The Reliability Design and Its Direct Effect on the Energy Efficiency

1. Introduction

Reliability refersto the ability of system or component to performa required function under stated environmental andoperational conditions for a specified period of time. Traditionally,the reliabilityover the product life can be illustrated by a bathtub curve that has three regions: a decreasingrate of failure, a constant rate of failure, and an increasing rate of failure, as shown inFigure1(a). As the reliability of a product (or part) improves, failure of the part becomes less frequent in the field. The bathtubcurve may change into a straight line with the slope angle β. In a straight line there are two variables to be measured: product life L_B (or mean time between failures) and failure rate λ, as shown in Eq. (1):

$$\text{R}\left({\text{L}}_{\text{B}}\right)=e^{-\lambda L_B}\cong 1-\lambda L_B$$E1

Figure 1.

System life and failure rate

We can thus establish the reliability growth plan of parts with a constant failure rate.

A company generally designs its new products to (1) minimize initial failures, (2) reduce random failures during the expected product working period, and (3) lengthen product life. Such aims are met through the use of robust design techniques, including statistical design of experiment (SDE) and the Taguchi methods [1]. The Taguchi methods describe the robustness of a system for evaluation and design improvement, which is also known as quality engineering [2-3] or robust engineering [4]. Robust design processes include concept design, parameter design, and tolerance design [5].Taguchi’s robust design methods place a design in anoptimum position where random “noise” does not cause failure, which then and helps in determining the proper design parameters [6].

However, for a simple mechanical structure, the Taguchi methods’ robust design processes need to consider a large number of design parameters.They also have difficulty in predicting the product life, L_B (or MTBF).

In this study we present a new method for the reliability design of mechanical systems. This new method takes into account the fact that products with missing or improper design parameters can result in recalls and loss of brand name value.Based on the analysis of a failed refrigerator drawer and handle systems, we demonstrated our new reliability design method. The new method uses ALT; the new concept of product life,L_B; and sample size, as a novel means of determining proper design parameters [7-14].

1.1. Targeting the refrigerator B_X life and failure rate λ

The multi-unit refrigerator used as a case study for this method consists of a compressor, a drawer, a door, a cabinet, and other units. For the drawer, the B₁life of the new design is targetedto be over10 years with a yearly failure rate of 0.1%. The entire refrigerator’s B_X life can be obtained by summing up the failure rates of each refrigerator unit. The refrigerator’s B₁₂ life with the new design is targetedto be over10 years with a yearlyfailure rate of 1.2 % (Table 1) [19].

No	Units	Market Data		Design	Conversion	Expected	Target	Bx Life	Based Bx
Failure Rate	Bx Life	Failure Rate	Bx Life
1	Compressor	0.34	5.3	New	x5	1.70	0.10	10	B1.0
2	Door	0.35	5.1	Given	x1	0.35	0.15	10	B1.5
3	Cabinet	0.25	4.8	Modified	x2	0.50	0.10	10	B1.0
4	Drawer	0.20	6.0	New	x2	0.40	0.10	10	B1.0
5	Heat exchanger	0.15	8.0	Given	x1	0.15	0.10	10	B1.0
6	etc	0.50	12.0	Given	x1	0.50	0.50	10	B6.0
Sum	R-Set	1.79	7.4	-	-	3.60	1.10	10	B12.0

Table 1.

Total parametric ALT plan of refrigerator

1.2. Analysis. of the problemsidentified in field samples (loads analysis)

In the field, certain components in these refrigerators had been failing or making noise, causing consumers to replace their refrigerators. Data from the failed products in the field showed how common used the refrigerators under common usage conditions. Refrigerator reliability problems in the field occurwhen the parts cannot endure repetitive stressesdue to internal or external forces over a specified period oftime. The energy flow in a refrigerator (or other mechanical) systemcan generally be expressed as efforts and flows (Table 2)[15]. Thus, the stresses come from the efforts.

