**2. Parametric ALT for Mechanical System**

*2.1. Definition of BX lifetime for Putting a Whole Parametric ALT Plan*

To carry out a parametric ALT, the BX life as a measure of system lifetime is required. The BX life, LB, can be explained as the elapsed time at which X percent of a collection of a selected product might have failed. Otherwise, 'BX life Y years' is a good expression for product lifetime that helps to satisfactorily decide the cumulative failure rate of a product and respond to field circumstances. For instance, if the lifetime of a product has a B20 life of 10 years, then 20% of the population might have been unsuccessful in achieving one's goal for 10 years of the working period.

Reliability might be explained as the system's ability to work under specified conditions for a stated period of time [40]. Product reliability, as shown in Figure 2, is often illustrated with the "bathtub curve" that is composed of three sections [41]. First, there is a declining failure rate in the earlier product life (β < 1). Secondly, there is a constant failure rate (β = 1) in the middle of the product's life. Lastly, there is a growing failure rate at the end of the product life (β > 1). If a manufacturer produces a product whose failure rate follows the bathtub curve, it might have difficulties achieving success in the marketplace because of shorted lifetime and large failure rates due to design faults in the early product life. Manufacturers need to enhance the product design by increasing its reliability targets to (1) eliminate untimely failures, (2) lessen random failures over the product lifetime, and (3) lengthen system lifetime. As the design of a mechanical product improves, its failure rate in the marketplace should decease and the product lifetime should be extended. For such circumstances, the conventional bathtub curve might be changed to a straight line in Figure 2.

**Figure 2.** Bathtub curve and straight line.

The failure rate on the bathtub (or straight line) can be defined as

$$
\lambda = \frac{f}{R} = \frac{dF/dt}{R} = \frac{\left(1 - R\right)'}{R} = \frac{-R'}{R} \tag{1}
$$

where *λ* is the failure rate, *f* is the failure density function, *R* is reliability, and *F* is unreliability.

If Equation (1) is integrated over time, we can obtain the *X*% cumulative failure *F*(*LB*) at *BX* life, *LB*. That is, 

$$
\int \lambda dt = -\ln R \tag{2}
$$

That is to say, it can be expressed as:

$$A = \langle \lambda \rangle \cdot L\_B = \int\_0^{L\_B} \lambda(t) \cdot dt = -\ln \mathbb{R}(L\_B) = -\ln(1 - F) \cong F(L\_B) \tag{3}$$

where *LB* is the BX life, A is the area that can be obtained from the multiplication of failure rate, *λ*, and BX life, *LB.*

Consequently, if a product failure follows an exponential distribution, the reliability of a mechanical product can be defined as:

$$R(L\_B) = 1 - F(L\_B) = e^{-\lambda L\_B} \stackrel{\sim}{=} 1 - \lambda L\_B \tag{4}$$

Equation (4) is relevant for when there are less than approximately 20% of the cumulative failures for the system [42]. The mechanical system could be improved by obtaining the objective product lifetime, *LB*, and failure rate, *λ,* after optimally identifying the market failure by parametric ALT and modifying the problematic design (or material) of structures (Figure 3).

**Figure 3.** Parameter diagram of hinge kit system (HKS) (example).

In seeking to improve the lifetime target of a mechanical system through an ALT examination, there are three potential product modules: (1) An altered module, (2) a newly designed module, and (3) an alike module to the previous design base on demand in the marketplace. The newly designed HKS in the refrigerator examined here as a case study was a new module that had design faults that had to be rectified because customers asked for replacements with a new one because the product failed during its expected lifetime.

The new module *B* from the market data shown in Table 1 had a failure rate of 0.24% per year and a B1 life of 4.2 years. To answer customer requests, a new lifetime target for the HKS was set to have B1 life 10 years with a cumulative failure rate of one percent.

### *2.2. Failure Mechanics and Accelerated Testing for Design*

Mechanical systems typically move energy and power from one location to another through mechanical mechanisms. If there is a design fault in the structure that causes an inadequate strength (or stiffness) when the loads are exerted, the mechanical system may suddenly fail before its anticipated lifetime. Fatigue due to design flaws can be characterized by two factors: (1) the stress due to loads on the structure and (2) the type of materials (or shape) used in the product. In reproducing the system failure by a parametric ALT, a designer could optimally design components with proper shapes and materials. The product could sustain repetitive loads over its lifetime so that it could achieve the targeted reliability (Figure 4).


**Table 1.** Whole ALT plan of mechanical system such as modules in a refrigerator.

The most important issue for a reliability test is how quickly the possible failure mode might be obtained. A failure model must be derived and its associated coefficients determined. The life-stress (LS) model also incorporates stresses and reaction parameters. The generalized life-stress (LS) model [1,43,44] might thus be defined as

$$TF = A \left[ \sinh(aS) \right]^{-1} \exp\left(\frac{E\_4}{kT}\right) \tag{5}$$

The sine hyperbolic expression [sinh(aS)]−<sup>1</sup> in Equation (5) can be expressed as:


An ALT is normally performed in the medium range, and Equation (5) might be defined as 

$$TF = A(S)^{-n} \exp\left(\frac{E\_d}{kT}\right) \tag{6}$$

As the stress of a mechanical system may not be easy to measure during testing, Equation (6) must be redefined. When the power is defined as the multiplication of flows and effort, stresses may come from effort in a multi-port system (Table 2) [45].


**Table 2.** Power definition in a multi-port system.

Stress is a physical quantity that indicates the internal forces that adjacent particles of a continual material apply on each other. For a mechanical system, because stress comes from effort, Equation (6) might be redefined as

$$TF = A(S)^{-n} \exp\left(\frac{E\_d}{kT}\right) = B(\varepsilon)^{-\lambda} \exp\left(\frac{E\_d}{kT}\right) \tag{7}$$

where A and B are constants

To derive the acceleration factor (AF) that can mainly enfluence the assessment of fatigue strength in product, expressed as the inverse of the stress ratio, R (=σmin/σmax), from Equation (7), AF might be expressed as the proportion between the adequate elevated stress amounts and normal working conditions. AF might be altered to incorporate the effort ideas:

$$AF = \left(\frac{S\_1}{S\_0}\right)^n \left[\frac{E\_d}{k}\left(\frac{1}{T\_0} - \frac{1}{T\_1}\right)\right] = \left(\frac{c\_1}{c\_0}\right)^\lambda \left[\frac{E\_d}{k}\left(\frac{1}{T\_0} - \frac{1}{T\_1}\right)\right] \tag{8}$$
