Patent Publication Number: US-2006009951-A1

Title: Method and apparatus for predicting failure in a system

Description:
The patent claims priority pursuant to 35 U.S.C. § 119(e)1 to provisional application 60/260,449 filed Jan. 8, 2001. 
    
    
     FIELD OF THE INVENTION  
      This invention relates to a method and apparatus for predicting failure of a system. More specifically it relates to a method and apparatus for integrating data measured from a system, and/or data referenced from other sources, with component failure models to predict component or overall system failure.  
     BACKGROUND OF THE INVENTION  
      Any product will eventually fail, regardless of how well it is engineered. Often failure can be attributed to structural, material, or manufacturing defects, even for electronic products. A failure at the component or sub-component level often results in failure of the overall system. For example, cracking of a piston rod can result in failure of a car, and loss of a solder joint can result in failure of an electronic component. Such failures present safety or maintenance concerns and often result in loss of market share.  
      A way to predict the impending failure of a system or component would be useful to allow operators to repair or retire the component or system before the actual failure, and thus avoid negative consequences associated from an actual failure.  
      Accurate prediction of impending structural, mechanical, or system failure could have great economic impact to industries within the aerospace, automotive, electronics, medical device, appliance and related sectors.  
      Engineers currently attempt to design products for high reliability. But it is most often the case that reliability information comes very late in the design process. Often a statistically significant amount of reliability data is not obtained until after product launch and warranty claims from use by consumers. This lack of data makes it common for engineers to add robustness to their designs by using safety factors to ensure that a design meets reliability goals.  
      Safety factors, however, are subjective in nature and usually based on historical use. Since modern manufacturers are incorporating new technology and manufacturing methods faster than ever before, exactly what safety factor is appropriate to today&#39;s new complex, state-of-the-art product is seldom, if ever, known with certainty. This complicates the engineering process. In addition, safety factors tend to add material or structural components or add complexity to the manufacturing process. They are counterproductive where industry is attempting to cut cost or reduce weight. Designing cost effective and highly reliable structures therefore requires the ability to reduce the safety factor as much as possible for a given design.  
      In attempting to reduce reliance on safety factors, designers have, over the years, developed models for the damage mechanisms that lead to failures. Failures can be attributed to many different kinds of damage mechanisms such as fatigue, buckling, and corrosion. These models are used during the design process, usually through deterministic analysis, to identify feasible design concept alternatives. But poor or less than desired reliability is often attributed to variability, and deterministic analysis fails to account for variability.  
      Variability affects product reliability through any number of factors including loading scenarios, environmental condition changes, usage patterns, and maintenance habits. Even a system response to a steady input can exhibit variability, such as a steady flow pipe with varying degrees of corrosion.  
      Historically, testing has been the means for evaluating effects of variability. Unfortunately, testing is a slow, expensive process and evaluation of every possible source of variability is not practical.  
      Over the years, probabilistic techniques have been developed for predicting variability and have been coupled with damage models of failure mechanisms to provide probabilistic damage models that predict the reliability of a population. But, given variability, a prediction of the reliability of a population says little about the future life of an individual member of the population. Safety factors are likewise unsatisfactory methods for predicting the life of an individual since they are based on historical information obtained from a population. Safety factors are also an unsatisfactory method for quickly and efficiently designing against failure since they rely on historical information obtained from test and component data. As a result, there exists a need for a method and apparatus for accurately predicting component and/or system failure that accounts for variability without the need for extensive test data on the component and/or system.  
     SUMMARY OF THE INVENTION  
      The present invention is a method and apparatus for predicting system failure, or system reliability, using a computer implemented model of the system. In an embodiment of the invention that model relies upon probabilistic analysis. Probabilistic analysis can incorporate any number of known failure mechanisms for an individual component, or components, of a system into one model and from that model can determine the critical variables upon which to base predictions of system failure. Failure can result from a number of mechanisms or combination of mechanisms. A probabilistic model of the system can nest failure mechanisms within failure mechanisms or tie failure mechanisms to other failure mechanisms, as determined appropriate from analysis of the inter-relationships between both the individual failure mechanisms and individual components. This results in a model that accounts for various failure mechanisms, including fatigue, loading, age, temperature, and other variables as determined necessary to describe the system. As a result of probabilistic analysis, the variables that describe the system can also be ranked according to the effect they have on the system.  
      Probabilistic analysis of a system predicts system and/or component failure, or reliability, based on acquired data in conjunction with data obtained from references and data inferred from the acquired data. This prediction of failure or reliability is then communicated to those using or monitoring the system. Furthermore, the analyzed system can be stationary or mobile with the method or apparatus of analysis and communication of the failure prediction being performed either on the system or remotely from the system. In addition, the apparatus may interface with other computer systems, with these other computer systems supplying the required data, or deciding whether and/or how to communicate a prediction.  
      An advantage of one embodiment of the invention is that it divides system variables into three types: directly sensed—those that change during operation or product use; referred—those that do not (significantly) change during operation or product use; and inferred—those that change during operation or use but are not directly sensed. This strategy divides the probabilistic approach into two broad categories, pre-process off-board analysis and near real time on-board or off-board analysis, allowing for prediction of a probability of failure based on immediate and historic use.  
      In one embodiment of the invention a computer implements a method for predicting failure in a system. This method comprises: measuring data associated with a system; creating a prediction of a failure of the system using a model of the system and the data; and communicating the prediction to a user or operator.  
      A second embodiment of the invention is an apparatus for predicting failure of a system. This apparatus comprises: sensors for acquiring data from the system and a computer, with the computer having a processor and memory. Within the memory are instruction for measuring the data from the sensors; instructions for creating a prediction of a failure of the system using a model and the data; and instructions for communicating the prediction. The apparatus also comprises communication means for communicating the prediction.  
      A third embodiment of the invention is a computer program product for predicting failure of a system for use in conjunction with a computer system. The computer program product comprises a computer readable storage medium and a computer program mechanism embedded therein. The computer program mechanism comprises: instructions for receiving data; instructions for storing the data; instructions for creating a prediction of failure of the system using a model and the data; and instructions for communicating this prediction. Furthermore, embodiments of these apparatuses and method use a system model developed with probabilistic methods. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      The foregoing and other aspects and advantages of the present invention will be better understood from the following detailed description of preferred embodiments of the invention with reference to the drawings, in which:  
       FIG. 1  is a schematic illustrating an embodiment of an apparatus of the present invention employed on a dynamic system and an indication of the process flow;  
      FIGS.  2 ( a )-( d ) illustrate a preferred embodiment of the off-board engineering portion of a n embodiment of a method of the present invention;  
      FIGS.  3 ( a ) and ( b ) illustrate an embodiment of the on-board failure prediction portion of the method also depicted in FIGS.  2 ( a )-( d );  
       FIG. 4  illustrates an embodiment of the invention employed in a static system; and  
      FIGS.  5 ( a )- 5 ( f ) illustrate an example of the method of  FIGS. 1, 2 , and  3  applied to a composite helicopter rotor hub. 
    
