Introduction

Welcome to the tutorial on Reliability Testing! In this tutorial, we will explore key concepts and models used in reliability testing, including the Crow-AMSAA model, Duane model, and Accelerated Life Testing (ALT) methods such as the Arrhenius and Power Law models. These models help engineers and reliability professionals assess and improve the reliability of products and systems over time.

Learning Objectives

By the end of this tutorial, learners will be able to:

• Define key reliability growth concepts, including the Crow-AMSAA and Duane models.
• Explain fundamental principles of accelerated life testing, including the Arrhenius and Power Law models.
• Fit reliability growth models to real-world datasets using R.
• Apply the Crow-AMSAA model to assess reliability growth.
• Conduct accelerated life tests with real-world datasets, utilizing R for analysis.
• Interpret accelerated life testing plots to identify trends.
• Analyze the impact of stress factors on product reliability using Arrhenius and Power Law models.

What is Reliability Testing?

Reliability testing is a process to ensure that a product or system performs its intended functions under specified conditions over a defined period. For example, in the manufacturing industry, we may want to design a machine that runs without breaking down for 5 to 10 years.

Why is Reliability Testing important?

Reliability testing is important for several reasons:

• Helps identify potential failure modes and weaknesses in a product or system.
• Ensures that products meet quality and safety standards.
• Enhances customer satisfaction by delivering reliable products.
• Reduces costs associated with warranty claims, repairs, and downtime.

Reliability Growth Analysis

Reliability Growth Analysis (RGA) is a method used to monitor and improve the reliability of a product or system over time. It involves analyzing failure data collected during testing phases to identify trends and patterns in reliability performance.

The Duane Model

The Duane Model, developed by Duane in 1964, is one of the earliest and most widely used models for reliability growth analysis. The model is particularly useful due to its simplicity and ease of interpretation. The Duane plot is a graphical representation that helps visualize whether failure rates are improving, stable, or worsening over time.

Specifically, the Duane Model is a log-log plot of the cumulative Mean Time Between Failures (MTBF) vs cumulative time, where the MTBF is calculated as the total operating time divided by the number of failures observed up to that time.

The slope of the line on the plot indicates the rate of reliability growth:

• A positive slope means that the system is improving (reliability is growing, the failure rate is decreasing).
• A zero slope means there is no change in reliability (the system is stable).
• A negative slope indicates that reliability is worsening (the failure rate is increasing).

ReliaGrowR

pak::pkg_install("ReliaGrowR")

times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)

fit <- duane(times, failures)
plot(fit, main = "Duane Model Example", xlab = "Cumulative Time", ylab = "Cumulative MTBF")

times    <- c(200, 450, 750, 1100, 1500, 2000)
failures <- c(3, 2, 2, 1, 1, 1)

# Use duane() with cumulative times and per-interval failure counts, then plot
# fit <- duane(time = times, event = failures)
# plot(fit)

times    <- c(200, 450, 750, 1100, 1500, 2000)
failures <- c(3, 2, 2, 1, 1, 1)
fit <- duane(time = times, event = failures)
plot(fit, main = "Duane Reliability Growth Plot",
     xlab = "Cumulative Test Time", ylab = "Cumulative Failures")

The Crow-AMSAA Model

The Crow-AMSAA Model is another widely used model for reliability growth analysis, developed by Larry Crow and the Army Materiel Systems Analysis Activity (AMSAA) organization in the 1970s. This model is particularly useful for systems that undergo continuous improvements and modifications during their development and testing phases. The Crow-AMSAA model is based on the concept of a Non-Homogeneous Poisson Process (NHPP), which allows for the modeling of failure behavior that changes over time.

Specifically, the Crow-AMSAA Model is a model of the cumulative number of failures vs cumulative time.

