Volume 10, Issue 2

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Using DOE Results in the Design of an Accelerated Life Test

Quantitative Accelerated Life Testing (QALT) is a powerful tool and one of the most interesting topics in reliability engineering. What’s unique about QALT is that it is a combination of statistics, engineering and physics of failure. Stress is increased during the test in order to excite the mechanism of failure and cause it to occur in a shorter period of time relative to the field operation. The underlying physics of failure need to be understood prior to performing the test; otherwise, the test could simply be useless. In fact, a successful accelerated life test requires:

  1. A firm understanding of the failure mechanisms to be accelerated.
  2. A firm understanding of the factors affecting the failure mechanisms.
  3. A determination of the intensity levels of these factors.

These three elements can be attained by investigating the underlying physics governing the mechanisms of failure or from engineering experience. The factors that determine a product's life behavior, such as the material, geometry and technology, can be correlated to the above mentioned elements, thus resulting in a well-executed accelerated life test. For instance, given the material of the product, it can be known that exceeding a certain stress level will result in foolish failures (i.e., failure modes that would not occur under normal use conditions). Let’s take a printed circuit board (PCB) as a specific example. It is well understood that temperature and thermal cycling are two of the contributing factors to PCB life expectancy. Based on the specific material used, we know that there is a maximum temperature not to be exceeded during a test; otherwise, nuisance failures will occur.

Even if all of the first three elements are taken into account, we still need to ensure that the accelerated test will yield failures within the allowable test time. So the fourth element in the list is:

  1. Sufficient test time.

Accelerated test planning methods and tools exist for assisting engineers to design successful accelerated life tests. But how can we ensure that engineers have the information they need for using these methods to create well-designed tests? For example, what if prior information about the factors influencing the life of the product are not known or well understood? What if the factors are known but their intensity levels are not? In such cases, experimentation can supplement engineering or physics-based knowledge. Depending on which questions need to be answered, different types of experiments can be performed prior to the accelerated life test. The most commonly used (and recommended) types are:

  • Design of Experiments (DOE). With the DOE methodology, the significant factors affecting the life of the product (i.e., stresses) can be determined. This will assist in designing more efficient accelerated life tests that do not waste time or resources by including unnecessary factors. [Note that as the number of factors included in an accelerated life test increases, the sample size requirements increase as well because units need to be tested at different level combinations of these factors.]
  • HALT/HASS. If the maximum levels of the stresses under consideration are not known (i.e., if the foolish failure levels are not known), then highly accelerated life/stress tests (HALT/HASS) can assist in determining them.

The results of such tests can provide the inputs for the accelerated test plans. In this article, we concentrate on the role of DOE and how we can use DOE results to plan a better accelerated life test. We will give brief introductions to the topics of QALT planning and DOE, then we will give an example to demonstrate how DOEs can complement QALT planning.

Introduction to QALT Plans

The purpose of a QALT plan is to minimize the uncertainty of the life and reliability extrapolations to normal use conditions by determining the optimum stress levels to be used in the test as well as the optimum allocation of test units at each level. QALT planning can be performed using simulation or analytical methods. In either case, a minimum amount of information needs to be provided in order to determine the optimum settings for the test. All QALT planning techniques begin with the assumptions that a) the underlying physics of failure are understood, b) the test will consider only the significant factors and c) the maximum stress levels are known for all stresses. We will review the analytical test planning methods here because they are more frequently used and provide less ambiguous solutions than simulation methods.

