A Degradation Model of Electrical Contact Performance for Copper Alloy Contacts with Tin Coatings Under Power Current-Carrying Fretting Conditions

Abstract

The fretting degradation of coating materials is complicated when the Joule heating effect of the load current is non-negligible. In this work, the fretting degradation behavior of tin-coated copper alloy contacts was examined across the current range from 5 A to 30 A and the fretting amplitude range from 200 μm to 400 μm. The uniform degradation process and associated mechanisms, including the wear-dominated stage, softening-dominated stage, and oxidation-dominated stage were recognized and interpreted explicitly. An innovative fusion model for the degradation of electrical contact performance, data-driven and based on physics mechanism, was proposed. The contact voltage was selected as the characterization variable and pre-processed by the threshold-crossing filter method and the Levenberg–Marquardt algorithm. The optimal degradation model, considering the coupled effects of current and fretting amplitude, was presented with the use of the support vector regression method. Finally, the prediction accuracy of the model was validated by experiment results.

1. Introduction

Tin-coated copper alloy contacts have gained popularity in electrical interconnection applications for their unique advantages of high electrical conductivity, formability and cost [1,2]. In particular, tin as a coating, belonging to non-noble metals, could efficiently prevent the base materials from oxidation and chemical corrosion in the atmosphere with lower cost. The development of electrical engineering and automation requires the participation of a large number of electrical and electronic components, including connectors, electromechanical relays and so on. However, fretting behavior occurring at the interface of two contacting components severely limits the durability of the surface coating and furthermore leads to the gradual increment of contact resistance and break-off phenomena of the electrical circuit [3,4,5,6,7]. Field data have demonstrated that more than 70% of intermittent electrical failure problems are related to fretting behaviors.

With the increasing use of electrical connectors, the reliability and remaining useful lifetime issues in the electrical contact performance of systems have brought wider attention from academia and industry. Contact resistance is considered as the preferred choice for characterizing the degradation process of electrical contact performance for coating materials and has been selected as the indicator of electrical connector and even system reliability [8,9,10]. So, the prediction method of the remaining life for coating materials based on the contact resistance variations is considered as the best candidate in system safety evaluation. Moreover, the estimation of the state of the health of connectors could assist in taking preventative measures through early replacement of the degraded contact component within the connectors.

Generally, the prognostic modeling methods in connectors have been classified into the physical model-based method, the data-driven prognostic method, and the fusion prognostic method [11]. The physical model-based degradation model offers a coherent theoretical explanation and reliable prediction of the degradation process. Archard [12] systematically linked wear volume to normal force, sliding distance, and material hardness, and developed a mathematical model to describe the physical wear process. Subsequently, the Archard wear law was extensively used in the tribological behavior modeling and fretting lifetime prediction of coatings [13]. Fouvry et al. [14] studied the relationship between coatings removal and the friction energy dissipated through the interface and proposed a friction energy density approach to predict electrical endurance under fretting conditions. Ji et al. [15] analyzed the physical mechanisms of oxide generation and peeling-off in tin-coated contacts and constructed a theoretical model to estimate the contact resistance during fretting. It is extremely difficult to establish a physics-based model for complex degradation processes, as it requires strong theoretical support.

Meanwhile, the data-driven modeling method has gained attention from researchers for its ability to function without requiring a deep understanding of complex failure mechanisms [16,17,18]. Wang et al. [19] conducted a statistical analysis of the experimental data and described the degradation path of the electrical connector with an inverse Gaussian degradation model. Furthermore, deep learning and artificial intelligence (AI) are often adopted for degradation modeling, which uses experimental data to train models to predict degradation trajectories [20,21]. However, data-driven modeling methods demand a large amount of experimental data and are prone to overfitting, which could result in prediction uncertainty.

The fusion modeling method benefits from the advantages of both physics-based and data-driven modeling methods while overcoming some of their limitations. The Wiener model is commonly employed to predict the non-monotonic degradation process of coating wear volume, with Bayesian parameter estimation [22] or maximum likelihood estimation [23] being utilized to calculate the model parameters from experimental data. Fouvry et al. [24,25,26,27] modified the Archard wear law and friction energy density approach through statistical analysis. Sun et al. [28] determined the degradation model for contact resistance based on the physical mechanism of increasing oxide film thickness and calculated the model parameters by particle filtering. Additionally, techniques like Kalman filtering [29] and regression analysis [30] have also been successfully applied in degradation modeling to calculate model parameters.

