerate failure distribution model for contact pairs

In the storage environment, the contact resistance will increase with time. If the contact resistance of a contact pair at time $t$ is expressed as $r (t)$ , the failure threshold is $D$ , then its life can be expressed as:

T_{e} = inf {t : r (t) \geq D; t \geq 0}

(17)

The lifetime of the contact pair can be obtained from equation (12) as:

r (T_{e}) = D \Rightarrow T_{e} = {(\frac{D - r_{0}}{α})}^{1 / β}

(18)

Since the effect of the randomness of $r_{0}$ on the estimated value of reliability can be ignored, so $r_{0}$ is regarded as a constant here.²⁰ Therefore, the degenerate failure distribution of the contact pair can be expressed as:

F_{e} (t) = P {T_{e} \leq t} = P {{[(D - r_{0}) / α]}^{1 / β} \leq t}

(19)

Since the degradation rate $α$ obey the lognormal distribution, the equation (19) can be further transformed as follows:

\begin{matrix} P {α \geq (D - r_{0}) / t^{β}} \\ = 1 - P {\ln α \leq \ln (D - r_{0}) - β \cdot \ln t} \\ = 1 - P {\frac{\ln α - μ_{α}}{σ} \leq \frac{\ln (D - r_{0}) - β \cdot \ln t - μ_{α}}{σ}} \end{matrix}

(20)

Thus, the degradation failure distribution model of the contact pair can be obtained as:

\begin{matrix} F_{e} (t) = 1 - Φ {\frac{\ln (D - r_{0}) - β \cdot \ln t - μ_{α}}{σ}} \\ = Φ {\frac{\ln t - [\ln (D - r_{0}) - μ_{α}] / β}{σ / β}} \end{matrix}

(21)

where $Φ {*}$ is the standard normal distribution function.

Then the reliability function of the contact pair at time $t$ can be expressed as:

\begin{matrix} R_{e} (t) = 1 - F_{e} (t) \\ = 1 - Φ {\frac{\ln t - [\ln (D - r_{0}) - μ_{α}] / β}{σ / β}} \end{matrix}

(22)

Life distribution model of electrical connectors

Assuming that there are $P$ groups of contact pairs with different apertures in the electrical connector, each group has $n_{g} (g = 1, 2 . . . P)$ contact pairs, and the life of the $j$ th contact pair of the $g$ th group is $T_{e, g}^{(j)} (j = 1, 2 . . . n_{g})$ , then the life $T_{E, g}$ of the contact pair of the $g$ th group can be expressed as:

T_{E, g} = \min {T_{e, g}^{(1)}, T_{e, g}^{(2)}, \dots, T_{e, g}^{(j)}, \dots, T_{e, g}^{(n_{g})}}

(23)

Because each contact pair is mutual independence and the $T_{e, g}^{(j)}$ in each subsystem follows the same distribution $F_{e, g} (t)$ , so the life distribution function of the contact pair of the $g$ th group can be expressed as:

\begin{matrix} F_{E, g} (t) = P {T_{E, g} \leq t} \\ = 1 - P {T_{E, g} \geq t} = 1 - [1 - F_{e, g} (t {)]}^{n_{g}} \end{matrix}

(24)

The multi-aperture electrical connector contains several groups of contact pairs with different apertures, and each contact pair group aslo contains multiple contact pairs with the same apertures. The failure of any one contact pair will lead to the failure of the entire electrical connector. Therefore, the life distribution model of electrical connector is the minimum extreme value problem of series system. According to the reliability model theory of series system, the life distribution function of electrical connector can be expressed as:

\begin{matrix} F (t) = 1 - Π_{g = 1}^{P} [1 - F_{E, g} (t)] \\ = 1 - Π_{g = 1}^{P} {[1 - F_{e, g} (t)]}^{n_{g}} \end{matrix}

(25)

Then at time $t$ , the reliability function of the electrical connector can be expressed as:

R (t) = 1 - F (t) = Π_{g = 1}^{P} {[1 - F_{e, g} (t)]}^{n_{g}}

(26)

