Accelerated Life Testing Module Accelerated Life Testing - Overview

Transcription

1 Accelerated Life Testing Module Accelerated Life Testing - Overview The Accelerated Life Testing (ALT) module of AWB provides the functionality to analyze accelerated failure data and predict reliability characteristics under normal use conditions. The ALT module allows data sets to be analyzed using a proportional hazards or accelerated failure time survival model. Standard stress profile models are provided (Arrhenius, Eyring and Power Law) together with the ability to define custom models. Constant, time-varying and multiple stress profiles may be specified for a single data set. The ALT module of AWB fits stressed data to exponential, Weibull, lognormal and normal distributions. Calculated parameters are produced in the form of the distribution parameters at use stress, B life and P parameter values and distribution plots (reliability, failure rate and pdf).

2 Accelerated Life Testing - User Interface To access the Accelerated Life Testing (ALT) module in Availability Workbench (AWB) select the Accelerated life testing pull-down option at the top of the left-hand window. If you have already defined one or more ALT sets in your project you can display the associated data (ALT set items) by selecting the appropriate ALT set node in the tree control. Ensure that either the Plot, Grid or Plot & Grid option is selected above the right-hand window. To add a new ALT set select the ALT Sets node in the tree control and use the right-button pop-up menu to add a new set. The grid in the right-hand window is used to display historical data. Data may be imported using the Windows clipboard (copy and paste) or may be typed into the grid by hand. The grid representing ALT set items has several columns. One column is used to specify the time value (the time to failure or time to suspension for a test). Other columns indicate whether the data is a failure or suspension (censored), whether the data item is currently disabled and how many components were failed or suspended at this time. As data items are entered into the grid, points will be plotted in the cumulative probability graph. These points represent the estimated cumulative probability values at each time point for failed data under stressed conditions. A typical ALT module screen display. The left window displays ALT sets in the project tree control. The top right window displays a stressed Weibull plot with calculated Weibull parameters at use stress. The bottom right window displays times to failure and other input parameters.

3 ALT Sets ALT Sets - Overview An ALT set is a collection of times to failure or suspensions that has been compiled from an accelerated life test. ALT set information may be automatically analyzed using Availability Workbench (AWB) to fit an appropriate distribution.

4 Defining or Importing ALT Set Items Users may assign ALT set items to an ALT set by typing in the data to a grid control or importing the data from an external source. Manual Data Definition If you have already defined one or more ALT sets in your project you can display the associated data (ALT set items) by selecting the appropriate ALT set node in the project tree control. Ensure that either the Plot, Grid or Plot & Grid option is selected above the right-hand window. To modify existing data simply type the new data into the grid control or select the appropriate check boxes. To add a new item you will need to access the Items tab of the ALT Set Properties dialog. One way of accessing this tab is to click the right mouse button over the grid area to reveal the grid pop-up menu. Then select the Add Record option. Alternatively, press the Ins key when the grid is displaying ALT set items. The Items tab of the ALT Set Properties dialog displays a grid control and a Delete button. To delete existing items select them in the grid control and then select the Delete button. To add new items type in the new data into the empty last row of the grid control. After selecting the OK button of the ALT Set Properties dialog new points will be plotted in the cumulative probability graph. These points represent the estimated cumulative probability values (under stress conditions) at each time specified by an ALT set item. Importing Data ALT set data may be imported directly from external databases or the Windows clipboard. To import data you will need to use the AWB import facility. The import facility may be accessed from the File, Import pull-down menu option. If you are importing data from the Windows clipboard you can quickly access the import facility by selecting the Paste from Clipboard option from the grid control right-button pop-up menu. The import facility will then enable you to match data on the clipboard to the appropriate columns in the ALT set items table. See Also Import

5 ALT Sets - General Properties ALT set general properties may be accessed from the General tab of the ALT Set Properties dialog. ID A unique identifier for the ALT set of no more than 40 alpha-numeric characters. Type The type assigned to the ALT set. ALT set types are used to organize ALT sets into groups. If a project contains many ALT sets then this will greatly assist in locating an ALT set in the tree control structure. Description A description for the ALT set of no more than 255 characters.

6 ALT Sets - Analysis Options ALT set analysis options may be accessed from the Analysis Options tab of the ALT Set Properties dialog. Survival Model There are two survival model options available - Accelerated failure time and Proportional hazards. The models indicate how applied stress factors affect the underlying failure probability distribution. If the Accelerated failure time model is selected AWB will assume that the applied stress factor directly affects the lifetime characteristic of the distribution. For the exponential distribution: For the Weibull distribution: For the Normal and lognormal distributions: If the Proportional hazards model is selected AWB will assume that the applied stress factor directly affects the failure rate of the distribution: If the Proportional hazards model is selected only exponential or Weibull distributions may be selected as the underlying distribution. For the exponential distribution: For the Weibull distribution: Fit model The model used to fit test data to the assigned distribution. Two options are available Least squares and Maximum likelyhood. The expression below indicates how the goodness of fit indicator is calculated for the Least squares model.