Hitherto, most of the available models mentioned above were developed to characterize the degradation processes of relatively simple physical mechanisms, such as fretting wear [12,13,14,24,25,26,27] and subsequent oxidation [15,28]. It is noted that the fretting degradation mechanism of tin-coated copper alloy contacts would become more complicated once the carrying current produces a substantial thermal effect. The degradation, which occurs under current-carrying fretting conditions, is a coupled mechanical, electrical, and thermal phenomenon. Thus, many efforts have been devoted to investigate the degradation phenomenon under current-carrying fretting conditions [31,32,33,34]. However, these studies are restricted to the mechanistic analysis under specific current-carrying conditions and there is currently a lack of systematic studies on the effects of current level, fretting intensity, and other factors on degradation tendencies. In the previous works by the present authors, the multiple fretting failure mechanisms of tin-coated copper alloy contact materials, including mechanical wear, softening, oxidation, melt erosion, and boiling at different current loads ranging from 0.1 to 12.5 A were clearly revealed [35,36]. However, to the best of our knowledge, a degradation model of electrical contact performance for tin coating under power current-carrying fretting conditions remains absent. Such situations make the reliable evaluation of power connectors under fretting conditions challenging.

The objective of this study is to develop a degradation model of electrical contact performance for tin-coated copper alloy contact materials under power current-carrying fretting conditions by using the fusion prognostics approach. This paper is organized as follows. First, in Section 2, the designed simulation test instrument and associated experimental conditions are described. The physical degradation mechanisms of the tin-coated copper alloy contact materials with varied load current and fretting amplitude conditions are illustrated individually in Section 3. For the sake of convenience in determining the physical status of the electrical contact interface, the contact voltage was selected as the main characterization indicator. Then, in Section 4, the segmented degradation model based on physical mechanisms and data-driven method is presented and verified. In addition, a novel support vector regression method was developed to optimize the unknown coefficients involved in the degradation model. Finally, the conclusion of the research is presented in Section 5.

2. Experimental Details

2.1. Test Rig

The designed test rig used in these experiments was described in detail in [36], which contained a brief summary thereof. Figure 1 presents a schematic sketch of the test rig used to simulate the fretting environment with controllable normal contact force, fretting amplitude, and fretting frequency. One end of the claw component is driven by an electric slide table (RCA2-TWA4NA, IAI, Shizuoka, Japan), whose movement direction is set aligning to Y direction. The other end is in contact with the blade component, which is attached to a force sensor (FA703, Fibos, Changzhou, China) capable of measuring three-dimensionally. Three-dimension linear translation stage combinations are used for the accurate adjustment of normal force and contact position.

During the fretting experiment, the power load current passes through the configured mechanical contact pairs all the time. The measurement and control unit is specially developed to acquire the contact current, contact voltage, normal force, friction force, and contact temperature signals simultaneously. As shown in Figure 1b, the carrying current is captured in real-time via a high-precision Hall sensor (100A-Y2, Xinghui, Zhuhai, China) with a measurement error of less than 0.5%. The voltage measurement module (Self-developed) incorporates a low-noise amplifier (Self-developed) and a low-pass filter (Self-developed) for handling the contact voltage signal effectively. The measurement accuracy of the contact voltage between the contact pair is within 1% after calibration. A K-type thermocouple (STTT-VK, ST, Beijing, China) is firmly glued onto the claw component to monitor the contact temperature. A high-resolution camera (SUB1600C, MindVision, Shenzhen, China) is positioned to facilitate the in situ observation of the contact pair. The signals mentioned above are collected using a data acquisition card (PCI-1706, Advantech, Taipei, China) with a sampling rate of 250 kHz. System control and data acquisition are performed with the help of custom-programmed LabVIEW software (LabVIEW 14.0, NI, Austin, TX, USA).

2.2. Experimental Conditions

The blade and claw samples (shown in Figure 2) are sourced from commercially available power connectors. The blade is made of C11000 electrolytic tough pitch (ETP) copper, which is known for its high electrical and thermal conductivity. And the claw is made of C18150 copper–chromium–zirconium alloy to achieve a higher normal force under limited interference in the mated state. The surface is plated with tin with a thickness of 2.5 µm, and a nickel interlayer (1.3 µm) is deposited in order to limit copper diffusion and the formation of intermetallic phases. This material combination is commonly used in heavy-duty applications because of its high electrical conductivity, oxidation protection, and economic efficiency. The surface roughness Ra is 0.1 μm.