By substituting equations (16) and (21) into equation (26), the mathematical relationship between the structural parameters of the contact and the contact reliability of the electrical connector can be obtained:

\begin{matrix} R (t) = Π_{g = 1}^{P} {[1 - Φ {\frac{\ln t - [\ln (D - r_{0}) - (E (\ln L^{3}) - E (\ln I_{z}) - E (\ln δ) - E (\ln N) + a + bx)] / β}{σ / β}}]}^{n_{g}} \end{matrix}

(27)

Accelerated test scheme and statistical analysis of test data

Constant stress accelerated test scheme

A type of multi-aperture electrical connector contains contact pairs with three aperture specifications, and the dimensions, numbers, and material parameters of contact parts are shown in Table 1.

Table 1.

The basic parameters of the contact parts.

Diameter (mm)	Materials	Elasticity modulus (GPa)	The number of reeds	The number of contact pairs
Φ1.1	QSn4-3-Y	110	2	206
Φ2.2			4	8
Φ3.5			8	8

According to the criterion that the test stress cannot change the failure mechanism, the three temperature levels of the test are determined as $x_{1} = 120^{°}$ , $x_{2} = 140^{°}$ , and $x_{3} = 158^{°}$ combined with the previous test experience and the temperature range provided by the test equipment. In order to ensure the accuracy of statistical analysis of test data, the sample size at each temperature level should not be less than 5. Considering the cost of the test, the number of samples at temperature levels $x_{1}$ , $x_{2}$ , and $x_{3}$ is 15, 10, and 5, respectively. According to the principle of short interval under high stress and long interval under low stress, the test time interval at each temperature level was determined. The specific test scheme is shown in Table 2.

Table 2.

Accelerated test scheme.

Serial number	Temperature level (°C)	Number of electrical connectors	The truncation time (h)	The test interval (h)
1	120	15	3500	72
2	140	10	1000	24
3	158	5	800	12

In order to determine whether the sample fails during the experiment, according to the product manual of this type of electrical connector, the failure criterion is determined as:

(a) The contact resistance of the contact pair of $Φ 1.1 mm$ is greater than $20 m Ω$ ;

(b) The contact resistance of the contact pair of $Φ 2.2 mm$ is greater than $10 m Ω$ ;

During the test, once any of the above phenomena occur, the test sample is judged to be invalid.

The measurement of low resistance value needs to eliminate the influence of wire resistance, so this paper adopts the Kelvin’s four-wire method to measure the contact resistance, and its testing principle as shown in Figure. 5. In the figure, $r_{1}$ , $r_{2}$ , $r_{3}$ , and $r_{4}$ is the resistance of the wire.

Figure 5.

The test principle of contact resistance.

The test device is shown in Figure 6. The contact pair of the electrical connector leads out two wires at both ends, forming two loops with the voltmeter and the ammeter. In one loop, the DC power supply is connected in series with the ammeter, which providing 5 V voltage and 1 A constant current in the loop, and the other loop is connected to the voltmeter to measure the voltage at both ends of the contact pair. According to the Ohm’s law, the value displayed by the voltmeter is the contact resistance. Each contact resistance measurement is performed three times, and the average value is taken as the final result.

Figure 6.

The test device for contact resistance.

The test results of contact resistance

In the test, the contact resistance of each sample was tested according to the test interval in Table 1. The degradation track of contact resistance of each contact pair group of a sample at 120°C is shown in Figure 7. The number in the legend indicates the number of the contact pair. Among them, there are 206 contact pairs with Φ1.1 mm aperture. In order to show the performance degradation clearly, only eight contact pairs are selected for display.

Figure 7.

Degradation trajectories of contact resistance for each contact pair group of a sample at 120°C: (a) degradation trajectory of Φ1.1 mm contact pairs, (b) degradation trajectory of Φ2.2 mm contact pairs, and (c) degradation trajectory of Φ3.5 mm contact pairs.