    
      Like reference numerals refer to corresponding elements throughout the several drawings.  
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
      An embodiment of the present invention uses sensed data combined with probabilistic engineering analysis models to provide a more accurate method for predicting the probability of failure of a component or a system. This embodiment uses probabilistic analysis models to address, on a component by component basis, the effects of the random nature associated with use, loading, material makeup, environmental conditions, and manufacturing differences. This embodiment assumes that the underlying physics of the system behavior is deterministic and that the random nature of the system response is attributed to the scatter (variability) in the input to the system and the parameters defining the failure physics.  
      The underlying physics of the system behavior is captured by developing a system response model. This model, which represents the nominal response of the system, uses random variables as input parameters to represent the random system behavior. The system response model may be based on the explicit mathematical formulas of mechanics of materials, thermodynamics, etc. Computational methods such as finite element analysis and computational fluid analysis, are sometimes used to assess the response of the system. Closely coupled with the system response models are failure models. The failure models, which address both initial and progressive damage, may be either in the form of maximum load interactive criteria, or more specific models, which have been developed by the system&#39;s original equipment manufacturers (OEMs), such as crack growth models.  
      Probabilistic analysis then determines the variation in the global system response as well as variation in the local system response. This probabilistic analysis also quantitatively assesses the importance of each of the random variables on the variation in the system response. This allows for development of a rational design framework for deciding which variables need to be controlled and how to increase the reliability of the system. The embodiment of the invention incorporating probabilistic analysis, therefore, provides for more accurate predictions of failure. Thus, this embodiment also provides a basis for more rational design decisions, while reducing expense and time to market.  
       FIG. 1  is a schematic illustrating an embodiment of an apparatus of the present invention employed on a dynamic system  22 . System  22  is this illustrative embodiment is an automobile with the embodiment described as a device in the automobile, but dynamic system  22  could be any dynamic system, such as a helicopter, airplane, automobile, rail car, tractor, or an appliance. On-board Prognostic Instrument Engineer (OPIE)  10 , generally includes a central processing unit (CPU)  18 ; a computer control  20 ; a user alert interface  26 ; and sensors  24 . The CPU  18  receives input in the form of criteria, equations, models, and reference data  14  derived from engineering analysis performed at step  12  and the OPIE  10  uses such input to make a failure prediction at step  16 .  
      Engineering analysis step  12  essentially comprises the preparatory steps that produce the criteria, equations, models, and reference data  14  that are used in failure prediction step  16  to assess the condition of the system or component of interest. Engineering analysis step  12  includes the steps: identify failure mechanisms  40 ; model failure mechanisms  42 ; formulate probabilistic strategy  46 ; and determine warning criteria  48 . Engineering analysis step  12  yields criteria, equations, models and reference data  14 , which are further described and shown in  FIG. 2 ( d ).  
      Continuing with  FIG. 1 , criteria, equations, models and reference data  14  are stored on a memory device  34  or incorporated into a computer program product within CPU  18  as a prediction analysis  30 . Desired criteria from criteria, equations, models and reference data  14  may also be programmed into overall system computer control  20 .  
      Sensors  24  send information to computer control  20 . Sensors  24  measure data on any number of conditions, such as temperature, speed, vibration, stress, noise, and the status and number of on/off cycles of various systems. Computer control  20  sends operation and sensor data  25  to CPU  18 . Operation and sensor data  25  includes data from sensors  24  in addition to other data collected by computer control  20 , such as ignition cycles, light status, mileage, speed, and numbers of activations of other sub-systems on system  22 . CPU  18  creates input  28  by combining operation and sensor data  25  with information from memory device  34  and information from previous output data  32  that was stored in memory device  34 .  
      CPU  18  analyzes input  28  as directed by prediction analysis  30  to produce the output data  32 . Output data  32  contains a prediction result  29  and possibly other information. Output data  32  is then saved in memory device  34  while prediction result  29  is sent to computer control  20 . Computer control  20  determines from criteria contained in criteria, equations, models and reference data  14 , or from criteria developed separately, whether and how to signal user alert interface  26  based on prediction result  29 . These criteria could be incorporated into CPU  18  instead, so that CPU  18  determined whether to activate user alert interface  26 .  
      User alert interface  26  is a number of individual components, with status, or alert indicators for each as is necessary for the systems being analyzed for failure, such as, for example, a yellow light signal upon predicted failure exceeding stated threshold value. A variety of user alert signal devices could be appropriate for the specific situation. Computer control  20  could also be configured to de-activate certain components upon receipt of the appropriate prediction result, e.g., vehicle ignition could be disabled should prediction result  29  indicate a brake failure.  
      FIGS.  2 ( a )- 2 ( d ) are flow charts depicting the operation of engineering analysis process step  12  ( FIG. 2 ( a )) that results in creation of criteria, equations, models, and reference data  14  ( FIG. 2 ( d )). In  FIG. 2 ( a ) engineering analysis step  12  begins by identifying failure mechanisms at step  40  through review of warranty and failure data (step  50 ) and research of literature (step  52 ) to determine which of the identified failure mechanisms are actual active failure mechanisms (step  54 ). This effort could incorporate discussions with component design staff. Determination of active failure mechanisms can include a variety of evaluations, discussions and interpretations of both component and system response.  
      Failure mechanisms describe how and why the component fails. For example, mechanisms for delamination in a multi-layered material could include shear forces between the layers, adhesive decomposition, or manufacturing defects. Failure mechanisms are then modeled at step  42  by evaluating failure physics (step  56 ) while also evaluating the inter-relationships between models (step  66 ). Evaluating failure physics (step  56 ) requires identifying models from the designer or open literature (step  58 ), identifying the significant random variables (step  59 ), evaluating and selecting the appropriate models (step  60 ), and developing models for unique failure mechanisms (step  62 ) if no existing models are appropriate. Identifying the significant random variables (step  59 ) requires determining whether variation in a particular variable changes the outcome of the system. If so, then that variable is significant to some extent.  