The blade and claw were degreased by using an ultrasonic cleaner (LT-05A, Longbiao, Huizhou, China) with distilled water, alcohol, and acetone then dried and firmly fixed in the test rig. The experiment was conducted at room temperature and humidity. The electric slide table was set up with a fretting cycle of 4 s and an acceleration of 2 m/s². According to the on-site measurement results of the fretting behavior, fretting amplitudes of 200, 250, 300, 350, and 400 μm were applied to the end of the claw. Figure 3 illustrates the waveforms of velocity and displacement during a single whole motion cycle. Furthermore, load currents of 5, 10, 15, 20, 25, and 30 A were selected based on the designed current-carrying capacity of the power connector. Once the measured average value of contact voltage surpasses 100 mV, the contact pair is marked as failed, then the experiment will be terminated. To confirm the results’ repeatability, each set of experiments was repeated five times. Scanning electron microscopy (Sigma 300, Zeiss, Oberkochen, Germany) and energy dispersive spectrometer (EDS) (Sigma 300, Zeiss, Oberkochen, Germany) were used to characterize the extent of fretting damage and the extent of oxidation at different degradation stages. The details of the experimental conditions used are listed in

Table 1. Testing conditions.

Parameters	Value
Load current (A)	5, 10, 15, 20, 25, and 30
Fretting amplitude (μm)	200, 250, 300, 350, and 400
Initial normal force (N)	3.5
Acceleration amplitude (m/s2)	2
Motion cycle (s)	4
Sampling time (μs)	4
Environment temperature (°C)	25
Humidity (% RH)	65

.

3. Results and Discussion

3.1. Typical Degradation Process of Contact Voltage, Friction Coefficient, and Temperature

Figure 4 shows the degradation trends of the contact voltage, friction coefficient, and temperature of tin-coated copper alloy contact pairs as a function of fretting cycles for different current and fretting amplitude combined conditions of 5 A and 300 μm, 15 A and 300 μm, 30 A and 300 μm, 15 A and 200 μm, and 15 A and 400 μm, respectively. Each data point represents the average of raw measurements during each fretting cycle.

Figure 4a shows the variations in contact voltage, friction coefficient, and temperature for the load current of 5 A and fretting amplitude of 300 μm. The contact voltage behavior appears to be characterized by a few distinct stages. In Stage I, there was a steady increase in contact voltage from 3 mV to 40 mV within the first 975 cycles, with a progressively accelerating rate. And the temperature rose from 23.5 °C to 29.2 °C simultaneously.

There was a slight increase in the contact voltage from 975 to 1246 cycles (from 40 mV to 75 mV) and the temperature correspondingly rose to 33.9 °C. It is worth noting that the rate of increase in contact voltage declines gradually with the fretting cycle, which is the noticeable difference from the already known behavior that has occurred in fretting experiments under low-current conditions.

In Stage III (from 1246 to 1262 cycles), the measured contact voltage surpassed 100 mV and the corresponding temperature was higher than 40 °C. Interestingly, the contact voltage began to grow rapidly with the continuing fretting.

A large fluctuation in the contact voltage was observed, which fluctuated from 68.8 to 240 mV repetitively after 1262 fretting cycles and the measured temperature oscillated correspondingly between 34 and 53 °C. Therefore, the initial appearance of unstable electrical contact performance at the 1262nd cycle was regarded as the lifespan of tin-coated copper alloy contact material experiencing power current-carrying fretting conditions.

As illustrated in Figure 4b–e, the degradation tendencies of contact voltages and temperatures for additional conditions of 15 and 30 A load currents and 200 and 400 μm fretting amplitudes uniformly conformed to the aforementioned three individual stages. The lifespan of tin-coated material under power current-carrying fretting conditions has close ties to load currents and fretting amplitudes.

3.2. Evolution of Microscopic Contact Behavior

The surface topographies and element distributions corresponding to contact voltages of 20 mV, 60 mV, 80 mV, and 100 mV were individually captured with a scanning electron microscope (Sigma 300, Zeiss, Oberkochen, Germany). Numerous sharp scratches were observed on the tin coating, as illustrated in Figure 5a. Only at Stage I was wear of the tin coating captured, with slight indications of the exposure of the nickel interlayer. With the help of an extended depth of field microscope (DSX1000, OLYMPUS, Tokyo, Japan), it was found that the surface roughness increased from 0.1 μm to 0.25 μm. This suggests that the observed increase in contact voltage was a consequence of wear on the surface coating. This is taken as the general rule for fretting or sliding behavior, regardless of carrying currents [8,9,37].

According to the well-known temperature–voltage equation [38], the maximum temperature T_m of contact a-spots within the current-carrying interface could be written as

$$T_m=\sqrt{T_0^2+{\displaystyle \frac{U_c^2}{4L}}}$$

(1)

where L = 2.4 ×10⁻⁸ (V/K)² is the Lorenz constant, T₀ is the bulk temperature (K), and U_c is the contact voltage (V).