Refrigerator Units; (or Parts)	Effort, e(t)	Flow, f(t)
Mechanical translation; (draws, dispenser lever)	Force component, F(t)	Velocity component, V(t)
Mechanical rotation; (door, cooling fan)	Torque component, τ(t)	Angular velocity component, V(t)
Compressor	Pressure difference, ΔP(t)	Volume flow rate, Q(t)
Electric; (PCB, condenser)	Voltage, V(t)	Current, i(t)

Table 2.

Effort and flow in the multi-port system

For a mechanical system, the time-to-failure approach employs a generalized lifemodel (LS model) [16], such as:

$$T_f=A{\left(S\right)}^{-n}\exp \frac{E_a}{kT}=A{\left(e\right)}^{-n}\exp \frac{E_a}{kT}$$E2

Repetitive stress can be expressed as the duty effectthat carries the on/off cycles and shortens part life [17]. Under accelerated stress conditions, the accelerationfactor (AF) can be described as:

$$AF={\left(\frac{S_n}{S_0}\right)}^n\left[\frac{E_a}{k}\left(\frac{1}{T_0}-\frac{1}{T_1}\right)\right]={\left(\frac{e_1}{e_0}\right)}^n\left[\frac{E_a}{k}\left(\frac{1}{T_0}-\frac{1}{T_1}\right)\right]$$E3

And n can be determined by multiple testings with different stress levels.

1.3. Parametric. ALT with B_X life and sample size

Traditionally, the characteristic life is defined as:

$${\eta }^{\beta }\equiv \frac{{\displaystyle \sum t_i^{\beta }}}{r}\cong \frac{n\cdot h^{\beta }}{r}$$E4

As the reliability of a product (or part) improves,failures of the product become less frequent in laboratory tests. Thus, it becomes more difficult to evaluate the characteristic life using Equation (4).The distribution of failed samples should follow the Poisson distribution for small samples [18]. For a 60% confidence level, the characteristic life can be redefined as

$${\eta }^{\beta }\cong \frac{1}{r+1}\cdot n\cdot h^{\beta }$$E5

In order to introducethe B_X life in the Weibull distribution, the characteristic life can be modified as

$$L_B^{\beta }\cong x\cdot {\eta }^{\beta }=\frac{x}{r+1}\cdot n\cdot h^{\beta }$$E6

where L_B = B_X life and x = 0.01X, on the condition thatx≤ 0.2.

B_Xis the time by which X% of the drawer and handle system installed in a particular population of refrigerators will have failed.In order to assess theB_X life with about a 60% confidence level, the number of test samples is derived in Eq. (7). That is,

$$n\cong \frac{1}{x}\cdot \left(r+1\right)\cdot {\left(\frac{1}{h^{*}}\right)}^{\beta }$$E7

with the condition that the durability target is defined as follows,

$$h^{*}=\left(AF\cdot h\right)/L_B\ge 1$$E8

Based on the customer usage conditions, the normal range ofoperating conditions and cycles of the product (or parts) are determined. Under the worst case, the objective number ofcycles and the number of required test cycles can be obtainedfrom Eq. (7). ALT equipment can then be conducted on thebasis of load analysis. Using ALT we can find the missing or improper parameters in the design phase.

1.4. Refrigerator unit L_Bx life and failure rate, λ, with the improved designs

The parameter design criterion of the newly designed samplescan be more than the target life of B_X= 10 years. From the field data and from a sample under ALT with a corrective action plans, we can obtain the missing or improper parameters of parts and their levels in the design phase.