      Inter-relationships between the selected models (step  66 ) are evaluated by literature review and designer interview (step  68 ) with the appropriate models tied together appropriately to simulate inter-relationships (step  70 ). Tying the models together as is appropriate to simulate inter-relationships (step  70 ) necessarily requires identifying inputs and outputs for each model (step  72 ) and a developing a sequencing strategy (step  74 ). Identifying inputs and outputs for each model also facilitates the developing a sequencing strategy (step  74 ).  
      FIGS.  2 ( a )- 2 ( c ) show how to formulate probabilistic strategy at step  46 . Formulating probabilistic strategy is a method for predicting the probability of failure that considers the variability of the input and system parameters. Still referring to  FIG. 2 ( a ), the first step is to characterize variables (step  76 ). Variables are classified as those that can be directly sensed  78  or that can be inferred  80  from directly sensed information. Otherwise, variable values must come from reference information  82 . A part of characterizing variables (step  76 ) is also to identify the randomness of each variable, i.e. determine the statistical variation of each variable.  
      Now referring to  FIG. 2 ( b ), formulation of probabilistic approach at step  84  requires identifying and selecting an appropriate probabilistic technique. Two primary probabilistic approaches may be appropriate for prediction analysis  30  ( FIG. 1 ): fast probability methods (FPM), or simulation techniques (ST). FPM include response surface FPM  88  and direct FPM  92  techniques. A response surface approximates the failure physics of the system with a single mathematical relationship. A direct method can have disjoint mathematical relationship and is more simplistic. ST include response surface ST  90  and direct ST  94  as well (FPM and ST techniques are discussed further with reference to  FIG. 2 ( c ) below, and see Ang and W. Tang,  Probability Concepts in Engineering Planning and Design , Vols. I and II, John Wiley &amp; Sons, 1975.). Several factors must be considered during selection of probabilistic strategy (step  46 ) including: CPU  18  computational capacity or limitations; whether it is possible to formulate a response surface equation; the mathematical form of the selected failure models (steps  60 ,  62 ) ( FIG. 2 ( a )); the needed prediction accuracy; the characteristics of the monitored system; and the desired update speed or efficiency, among others. All factors are weighed in the balance by one of skill in the art, recognizing that engineering analysis  12  ( FIG. 1 ) must determine which probabilistic technique is most appropriate for prediction analysis  30  ( FIG. 1 ) for the particular type of system  22  ( FIG. 1 ).  
      The system itself may dictate the approach. Of the primary probabilistic techniques available for prediction analysis  30 , direct FPM  92  and ST  94  methods will always provide a solution to the system that facilitates prediction analysis  30 . Response surface FPM  88  and ST  90 , however, do not always provide a workable solution. For example, a response surface cannot be formed when considering variables that vary with time and present discontinuities. Direct methods are then necessary. Potentially, such a situation could be handled using multiple nested response surface equations, but a single response surface equation will not suffice. Where a response surface may be used, however, its use can increase the efficiency of the prediction calculations.  
      Referring to  FIG. 2 ( c ), FPM optional approaches include first order reliability methods (FORM), second order reliability methods (SORM), advanced mean value (AMV) methods and mean value (MV) methods. ST optional approaches include Monte Carlo (MC) methods and importance sampling methods. These different methods are also discussed in further detail in an Example within.  
      Response surface techniques, whether response surface FPM  88  or ST  90  are divided into capacity and demand segments (steps  112 ,  118 ) respectively. For response surface FPM  88 , one of the approaches of FORM, SORM, AMV methods, or MV methods is used to produce a full cumulative distribution function (CDF) for the capacity portion of the response surface equation (step  114 ). A CDF is a plot describing the spread or scatter in the results obtained from only the capacity portion. For response surface ST  90 , either MC or importance sampling methods are used to produce a full CDF for the capacity portion of the response surface equation 120. An equation is then fit to the CDF plots (steps  116 ,  122 ).  
      Often the capacity section is based on referenced data  82  ( FIG. 2 ( a )), while the demand section is based on sensed data  78  and inferred data  80 . In such a case the equation from steps  116  and  122  produces a failure prediction for data representing referenced data  82 , the capacity section of the response surface. Example 1, within, further illustrates this situation.  
      Direct techniques FPM  92  or ST  94  also have both capacity and demand designations, but no response surface is involved. Direct methods are therefore most often appropriate when a response surface cannot be created. The first step in direct FPM is to establish a method for generating random variables and calculating the corresponding random variable derivatives (step  124 ). The next step is to establish a scheme for using the random variable derivatives in a failure model (step  126 ). The failure model is the one developed in model failure physics (step  42 ) (FIGS.  1 ,  2 ( a )). The scheme established in step  126  serves to produce many random variable derivatives for input into the failure model from step  42  (FIGS.  1 ,  2 ( a )). Then one must determine the convergence criteria (step  128 ) to know when to cease inputting the random variable derivatives into the failure model.  
      Similarly, direct ST  94  uses the failure model from model failure physics (step  42 ). As with direct FPM, direct ST  94  must also create a random variable generation method (step  130 ). But direct ST  94  does not calculate derivatives of these random variables. The next step using direct ST  94  is to establish a method for using the random variables themselves in the failure model (step  132 ). And the last step is to determine the number of simulations to be conducted (step  134 ), which sometimes requires trial and error to determine the number of simulations necessary to give a failure prediction with the desired precision.  
      Returning to  FIG. 2 ( b ), the step  46  of formulating probabilistic strategy continues with a determination of the analysis frequency (step  96 ), or the frequency with which prediction analysis  30  ( FIG. 1 ) analyzes input  28  ( FIG. 1 ). To determine analysis frequency (step  96 ) one must determine how often relevant direct sensed data is acquired and processed (step  98 ), determine the fastest update frequency required (step  100 ) and determine the appropriate analysis frequency (step  102 ) for prediction analysis  30  ( FIG. 1 ).  