In [39], the calculated softening voltage of tin is taken as 70 mV, which is derived from the softening temperature of 100 °C. The instantaneous contact voltages in response to actuator positions in different cycles are presented in Figure 6. A rising fluctuation in transient contact voltage was detected across the actuator position from −155 μm to −100 μm. The contact material is softened once the instantaneous voltage exceeds 70 mV even though the average value is much less than the threshold value of tin material softening. As shown in Stage II, the tin-coating softening was exacerbated with the increment of contact voltage and in turn, limited contact-voltage growth. Morphological examination results in Figure 5b illustrated the wear-out phenomenon of the thin coating and the extensive exposure of the nickel interlayer. The measured surface roughness of 0.1175 μm across the fretting region indicates that the thermal effect caused by the power current softened the wear debris and produced a smoother profile compared to Stage I.

As shown in Figure 6a, the contact voltage remained above the tin-softening voltage in Stage III. The observed fretting zone became progressively smoother (Figure 5c,d), with surface roughness decreasing further to 0.054 μm and 0.019 μm as the contact voltage rose to 80 mV and 100 mV, respectively. In order to investigate the rapid increase phenomenon of contact voltage during Stage III, the EDX surface scanning across the fretted zone plotted in Figure 7 confirms that the intensity of oxygen increased gradually when the contact voltage changed from 20 mV to 80 mV. Within the nickel-exposed zone marked in Figure 5d, the proportions of tin, oxygen, and nickel were 5.33 at%, 7.71 at%, and 86.95 at%, respectively. It proves that nickel shows a less oxidation tendency. However, the sharply elevated oxygen content (55.61 at%) observed when the contact voltage reached 100 mV (corresponding to Stage III) substantiates that the remaining tin underwent severe oxidation at high temperatures. This is the root cause of the rapid increase in contact voltage and subsequent electrical contact failure. In addition, the intensity of Ni exhibited a notable rise from 2.88 at% to 11.14 at% as the contact voltage increased from 20 mV to 40 mV. It proves that the wear of the tin coating continued to intensify during Stage I. Subsequently, the intensity of Ni slightly decreased to 10.4 at%, which indicates a reduction in the exposure of the interlayer. This indirectly confirms the softening and spreading of the tin coating.

As plotted in Figure 6b, the instantaneous contact voltages were observed to regularly exceed 130 mV after 1262 fretting cycles, which is believed to be the melting voltage of tin coating [39]. As expected, the contact material experienced a series of complex and coupled physical behaviors including melting, solidification, oxidation, wear, and remelting under power current-carrying fretting conditions [35,36]. The noticeable fluctuations in the friction coefficient between 1.07 and 1.15 indirectly reflect the variations in the physical state of the contact interface; the melting of the contact materials reduced the obstruction to relative motion between the contact pair, which in turn caused a rapid decrease in the friction coefficient and the cooled and solidified material obstructed the relative motion as the contact position shifted due to fretting behavior, which led to a sudden increase in the friction coefficient.

3.3. Physical Degradation Mechanisms

Based on the above analysis, the physical degradation of the tin-coated copper alloy contact pair under current-carrying fretting conditions involves a multi-physics coupling of mechanical, electrical, and thermal effects. As shown in Figure 8, the tin coating is rapidly worn by the tangential fretting behavior in Stage I. The reduction in the real areas for current conduction is the significant factor for the increase in contact voltage and temperature. Once the contact temperature exceeds the softening point of the tin coating, the softening and expansion of the contact material help mitigate the rise in contact voltage during Stage II. The increasing temperature accelerates the oxidation of the residual tin coating, and then causes a rapid rise in both contact voltage and temperature in Stage III. Finally, the contact pair fails with an unstable electrical contact performance when the instantaneous contact temperature exceeds the melting point of the tin coating. In summary, contact voltage is identified as an appropriate characterization variable for evaluating electrical contact performance under current-carrying fretting conditions. And the physical degradation of contact voltage uniformly follows three distinct stages: the tin wear-dominated stage, the tin soften-dominated stage, and the tin oxidation-dominated stage.

4. Degradation Modeling

4.1. The Degradation Model of Contact Voltage Based on Levenberg–Marquardt Algorithm

The degradation mechanisms presented above indicate that the contact voltage is an appropriate parameter for characterizing the degradation of electrical contact performance for tin-coated copper alloy contact materials under power current-carrying fretting conditions. The physical degradation mechanism under any combination of currents ranging from 5 A to 30 A and fretting amplitudes ranging from 200 μm to 400 μm is assumed to follow the three stages identified above. It is further assumed that these three stages are independent of each other, and the fluctuations in contact voltage are temporarily ignored (it will be eliminated by the data preprocessing method proposed subsequently).

For the tin coating wear-dominated stage, just as mentioned in [26], the contact resistance or contact voltage degradation process conforms to the exponential laws during the initial coating wear stage. Thus, the variation in contact voltage as a function of fretting cycles could be expressed as

$$U_c=A_1\cdot {\mathrm{e}}^{B_1\cdot N}+C_1 , 0\le U_c\le 40$$

(2)

where A₁, B₁, and C₁ are the undetermined coefficients.