The statistical analysis of test data

The contact resistance value of the $j$ th sample of the $g$ th group contact pair in the temperature level $x_{i}$ at $t$ time can be expressed as:

r_{ij, g} (t) = r_{0 ij, g} + α_{ij, g} t^{β_{ij, g}}

(28)

where $r_{0 ij, g}$ is the initial value of contact resistance of the $j$ th sample of the $g$ th group contact pair at the temperature level $x_{i}$ , ${α_{ij}}_{, g}$ is the degradation rate of the $j$ th sample of the $g$ th group contact pair at the temperature level $x_{i}$ , $α_{ij, g} = ξ_{ij, g} \exp {E (\ln L_{g}^{3}) - E (\ln I_{z, g}) - E (\ln δ_{g}) - E (\ln N_{g}) + a_{g} + b_{g} x_{i}}$ , $ξ_{ij, g} ~ LN (0, σ_{g}^{2})$ , where $r_{0 ij, g}$ , $α_{ij, g}$ , $β_{ij, g}$ , $a_{g}$ , $b_{g}$ , and $σ_{g}$ is the unknown parameter to be estimated.

According to equation (29), the estimated values of the degradation model parameters $r_{0 ij, g}$ , $α_{ij, g}$ , and $β_{ij, g}$ of the $j$ th sample of the $g$ th group contact pair at the temperature level $x_{i}$ can be obtained:

SSE (r_{0 ij, g}, α_{ij, g}, β_{ij, g} | x_{i}) = \sum_{u = 1}^{l_{ij, g}} {[H_{ss} (t_{w} | x_{i}) - r_{ss} (t_{w} | x_{i})]}^{2}

(29)

where $H_{ss} (t)$ is the degradation data from actual measurements, $r_{ss} (t)$ is the theoretical degradation data, $l_{ij, g}$ is the number of tests of the $j$ th sample of the $g$ th group contact pair at the temperature level $x_{i}$ .

The estimated values of $r_{0, g}$ and $β_{g}$ can be obtained by the following equation:

{\hat{r}}_{0, g} = \frac{1}{\sum_{i = 1}^{M} n_{i, g}} \sum_{i = 1}^{M} \sum_{j = 1}^{n_{i, g}} {\hat{r}}_{0 ij, g}

(30)

{\hat{β}}_{g} = \frac{1}{\sum_{i = 1}^{M} n_{i, g}} \sum_{i = 1}^{M} \sum_{j = 1}^{n_{i, g}} {\hat{β}}_{ij, g}

(31)

The estimated value ${\hat{α}}_{ij, g}$ of $α_{ij, g}$ is obtained from equation (29). Because of $α ~ LN (μ_{α}, σ^{2})$ , the parameters $a_{g}$ , $b_{g}$ , and $σ_{g}$ are estimated by using the maximum likelihood estimation. Establishing the likelihood function and taking the logarithm of both sides can be expressed as:

\begin{array}{l} L_{g} (a_{g}, b_{g}, σ_{g} | {\hat{α}}_{i j, g}) = \sum_{i = 1}^{M} \sum_{j = 1}^{n_{i, g}} {- \frac{1}{2} \ln (2 π) - \frac{1}{2} \ln σ_{g} \\ - \frac{{(\ln {\hat{α}}_{i j, g} - E (\ln L_{g}^{3}) + E (\ln I_{z, g}) + E (\ln δ_{g}) + E (\ln N_{g}) - a_{g} - b_{g} x_{i})}^{2}}{2 σ_{g}^{2}}} \end{array}

(32)

According to equation (32), the estimated values of the parameters ${\hat{a}}_{g}$ , ${\hat{b}}_{g}$ , and ${\hat{σ}}_{g}$ for the degradation model can be obtained by using the fminsearch function in matlab.

Model validation and reliability evaluation

Validation of the degradation trajectory model

According to the performance degradation model in equation (12), a nonlinear regression model is established:

y = y_{0} + α t^{β} + ε

(33)

where $β$ is the constant. Let $τ = t^{β}$ , then equation (33) can be transformed into a linear regression model, that is:

y = y_{0} + α τ + ε

(34)

The validation of the degradation trajectory model is to test the goodness of fit of the model to the degradation data. The linear regression model was tested by residual analysis, and the fit of the linear regression model to the test data can be measured by the ratio $R$ between the regression sum of squares $SSR$ and the total sum of squares $SST$ .