Similar to the Duane Model, the shape of the model indicates the rate of reliability growth:

• A model fit with downward curvature means that the system is improving (reliability is growing, the failure rate is decreasing).
• A linear model means there is no change in reliability (the system is stable).
• A model fit with upward curvature indicates that reliability is worsening (the failure rate is increasing).

times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)

result <- rga(times, failures)
plot(result, main = "Crow-AMSAA Model Example", xlab = "Cumulative Time", ylab = "Cumulative Failures")

times    <- c(50, 120, 220, 360, 530, 740, 1000)
failures <- c(4, 3, 3, 2, 2, 1, 1)

# Use rga() with model_type = "Crow-AMSAA", then plot
# fit <- rga(time = times, event = failures, model_type = "Crow-AMSAA")
# plot(fit)

times    <- c(50, 120, 220, 360, 530, 740, 1000)
failures <- c(4, 3, 3, 2, 2, 1, 1)
fit <- rga(time = times, event = failures, model_type = "Crow-AMSAA")
plot(fit, main = "Crow-AMSAA Reliability Growth",
     xlab = "Cumulative Test Time", ylab = "Cumulative Failures")

times <- c(25, 55, 97, 146, 201, 268, 341, 423, 513, 609, 710, 820, 940, 1072, 1217)
failures <- c(1, 1, 2, 4, 4, 1, 1, 2, 1, 4, 1, 1, 3, 3, 4)
breaks <- 500

result <- rga(times, failures, model_type = "Piecewise NHPP", breaks = breaks)
plot(result, main = "Piecewise NHPP Model Example", xlab = "Cumulative Time", ylab = "Cumulative Failures")

times <- c(25, 55, 97, 146, 201, 268, 341, 423, 513, 609, 710, 820, 940, 1072, 1217)
failures <- c(1, 1, 2, 4, 4, 1, 1, 2, 1, 4, 1, 1, 3, 3, 4)

result <- rga(times, failures, model_type = "Piecewise NHPP")
plot(result, main = "Piecewise NHPP with Change Point Detection", xlab = "Cumulative Time", ylab = "Cumulative Failures")

Accelerated Life Testing

Accelerated Life Testing (ALT) is a method used to estimate the life of a product or system under normal operating conditions by subjecting it to elevated stress levels in a controlled environment. The goal is to identify potential failure modes and predict how long the product will last under normal conditions without having to wait for failures to occur naturally over a longer time period.

The Arrhenius Model

The Arrhenius Model is one of the most commonly used models in ALT. It describes the relationship between temperature and the rate of chemical reactions, which can be directly related to the degradation or failure rate of materials. The model is based on the Arrhenius equation, which states that the rate of a chemical reaction increases exponentially with an increase in temperature.

The Acceleration Factor (AF) for the Arrhenius model is given by:

\[ AF = \exp\left(\frac{E_a}{k} \left( \frac{1}{T_{\text{use}}} - \frac{1}{T_{\text{stress}}} \right) \right) \]

Where:

• \(E_a\) = Activation energy in electron volts (eV). The activation energy is a measure of the energy barrier that must be overcome for a chemical reaction to occur. Higher activation energies indicate that the reaction is more sensitive to temperature changes.
• \(k\) = Boltzmann constant (\(8.617 \times 10^{-5} \, \text{eV/K}\)). The Boltzmann constant relates the average kinetic energy of particles in a gas with the temperature of the gas.
• \(T_{\text{use}}\) = Temperature at normal use condition in Kelvin
• \(T_{\text{stress}}\) = Temperature at accelerated stress condition in Kelvin

By using the Arrhenius model, you can determine how much faster a product will fail at an elevated temperature compared to its normal operating temperature.

WeibullR.ALT

For this tutorial, we will use WeibullR.ALT, an extension of the WeibullR package for Accelerated Life Testing.

First, check if WeibullR.ALT is installed in R and install if not.

pak::pkg_install("WeibullR.ALT")

To illustrate the Arrhenius Model, we will use the data set MeekerData that comes prepackaged with WeibullR.ALT. This data set contains time to failure data for carbon-film resistors tested at different temperature levels. The testing was conducted at 4 increasing temperature levels to accelerate the failure process. The temperature levels were 10, 40, 60, and 80 degrees Celsius.

First, load the data set and use the head function to view the first few rows.

data <- MeekerData("table10")
head(data)

The data set contains columns for the time to failure (time), event type (event) (1 = failure, 0 = suspension), quantity of samples (qty) and temperature as the acceleration factor (TempC).

