*2.3. Parametric ALT of Mechanical Systems*

To obtain the mission cycle of ALTs from the objective BX lifetime on the experiment scheme in Table 1, the sample size formulation integrated with the AF should be obtained [1]. Until now, numerous methodologies have been suggested to decide sample size. The Weibayes model for Weibull analysis is a popularly recognized method of examining reliability data. However, it is hard to directly use because of the mathematical complication. The whole cases as failures (*r* ≥ 1) and no failures (*r* = 0) need to be separated. Consequently, it is possible to acquire a comprehensible sample size equation that might provide the mission cycle after proper assumptions.

In choosing the model parameters to maximize the likelihood function, the maximum likelihood estimation (MLE) statistic is a widespread way of approximating the parameters of a model. The characteristic life *ηMLE* can be expressed as:

$$\eta\_{MLE}^{\mathcal{S}} = \sum\_{i=1}^{n} \frac{t\_i^{\mathcal{S}}}{r} \tag{9}$$

where *ηMLE* is the maximum likelihood estimate of the characteristic life, *n* is the total number of samples, *ti* is the test duration for each sample, and *r* is the number of failures.

If the number of failures is *r* ≥ 1 and the confidence level is 100(1 − *α*), the characteristic life, *ηα*, can be approximated from Equation (9),

$$
\eta\_a^\beta = \frac{2r}{\chi\_a^2(2r+2)} \times \eta\_{MLE}^\beta = \frac{2}{\chi\_a^2(2r+2)} \times \sum\_{i=1}^n t\_i^\beta \text{for} \\
r \ge 1 \tag{10}
$$

where *<sup>χ</sup>*2*α*() is the chi-square distribution when the *p*-value is *α.*

Assuming there are no number of failures, ln (1/*α*) is mathematically identical to the chi-square value, *<sup>χ</sup>*2*α*(2) 2 [46]. In other words,

$$p - value: a = \int\_{\chi^2(2)}^{\infty} \left( \frac{e^{-\frac{\pi}{2}} \mathbf{x}^{\frac{\chi}{2} - 1}}{2^{\frac{\nu}{2}} \Gamma\left(\frac{\nu}{2}\right)} \right) d\mathbf{x} = \int\_{2\ln a^{-1}}^{\infty} \left( \frac{e^{-\frac{\pi}{2}} \mathbf{x}^{\frac{\nu}{2} - 1}}{2^{\frac{\nu}{2}} \Gamma\left(\frac{\nu}{2}\right)} \right) d\mathbf{x} \text{for} \mathbf{x} \ge 0 \tag{11}$$

where Γ is the gamma function and *ν* is the shape parameter

For *r* = 0, the characteristic life *ηα* from Equation (10) can be defined as:

$$\eta\_a^\S = \frac{2}{\chi\_a^2(2)} \times \sum\_{i=1}^n t\_i^\S = \frac{1}{\ln\frac{1}{a}} \times \sum\_{i=1}^n t\_i^\S \tag{12}$$

As Equation (10) is proved for all cases *r* ≥ 0, characteristic life, *ηα*, can be expressed as follows:

$$\eta\_a^\beta = \frac{2}{\chi\_a^2(2r+2)} \times \sum\_{i=1}^n t\_i^\beta \text{ for} \\ r \ge 0 \tag{13}$$

If the logarithm in the Weilbull distribution is taken, the connection between characteristic life and BX life, *LB*, can be defined as:

$$L\_B^\S = \left(\ln\frac{1}{1-x}\right) \times \eta^\S \tag{14}$$

If the approximated characteristic life of the *p*-value α, *ηα*, in Equation (13), is changed into Equation (17), we obtain the BX life formulation:

$$L\_B^\S = \left(\ln\frac{1}{1-\chi}\right) \times \frac{2}{\chi\_a^2(2r+2)} \times \sum\_{i=1}^n t\_i^\S \tag{15}$$