For the tin softening-dominated stage and oxidation-dominated stage, two different logarithmic functions are selected to describe them, based on the collected raw data and ever-presented models [15]. So, the variation in contact voltage as a function of fretting cycles could be written as

$$U_c=A_2\cdot \ln (N-B_2)-C_2 , 40<U_c\le 75$$

(3)

$$U_c=C_3-A_3\cdot \ln (B_3-N) , 75<U_c\le 100$$

(4)

where A₂, B₂, C₂, A₃, B₃, and C₃ are the undetermined coefficients.

As shown in Figure 4, there were different levels of fluctuations in the contact voltage during the softening-dominated stage within the whole fretting process. As interpreted above, it could be considered as the self-equilibrium process of the contact interface area. Although the continuous fretting wear effect and current lubricating effect alternately dominated the change in contact voltage, the peak value of the contact voltage among the already appearing values could represent the worst of the electrical contact interface situation. Therefore, only the local peak value was reserved in the degradation modeling process in order to make the variations in contact voltage as a function of fretting cycles monotonic.

In this work, a threshold crossing filter (TCF) method is used to pre-process the collected contact voltage results plotted in Figure 4. The operation procedure is scheduled as follows:

(a)

Define a series of voltage thresholds U_ck (k = 1, 2, 3,…). The initial U_c1 is 10 mV, and the end value U_c20 is 100 mV. The interval is 5 mV.

(b)

Data sequence and identify the fretting cycles N_kj at which the contact voltage first exceeds U_ck, i.e., N_kj = min{N∣U_Nj ≥ U_ck }, where j represents the j-th of parallel tests, j ∈ [1, 5].

(c)

Repeat steps (a) and (b) for 5 sets of parallel experimental data, then calculate the average value $\overline{N_k}=1/5{\displaystyle \sum _{j=1}^5{N}_{kj}}$ and the standard deviation $\sigma =\sqrt{{\displaystyle \frac{1}{5}}{\displaystyle \sum _{j=1}^5{(N_{kj}-\overline{N_K})}^2}}$.

The processed data under the conditions of current ranging from 5 A to 30 A and fretting amplitude ranging from 200 μm to 400 μm are listed in

Table A1. Processed data under the conditions of current ranging from 5 A to 30 A and fretting amplitude ranging from 200 μm to 400 μm.