With the improved design parameters, we can derive theexpected L_Bx life of the final design samples using Equation (6).

$$L_B{}^{\beta }\cong x\cdot \frac{n\cdot {\left(h\cdot AF\right)}^{\beta }}{r+1}$$E9

Let x = λ L_B in Equation (9). The failure rate of the final design samples is derived in Equation (10)

$$\lambda \cong \frac{1}{L_B}\cdot \left(r+1\right)\cdot \frac{L_B^{\beta }}{n\cdot {\left(h\cdot AF\right)}^{\beta }}$$E10

2. Case study: Reliability design of arefrigerator drawer and handle system

Figure2 shows a refrigerator with the newly designed drawer and handle system and its parts. In the field, the refrigerator drawer and handle system had been failing, causing consumers to replace their refrigerators (Figure 3). The specific causes of failures of the refrigerator drawers during operation wererepetitive stress and/or the consumer improper usage. Field data indicated that the damaged products had structural design flaws, including sharp corner angles and weak ribs that resulted in stress risers in high stress areas.

A consumer stores food in a refrigerator to have convenient access to fresh food. Putting food in the refrigerator drawer involves opening the drawer to store or takeout food, closing the drawer by force. Depending on the consumer usage conditions, the drawer and handle parts receive repetitive mechanical loads when the consumer opens and closes the drawer.

Figure 4 shows the functional design concept of the drawer and handle system. The stress due to the weight load of the food is concentrated on the handle and support slide rail of the drawer. Thus, the drawer must be designed to endure these repetitive stresses.

The force balance around the drawer and handle system cans be expressed as:

$$F_{draw}=\mu W_{load}$$E11

Figure 2.

Refrigerator and drawer assembly. (a) French refrigerator (b) Mechanical parts of the drawer: handle (1), drawer (2), slide rail (3), and pocket box (4)

Figure 3.

A damaged product after use

Figure 4.

Functional design concept of the drawer and handle system

Because the stress of the drawer and handle system depends on the food weight, the life-stress model (LS model) can be modified as follows:

$$T_f=A{\left(S\right)}^{-n}=A{\left(F_{draw}\right)}^{-n}=A{\left(\mu W_{load}\right)}^{-n}$$E12

where A is constant. Thus, the acceleration factor (AF) can be derived as

$$AF={\left(\frac{S_1}{S_0}\right)}^n={\left(\frac{F_1}{F_0}\right)}^2={\left(\frac{\mu W_1}{\mu W_0}\right)}^2={\left(\frac{W_1}{W_0}\right)}^2$$E13

3. Laboratory experiments

The normal ranges of the operating conditions for the drawer system and handle were 0 to 50℃ ambient temperature, 0 to 85% relative humidity and 0.2 to 0.24G vibration. The normal number of operating cycles for one day was approximately 5; the worst case was 9. Under the worst case, the objective drawer open/close cycles for ten years would be 32,850 cycles (Table 3).

Item	Number of operations (times)
1 day		10 years
Normal	Worst	Normal	Worst
Drawer	5	9	18,250	32,850

Table 3.

Operating number of a drawer

For the worst case, the food weight force on the handle of the drawer was 0.34 kN. The applied food weight force for the ALT was 0.68 kN. With a quotient, n, of 2, the total AFwas approximately 4.0 using equation (13).

The parameter design criterion of the newly designed drawer can be more than the target life of B₁ = 10 years. Assuming the shape parameter β was 2.0 and xwas 0.01, the test cycles and test sample numbers calculated in Equation (7) were 67,000 cycles and 3units, respectively. The ALT was designed to ensure a B₁ life of 10 years with about a 60% level of confidence that it would fail less than once during 67,000 cycles.

Figure 5.

ALT equipment and duty cycles.

Figure 5 shows ALT equipment and duty cycles for the repetitive food weight force, F_draw. For the ALT experiments, the control panel on top of the testing equipment started and stopped the drawer during the mission cycles. The food load, F, was controlled by the accelerated weight load in the drawer storage. When a button on the control panel was pushed, mechanical arms and hands pushed and pulled the drawer.