      The last step  48  in engineering analysis step  12  ( FIG. 1 ) is to develop warning criteria ( FIG. 1 ). Continuing with  FIG. 2 ( b ), determining warning criteria  48  requires establishing the reliability or probability of failure (POF) threshold for sending a warning (step  104 ) based on prediction analysis  30  ( FIG. 1 ). The next step is to set the level of analysis confidence needed before a warning signal is to be sent (step  106 ) and then to develop a method for confidence verification prior to sending the warning (step  108 ). At some point, listed last here, one must determine a type of warning appropriate for the system or user (step  110 ).  
      Now referring to  FIG. 2 ( d ), the results of the previous steps are programmed at step  136  into memory device  34  ( FIG. 1 ) and CPU  18  ( FIG. 1 ) as appropriate criteria, equations, models, and reference data. For response surface FPM  88  or ST  90 , the appropriate criteria, equations, models, and reference data  14  include: a mapping strategy for each variable and response surface equation; a statistical distribution, or CDF, of the capacity portion of response surface equation; and an analysis frequency strategy and warning criteria  138 . The mapping strategy essentially relates sensed, inferred, and referenced data to the variable in the analysis that represents that data. For direct FPM  92  the appropriate criteria, equations, models, and reference data  14  include: a variable derivative method for FORM, SORM, AMV methods, or MV methods analysis; a convergence criteria; and an analysis frequency strategy and warning criteria  140 . And for direct ST  94  the appropriate criteria, equations, models, and reference data  14  include: a random variable generation method for MC or importance sampling analysis; a number of simulations to be conducted; and an analysis frequency strategy and warning criteria  142 . One of ordinary skill in the art will know to mesh the invention with the system of interest in a way that allows both the invention and system to operate correctly.  
      FIGS.  3 ( a ) and  3 ( b ) are flow charts that illustrate the operation of the failure prediction step  16  depicted schematically in  FIG. 1 . Referring to  FIG. 3   a , the step of prediction analysis  30  ( FIG. 1 ) on CPU  18  ( FIG. 1 ) receives the equations from criteria, equations, models, and reference data  14 . Failure prediction is performed by CPU  18  in response to operations and sensor data  25  received from computer control  20 . CPU  18  reads or receives operation and sensor data  25  from control computer  20  according to the frequency strategy. Operation and sensor data  25  are combined with referenced data  82  ( FIG. 2 ( a )) from memory  34  to create input  28 . CPU  18  maps the data in input  28  to the appropriate variables for prediction analysis  30 .  
      Continuing with  FIG. 3 ( a ), prediction analysis  30  follows different paths depending upon the technique chosen: probabilistic response surface FPM  88 , or ST  90 ; probabilistic direct FPM  92 ; or probabilistic direct ST  94 .  
      For direct FPM  92 , POF is determined at step  152  using FORM, SORM, AMV methods or MV methods as previously determined (see  FIG. 2 ( d )). Then POF is compared at step  160  to exceedence criteria and verified per confidence criteria. Exceedence criteria for direct FPM  92  can be defined as the state when POF exceeds the established reliability or POF warning criteria threshold established at step  104  ( FIG. 2 ( b )).  
      For direct ST  94 , POF is determined at step  156  using MC or importance sampling methods as previously determined (see  FIG. 2 ( d )). Then POF is compared at step  160  to exceedence criteria and verified per confidence criteria. Exceedence criteria can be defined as the state when POF exceeds the established warning criteria threshold value established at step  104 . An example applicable to direct techniques  92  or  94  is where prediction analysis  30  determined POF at steps  152 ,  156  at 1.2 percent which was compared to POF threshold  104  of 1.0 percent, thus establishing the need for a warning signal.  
      For response surface FPM  88  or ST  90 , the demand portion of the response surface is calculated at step  146  and the POF is determined at step  148  using the CDF equation. POF is then compared at step  160  to exceedence criteria and verified per confidence criteria. Exceedence criteria can be defined as the state when the demand portion of the response surface exceeds the capacity portion of the response surface that is determined during engineering analysis step  12  ( FIG. 1 ).  
      An example applicable to response surface FPM  88  or ST  90  is where the CDF is represented by the simple equation POF=(constant)*(demand). The demand portion of the response surface calculated at step  146  yields at step  148  a POF that is then compared to POF threshold  104 . POF is then verified using the method for confidence verification  108  ( FIG. 2 ( b )) with memory device  34  ( FIG. 1 ). For these analysis methods, if POF as determined at steps  148 ,  152 ,  156  is compared and verified at step  160  and meets the exceedence criteria, then in step  162  the warning criteria are followed and a warning included in output data  32 .  
      Output data  32  includes the variable readings; POF; selected warning criteria; and warning information. For example, output warning criteria could be to turn on a light when the calculated POF is greater than 1 percent. The demand variable readings; calculated values; POF; and selected warning criteria are stored at step  164  in memory device  34  and the appropriate warning information is communicated at step  166  as prediction results  29  to the vehicle computer control  20 . Prediction results  29  may contain only a portion of the information in output data  32 . The stored variable readings, POF, selected warning criteria and warning information  164  serve as input for subsequent cycles.  
      Now referring to  FIG. 3 ( b ), at step  168  computer control  20  ( FIG. 1 ) receives information from on-board sensors  24  and systems and sends the appropriate operation and sensor data ( FIG. 1 ) to CPU  18  ( FIG. 1 ), forming part of input  28  ( FIG. 1 ). Operation and sensor data  25  includes data from sensors  24  in addition to other data collected by computer control  20 , such as ignition cycles, brake light status, mileage, speed, and numbers of activations of other systems on dynamic system  22 . Computer control  20  also collects at step  172  warning signal information as produced by CPU  18  and decides at step  178  if a signal should be sent to user alert interface  26 . At step  176  user alert interface  26  receives the warning signal information from overall system computer control and at step  178  activates alerts as appropriate. User alert interface  26  shows a number of individual components, with status, or alert, indicators for each as is necessary for the systems being analyzed for failure, such as, for example, yellow light  27 .  
       FIG. 4  is a schematic illustrating an embodiment of an apparatus of the present invention employed on a static system  22  and an indication of the process flow. Prognostic Instrument Engineering System (PIES)  11  would be used where system  22  is a structure such as a bridge or a moving structure such as an airplane where the on-board information (from operation and sensor data  25 ) is used for predictions analysis  30  using a CPU  18  that is not on the system  22 . PIES  11  generally includes a central processing unit (CPU)  18 ; a computer control  20 ; a user alert interface  26 ; and sensors  24 . The CPU  18  receives input in the form of criteria, equations, models, and reference data  14  derived from engineering analysis performed at step  12  and the PIES  11  uses such input to make a failure prediction at step  16 . PIES  11  is substantially similar to OPIE  10  ( FIG. 1 ), a difference being that CPU  18  resides off-board and thus communication device  23  is needed to transmit data from sensors  24  to overall system computer control  20 .  