For analytical methods, the minimum input requirements are:

  1. Number of stresses under consideration. This is the total number of factors that will be included in the test.
  2. Life-stress relationship. This is the physics-based model that relates life with stress. This relationship is determined by the underlying physics of failure and allows us to compute the reliability of the product at different stress conditions. For example, the use of the power law equation on S-N curves (or Wöhler curves) in fatigue is such a relationship. The Arrhenius model for thermal stressing is another. The relationship needs to be specified for every stress under consideration.
  3. Use stress level. This is the stress level at which the design is expected to operate under normal use conditions.
  4. Highest (or maximum) stress level. This is the maximum stress level that the design can withstand without introducing foolish failure modes.
  5. Use level unreliability criterion. This represents the percentile of the distribution at use conditions that needs to be estimated from the QALT data analysis. In other words, this is the objective point of the use level distribution where extrapolations will be performed. This is needed in order to have a criterion such that, using the failure data obtained from the test, we can make predictions with a relatively small uncertainty.
  6. Test duration. This is the length of time for which the units will be tested.
  7. Probability of failure for the specified test duration at the specified use stress level. This is an estimate of the percentage of the units that would fail if the test were performed at the normal use conditions for the specified test duration.
  8. Probability of failure for the specified test duration at the specified highest stress level. This is an estimate of the percentage of the units that would fail if the test were performed at the maximum stress level conditions for the specified test duration.

The last two inputs are needed in order to determine the optimum stress levels at which to run the test. These levels are determined by ensuring that a) enough failures are observed within the specified test duration and b) the test stress levels are not too far from the use conditions, thus reducing the extrapolation error. The extrapolation error is computed in conjunction with the use level unreliability criterion.

It can been seen that if these inputs are not known a priori, a lot of guesswork will be involved, thus increasing the probability of an unsuccessful QALT (e.g., not enough failures observed during the test or observing unwanted failure modes). So how can we obtain better estimates for these inputs? As described in the next section, the analysis of DOE test results is one possibility.

Introduction to DOE Principles

The design and analysis of experiments revolves around the understanding of the effects of different variables on another variable. In mathematical jargon, the objective is to establish a cause-and-effect relationship between a number of independent variables and a dependent variable of interest. The dependent variable, in the context of DOE, is called the response and the independent variables are called factors. Experiments are run at different factor values, called levels. Each run of an experiment involves a combination of the levels of the investigated factors. Each of the combinations is referred to as a treatment. When the same number of response observations are taken for each of the treatments of an experiment, the design of the experiment is said to be balanced. Repeated observations at a given treatment are called replicates. [1]

The number of treatments of an experiment is determined on the basis of the number of factor levels being investigated in the experiment. For example, if an experiment involving two factors is to be performed, with the first factor having x levels and the second having z levels, then x x z treatment combinations can possibly be run and the experiment is an x x z factorial design. If all x x z combinations are run, then the experiment is a full factorial design. If only some of the x x z treatment combinations are run, then the experiment is a fractional factorial design. In full factorial experiments, all of the factors and their interactions can be investigated, whereas in fractional factorial experiments, certain interactions are not considered because not all treatment combinations are run.

To summarize, the objective of DOE is to identify the "vital few" factors in the most efficient way. In other words, this technique allows the determination of the significant factors affecting a response (dependent variable) with the minimum number of runs. In the context of QALT planning, DOEs can assist in determining the significant stresses affecting the life of the product and, in addition, provide better estimates for some of the inputs required for QALT plans. Specifically:

  • The most obvious outcome is the determination of which stresses to consider during the accelerated life test. In the absence of engineering or prior knowledge, a DOE can be set up to test which stresses have the most significant impact on the life of the product (input #1 in the list of requirements for QALT planning). The response in this case is life (or a degradation metric indicative of life) and the factors are the stresses.
  • A less obvious use of DOE is the possibility of obtaining better estimates for some of the additional inputs required for an accelerated test plan. These include #4, #6, #7 and #8 in the list of QALT planning inputs. [Note that DOE is not the ideal technique for determining the maximum stress level (#4). If foolish failure modes were observed due to inappropriate stress settings, additional tests with lower stress levels may need to be conducted.]
  • DOEs can also assist in determining the form of the life-stress relationship (#2 in the list of QALT planning inputs). Specifically, after conducting the DOE analysis, if it is determined that the interactions between the factors are significant, then a physics-based model that includes the interaction terms would have to be used.