Next, use the alt.data function to create a separate object for the data at each temperature level. Each data set is created by subsetting the original data set for each temperature level and selecting only the relevant columns (time, event, qty). The stress parameter is set to the corresponding temperature level. The which function is used to filter the rows based on the temperature level.

data10C <- alt.data(data[which(data$TempC==10), 1:3], 
                    stress = data[which(data$TempC==10), 4][1])
data40C <- alt.data(data[which(data$TempC==40), 1:3], 
                    stress = data[which(data$TempC==40), 4][1])
data60C <- alt.data(data[which(data$TempC==60), 1:3], 
                    stress = data[which(data$TempC==60), 4][1])
data80C <- alt.data(data[which(data$TempC==80), 1:3], 
                    stress = data[which(data$TempC==80), 4][1])

Then, use the alt.make function to fit an Arrhenius-Weibull model to the data. Each data set is passed as a list to the alt.make function, along with the distribution type (dist = "weibull"), the acceleration model (alt.model = "arrhenius"), and the fitting method (method.fit = "mle"). The resulting object arrobj contains the fitted model.

arrobj <- alt.make(list(data10C, data40C, data60C, data80C), dist = "weibull",
                alt.model = "arrhenius", method.fit = "mle")

The plot shows an Arrhenius-Weibull model fit to the data. The different colors represent the different temperature levels. Starting with a temperature level of 40 degrees C on the right, the curves shift to the left as the temperature increases, indicating that higher temperatures lead to shorter life.

Finally, calculate the AF at 80 degrees C using the Arrhenius Model. Assume that the Activation Energy is 0.7 eV. The code below calculates the AF using the formula provided earlier, with \(T_{\text{use}}\) set to 10 degrees C (283K) and \(T_{\text{stress}}\) set to 80 degrees C (353K).

E_a <- 0.7
k <- 8.617E-5
T_use <- 10 + 273
T_stress <- 80 + 273
AF <- exp(E_a / k * (1 / T_use - 1 / T_stress))
cat(paste0("The Acceleration Factor at 80 degrees C is ", round(AF, 2)))

The Acceleration Factor at 80 degrees C is 296.54

The AF at 80 degrees C is approximately 300. This means that the resistors tested at 80 degrees C are expected to fail 300 times faster than those tested at 10 degrees C.

E_a      <- 0.65        # activation energy (eV)
k        <- 8.617e-5    # Boltzmann's constant (eV/K)
T_use    <- 25 + 273.15 # use temperature in Kelvin
T_stress <- 85 + 273.15 # stress temperature in Kelvin

# AF = exp((E_a / k) * (1/T_use - 1/T_stress))

E_a      <- 0.65
k        <- 8.617e-5
T_use    <- 25 + 273.15
T_stress <- 85 + 273.15
AF <- exp((E_a / k) * (1 / T_use - 1 / T_stress))
cat("Acceleration Factor:", round(AF, 1), "\n")

The Power Law Model

The Power Law Model is another commonly used model in ALT, particularly for mechanical systems and components. This model describes how a product’s life is affected by a certain stress factor (such as temperature, voltage, or pressure). Unlike the Arrhenius model, which is primarily used for temperature-based stresses, the Power Law Model can be applied to a wide variety of stress factors.

The AF for the Power Law model is given by:

\[ AF = \left( \frac{S_{\text{stress}}}{S_{\text{use}}} \right)^{n} \]

Where:

• \(S_{\text{stress}}\) = Stress level during accelerated testing (e.g., voltage, load)
• \(S_{\text{use}}\) = Stress level during normal use conditions
• \(n\) = Stress exponent, indicating the sensitivity of the failure mechanism to the applied stress

To illustrate the Power Law Model, we will use the data set NelsonData that comes prepackaged with WeibullR.ALT. This data set contains time to failure data for electrical insulating fluids tested at different voltage stress levels. The testing was conducted at 7 increasing voltage levels to accelerate the failure process.

First, load the data set..

data <- NelsonData("table3.1")

The data is a list of time to failure data at 7 different voltage stress levels, 26 kilovolts (KV), 28kV, 30kV, 32kV, 34kV, 36kV, and 38kV.