	5 A and 300 μm		10 A and 300 μm		15 A and 300 μm		20 A and 300 μm		25 A and 300 μm
Uc (mV)	Cycle	σ	Cycle	σ	Cycle	σ	Cycle	σ	Cycle	σ
5	335.33	62.94	——	——	——	——	——	——	——	——
10	588.67	155.84	234.60	71.94	49.89	15.51	——	——	——	——
15	687.00	176.41	381.80	130.33	232.00	60.22	115.80	13.45	——	——
20	740.33	173.91	497.60	173.52	310.00	79.94	251.60	55.12	147.40	53.04
25	791.33	173.58	581.60	226.81	357.33	94.10	322.00	61.70	242.60	114.65
30	830.00	185.38	620.20	235.67	409.33	127.70	369.80	72.55	315.60	143.79
35	864.00	199.17	684.20	211.76	443.44	132.57	406.00	74.17	361.00	164.60
40	899.33	205.87	710.20	226.11	470.78	142.71	450.40	68.83	397.00	175.93
45	927.75	204.00	724.40	233.55	500.22	147.56	481.00	78.65	423.40	187.95
50	956.00	228.87	758.60	231.92	526.89	149.76	500.00	89.70	452.80	204.44
55	992.50	242.01	772.60	234.20	552.22	154.14	527.80	102.02	475.60	221.62
60	1030.00	249.18	793.60	230.21	594.33	164.09	551.20	96.78	495.20	221.26
65	1070.25	216.80	835.60	214.99	622.67	176.80	578.00	109.82	529.20	217.50
70	1115.67	234.65	864.00	216.45	656.33	168.87	610.80	122.96	569.00	222.03
75	1164.00	228.87	919.50	234.19	709.67	186.41	662.00	147.57	605.40	238.95
80	1196.50	208.38	964.50	188.34	740.78	201.32	719.20	201.98	616.60	241.43
85	1221.50	219.99	999.25	219.30	770.44	191.40	752.60	219.27	643.80	259.25
90	1232.75	225.05	1010.75	211.07	790.00	191.83	764.60	213.34	657.00	251.54
95	1234.75	222.34	1019.75	204.52	800.56	188.34	770.00	213.29	657.80	250.76
100	1237.75	224.26	1023.50	224.86	813.78	197.08	778.80	212.98	660.40	248.28
	30 A and 300 μm		15 A and 200 μm		15 A and 250 μm		15 A and 350 μm		15 A and 400 μm
Uc(mV)	Cycle	σ	Cycle	σ	Cycle	σ	Cycle	σ	Cycle	σ
5	——	——	——	——	——	——	——	——	——	——
10	——	——	92.50	3.72	104.75	25.32	87.20	21.24	25.33	9.48
15	——	——	797.00	208.91	352.00	112.31	230.60	31.36	195.00	52.63
20	44.00	7.82	1278.25	353.31	467.00	187.94	284.60	36.72	334.67	109.13
25	144.60	17.51	1532.75	417.44	534.00	190.73	317.00	40.78	424.00	106.09
30	209.00	48.57	1663.50	444.87	592.25	192.95	342.00	43.95	478.33	125.53
35	252.60	68.80	1745.50	465.86	634.00	196.94	378.60	58.09	508.00	120.54
40	288.80	78.56	1795.75	473.32	682.25	205.17	413.60	66.49	534.67	109.22
45	318.20	87.83	1838.75	480.12	709.75	208.59	436.20	69.15	550.00	108.59
50	349.20	102.23	1874.25	487.43	726.25	215.47	460.40	74.99	569.33	106.30
55	382.00	123.72	1944.00	494.54	764.25	201.42	488.00	89.87	588.67	100.18
60	420.20	133.10	2001.00	509.35	794.75	197.09	519.00	88.60	608.00	107.81
65	456.80	161.05	2082.50	521.48	962.25	191.09	553.00	85.73	631.33	121.30
70	495.80	183.50	2138.50	520.14	1046.50	253.90	597.40	103.07	659.33	141.65
75	530.80	185.67	2209.00	521.16	1121.00	318.78	642.40	140.98	686.33	153.76
80	564.60	209.11	2331.00	561.50	1184.75	304.54	671.40	152.03	697.33	157.70
85	573.60	214.37	2403.25	578.33	1258.25	283.76	684.00	157.41	697.67	157.98
90	589.00	226.85	2587.75	648.86	1287.25	294.94	689.00	155.45	702.67	156.56
95	599.60	237.42	2632.25	644.55	1300.50	308.45	697.60	151.51	703.67	157.01
100	603.20	241.58	2667.50	654.15	1316.50	309.66	699.00	151.74	704.00	157.29

. The Levenberg–Marquardt (L–M) algorithm [40], combining the strengths of both gradient descent and the Gauss–Newton method, are used to calculate the model parameters (A1, …, C3) of the pro-processed contact voltage data with the minimal sum of squared errors (S(β)). The specific steps of the iterative calculation process are illustrated in Figure 9. Based on the pre-processed dataset ($\overline{N_k}$, U_ck) and the model function U_c = f (N; β), the L–M algorithm minimizes the sum of squared errors (S(β)) by continuously adjusting the parameter vector β:

$$S(\beta )={\displaystyle \sum _{k=1}^K{[U_{ck}-f(\overline{N_k};\beta )]}^2}$$

(5)

where k ∈ [1, K] is the calculated point in the above TCF method, U_ck is the voltage threshold used in the above TCF method, and f ($\overline{N_k}$; β) is the calculated value of Equations (2)–(4) with the parameters regulated by the parameter vector β.

Firstly, the partial derivative $J_{\mathit{k\; m}}={\displaystyle \frac{\partial f(\overline{N_k};\beta )}{\partial {\beta }_m}}$ of the objective function with respect to β is calculated to form the Jacobian matrix. Then, the difference between the captured contact voltage U_ck and the calculated value $f(\overline{N_k};\beta )$ of the objective function is input into the equation $(J^{T}J+\lambda I)\Delta \beta =J^T[U_{ck}-f(\overline{N_k};\beta )]$ to calculate the increment Δβ. The sum of squared errors is calculated again using the updated β value. This iterative calculation is continued until the allowable error is satisfied.

The detailed calculated model parameters for the combinations of current ranging from 5 A to 30 A and fretting amplitude ranging from 200 μm to 400 μm are listed in

Table A2. Model parameters calculated for the combinations of current ranging from 5 A to 30 A and fretting amplitude ranging from 200 μm to 400 μm.