The following case study demonstrates some of these uses of DOE to design an optimum accelerated life test plan.

Case Study

This case study involves an effort to design an accelerated life test for a mechanical component. The component under consideration was a joint between two shafts. The whole assembly was rotating ±360o from the resting position. The failure mechanism was the wearout of the joint components as cycles were accumulated over time. Based on engineering judgment, it was believed that there were three factors affecting the life of this joint; namely, the applied torque, the speed of the rotation (RPM) and the stroke (i.e., the linear movement, in either direction, of a reciprocating mechanical part). The purpose of the testing was to determine the life of the product and to use the results to create a demonstration test that could be used in future iterations of the product for years to come.

In order to determine whether the three identified factors were indeed significant, a DOE was planned. The engineering team decided to run a full factorial design due to the importance of the study for establishing a baseline for future demonstration testing and also in order to investigate any interactions between the factors that could be significant.

One of the challenges in this case was the expected high durability of the design. It was expected that the design could withstand hundreds of thousands of cycles without a failure, even at accelerated conditions (although the exact reliability under accelerated conditions was unknown). However, the allowable test time was on the order of thousands of cycles. Since the response in this experiment was life (or more specifically, cycles-to-failure), the concern was that measurement of the response probably would not be possible within the allowable test time. To resolve this issue, the team chose to measure the physical degradation of the component, which is correlated with the eventual failure. Thus, if no failures were observed during the testing, the degradation measurements (i.e., the wearout) could be used as a response instead.

One more element in the design of this experiment was the issue of manufacturing and material variation. In order to account for such variability and to make the experiment more robust, it was decided to run 3 replicates per treatment. Thus, the final design was a 2^3 design since it was decided to test at 2 levels for each factor. This results in 8 combination runs and, with 3 replicates to be considered for each run, the total number of runs was 24. Figure 1 shows the experiment design in ReliaSoft's DOE++ software.

Figure 1: Full factorial experiment design in DOE++

DOE Results

As expected, no failures were observed by the end of the experiment. However, since the samples were checked at predefined intervals and the wearout at the joint was measured each time, the values of the degradation at the end of the test (in this case, in mm) could be used as the response. The analysis of the data demonstrated at the 10% significance level that only the torque and stroke were significant factors, with the speed and all interactions not being significant. The Pareto chart in Figure 2 shows these results.

Figure 2: Pareto chart of factors

Based on this outcome, the analysts could proceed with the design of the accelerated test by considering only these two significant factors and, since the interaction terms were also found to be not significant (with statistical vigor), using a physics-based model that does not include an interaction term.

QALT Planning Inputs

From the DOE analysis, it was determined that only torque and stroke were the significant factors, without any interactions. This is enough information to start planning the QALT but it is not enough for a robust plan. The following picture displays the inputs required for a more accurate QALT plan. This picture shows the test planning tool from ReliaSoft's ALTA software, which provides several analytical methods for QALT planning. [2,3]

The information available up to this point is enough to complete the first portion of ALTA's test planning tool. Two stresses will be considered in this test, where the maximum stress levels are determined based on the physical characteristics of the design as well as the test equipment limitations, and the use stress levels are determined based on the operating profile of this assembly in the field. The stress relationship is a power law model for each of the factors (based on physics for a wearout mechanism).

Determining most of the rest of the inputs will require an understanding of (or at least an educated guess about) the life distribution and its behavior as a function of the stress factors. This will provide the distribution input for this method as well as the values of the three probabilities needed. In this particular case study, this understanding of the life distribution as a function of the stresses does not need to be guessed because sufficient data are available from the DOE analysis to estimate it using traditional QALT analysis methods. In other words, instead of "guestimating" the distribution and the three probabilities required for a more accurate test plan, the analysts can actually estimate them from the DOE data.

