Title: Predictive Reliability of Tactical Missiles Using Health
Monitoring Data And Probabilistic Engineering Analysis

> Authors: Stephen A. Marotta, US Army, Aviation and Missile Research
> Development and Engineering Center (AMRDEC), Presenter

Abdul Kudiya, US Army AMRDEC

> Teng K. Ooi, US Army AMRDEC

Houssam A. Toutanji, University of Alabama in Huntsville

> John A. Gilbert, University of Alabama in Huntsville

#  

# ABSTRACT

Tactical missiles within the U.S. Army are regularly subjected to severe
stresses such as long-term exposure in harsh environments and
transportation handling. These stresses factor into the aging,
deterioration, and eventual decommissioning of some of the Army's
critical warfighting assets. The negative reliability impacts associated
with long-term aging and deterioration significantly affect the total
life cycle cost of fielding these weapons in a high state of readiness.

Reliability evaluation of past data has indicated failures in missile
structural, energetic, and electronic components associated with the
long-term exposure to heat, humidity, and transportation shocks. Unlike
strategic missiles, tactical missiles undergo a very minimum of field
checks and non-destructive evaluation on a routine basis.

The Army Aviation and Missile Research and Development and Engineering
Center have been developing a health monitoring system called Remote
Readiness Asset Prognostics and Diagnostics System (RRAPDS) to assess
and improve reliability of the missiles during storage and field
exposures \[3\]. RRAPDS will use external and internal sensors to
provide data to assess missile conditions and predict reliability.

This paper describes the approach to predict reliability of missile
components like propellant, nozzles, and thermal batteries using sensor
data from RRAPDS, and prognostic models for structural integrity and
damage mechanisms. Probabilistic models will quantify all the
uncertainties present in the health monitoring data and finite element
models, to provide a realistic reliability evaluation.

# INTRODUCTION

The US Army fields tactical missiles of various types all over the
world. These missiles are exposed to different environments during
storage, transportation, and operation. Typical environments include
cyclic exposure to temperature and humidity extremes, vibration and
shocks, and corrosive atmospheric conditions. Long-term environmental
exposures affect a missile's performance and reliability as component
material properties degrade, and this degradation negatively impacts
critical performance parameters.

The reliability of the Army's missile stockpile and individual missile
shelf life are monitored through dedicated programs of surveillance and
testing. After fielding, the Army collects pertinent reliability data
over a missile's lifecycle utilizing a variety of test methodologies
both destructive and non-destructive. This data is analyzed for trends
associated with age, manufacturing strata, and or unique environmental
exposures.

If a missile system continued to perform reliably and safely based on
surveillance data, then an extension of shelf life for that type missile
is recommended to major decision makers and coordinated with the user
and logistics community. If surveillance analysis indicates undesirable
trends, then whole missile populations or subsets of populations are
suspended for use or restricted for special use only. Obviously, the
degradation and ageing of missile populations and their effects on
readiness of the stockpile can have major economic implications if new
procurements are warranted.

A current surveillance and periodic test program is an essential element
to maintain reliability of the missile stockpile by removing suspect
assets before deployment. Analysis of test data identifies failure
scenarios that include manufacturing defects, contamination during
manufacturing and, most importantly, the degradation due to aging
exposure and environmental exposure. In the case of environmental
degradation and aging effects, the failure mechanism points towards
accumulated damage resulting from exposure to temperature, humidity, and
shock and vibration. It is evident from the failure mode analysis that
real time monitoring and analysis of data may provide tools to predict
the reliability of the missiles in storage and determine ways to improve
it.

**MISSILE FAILURE MODES**:

Table 1 provides a snapshot of generic missile failure modes identified
during storage. Fault tree analyses have been conducted to determine
pathways from basic storage conditions to the top failure events. Basic
events like thermal and humidity cycling, shocks and vibration during
handling and transportation and corrosion are major causes of components
failures during long-storage.

Table 1 shows generic failure modes classified under temperature,
humidity, shock and corrosion related causes. RRAPDS provides data on
these variables, which are then used and predict missile reliability,
improve the shelf life and provide reliable hardware for operational
use.