      Engineering analysis step  12  essentially comprises the preparatory steps that produce the criteria, equations, models, and reference data  14  that are used in failure prediction step  16  to assess the condition of the system or component of interest. Engineering analysis step  12  includes the steps: identify failure mechanisms  40 ; model failure mechanisms  42 ; formulate probabilistic strategy  46 ; and determine warning criteria  48 . Engineering analysis step  12  yields criteria, equations, models and reference data  14 , which were further described and shown in  FIG. 2 ( d ).  
      Continuing with  FIG. 4 , criteria, equations, models and reference data  14  are stored on a memory device  34  or incorporated into a computer program product within CPU  18  as prediction analysis  30 . Desired criteria from criteria, equations, models and reference data  14  may also be programmed into overall system computer control  20 .  
      Sensors  24  measure data on any number of conditions, such as temperature, speed, vibration, stress, noise, and the status and number of on/off cycles of various systems. Data acquired by sensors  24  are transmitted via communication device  23  (for example: hard wire, satellite, and cell phone systems)  23  to computer control  20 . Computer control  20  sends operation and sensor data  25  to CPU  18 . Operation and sensor data  25  includes data from sensors  24  in addition to other data collected by computer control  20 , such as weather conditions. CPU  18  creates input  28  by combining operation and sensor data  25  with information from memory device  34  and information from previous output data  32  that was stored in memory device  34 .  
      CPU  18  analyzes input  28  as directed by prediction analysis  30  to produce the output data  32 . Output data  32  contains a prediction result  29  and possibly other information. Output data  32  is then saved in memory device  34  while prediction result  29  is sent to computer control  20 . Computer control  20  determines from criteria contained in criteria, equations, models and reference data  14 , or from criteria developed separately, whether and how to signal user alert interface  27  based on prediction result  29 . These criteria could be incorporated into CPU  18  instead, so that CPU  18  determined whether to activate user alert interface  27 .  
      User alert interface  27  is a number of individual components, with status, or alert indicators for each as is necessary for the systems being analyzed for failure, such as, for example, a yellow light signal upon predicted failure exceeding stated threshold value. A variety of user alert signal devices could be appropriate for the specific situation. Computer control  20  could also be configured to de-activate certain components upon receipt of the appropriate prediction result. For example, if a POF for a bridge structure exceeded exceedence criteria, the State Department of Transportation might request that a team of engineers visually inspect the bridge. Another example might be that PIES has predicted increased POF due to continuous heat cycling that may have degraded solder connections within an electronic component. Here a signal would be sent to the overall system computer control  20  and a flash message would be sent as signal  38  to the system operator user alert interface  26 .  
      The principles of the present invention are further illustrated by the following example. This example describes one possible preferred embodiment for illustrative purposes only. The example does not limit the scope of the invention as set forth in the appended claims.  
     EXAMPLE 1  
      The following example describes the modeling and prediction of failure in an exemplary embodiment according to the present invention.  
      FIGS.  5 ( a )- 5 ( f ) illustrate a preferred embodiment of the invention applied to a single dynamic component, namely a composite helicopter rotor hub. Reference numerals refer to the elements as they were discussed with respect to  FIGS. 1-4 . In this example engineering analysis step  12  first incorporates a probabilistic approach using response surface FPM  88  techniques. Thereafter, the same example is used to demonstrate any difference that response surface ST  90 , direct FPM  92 , or direct ST  94  would have yielded.  
      A helicopter rotor hub is a structure to which the blades of the helicopter are attached. The rotor hub is a composite laminate structure, which means that it is manufactured by laying plies of composite sheets together and joining them (with an adhesive resin) to form an integral structure. Each composite sheet is called a ply. During flight, the rotor hub experiences continuous cyclic loading due to rotation of the helicopter blades, which causes structural fatigue failure. Upon inspection of failed hubs, it was determined that the initial cause was a cracking problem in the composite rotor hub. Thus, an identified failure mechanism was the cracking in the rotor hub.  FIG. 5 ( a ) shows a one-half schematic finite element model (FEM) of the hub. Upon closer examination, it was observed that cracking was occurring at the laminate ply interfaces as depicted in  FIG. 5 ( b ). After reviewing literature (failure reports in this case) and discussions with the part designer (step  52 ), the active failure mechanisms were determined (step  54 ) to be the cracking at the laminate ply interfaces. This was causing composite ply delamination. Thus, in general, an identified failure mechanism from steps  40 ,  50 ,  52 , and  54  generally illustrates how and why a part failed.  
      The next step was to model the failure mechanism  42 . The first step in modeling was to evaluate the failure physics (step  56 ). Discussions with the part designer identified a model (step  58 ) used to model the failure of similar parts; virtual crack closure technique (VCCT). VCCT was selected (step  60 ) to model the physics of delamination. VCCT was used to calculate the strain energy release rate (G) at the delamination (crack) tip. If the calculated strain energy release rate exceeded the critical strain energy release rate (G cis ) obtained from material tests, delamination failure was assumed to have occurred. VCCT was used to calculate the strain energy release rate (G) at the delamination tip such that:  
             G   =       G   I     +     G   II               Eq   .           ⁢     (   1   )               where                             G   I     =     -       1     2   ⁢           ⁢   Δ       ⁡     [         F   ni     ⁡     (       v   k     -     v   k   ·       )       +       F   nj     ⁡     (       v   m     -     v   m   ·       )         ]           ;   and           Eq   .           ⁢     (   2   )                   G   II     =     -       1     2   ⁢   Δ       ⁡     [         F   ti     ⁡     (       u   k     -     u   k   ·       )       +       F   nj     ⁡     (       u   m     -     u   m   ·       )         ]                 Eq   .           ⁢     (   3   )               
 