7 where and are the fitted distribution unreliability values and estimated unreliability point values respectively. N is the total number of points plotted. The Maximum likelyhood fit model computes the Log Likelyhood function and selects the distribution parameters that provide the highest likelyhood value. The Log Likelyhood (LL) value is given by B Life When an analysis is performed Availability Workbench (AWB) calculates the time for the specified 'B' life at use stress. If a B life of 10 is specified by the user (the default) then the program calculates the time by which 10% of component failures would have occurred. Another way of expressing this is that the program will calculate the time at which the unreliability of the component is 0.1. A B life of 30% would prompt the program to determine the time by which 30% of component failures are likely to have occurred or the time at which the unreliability is 0.3. Times calculated for B values are shown to the right of the cumulative probability graph. P parameter The P parameter indicates the time at which the unreliability (at use stress) is to be calculated during an analysis. For example, if a value of 8760 hours is specified then the program will calculate the unreliability at 8760 hours. The calculated value is shown to the right of the cumulative probability graph. Reliability estimation Users may choose the Median rank, 90% rank, 95% rank or Nelson methods for estimating reliability values from times to failure. Median Rank Method Median ranks may be used to obtain estimates of unreliability at the time point represented by each failure. This method first calculates rank numbers for each failure based on the following expression: Median ranks may then be obtained by applying where j = rank number, N = population The table below illustrates how unreliability is estimated by the median rank number. Time Status Rank No. Median Rank, 5000 Failure

8 90% and 95% Rank Methods 9600 Suspended Failure Suspended Failure The 90% and 95% rank methods are alternative ways of estimating the component unreliability. They provide a more conservative estimate (higher unreliability) than the median rank method to 90% and 95% confidence levels respectively. Nelson s Method At any time at which a failure occurs a hazard rate function,, is calculated by dividing the number of failures occurring at that time by the number of survivors immediately before that time: where = number of failures = number of survivors The cumulative sum of these values gives a sample estimate of the cumulative hazard function,. An estimate of the unreliability,, is then given by The table below illustrates how the hazard rate function and cumulative hazard function are calculated with some example data. Time x S Suspended Suspended Parameter precision The user may specify a parameter precision value of 2, 3, 4, 6 or 9. The precision value determines the number of significant figures calculated when determining the parameters of the Weibull and other distributions. Consider a beta parameter that has been generated with the value The table below illustrates how the beta parameter will be rounded for a variety of precision values. Precision Rounded Number

9 ALT Sets - Notes ALT set notes may be accessed from the Notes tab of the ALT Set Properties dialog. Up to 4 descriptive notes may be assigned to each ALT set. Users may customize note headers (the labels used to identify a note category) using the Notes tab in the Project Options dialog.

10 ALT Sets - Stress Profiles A stress profile defines the stresses applied during an accelerated life test. Multiple stress profiles may be assigned to a single test. Each stress profile may have a different stress model associated with it. Select the Add Profile button in the ALT Set Properties dialog (Stress Profile tab) to define each stress profile to be associated with a test. Select the Delete Current Profile button to delete the current displayed profile. Model The Model list allows the user to select the appropriate model for the current stress profile. You may select from four model options: Arrhenius Power Eyring Custom Arrhenius The Arrhenius model uses the following expression to determine the stress factor: Power The Power model uses the following expression to determine the stress factor: Eyring The Eyring model uses the following expression to determine the stress factor: Custom You may create your own single-parameter custom stress model using simple mathematical expressions and library functions. The custom expression is provided in the Custom expression field. Model parameter For the Arrhenius model this is the value of the activation energy in units of ev (electron Volts). For the Power model this is the exponent. For the Eyring model this is the exponential constant. For the custom model this is the value of p in custom expressions. Use stress

11 The value of the use stress. This value will be used by AWB to determine the underlying distribution parameters at use stress conditions. Examples of use stresses are normal ambient temperature normal operating voltage Custom expression This field only applies when the Custom model is selected. A custom expression may only use the variables (case-sensitive): stress usestress p These variables are the stress, the use stress and the model parameter values respectively. The operators * / + - represent multiplication, division, addition and subtraction respectively. An example of a simple expression for the stress factor is: stress / (usestress - p) A more complex example is Math.Exp((p / 8.62E-05) * (1.0 / (usestress + 273) / (stress + 273))) The second example uses a Math Library function. All Math Library functions must be preceded by Math. Examples of common Math functions are: Abs(x) Returns the absolute value of x. Exp(x) Returns e raised to the power x. Log(x) Returns the natural logarithm of x Log10(x) Returns the base 10 logarithm of x Pow(x,y) Returns x raised to the power y Sqrt(x) Returns the square root of x Description A description of the stress profile Time and stress value grid The values entered in the time and stress values grid define the stress profile. If the accelerated life test was performed at a single elevated stress then you need only enter the single stress value with a time value equal to the length of the test. If the stress was varied during the test, enter the different stress values with the time values indicating the end of each stress period. For example, the table below indicates that the stress applied (temperature in Kelvin) was 353 from the beginning of the test until 96 hours, was then increased to 373 from 96 to 120 hours and finally increased to 393 from 120 to 144 hours. Note that AWB assumes the stress falls to the use stress after the final specified time.