##### Table 1: Snapshot of Generic Missile Failure Modes due to Environmental Exposure

+---------------------+-----------------+--------------+--------------+
| **Temperature**     | **Humidity**    | **Shock**    | *            |
|                     |                 |              | *Corrosion** |
+---------------------+-----------------+--------------+--------------+
| -   Propellant      | -   Degradation | -   Case     | -   Failure  |
|     Grain Cracking  |     of          |     Damage   |     of       |
|                     |     ballistic   |              |     solder   |
| -   Liner Unbond    |     properties  | -   Liner    |     joints   |
|                     |                 |     Unbond   |              |
| -   Voids and       | -   Degradation |              | -            |
|     Porosity in     |     of          | -            |    Corrosion |
|     Propellant      |     composite   |   Propellant |     of       |
|                     |     case        |     Cracking |     metallic |
| -   Case Strength   |                 |              |              |
|     Degradation     | -   Degradation | -   Material |   components |
|                     |     of O-rings  |     Fatigue  |              |
| -   Failure of      |                 |     due to   | -   Stress   |
|     Electronics     | -   Degradation |              |              |
|     Assemblies      |     of          |    vibration |    corrosion |
|                     |     Propellant  |              |     cracking |
| -   Cracks in       |     properties  | -            |     at       |
|     Packaging and   |                 |    Loosening |              |
|     Seals           | -   Failure of  |     of       | interconnect |
|                     |     Electronics |              |     /bends   |
| -   Failure in      |     Assemblies  |   components |     in       |
|     Guidance        |                 |              |              |
|     Components      | -   Cracks in   |              |   electronic |
|                     |     Packaging   |              |              |
|     -   Gyros       |     and Seals   |              |   assemblies |
|                     |                 |              |              |
|                     | -   Failure in  |              | -            |
|  -   Accelerometers |     Guidance    |              |    Corrosion |
|                     |     Components  |              |     of       |
|                     |                 |              |              |
|                     | -   Gyros       |              |    actuators |
|                     |                 |              |     and      |
|                     | -               |              |     bearings |
|                     |  Accelerometers |              |              |
|                     |                 |              | -            |
|                     |                 |              |    Corrosion |
|                     |                 |              |     of       |
|                     |                 |              |     contacts |
|                     |                 |              |              |
|                     |                 |              | -   Failure  |
|                     |                 |              |     in squib |
|                     |                 |              |     wires    |
+---------------------+-----------------+--------------+--------------+

**PROBABILISTIC PROGNOSTICS MODELING**

Temperature, humidity, shock, vibration, and corrosion (chemicals)
parameters can be measured in real time with an integrated health
monitoring system. The data from this system can then be utilized to
develop diagnostic and predictive models for components' health and
integrity and to determine if a missile will operate successfully when
fired.

The US Army has designed a system to monitor missile storage and
transportation environments on a real time basis. The system, called the
remote readiness asset prognostics and diagnostics system (RRAPDS),
utilizes temperature, humidity, and shock sensors as an integral part of
the weapon to monitor and perform diagnostic/prognostics analysis of the
stockpiles during long-term storage \[12\].

RRAPDS is currently being field-tested and it is providing data to be
used by probabilistic engineering models sufficient to predict the
reliability of a weapon system at any point of time.
Prognostic/predictive models are being developed to assess the
reliability and structural integrity of the weapon system components and
they can be used as a decision making tool for field deployment.
Diagnostic and prognostic models will be utilized to translate health
monitoring data into an assessment of reliability and performance of the
weapon. The models are developed to determine if the component has or
will degrade to a point where it cannot withstand the anticipated
operating loads.

The models are developed to compute degradation in material properties
as a result of exposure to thermal and humidity cycling, shock and
vibration, and or a corrosive environment. The material properties data
are determined using sensor information \[1\] that is then correlated
with chemical kinetics or age-related relationships to determine change
in modulus, strain energy, or other properties of this nature. Degraded
material properties are then used in a finite-element method or other
similar mathematical technique to evaluate induced internal stresses and
predict current and future factors of safety. The factor of safety
provides criteria for survivability of a component or weapon system in
the actual field environment.

The prognostics and diagnostics models based on the deterministic
approach stated above may not provide the actual quantification of
uncertainty and variability presented in the health data and
mathematical models. The real-time health monitoring data would consist
of large variations in parameter values over time and the application of
an average or worst-case value may overlook the occurrence of the
failure frequency. Furthermore modeling uncertainty may not provide high
confidence in the reliability assessment of the weapon system. The
assumption of deterministic variables is an idealization that is not
true in the real world. The extrapolation of the deterministic data to
predict failure over time will be suspect and it adds another dimension
of uncertainty.

A sound approach to modeling for prognostic and diagnostic analysis of
the weapon system will be based on probabilistic engineering analysis.
The probabilistic approach will attempt to quantify variability in the
health monitoring data and modeling uncertainties and forecast the true
failure frequency for decision-making purposes.

In this approach, the parameters of the prognostic and diagnostic models
are specified as statistical distributions. These distributions are
determined using statistical analysis of the health monitoring data. The
model output response that includes the induced loads and component
capabilities are also output as a statistical distribution. The
synthesis of induced loads and component capabilities generate a failure
function that can be analyzed to predict current and future reliability
of the weapon system. The probabilistic approach will quantify increased
variability in the failure function as data are extrapolated for future
reliability assessment.

Several methods are available to analyze failure modes using the
probabilistic engineering approach. The methods range from simple
synthesis of material capability (strength) distribution with the
applied (induced load) distribution to complex Monte Carlo simulation
and sensitivity analysis. All of the analyses require the statistical
analysis of all the input data to the failure function. The methods that
are being evaluated for the army tactical systems are classified under
three different categories: 1) probabilistic engineering evaluation
using strength and stress interference; 2) probabilistic evaluation of
the cumulative damage function; and, 3) prediction of component life
based on Weibull analysis.

RRAPDS will provide data with information on exposed temperature,
humidity, shock, vibration, and chemicals environment. Significant
variables and trends can be identified using data mining techniques.
Data collected can be analyzed to update design parameters such as
failure rate of components, test costs, environmental thresholds, etc.,
and to predict spare parts requirements.