 In Eq. 2 and 3, u and v are tangential and perpendicular nodal displacements respectively and F i  and F n  are the tangential and perpendicular nodal forces respectively. Delamination onset was assumed to occur when the calculated G exceeded the G crit  derived from material delamination tests. Since VCCT was adequate for modeling this failure mechanism no unique model needed to be developed as in step  62 . 
 
      In this case, seven significant random variables were identified at step  59  and are shown in Table I, where: 
          E 11 , Msi Longitudinal Young&#39;s modulus     E 22 , Msi Transverse Young&#39;s modulus     G 13 , Msi Shear modulus     v 13  Poisson&#39;s ratio     P, kips Tensile load     Φ, degrees Bending angle     G crit  Critical strain energy release rate        

      N Fatigue cycle  
               TABLE I                          The Significant Random Variable for the Response Surface Fpm Example       Random Variables                                 Property   Mean   Std. Dev.                                             E 11 , Msi   6.9   0.09           E 22 , Msi   1.83   0.05           G 13 , Msi   0.698   0.015           ν 13     0.28   0.01           P, kips   30.8   3.08           Φ, degrees   12   1.67           G crit     448.56-58.57 Loge(N)   36.6 J 2 /m                      
 
 Computation of strain energy release rate, G, required determination of nodal forces and displacements at the delamination tip as shown in  FIG. 5 ( b ). Determination of the nodal forces and displacements required development of a finite element model (FEM) for the rotor hub with the appropriate loads and material properties of the hub. 
 
      Referring to  FIG. 5 ( d ), the physics of failure for the rotor hub required a combination of different models. Once the models were selected at step  60  or developed at step  62 , the next step was to evaluate inter-relationships between models at step  66 . This involved identifying the inputs and outputs of each model (step  72 ) as well as identifying inter-relationships from the literature and designer interviews in step  72 . Then the models were tied at step  70  and the overall model sequencing strategy developed at step  74 . Since the material properties of the rotor hub were not readily available, they had to be derived from the material properties of the individual composite plies  180  using a laminate model  182  to give the laminate material properties  186 . Laminate properties  186  and load data  184  were input into FEM  188  to yield nodal forces and displacements  190 . Nodal forces and displacements  190  were input into VCCT  192  to yield strain energy rate (G)  194 .  
       FIG. 5 ( d ) shows that the calculated strain energy release rate, G was determined from the ply material properties  180  and the loads  184 . G was the dependent variable and ply material properties and the loads were the independent variables. The next step was to develop a probabilistic strategy (step  46 ). First all the variables were characterized in step  76  in terms of randomness and as directly sensed  78 , inferred  80 , or referenced  82 . P max  and Φ are the directly sensed variables and the material properties, including E 11 , E 22 , G 13 , and v 13 , and G crit  are inferred variables whose randomness is presented in Table I. The part designer had gathered test data on G crit  versus the number of fatigue cycles (N). Based on the statistical analysis of this G crit  vs. N data (see  FIG. 5 ( c )), it was determined that G crit  is a Gaussian (normal) random variable with its mean value and standard deviation shown in Table I. There are several different probabilistic assessment approaches (step  84 ) available. Direct ST  94  and FPM  92  and response surface ST  90  and FPM  88  techniques were discussed earlier and this example will apply each to the rotor hub.  
      One such approach is to use the first order reliability method (FORM), which is an example of a fast probabilistic method (FPM), in conjunction with the response surface, referred to previously as a response surface FPM (88) approach. First a response surface must be developed relating G to the independent variables. Developing a response surface is widely discussed in the open literature. See A. Ang and W. Tang,  Probability Concepts in Engineering Planning and Design , Vol. 1, John Wiley &amp; Sons, 1975. Based on the seven random variables a Design of Experiment (DOE) scheme was chosen as shown in Table II.  
               TABLE II                          Design of Experiments Scheme for Response Surface - FPM Approach                                             Variable   Trial 1   Trial 2   Trial 3   Trial 4   Trial 5   Trial 6   Trial 7               E 11     1   0   0   0   0   0   0       E 22     0   1   0   0   0   0   0       G 13     0   0   1   0   0   0   0       ν   0   0   0   1   0   0   0       P   0   0   0   0   1   0   0       Φ   0   0   0   0   0   1   0       Gcrit   0   0   0   0   0   0   1           ↓   ↓   ↓   ↓   ↓   ↓   ↓       Strain Energy Rate   E11   E 22     G 13     ν   P   Φ   Gcrit       Sensitivity to:                  
 