	5 A and 300 μm	10 A and 300 μm	15 A and 300 μm	20 A and 300 μm	25 A and 300 μm
A1	0.42474	2.64146	2.26005	3.07944	3.81658
B1	0.00500	0.00366	0.00569	0.00515	0.00502
C1	2.46277	3.71963	6.96544	9.17703	11.81203
A2	52.80663	34.58883	59.91668	55.57935	64.82063
B2	616.59758	593.03751	178.44899	225.39250	114.11079
C2	258.02616	124.62783	300.50292	261.88600	326.42660
A3	6.66216	7.75326	17.25970	9.05224	6.28440
B3	1239.62501	1028.01982	845.51386	785.88479	661.84623
C3	104.29731	111.62254	159.87916	118.13538	102.25083
kσ	0.24408	0.36801	0.26371	0.23481	0.40859
	30 A and 300 μm	15 A and 200 μm	15 A and 250 μm	15 A and 350 μm	15 A and 400 μm
A1	4.52267	0.60671	4.28896	2.45438	1.95619
B1	0.00604	0.00215	0.00250	0.00646	0.00515
C1	14.10424	9.71045	4.85973	5.66467	8.27086
A2	113.89099	54.59782	29.38419	39.05131	53.41127
B2	−394.99957	1317.24200	670.21730	256.77320	370.59340
C2	703.25948	296.44660	110.47500	157.52920	232.39330
A3	12.47878	13.45950	10.63947	9.46488	6.11403
B3	614.96786	2771.16300	1581.74800	703.66950	704.36880
C3	130.19653	161.78190	137.26320	113.68390	93.53919
kσ	0.35652	0.25117	0.25498	0.24353	0.27189

. By substituting these parameters into Equations (2)–(4), the degradation curves of contact voltage as a function of fretting cycles under varying load currents and fretting amplitudes are calculated and illustrated in Figure 10.

As shown in Table A1, the calculated standard deviation σ was found to be nearly linear dependent on the fretting cycles. The cumulative deviation inevitably results in a reduction in the fitting accuracy during the oxidation-dominated stage. If we assume that σ varies approximately proportional to the fretting cycle N, that is

$$\sigma =k_{\sigma }N$$

(6)

where k_σ is the proportional coefficient, whose detailed values are listed in Table A2.

Meanwhile, if the probability distribution of fretting cycles N follows a normal distribution with N_p (derived from Equations (2)–(4)) as the mean and σ_p (calculated from Equation (6)) as the standard deviation, then for any specified contact voltage, it could be given by

$$f(N)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}\exp (-{\displaystyle \frac{{(N-N_{100\mathrm{mV}})}^2}{2{\sigma }^2}})$$

(7)

And the cumulative distribution function F(N) of the failure probability and the reliability function R(N) of the tin-coated copper alloy contact pair could be expressed as

$$F(N)={\displaystyle {\int }_{-\infty }^{N}f(N)\mathrm{d}N=\frac{1}{\sqrt{2\pi }\sigma }{\displaystyle {\int }_{-\infty }^N\exp (-\frac{{(N-N_{100\mathrm{mV}})}^2}{2{\sigma }^2})}}\mathrm{d}N=\Phi ({\displaystyle \frac{N-N_{100\mathrm{mV}}}{\sigma }})$$

(8)

$$R(N)=1-F(N)=1-\Phi ({\displaystyle \frac{N-N_{100\mathrm{mV}}}{\sigma }})$$

(9)

in which Φ denotes the cumulative distribution function of the standard normal distribution. Based on Equation (8), the 80% confidence interval for the prediction results could be further calculated as N ∈ [N_p − 1.28σ_p, N_p + 1.28σ_p].

4.2. Degradation Model Considering the Coupled Effects of Current and Fretting Amplitude

In order to build the complete degradation model of contact voltage considering the coupled effects of carrying load current and fretting amplitude, a Support Vector Regression (SVR) method based on grid search is proposed in this section. The SVR method is widely used for predicting parameters in high-dimensional feature spaces and small sample size problems. The specific steps are described as following six steps:

(a)

Input the varying conditions and the corresponding model parameters from Table A2 as the training data.

(b)

Normalize data.

(c)

Perform grid search and 5-fold cross-validation on all combination conditions within the three-dimensional space defined by the kernel function and two hyperparameters. The available kernel functions for selection include the linear, the polynomial, and the radial basis kernel functions (RBF). And the two hyperparameters are used to control the training error and the prediction error.

(d)

Calculate the mean squared error and select the optimal combination of the kernel function and the two hyperparameters that minimize this error.

(e)

Predict the model parameters under the specified current and fretting amplitude conditions using the optimal model determined in step (d).

(f)

Output the prediction results.

The above prediction method, which systematically traverses and optimizes combinations of kernel functions and hyperparameters, can significantly enhance both the generalization ability and the accuracy of the prediction. The 5-fold cross-validation method makes full use of the limited training data and effectively avoids both overfitting and underfitting. The degradation model for tin-coated copper alloy contacts under specified conditions can be obtained by substituting the predicted parameters into Equations (2)–(4).