As nearly all life testing commonly has inadequate samples to approximate the lifetime for the assigned number of failures that might be less than that of the sample size, the plan testing time can begin as:

$$nh^{\beta} \ge \sum t\_i^{\beta} \ge (n - r) \times h^{\beta} \tag{16}$$

If Equation (16) is exchanged with Equation (15), the BX life equation can be redefined as:

$$\frac{1}{2}L\_B^6 \cong \left(\ln\frac{1}{1-\mathbf{x}}\right) \times \frac{2}{\chi\_a^2(2r+2)} \cdot nh^6 \ge \left(\ln\frac{1}{1-\mathbf{x}}\right) \times \frac{2}{\chi\_a^2(2r+2)} \times (n-r)h^6 \ge L\_B^{\*\frac{\alpha}{6}} \tag{17}$$

If Equation (17) is rearranged, the sample size formulation with the failure numbers can be defined as:

$$m \geq \frac{\chi \chi\_a^2 (2r + 2)}{2} \times \frac{1}{\left(\ln \frac{1}{1 - x}\right)} \times \left(\frac{L\_B^\*}{h}\right)^{\beta} + r \tag{18}$$

Because *<sup>χ</sup>*2*α*(<sup>2</sup>*r*+<sup>2</sup>) 2 ∼= (*r* + 1) for *α* = 0.6 and ln(1 − *x*)−<sup>1</sup> = *x* + *x*22 + *x*33 + ··· ∼= *x*, the sample size Equation (21) can be simply close to:

$$m \ge (r+1) \times \frac{1}{\mathfrak{x}} \times \left(\frac{L\_B^\*}{h}\right)^{\mathfrak{f}} + r \tag{19}$$

where the sample size equation can be restated as *n* ~ (failure numbers + 1)·(1/cumulative failure rate)·((target lifetime/(plan testing time)) ˆ *β* + *r.*

If Equation (8) is attached to the plan testing time *h*, Equation (19) can replaced as:

$$m \ge (r+1) \times \frac{1}{\mathfrak{x}} \times \left(\frac{L\_{\rm B}^\*}{AF \cdot h\_{\rm d}}\right)^{\beta} + r \tag{20}$$

If the lifetime target of a mechanical system such as the HKS in a domestic refrigerator is assigned to be B1 life 10 years, the mission cycles might be attained for an assigned set of samples subjected to the food loading. In ALTs, the design flaws of the new product might be recognized to fulfill the lifetime target [47–49].

### *2.4. Case Study—Reliability Design of a Newly Designed HKS in Domestic Refrigerator*

When a consumer operates a refrigerator door, they want to comfortably close the door. A new HKS was designed for the refrigerator (see Figure 5) to enhance the ease of opening and closing the door for the consumer. When opening/closing the door, the HKS was subjected to repeated impact loads over the lifetime of the domestic refrigerator. To endure the loads of the HKS, new metals—standard austenitic ductile iron (18 wt% Ni)—for the torsional shaft were a key metal component [50] used. Due to their cheap cost as well as outstanding workability, ductile cast irons have been utilized for numerous mechanical parts. They have fine monotonic strength and high ductility compared to malleable cast irons and gray cast irons. The fatigue strength of ductile cast irons is comparatively lower than those of the steels and alloys with the identical quantity of monotonic strength because of their distinctive microstructure holding graphite particles and casting defects [51]. The fatigue strength of a ductile cast iron in the current HKS design was evaluated through parametric ALT.

**Figure 5.** A domestic refrigerator (**a**) and HKS (**b**) and its parts: (1) kit cover, (3) support, (4) torsional shaft (cast iron), (5) spring, and (6) kit housing (high-impact polystyrene, HIPS).

The HKS shown in Figure 5b consisted of a kit cover, torsional shaft (ductile iron), spring, and kit housing. To suitably work its function for a product lifetime, the HKS should be designed to endure the working circumstances subjected to it by the customers who utilize the refrigerator. In the Korean domestic market, the representative customer opened and closed the refrigerator door from three to ten times per day. Stocking food in the refrigerator had some repeated working procedures: (1) Open the door of refrigerator, (2) put the food into it, and then (3) close it. The HKS had different mechanical impact loadings when the customer utilized it.

The HKS in the marketplace had been fracturing, causing customers to demand the refrigerator be replaced. As subject to repeated impact stresses in using the refrigerator door, it was determined that the problematic HKS originated from several design defects. Market data also indicated that the returned products had crucial design problems on the structure, including stress risers—sharp corner angles and thin ribs. These design defects

prohibited the HKS from enduring the repeated impact loads during the openings/closings and resulted in a crack that propagated to its end. The HKS was originally designed to endure repeated impact loading under customer working conditions (Figure 6).