      In Table II, Trial 1, E 11  was changed from its nominal value (mean value, indicated as 1), while all the remaining six variables were kept at their respective mean values (indicated as 0) and the value of G  194  was calculated. This process was repeated for each of the six other variables. Following this step, a regression analysis was performed and an initial response surface was developed that related G to all the seven significant random variables. After this, an Analysis of Variance (ANOVA) was performed to determine if all the seven significant random variables needed to be included in the response surface. The ANOVA results yielded that out of the seven random variables only 4 random variables (G crit , E 11 , P and Φ) needed to be included in the response surface. Based on this, an updated DOE scheme was adopted as shown in  FIG. 5 ( e ) to create a quadratic response surface equation. Regression analysis yielded the final response surface equation shown in Eq. (4). This strategy was verified by input of data published in the open literature and comparing the output results with results published in the open literature. 
 
 g=G   crit −175.344*(0.569−0.0861  E   11 +0.023 P   max −0.117Φ−0.000546 P   2   max +0.00376Φ 2 +0.0046 P   max Φ)  Eq. (4) 
 
      The next step in response surface FPM  88  ( FIG. 2 ( b )) approach is to divide the response surface into the capacity and demand segments (step  112 ). The separation was as follows: Eq. (5) represents the capacity segment of Eq. (4) and Eq. (6) represents the demand segment of Eq. (4). 
 
Capacity= G   crit −175.344*(0.569−0.0861 E   11 )  Eq. (5) 
 
Demand= G   crit −175.344*(0.023 P   max −0.117Φ−0.000546 P   2   max +0.00376Φ 2 +0.0046 P   max Φ)  Eq. (6) 
 
      For this particular example, the variables in the capacity section of the response  30  surface equation are the material property E 11  and G crit . The variables in the demand portion of the response surface equation are the load (P) and the angle of the load (Φ). Eq. (5) was then used to produce a full CDF for the capacity portion of the response surface equation (step  114 ). This CDF is shown in  FIG. 5 ( f ) with capacity equated to the probability of failure.  
      Using FORM all the variables in the capacity portion of the response surface (E 11  and G crit ) are transformed to equivalent uncorrelated standard normal variables (Y1 and Y2). In the transformed uncorrelated standard normal space, a linear approximation is constructed to the capacity portion of the response surface and is given by the equation: 
 
 y =(9 E− 14) x   6 −(9 E− 11) x   5 +(3 E− 8) x   4 −(5 E− 6) x   3 +0.0004 x   2 −0.0087 x   Eq. (7) 
 
 To estimate the CDF using FORM a constrained optimization scheme is adopted to search for the minimum distance from the origin to the transformed response surface. Mathematically, the problem can be formulated as: 
 
Minimize such that  g ( Y )=0  Eq. (8) 
 
 where, β is the minimum distance and g(Y) is the transformed capacity portion of the response surface. Several optimization routines are available to solve the above-constrained optimization problem. The method used in this example was formulated by Rackwitz and Fiessler. See Rackwitz, R. and Fiessler, B.,  Reliability Under Combined Random Load Sequences , Computers and Structures, Vol. 9, No. 5, pp. 489-494, 1978. A first order estimate of the failure probability is then computed as: 
 
 CDF= 1 −F (−β)  Eq. (9) 
 
 where F(−β) is the cumulative distribution function of a standard normal variable (i.e., a normal variable with zero mean value and unit standard deviation). 
 
      A graph of the resultant CDF is shown in  FIG. 5 ( f ). Although mathematical expressions exist to determine the CDF, these expressions involve multiple integrals, which can be quite cumbersome to evaluate. Hence to make the process of CDF computation faster and more tractable, an equation was fit to the CDF plot in step  116  using traditional curve fit methods. For this example relevant direct sensed data was acquired and processed (step  98 ) and POF could be predicted every flight cycle. Sensor data was also collected continuously during flight, but it was decided that POF would only be reviewed after every 2 flight cycles (step  102 ). It was then determined in step  104  that a POF greater than 1 percent would trigger a warning “No-Go” signal that would in turn activate a yellow light within user alert interface  26 . Also it was decided that the method of confidence verification (step  108 ) was that, within the same flight cycle, a second POF will be determined based on updated sensor data. If prediction analysis  30  returned a POF greater than 1 percent two successive times within the same flight cycle a warning should be sent in step  110 . That warning would be the “No-Go” signal that activated yellow light  27 . This completed engineering analysis step  12 .  
      Criteria, equations, models, and reference data  14 , consisting of the variable mapping strategy, response surface equation, statistical distribution of capacity portion of the response surface equation, analysis frequency, and warning criteria were programmed into memory  34  in step  138 .  
      Failure prediction (step  16 ) began by sending sensor data on the two directly sensed variables (step  25 ), which for this example were P max  and Φ. The next step  146  was to compute the demand portion of the response surface equation. The result from the demand portion of the response surface (Eq. (6)) was then input into the CDF equation derived from the capacity portion of the response surface equation (Eq. (7)). Thus, the current POF at demand was computed in step  148  based on the directly sensed data. The POF was calculated after every second flight cycle based on P max . For this example the calculated demand (or sensed/inferred data) contribution from step  146  to the POF is shown in Table III for the selected cycle numbers after acquisition and analysis of the appropriate sensed and inferred data. In this example the capacity contribution is based on referenced data  82  and the demand contribution is based on sensed data  78  and inferred data  80 , but as discussed earlier in the specification with reference to steps  112  and  118 , the capacity and demand sections are not always based on the same data types.  
               TABLE III                          Response Surface - FPM Prediction Results                                     Cycle   P   θ   (demand)   POF   Warning                                             Cycle-1   30.8   12   68.52277995   0%   Go       Cycle-3   33.88   10.33   44.54782678   0%   Go       Cycle-5   27.72   13.67   86.06125102   0%   Go       Cycle-7   36.96   15.33   181.6366807   15%    Go       Cycle-7   37   16   201.9535041   17%    No-Go                  
 