4. Parametric ALTs with corrective action plans

Figures 6(a) and 6(b) show the failed product from the field and the 1^st accelerated life testing, respectively. The failure sites in the field and the first ALT occurred at the drawer handle as a result of high concentrated stress. Figure 7 shows a graphical analysis of the ALT results and field data on a Weibull plot. For the shape parameter, the estimated value on the chart was 2.0. For the final design, the shape parameter was determined to be 3.1. These methodologies were valid for pinpointing the weak design responsible for failures in the field and 1^st ALT.

Figure 6.

Failed products in field and first ALT; (a) Failed product in field and (b) Failed sample in first accelerated life testing

Figure 7.

Field data and results of 1^stALT on Weibull chart.

Figure 8.

Failed slide rails in second ALT

The fracture of the drawer in the first and second ALTs occurred in the handle and slide rails (Figure 6(b) and Figure 8). The missing or improper parameters of the handle and slide rails in the design phase arelisted in Table 4. These design flaws can result in a fracture when the repetitive food load is applied.

CTQ	Parameters			Unit
Fracture	KNP	N1	Consumer food loading	kN
KCP	C1	Reinforced handle width	-
C2	Handle hooker width	-
C3	Fastening screw number	-
C4	Slide rail chamber	mm
C5	Slide rail boss thickness	mm
C6	New added rib	-

Table 4.

Vital parameters based on ALTs

To prevent the fracture problem and release the repetitive stresses, the handle and slide rails were redesigned. The corrective action plan for the design parameters included: (1) increasing the width of the reinforced handle, C1, from 90mm to 122mm; (2) increasing the handle hooker size, C2, from 8mm to 19mm; (3) increasing the rail fastening screw number, C3, from 1 to 2; (4) adding an inner chamber and plastic material, C4, from HIPS to ABS; (5) thickening the boss, C5, from 2.0mm to 3.0mm; (6) adding a new support rib, C6 (Table 5).

The parameter design criterion of the newly designed samples was more than the target life, B₁, of ten years. The confirmed value,β,on the Weibull chart was 3.1. For the second ALT, the recalculated test cycles and sample size in Equation (7) were 32,000 and 3units, respectively. In the third ALT, no problems were found with the drawerafter 32,000 cycles and 65,000 cycles. We therefore concluded that the modified design parameters were effective.

Table 6 provides a summary of the ALT results. Figure 9 shows the results of the 1^stALT and 3^rd ALT plotted in a Weibull chart. With the improved design parameters, the B₁ life of the samples in the third ALT was lengthened tomore than 10.0 years.

Handle	Right/left slide rail
C1: Width L90mm → L122mm (1^st ALT); C2: Width L8mm → L19mm (1^st ALT)	C3: Rail fastening screw number 1→2 (2^nd ALT); C4: Chamfer: Corner chamfer; Plastic material HIPS → ABS (2^nd ALT); C5: Boss thickness 2.0 → 3.0 mm (2^nd ALT); C6: New support rib (2^nd ALT)

Table 5.

Redesigned handle and right/left slide rail

Figure 9.

Results of 1^stALT and 3^rd ALT plotted in Weibull chart

	First ALT	Second ALT	Third ALT
Initial design	First design iteration	Final design
In 32,000 cycles, Fracturing is less than 1	7,500 cycles: 2/3 Fail; 12,000 cycles: 1/3 OK; λ = 26.6%/year; B1 = 3.4 year	16,000 cycles: 2/3 Fail; λ= 2.46%/year; B1 = 7.3 year	32,000 cycles: 3/3 OK; 65,000 cycles: 3/3 OK; λ = 0.1%/year; B1 = 10 year
Drawer structure
Material & Spec.	Width1: L90 →L122; Width2: L8→L19.0	Rib1: new support rib; boss: 2.0 → 3.0 mm; Chamfer1: Corner; Material: HIPS →ABS

Table 6.