Next, use the alt.data function to create a separate object for the data at each stess level. The stress parameter is set to the corresponding voltage level.

data26 <- alt.data(data$kV26, stress = 26)
data28 <- alt.data(data$kV28, stress = 28)
data30 <- alt.data(data$kV30, stress = 30)
data32 <- alt.data(data$kV32, stress = 32)
data34 <- alt.data(data$kV34, stress = 34)
data36 <- alt.data(data$kV36, stress = 36)
data38 <- alt.data(data$kV38, stress = 38)

Then, use the alt.make function to fit a Power-Weibull model to the data and plot the results. Similar to the previous example, each data set is passed as a list to the alt.make function, along with the distribution type (dist = "weibull") and the acceleration model (alt.model = "power"). The resulting object pwrobj contains the fitted model.

pwrobj <- alt.make(list(data26, data28, data30, data32, data34, data36, data38), 
                   dist = "weibull", alt.model = "power")

The plot shows a Power-Weibull model fit to the data. The different colors represent the different voltage stress levels. Starting with a stress level of 28kV on the right (dark green curve), the curves shift to the left as the stress increases, indicating that higher stress leads to shorter life.

Multiple Stress Factors

The Eyring Model is an extension of the Arrhenius model and is commonly used in ALT to account for the combined effects of multiple stressors such as temperature, voltage, and humidity on the failure mechanisms of a product. This model is particularly useful when the product is exposed to environmental factors beyond just temperature.

The general form of the Eyring Model is:

\[\text{Life}(S) = A \cdot \exp\left(\frac{B}{S} + C \cdot S\right)\]

Where:

• \(\text{Life}(S)\) is the expected life under stress level \(S\),
• \(A\) is a constant,
• \(B\) and \(C\) are model parameters,
• \(S\) is the applied stress factor (e.g., temperature, voltage, humidity).

This module does not include an example of the Eyring Model, but the WeibullR.ALT package does support it. See the package documentation for more information.

Parallelization

Parallelization is a technique used in ALT to simplify the analysis of data collected at multiple stress levels. By assuming that the shape factor (slope) of the life distribution is the same across all stress levels, the analysis can be streamlined, making it easier to interpret the results and draw conclusions about the relationship between stress and life.

To fit a parallel model to the Meeker data, use the alt.parallel function.

prlobj <- alt.parallel(arrobj)

The plot shows a Parallel model fit to the data. All curves have the same slope, indicating that all subsets have the same shape factor.

Similarly, to fit a parallel model to the Nelson data, use the alt.parallel function.

expobj <- alt.parallel(pwrobj)

Relationship Plots

Another useful tool in ALT is the relationship plot. The relationship plot provides a visual representation of how the life of a product changes as the stress factor increases. A relationship plot is a log-linear plot with the stress factor on the x-axis and the log of the unreliability (failure probability) on the y-axis.

A linear model is fit to the data, which shows patterns indicating how life changes as stress increases. The slope of the line provides insights into the relationship between stress and life:

• A negative slope (decreasing life with increasing stress) indicates that higher stress leads to shorter life.
• A positive slope (increasing life with increasing stress) is less common but indicates that life improves with higher stress.
• A flat slope indicates that higher stress does not have an impact on life.

To fit a relationship plot to the Meeker data, first fit a linear model to the data using the alt.fit function and then plot the results using the plot function.

lnrobj <- alt.fit(prlobj)
plot(lnrobj, suppress.dev.new = TRUE)

The relationship plot shows a negative slope, indicating that higher temperature stress tends to lead to shorter life. Each line indicates a different probability of failure. For example, the P63.2 line indicates the stress vs life relationship for a 63.2% probability of failure. The P90 line indicates the stress vs life relationship for a 90% probability of failure.

Similarly, to fit a relationship plot to the Nelson data, first fit a linear model to the data and then plot the results.

lnrobj2 <- alt.fit(expobj)
plot(lnrobj2, suppress.dev.new = TRUE)

Similarly, the relationship plot shows a negative slope, indicating that higher voltage stress tends to lead to shorter life.

Summary

Congratulations on completing the Reliability Testing tutorial!

In this tutorial, we’ve introduced the concepts of Reliability Growth Analysis and Accelerated Life Testing. You have learned to calculate and visualize reliability using different types of models.

?rga

?alt.make

help(package="ReliaGrowR")

help(package="WeibullR.ALT")