R = \frac{SSR}{SST}

(35)

where $SSR = \sum_{i = 1}^{n} {({\hat{y}}_{i} - \bar{y})}^{2}$ , $SST = \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}$ , $y = \frac{1}{l} \sum_{i = 1}^{l} y_{i}$ , $y_{i}$ is the test data of the kth, ${\hat{y}}_{i}$ is the fitting data of the kth.

According to the data of the accelerated degradation test, the $R$ values of each aperture contact pair at different temperature levels are obtained as shown in Table 3.

Table 3.

The $R$ values for each aperture contact pair at different temperature levels.

Temperature (°C)	Diameter (mm)	R
120	1.1	0.910
	2.2	0.968
	3.5	0.884
140	1.1	0.859
	2.2	0.852
	3.5	0.896
158	1.1	0.918
	2.2	0.862
	3.5	0.927

The closer the $R$ value is to 1, the better the fitting effect of the linear regression model is. Table 5 shows that the $R$ values of different contact pairs are greater than 0.85 and closer to 1 at all temperatures. Therefore, it can be considered that the linear regression model fits the test data well, and the degradation trajectory model is correct.

The validation of accelerated model

The validation of the accelerated model is to test whether the linear regression between the log mean $μ_{α}$ of the degradation rate $α$ and temperature $x$ is significant. According to equation (16), a linear regression model was established:

\begin{matrix} μ_{α} = E (\ln α) = E (\ln L^{3}) - E (\ln I_{z}) \\ - E (\ln δ) - E (\ln N) + η_{0} + η_{1} x + ε \end{matrix}

(36)

where $ε$ is the random error and satisfies the Guess-Markov assumption.

The estimated value $μ_{α, i}^{*}$ of the log mean of the degradation rate can be obtained by the statistical analysis method in Section “The statistical analysis of test data,” then the estimated value ${\hat{η}}_{0}$ and ${\hat{η}}_{1}$ of the regression coefficients in equation (36) can be obtained by using the least square method.

The significance of equation (36) can be tested by the residual analysis, and the test statistic can be expressed as:

F = \frac{SSR}{SSE / (M - 2)}

(37)

where $SSR = \sum_{i = 1}^{3} {({\hat{μ}}_{α, i} - {\bar{μ}}^{*})}^{2}$ , $SSE = \sum_{i = 1}^{3} {(μ_{α, i}^{*} - {\hat{μ}}_{α, i})}^{2}$ , ${\bar{μ}}^{*} = \frac{1}{3} \sum_{i = 1}^{3} μ_{α, i}^{*}$ , $M$ is the number of temperature levels, here $M = 3$ .

Here the significance level is $θ = 0.05$ and the quantile $F_{0.95} (1, 1) = 161$ is determined by looking up the F-distribution table. The test statistics $F$ values for contact pairs with different apertures are shown in Table 4.

Table 4.

The test statistic F values for contact pairs with different apertures.

Diameter (mm)	1.1	2.2	3.5
Statistic F	724	528	614

Table 4 shows that the test statistic $F$ is greater than 161 for each aperture contact pair. Therefore, it can be considered that there is a significant linear relationship between the log mean $μ_{α}$ of the degradation rate and the temperature $x$ , and the acceleration model is correct.

Contact reliability assessment of electrical connectors

The estimated values of the degradation model parameters can be obtained by using the statistical analysis method in Section “The statistical analysis of test data” to deal with the test data, as shown in Table 5.

Table 5.

The estimated values of the degradation model parameters.

Diameter (mm)	${\hat{r}}_{0}$	$\hat{a}$	$\hat{b}$	$\hat{σ}$	$\hat{β}$
Φ1.1	4.682	19.440	−16.184	0.067	1.788
Φ2.2	1.487	20.522	−16.317	0.159	1.839
Φ3.5	1.094	20.406	−16.227	0.114	1.938

By substituting the estimated values in Table 3 into equation (24), the reliability function of each contact pair group in the storage environment (25°C) can be obtained, and its reliability curve is shown in Figure 8.