12 Time Stress

13 ALT Sets - Fitting a Distribution Availability Workbench (AWB) allows the user to choose from a number of different fit distributions for the test data associated with an ALT set. To select a different distribution simply choose the appropriate option from the Distribution drop-down list in the main toolbar. Alternatively use the Select Distribution Automatically option on the Analysis pull-down menu to prompt the program to automatically select the 'best-fit' distribution. AWB determines 'best-fit' by taking the smallest goodness of fit indicator or highest likelyhood value for all the available distributions. The goodness of fit is determined from the following expression: where and are the fitted unreliability values and estimated unreliability point values respectively. N is the total number of points plotted. The Log Likelyhood (LL) value is given by AWB allows the user to fit a distribution to an ALT set manually as well as using automatic data fitting algorithms. To access this facility select the Analysis, Set Distribution Parameters Manually pull-down menu option in the ALT module. A Distribution Parameters dialog will be revealed allowing the appropriate parameters to be modified by the user. As the parameters are modified, the graph will change to reflect the changing shape of the distribution. When fitting data manually, users may find it useful to first display the failure rate graph. It is often easier to fit data manually to the estimated regionalized rate in this graph. Exponential Distribution The exponential distribution should be used to model the failure characteristics of components that do not exhibit any ageing. The distribution represents a constant failure rate. The expressions below represent the use of the distribution for failures. Probability Density Function, Unreliability, Failure Rate, Mean Time to Failure, MTTF

14 1-Parameter Weibull The Weibull distribution is used to model the failure characteristics of components with time-dependent failure rates. A common use is to model the ageing characteristics of mechanical components. The 1-Parameter Weibull calculation method requires the user to specify the shape parameter of the distribution. When fitting the Weibull distribution to historical data AWB varies the characteristic life parameter to obtain the best fit. Probability Density Function for 1-Parameter Weibull, where = characteristic life parameter = shape parameter Unreliability, Failure Rate, Mean Time to Failure, MTTF where 2-Parameter Weibull Expressions for the 2-Parameter Weibull are identical to the expression given above for the 1-Parameter Weibull method. The only difference is that the user does not specify the value of the shape parameter. The program will assign the shape parameter as well as the characteristic life when fitting the distribution to data. 3-Parameter Weibull Probability Density Function for 3-Parameter Weibull, where = characteristic life parameters = shape parameter = location parameter

15 Unreliability, Failure Rate, Mean Time to Failure, MTTF where = gamma function Lognormal The lognormal distribution is defined by the following expressions. Probability Density Function, Mean Time to Failue, MTTF Standard deviation, Std Median (peak of distribution) Mode (most probable value) Normal The normal distribution is defined by the following expressions. Probability Density Function, where = mean time to failure = standard deviation of mean time to failure

16 ALT Sets - Goodness of Fit, Log Likelyhood and Correlation Coefficients When assigning distribution parameters using the straight line fit method, Availability Workbench (AWB) calculates a correlation coefficient. The correlation coefficient is a measure on how well the linear model fits the data. The closer the absolute value is to 1, the better the fit. The correlation coefficient is calculated from where and are the x and y values of points in the cumulative probability plot. N is the total number of points plotted. Goodness of Fit Indicator The expression below indicates how the goodness of fit indicator is calculated. where and are the fitted Weibull unreliability values and estimated unreliability point values respectively. N is the total number of points plotted. Log Likelyhood The Log Likelyhood (LL) value is given by

17 ALT Plots ALT plots will be displayed when you have the Plot or Plot & Grid option selected above the right-hand window. There are three different ALT plot types. The Cumulative Probability plot displays estimated unreliability values versus time under stress conditions. The Failure Rate plot displays estimated and distribution-derived failure rate values over time at use stress and stress test conditions. The Probability Density Function plot displays distribution-derived probability density function values at use stress and stress test conditions. Cumulative Probability Plot The cumulative probability plot displays estimated unreliability values versus time under stress conditions. Estimated values are determined using the reliability estimation method specified by the user in the ALT Set Properties dialog and are displayed as red dots in the diagram. Each dot represents an enabled and nonsuspended item in the ALT set. Availability Workbench (AWB) automatically fits the selected stressed distribution to the estimated unreliability points and displays the distribution-derived unreliability as a continuous red line. Unreliability is defined as the probability that a first failure has occurred before or at the specified point in time. Time corresponds to the age of an equipment relative to its as good as new condition. Failure Rate Plot The failure rate plot displays estimated and distribution-derived failure rate values over time at use stress and stress condtions. The continuous red line represents the distribution-related failure rate value under stress conditions. The blue line represents the distribution-related failure rate value under use stress conditions. The failure rate is defined as the probability of first failure per unit time conditional on the equipment being operational at a given time. Time corresponds to the age of an equipment relative to its as good as new condition. Probability Density Function The probability density function (pdf) plot displays estimated and distribution-derived pdf values over time at use stress and stress condtions. The continuous red line represents the distribution-related pdf value under stress conditions. The blue line represents the distribution-related pdf value under use stress conditions. The probability density is defined as the probability of first failure per unit time. Time corresponds to the age of an equipment relative to its as good as new condition.