**Figure 6.** Damaged HKS in field after use.

When customers operated the refrigerator door, they could take out and put in food. Relying on the end-user working conditions, the HKS experienced repeated impact loading in the process. To correctly work the HKS, many mechanical structural parts in the HKS assembly needed to be designed robustly. As the concentrated stress in the mechanical system was revealed at stress raisers such as sharp corner angles, it was crucial to demonstrate these design flaws experimentally. As a result, engineers could then modify the design.

As seen in Figure 7, from the functional design ideas of a mechanical HKS, we knew that the impact force on the HKS came from the door weight. That is, the moment balance around HKS can be stated as

$$M\_0 = \mathcal{W}\_{dcor} \times R \tag{21}$$

$$(2\,1) = T\_0 = F\_0 \times R \tag{22}$$

where *b* is distance from the HKS to the center of gravity (CG) of the door.

**Figure 7.** Functional design ideas of a mechanical HKS.

To increase the impact on the HKS, additional accelerated weight was added. The moment balance around the HKS with an accelerated weight can be stated as

$$M\_1 = M\_0 + M\_A = W\_{door} \times b + M\_A \times a \tag{23}$$

$$(\text{23}) = T\_1 = F\_1 \times R \tag{24}$$

where *a* is distance from the HKS to the accelerated weight

Because the time to failure depended on the impact force due to moment, the impact was controlled during the accelerated life testing. Under the same working conditions, the life-stress model (LS model) in Equation (7) can be restated as

$$TF = A(S)^{-n} = AT^{-\lambda} = A(F \times R)^{-\lambda} = B(F)^{-\lambda} \tag{25}$$

where A and B are constant

> Therefore, the AF in Equation (8) can be restated as

$$AF = \left(\frac{S\_1}{S\_0}\right)^n = \left(\frac{T\_1}{T\_0}\right)^\lambda = \left(\frac{F\_1}{F\_0}\right)^\lambda \tag{26}$$

For a refrigerator including the HKS, the environmental (or working) customer conditions were roughly 0–43 ◦C with a relative humidity varying from 0 to 95%, and 0.2–0.24 g of acceleration. As previously mentioned, the number of openings/closings of the HKS per day varied from 3 to 10 times. With a design criterion of a product lifetime for 10 years, *<sup>L</sup>*<sup>∗</sup>*B*, the HKS has 36,500 usage cycles in the worst case.

Under a lifetime target—B1 life 10 years—if the number of lifetime cycles *L*∗*B* and AF are computed for the assigned sample size, the actual mission cycles, *ha*, might be acquired from Equation (20). Then, the ALT equipment can be made and performed in accordance with the working course of the HKS. Through parameter ALTs, the design missing parameters (or design flaws) for the new mechanical system can be identified.

The greatest impact force due to the door weight exerted by the customer in utilizing the refrigerator, F1, was 1.1 kN. To determine the stress level for ALT, we used the step-stress life test that can assess the lifetime under constant used-condition for various accelerated weights [52]. As the stress level to a different level was changed, the failure times of the HKS at a particular stress level was observed. Finally, for an ALT with an accelerated weight, we determined that the exerted impact force, F2, was 2.76 kN. With a cumulative damage exponent, *λ*, of 2, the AF was 6.3 from Equation (26). To obtain the missing design parameters of a newly designed HKS, a lifetime target should be more than B1 life 10 years. If the shape parameter *β* was 2.0, the number of test cycles computed from Equation (20) would be 23,000 cycles for 6 sample units. If the parametric ALT failed less than once for 23,000 cycles, the lifetime for the HKS would be assured to be B1 life 10 years (Figure 8).

The control console was used to run the testing apparatus—the number of test cycles, beginning or ending the equipment, etc. As the start knob on the controller console gave the starting signal, the straight hand-shaped arms clasped and raised the refrigerator door. When the door was shut, it was exerted to the HKS with the greatest mechanical impact force due to the accelerated load (2.76 kN).

**Figure 8.** ALT (**a**) equipment; (**b**) duty cycles of repeated impact load F.