Figure 8.

The reliability curves for each contact pair group at 25°C.

It can be seen from Figure 8 that the reliability level of each contact pair group is significantly different, and the reliability of Φ1.1 mm contact pair group is the highest, followed by the Φ2.2 mm contact pair group, and the reliability of Φ3.5 mm contact pair group is the lowest reliability.

According to equation (27), the contact reliability function of the electrical connector can be obtained, and its reliability curve under the storage environment temperature (25°C) is shown in Figure 9.

Figure 9.

The contact reliability curve of electrical connector at 25°C.

The contact reliable life of the multi-aperture electrical connector at 25°C is shown in Table 6. The evaluation results meet the reliability requirements of this type of electrical connector.

Table 6.

The contact reliable life of the electrical connectors at 25°C.

Temperature (°C)	t _0.5	t _0.9	t _0.99	t _0.999
25	516,787 h	491,896 h	469,314 h	451,958 h

Conclusion

This paper focuses on the reliability evaluation of multi-aperture electrical connectors. Considering the influence of the aperture difference between the contact pairs on the contact reliability of the electrical connector, a reliability model of the multi-aperture electrical connectors is established by analyzing the structure and failure mechanism of contact pairs. Through the mathematical model of contact force, the main structural parameters of the contact pair are related to the contact reliability model of the electrical connector, which is helpful to explore the influence law of the structural parameters of the contact pair such as pin radius, reed length, curvature radius of reed inner and outer surface, and reed number on the contact reliability of the electrical connector.

In order to validate the proposed model, this paper conducts an accelerated test on a type of multi-aperture electrical connector. Through the test data, the correctness of the established model is verified, and the unknown parameters in the model is determined by using the least square estimation method and the maximum likelihood estimation method. The reliability levels of each aperture contact pair group in the electrical connector is compared, and the results showed that there were large differences between the contact pair groups, and the contact reliability of the multi-aperture electrical connector is evaluated in the final.

In conclusion, this paper presents a comprehensive approach for modeling and evaluating the contact reliability of multi-aperture electrical connectors, this approach is applicable when the contact pair groups of electrical connectors are independent of each other. Further research can be carried out on the basis of this paper when the contact pair groups of electrical connectors interfere with each other.

Footnotes

Appendix

The detailed derivation process of equation (3) is as follows. The area of the transverse section in Figure 4 can be expressed as:

(A1)

A_{z} = α_{3} (R_{1}^{2} - R_{2}^{2})

The static moment $S_{z_{0}}$ of the section graph on the $Z_{0}$ axis can be expressed as:

(A2)

S_{z_{0}} = \frac{2 (R_{1}^{3} - R_{2}^{3}) \sin α_{3}}{3}

The moment of inertia $I_{z_{0}}$ of the section graph on the $Z_{0}$ axis can be expressed as:

(A3)

I_{z_{0}} = (2 α_{3} + \sin 2 α_{3}) \frac{R_{1}^{4} - R_{2}^{4}}{8}

The distance $y$ from the centroid of the section graph to the $Z_{0}$ axis can be expressed as:

(A4)

y = \frac{S_{z_{0}}}{A_{z}}

According to the parallel-axis formula, the moment of inertia $I_{z}$ of the section graph on the $Z$ axis can be obtained:

(A5)

\begin{matrix} I_{z} = I_{z_{0}} - y^{2} \cdot A_{z} \\ = \frac{2 α_{3} + \sin 2 α_{3}}{8} (R_{1}^{4} - R_{2}^{4}) - \frac{4 (R_{1}^{3} - R_{2}^{3}) \sin^{2} α_{3}}{9 α_{3} (R_{1}^{2} - R_{2}^{2})} \end{matrix}

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by Zhejiang Provincial Key Research and Development Program (2021C01133).

ORCID iD

Ping Qian

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