## 2 Parametric ALT for a System Operated by a Machinery
### Case Study--Increasing the Life of a New Water Dispenser System in a BMF Refrigerator  
Because consumers want to have the convenience of cold water being dispensed from a domestic refrigerator, a water dispenser system, including a lever, a cover, and a spring, was designed into the front of the refrigerator. The dispenser lever was also changed from stainless steel to polypropylene (PP) because of market requirements, such as material cost. When the consumer uses a cup to exert force on the dispenser lever, cold water is dispensed to the cup. They require high-strength fatigue at room temperature because of the repeated impact stresses (Figure 7). First, the customer usage conditions should be investigated. In the dispensing operation, components, such as a lever made of stainless steel or polypropylene (PP), in a dispenser system experience various mechanical loads and are required to have enough strength not to fail due to fatigue until the anticipated lifetime. Domestic refrigerators in the United States are designed to distribute water from four to twenty times per day. As the dispenser is pressed, it is repeatedly subjected to (impact) loads on the lever. In the field, water dispensing systems in a domestic refrigerator failed under unspecified consumer operations after some period. As investigated, field data demonstrated that the products returned from the field might have had design defects, such as stress raisers (sharp corner angles and thin ribs) in the dispenser. As the consumer requested it to be replaced because the lever systems suddenly fractured and no longer worked, engineers needed to find the root causes by failure analysis or reliability testing and correct them (Figure 8). By employing failure analysis (and laboratory tests) for returned field products, a crack was observed that started in the front corner of the lever and propagated to the end of the lever. To keep it functioning for its anticipated lifetime, an engineer was required to redesign the product with imperfections (sharp corner angles, thin ribs, etc.). To reproduce the problematic part(s) and alter them, an engineer needed to carry out parametric ALT for the newly designed product by targeting product lifetime, \(L_{B}\)--a B1 life of 10 years in Equation (6) in Sections 2.1 and 2.2. The systematic method was made up of (1) a load examination for the problematic returned product in the subsequent Equations (37)-(39), which is connected in Equation (23) in Section 2.3, (2) the action of making the practical and effective use of ALTs with design alternations, and (3) the evaluation of whether the lifetime objective of designs had been fulfilled by attaining from the actual mission cycle, \(l_{a}\), in Equation (35) in Section 2.4. From the free-body illustration of the water dispenser (Figure 9), the balance of momentum and force can be stated as:  
\[\sum M_{z}=aF_{X}-bF_{Y}=0 \tag{37}\]  
\[F_{X}=(b/a)F_{Y} \tag{38}\]  
Figure 8: Damaged dispenser lever after usage. (**a**) Water dispenser and (**b**) failed dispenser lever in the field. Figure 7: New water dispenser assembly in a refrigerator. (**a**) A domestic refrigerator and (**b**) mechanical components of the water dispenser. (1) Dispenser lever cost-downed from stainless steel, (2) spring, and (3) cover. As the lever stress depends on the impacted force, \(F\), of the cup, Equation (23) might be stated as:  
\[TF=A(S)^{-n}=A(F_{X})^{-\lambda}=B(F_{Y})^{-\lambda} \tag{39}\]  
where \(A\) and \(B\) are constant quantities. The acceleration factor (\(AF\)) from Equation (24) can be expressed as:  
\[AF=\left(\frac{S_{1}}{S_{0}}\right)^{n}=\left(\frac{F_{1}}{F_{0}}\right)^{\lambda} \tag{40}\]  
where \(S_{1}\) (or \(F_{1}\)) is structural stress (or impact force) under accelerated circumstances and \(S_{0}\) (or \(F_{0}\)) is structural stress (or impact force) under normal circumstances. Parametric ALT from Equation (35) can be performed until the assigned cycles that supply the aimed life-- B1 life of 10 years--are fulfilled.