Using the prediction for parameter A1 under the condition of 30 A and 400 μm as an example, we present the average training errors and average validation errors for three kernel functions during 5-fold cross-validation, as shown in

Table 2. Average training errors and average validation errors for three kernel functions during 5-fold cross-validation.

	Liner Training	Liner Validation	Polynomial Training	Polynomial Validation	Rbf Training	Rbf Validation
1	1.62	5.31	1.69	0.67	2.11	0.08
2	1.89	0.58	1.80	0.23	1.69	5.36
3	1.90	0.44	1.72	0.54	1.62	1.63
4	1.24	4.00	0.88	0.49	2.02	0.09
5	1.69	2.70	0.57	0.21	1.33	2.79
Average for 5 folds	1.67	2.60	1.33	0.43	1.75	1.77

. Both errors were minimized with the polynomial kernel function, yielding average values of 1.33 and 0.43, respectively. Thus, the polynomial kernel function was selected to predict the A1 value.

Compared with traditional prediction methods that utilize a single kernel function or a single training dataset, the proposed approach, which integrates grid search and 5-fold cross-validation techniques, demonstrates superior adaptability to various parameters. Furthermore, it effectively mitigates overfitting issues arising from improper training data selection, as exemplified by the second fold of the RBF kernel function in Table 2. However, the implementation of grid search and 5-fold cross-validation results in a substantial increase in computational complexity. This trade-off, nevertheless, is essential for achieving improved prediction accuracy.

To quantify the prediction error of the degradation model, the mean absolute percentage error (MAPE) of the predictions during the degradation process

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{\left|N_{pi}-N_{ei}\right|}{N_{ei}}}\cdot 100\%$$

(10)

and the fretting life percentage error (FLPE)

$$FLPE={\displaystyle \frac{\left|N_{pf}-N_{ef}\right|}{N_{ef}}}\cdot 100\%$$

(11)

were calculated as evaluation metrics in which n is the number of statistical points, i represents the i-th experimental value, N_ei is the average value of the experimental fretting cycles, N_pi is the average value of the prediction fretting cycles, N_ef is the experimental failure cycles, and N_pf is the prediction failure cycles.

To verify the accuracy of the presented degeneration model, validation experiments were conducted under the combined conditions of 5 A and 200 μm, 5 A and 400 μm, 30 A and 200 μm, and 30 A and 400 μm. The comparisons between experimental results and prediction results are presented in Figure 11. There, the blue line represents the experimental results preprocessed by the proposed TCF method. The black line represents the prediction results calculated by the degradation model (Equations (2)–(4)), which used the parameters predicted by the SVR method. The red line represents the prediction probability density (calculated by Equation (7)) for the fretting cycle corresponding to the contact voltage rising to 100 mV. The yellow shaded area, surrounded by a red dashed line, represents the 80% confidence interval (calculated by Equation (8)) for the prediction results.

The experimental results uniformly fell within the 80% confidence interval of the prediction results. The prediction errors between the experimental results and the prediction results are presented in

Table 3. Prediction errors between experimental results and prediction results under combined conditions of carrying current and fretting amplitude.

Condition	MAPE (%)	FLPE (%)
5 A, 200 μm	2.97	7.91
5 A, 400 μm	7.22	6.41
30 A, 200 μm	6.46	1.53
30 A, 400 μm	9.32	4.42

. The maximum MAPE was 9.32% (at carrying current of 30 A and fretting amplitude of 400 μm) and the maximum FLPE was 7.91% (at carrying current of 5 A and fretting amplitude of 200 μm). The good fit between the prediction and experimental results demonstrates that the proposed degradation model can accurately predict the degradation process and remaining useful lifetime of tin-coated copper alloy contact pairs under power current-carrying fretting conditions.

5. Conclusions

The fretting degradation behavior of tin-coated copper alloy contacts was studied and modeled for a combination of conditions of current, ranging from 5 A to 30 A and fretting amplitude ranging from 200 μm to 400 μm. It is confirmed that the degradation process under current-carrying fretting conditions uniformly conforms to three distinct physical stages, that is, tin wear-dominated stage, tin softening-dominated stage, and tin oxidation-dominated stage. Contact voltage was identified as an appropriate characterization variable for evaluating electrical contact performance, with the above stages demarcated by voltage thresholds of 40 mV and 75 mV, respectively. An innovative fusion model, based on the determined physics mechanism and the experimental data-driven method, was proposed to describe the variation in contact voltage as a function of fretting cycles. The grid search and 5-fold cross-validation techniques were used to optimize the SVR method to predict the model parameters for any combination of current and fretting amplitude. The maximum MAPE for degradation process and the maximum FLPE for fretting life in the validation experiments were 9.32% and 7.91%, respectively. The identified physical degradation mechanisms and the proposed degradation model provide valuable insights for health status assessment and predictive maintenance.