      Table III also shows the POF determined at step  148  that is compared to the exceedence criteria at step  160 . When POF exceeded one percent twice consecutively within the same cycle (cycle  7 ) the warning criteria was followed in step  162  and a “No-Go” warning was issued as part of output data  32 . Output data  32  included all the values from Table III. These were stored in step  164  in memory  34  of CPU  18 . Thus memory  34  stored cycle data that served as input for subsequent cycles. Of that data in Table III, in this example, only the warning or lack of warning of “No-Go” or “Go” was sent in step  166  to the equivalent of control computer  20 . After collecting the warning signal of “No-Go” in step  172 , control computer  20  decided in step  174  that the warning required further communication to user alert interface system  26 . Upon receipt of the warning at step  176 , user alert interface  26  activated a yellow cockpit indicator light and highlighted “check rotor hub” on a malfunction monitor.  
      A difference between the response surface FPM  88  and ST  90  approaches is the method used to create the CDF. This response surface ST approach used Monte Carlo (MC) methods to produce the CDF. Like the response surface FPM approach  88 , the first step in the response surface ST  90  approach was to separate the response surface equation into capacity and demand portions at step  118 . Following the division of the response surface, Monte Carlo simulation methods were used to develop the full CDF of the capacity portion of the response surface at step  120 . For each MC simulation, random values of G crit  and E 11  were generated based on their respective statistical distribution types and respective statistical parameters. With each set of G crit  and E 11  values generated, the capacity portion of the response surface equation was computed. Following that, a histogram analysis was performed to develop the CDF curve for the capacity portion of the response surface equation.  
      Once the CDF curve fit was developed at step  122  for the capacity portion of the response surface equation the failure prediction method followed the steps outlined in the Response Surface FPM  88  approach following this embodiment of the invention. Table IV shows the results of estimating the probability of failure using the response surface ST  90  approach.  
               TABLE IV                          Response Surface - ST Prediction Results                                     Cycle   P   θ   (demand)   POF   Warning                                             Cycle-1   30.8   12   68.52277995   0%   Go       Cycle-3   33.88   10.33   44.54782678   0%   Go       Cycle-5   27.72   13.67   86.06125102   0%   Go       Cycle-7   36.96   15.33   181.6366807   15%    Go       Cycle-7   37   16   201.9535041   16%    No-Go                  
 
      The direct FPM approach  92  does not require the development of a response surface to predict the probability of failure. This example used direct FORM to transform the seven random variables (in this example these variables are the material properties, G crit , P max  and Φ) to equivalent uncorrelated standard normal variables (represented by vector Y). After transformation, a numerical differentiation scheme was employed at step  124  to determine the derivatives of the random variables. In the transformed uncorrelated standard normal space, a linear approximation was constructed to the final failure equation, which in this case is G&gt;G crit . The derivatives of the random variables were used at step  126  to determine the perturbed values of the random variables. To estimate the probability of failure using FORM a constrained optimization scheme was adopted to search for the minimum distance from the origin to the transformed failure equation. Mathematically, the problem was formulated the same at Equation (8) where β was the minimum distance, but where g(Y) was the transformed failure equation. The method used in this example was formulated by Rackwitz and Fiessler optimization scheme and was used to solve the above constrained optimization scheme. See Rackwitz, R. and Fiessler, B.,  Reliability Under Combined Random Load Sequences , Computers and Structures, Vol. 9, No. 5, pp. 489494, 1978. The constrained optimization scheme is an iterative process to estimate the probability of failure. A convergence criterion was determined at step  128  ( FIG. 2 ( c )) to force the iterations to converge on a failure probability estimate. After the appropriate criteria, equations, models, and reference data were programmed at step  136  into memory device  34 , a first order estimate of the POF was determined at step  152  using FORM as: 
 
 POF=F (−β)  Eq. (10) 
 
      where F(−β) was the CDF of a standard normal variable (i.e., a normal variable with zero mean value and unit standard deviation). Table V shows example results from estimating the probability of failure using Direct FPM approach.  
               TABLE V                          Direct - FPM Prediction Results                                         Cycle   P   θ   POF   Warning                                                     Cycle-1   30.8   12   0%   Go           Cycle-3   33.88   10.33   0%   Go           Cycle-5   27.72   13.67   0%   Go           Cycle-7   36.96   15.33   14%    Go           Cycle-7   37   16   14%    No-Go                      
 
      Like the direct FPM approach, the direct ST  94  approach also does not require the development of a response surface. This example also used Monte Carlo (MC) methods within direct ST  94 . The same seven significant variables from Table I were selected. Based on the analysis frequency, previously determined to be two flight cycles, once the sensors gathered the values of the directly sensed variables, values of the inferred variables were randomly generated in step  130  using MC methods and random values of G crit  and E 11  were generated based on their respective statistical distribution types and respective statistical parameters. For each set of directly sensed data, several sets of the inferred variables were generated. For each set of inferred variables generated, the value of the strain energy release rate G was computed in step  194  as shown in  FIG. 5 ( d ). The number of sets of referred variables was based on the number of simulations to be conducted from step  134 . Appropriate criteria, equations, models, and reference data were stored at step  136  in memory  34  of CPU  18 .  
      For each simulation, if G&gt;G crit  a failure counter was incremented by one. For example, let us assume that for each set of P max  and Φ sensed, M sets of the inferred variables were generated. Among those M sets, for n sets (n≦M), G was greater than G crit . Then the probability of failure would be n/M. Table VI shows example results from estimating the probability of failure using Direct ST approach.  
               TABLE VI                          Direct - ST Prediction Results                                         Cycle   P   θ   POF   Warning                                                     Cycle-1   30.8   12   0%   Go           Cycle-3   33.88   10.33   0%   Go           Cycle-5   27.72   13.67   0%   Go           Cycle-7   36.96   15.33   16%    Go           Cycle-7   37   16   18%    No-Go                      
 
      While the foregoing description and drawings represent embodiments of the present invention, it will be understood that various additions, modifications and substitutions may be made therein without departing form the spirit and scope of the present invention as defined in the accompanying claims. In particular, it will be clear to those skilled in the art that the present invention may be embodied in other specific forms, structures, arrangements, proportions, and with other elements, materials, and components, without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims, and not limited to the foregoing description.