QALT Analysis of the DOE Data

From the DOE test, the 24 runs provide sufficient data for fitting a life distribution and a life-stress relationship. Since the experiment was a 2^3 design, it provides 8 combinations of the stresses. Also, since 3 replicates were performed at each treatment level, we end up with 3 data points at each combination. In addition, since the speed was determined to be not statistically significant from the DOE analysis, we end up with 4 combinations of the 2 stresses under consideration (torque and stroke) and 6 data points at each of these combinations. Therefore, enough data points are available to perform a QALT analysis. [Note that even if speed were considered as a factor, there still would be enough data for such an analysis.]

The results obtained from the DOE were not failure data because no failures occurred during the test. Essentially, the data set consists of right censored data, which is not sufficient to build a life distribution as a function of torque and stroke. However, this can be addressed by fitting a degradation model to the data because the wear was measured on each test specimen at different age intervals (in cycles). Based on the degradation model, we can extrapolate and predict the cycles-to-failure (i.e., the point at which the predicted model intercepts the critical degradation level). Figure 3 shows the results of this degradation analysis.

Figure 3: Degradation Analysis

The predicted failure data can now be analyzed and a life distribution with an underlying life-stress relationship can be fitted. In this case, we use the Weibull distribution with power law relationships for both torque and stroke. Figure 4 shows these results.

Figure 4: QALT analysis of the degradation data

From the QALT analysis, the following probabilities can be determined:

  • P(3000, 25, 3) = 0.05
  • P(3000, 80, 3) = 0.11
  • P(3000, 25, 6) = 0.87

After including this information in the QALT planning tool, Figure 5 shows the optimum test plan that is generated. Under this plan, 13% of the available samples should be tested at Torque = 80Nm and Stroke = 6mm, 28% at Torque = 80Nm and Stroke = 3.4mm and 59% at Torque = 25Nm and Stroke = 3.9mm.

Figure 5: Optimum accelerated life test plan

At this point, it could be argued that there is no need to proceed with the accelerated life test because the analysis of the DOE data can provide the reliability predictions at use conditions. However, it must be noted that the failure times used for this analysis were not actual but predicted from the degradation analysis. Even though this is a perfectly acceptable technique, one may wish to achieve more accuracy (i.e., less uncertainty) in the results by reducing the number of assumptions and extrapolations. [Note that, in essence, a "double extrapolation" was performed here by first extrapolating to solve for the cycles-to-failure (degradation model) and then extrapolating again to the use level conditions (life-stress relationship).] In this case, the engineers required a higher confidence in the results. Therefore, they proceeded with the optimized accelerated test plan and were able to observe actual failures for the majority of the samples. The QALT results were very close to the expectations from the first phase of the analysis and did help the engineers to solidify the procedure for future applications.

Summary and Conclusions

In this article, we presented a case study that demonstrates how DOE data can be used to design better accelerated tests. Specifically, DOE data allowed us to:

  • Eliminate non-significant factors, thus reducing wasted time and resources from testing for insignificant stresses.
  • Perform a "quick and dirty" QALT analysis (i.e., fit a distribution and a life-stress relationship to the DOE response data). This analysis provided more accurate estimates for the inputs to QALT planning techniques.

In the case study described in this article, there were enough replicates at each treatment level in the DOE testing to make it possible to perform a QALT analysis with 6 data points per stress combination. However, it should be noted that this methodology could be used even if fewer replicates were performed and, in some cases, even with no replicates. For example, in this case study, the analysis method could have been performed even without any replicates because a full factorial experiment was performed and one of the factors was dropped as non-significant, thus still providing 2 data points per combination.

References

[1] ReliaSoft Corporation, Experiment Design and Analysis Reference, ReliaSoft Publishing, 2008.

[2] ReliaSoft Corporation, "Accelerated Life Test Plans: Analytical Solution and Comparison with Simulation," Reliability Edge, Volume 8, Issue 1, 2007.

[3] ReliaSoft Corporation, Accelerated Life Testing Reference, ReliaSoft Publishing, 2007.

End Article

 

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