Results of ALTs

6. Conclusions

We developed a new reliability design method based on a study of a defective refrigerator drawer and handle system that was failing under field use conditions. The failure modes and mechanisms for the drawer in the field and in the ALTs were identified. Important design parameterswere studied and improvements were evaluated using ALTs.

Based on the products returned from the field and the results of the firstALT, we found that the handles were fracturing because of design flaws. The handle design was corrected by increasing the handle width. During the second ALT, the slide rails fractured because they did not have enough strength to endure the repetitive food storage loads. The slide rails were corrected by providing additional reinforced ribs, reinforced boss, and an inner chamber. As a result these modified design parameters, there were no problems in the third ALT. We therefore concluded that the values for the design parameterswere effective to meet the life cycle requirements. The yearly failure rate and B₁ life of the redesigned drawer and handle system, based on the results of ALT, were under 0.1% and more than 10 years, respectively. The study of the missing or improper design parameters in the design phase, through the inspection of failed products in the field, load analysis, and ALTs was very effective in redesigning more reliableparts with significantly longer life.

The casestudy focused on a mechanical structure consisting ofseveral parts subjected to repetitive stresses under consumer usage conditions. The same principles developed for the new reliability design methodology could be applied to other mechanical systems, including construction equipment, automobile gear trains and engines,forklifts, washing machines, vacuum cleaners, and motor fansystems. We recommend that the missing or improper controllable design parameters on these systemsalso be studied for reliability design. These parameter studies would also include failure analysis, load analysis, and a tailored series of accelerated life tests. Thesemethodologies could then predict part life quantitatively through accelerated factors and exact sample size.

7. Nomenclature

1. AF - acceleration factor

2. B_X - durability index

3. C1 - width of reinforced handle, mm

4. C2 - width of handle hooker, mm

5. C3 - back rib of slide rail

6. C4 - screw boss height of slide rail, mm

7. C5 - inner chamber of slide rail

8. C6 - material of slide rail

9. C7 - screw number of slide number

10. e - effort

11. e₀ - effort under normal stress conditions

12. e₁ - effort under accelerated stress conditions

13. E_a - activation energy

14. f - flow

15. F(t) - unreliability

16. F_draw - open/close force of the freezer drawer system, kN

17. F₁ - weight force under accelerated stress conditions, kN

18. F₀ - weight force under normal conditions, kN

19. h - testing time (or cycles)

20. h^* - non-dimensional testing cycles, $$h^{*}=h/L_B\ge 1$$

21. i - current, A

22. k - Boltzmann’s constant, 8.62 x 10^-5eV/deg

23. KCP - Key Control Parameter

24. KNP - Key Noise Parameter

25. L_B - the target B_X life and x = 0.01X, on the condition thatx≤ 0.2

26. n - the number of test samples

27. N1 - consumer freezer door drawer open/close force, kN

28. ΔP - pressure difference, MPa

29. r - failed numbers

30. R - reliability function

31. S - stress

32. S₀ - mechanical stress under normal stress conditions

33. S₁ - mechanical stress under accelerated stress conditions

34. t_i - test time for each sample

35. T - absolute temperature, K

36. T₁ - absolute temperature under accelerated stress conditions, K

37. T₀ - absolute temperature under normal stress conditions, K

38. T_f - time to failure

39. V - velocity, m/s

40. V - voltage, volt

41. W₁ - food weight force under accelerated stress conditions, kN

42. W₀ - food weight force under normal stress conditions, kN

43. W_load - total food weight force in the freezer door drawer, kN

44. X - accumulated failure rate, %

45. x - x = 0.01 X, on condition thatx≤ 0.2.

Greek symbols

1. η - characteristic life

2. λ - failure rate

3. μ - friction coefficient

Superscripts

1. β - shape parameter in a Weibull distribution

2. n - stress dependence, $$n=-{\left[\frac{\partial \ln (T_f)}{\partial \ln (S)}\right]}_T$$

Subscripts

1. 0 - normal stress conditions

2. 1 - accelerated stress conditions