STRESS AND STRENGTH INTERFERENCE METHOD

In this approach, the material capability ($\overline{C}$) and the
induced load distributions ($\overline{R}$) are used to compute the
probability of failure at a point in time. If both parameters are
normally distributed, the probability of failure is given by, \[5\]

> $P_{f} = 1 - \varphi\left\{ \frac{\overline{C} - \overline{R}}{\sqrt{(S_{C}^{2} + S_{R}^{2})}} \right\}$
> (1)

where $\overline{C}$ and $\overline{R}$are average capabilities and s~c~
and s~r~ are standard deviation of the capability and induced loads,
respectively. Ф is the normal probability function determined from
standard normal table.

Figure 1 \[5\] shows the probability distributions as the material
capability degrades while the induced stress due to long-term
environmental effects increases.

![](media/image1.jpeg){width="3.8159722222222223in"
height="2.084722222222222in"}

Figure 1: Component reliability with age.

In the stress and strength model, the capability or material strength
(properties) is determined using a degradation mechanism resulting from
humidity or temperature cycling, or shock and vibration. Some of the
degradation functions include empirical Coffin-Manson functions, the
Arrhenius law, or Boltzman formula \[6\]. The induced loads are
determined using structural engineering models and input distribution
from material properties (modulus, temperature) to compute the
distribution of r. Statistics from these distributions are applied at
any point of time to evaluate P~f~ .

An appropriate example for the stress and strength approach is to
determine the reliability of the propellant/casebond system in the
missile ordnance components like rocket motors and gas generators.
Propellant modulus changes with age which impacts both the strength
properties and the induced stress. Knowing propellant modulus at a given
point of time, the propellant reliability can be determined using a
finite element analysis model of induced stress and data on degraded
material properties.

CUMULATIVE DAMAGE FUNCTION METHOD

Cumulative damage models evaluate the aggregate of small or microscopic
damage within the component due to stress induced by environmental
conditions over a time period. The incremental stresses are accumulated
over time to determine the degradation in the component strength and to
make predictions on whether the missile will withstand operational loads
without failure. The rationale of the cumulative damage function is that
eventually microscopic damage will accumulate and lead to failure.
According to this theory, missile component failure will occur if the
following is true:

![](media/image2.wmf) ≥ 1.0 .

The simplest form of the cumulative damage law is described by the
Palmgren-Miner \[7\] rule as,

> $D = \sum_{}^{}\frac{n_{i}(t)}{N_{f}}$ (2)

where d denotes the fatigue damage and n~i~ are the number of actual
applied cycles at time t, and N~f~ is the total number of cycles to
failure. At failure, d = 1 and n~i~ (t) = N~f~.

Equation (2) represents a linear damage function that was originally
proposed for the life prediction of metallic components undergoing
fatigue. However, the linear damage function was found to give
non-conservative results, as it predicts lives greater than those
observed experimentally. Its main deficiencies are: 1) load level
interdependence; 2) load sequence interdependence; and, 3) lack of load
interaction accountability.

To make the cumulative damage law more appropriate to the real world, a
simple non-linear form of Eq. 2 was presented using the definition of
the power law as

$D = \sum_{}^{}\left\lbrack \frac{n_{i}(t)}{N_{f}} \right\rbrack^{\beta}$
(3)

where the power is the load-dependent variable.

The cumulative damage \[(D(t)\] resulting from small steps in induced
strength due temperature and humidity cycling or shock and vibration
over a period of time is described by a damage function \[2\] as

$D(t) = \int_{0}^{t_{f}}\frac{1}{t_{f}}\text{dt}$ (4)

$t_{f} = \left( \frac{\sigma_{0}}{\sigma(t)} \right)^{\beta}$ (5)

$D(t) = \int_{}^{}\left( \frac{\sigma(t)}{\sigma_{0}} \right)^{\beta}\text{dt}$
(6)

where t~f~ is the time to failure, (t) is the induced stress as a
function of time, ~0~ is the material strength, and is the power law
exponent showing the interaction between stress and strength parameters.

Equation 6 is shown graphically in Fig. 2 \[2\].

![](media/image3.jpeg){width="3.8194444444444446in"
height="1.6756944444444444in"}

Figure 2: Time-to-failure versus damage due to increased stress.

Equation (6) can be evaluated using calculated stress values induced by
temperature, humidity cycling, or shock and vibrations over time t. The
value of the damage function at time (t) must be less than 1.0 to ensure
the structural integrity of the component. Stress values are determined
using finite element modeling or similar techniques.

The damage function D(t) evaluated in Eq. (6) represents a precise value
and does not show any variability in D(t). It is calculated using
specified values of temperature, humidity, or shock.

The data from the health monitoring systems will show a large variation
in the measured parameters and the incorporation of average values to
calculate stress will not provide an appropriate failure scenario. Since
the damage function D(t) is very sensitive to applied stress, any
variation in mechanical properties or other data could provide
uncertainties in the results.

A probabilistic approach will be more appropriate to forecast the true
failure frequency \[10\]. Figure 3 shows the comparison of mean damage
(deterministic value) versus the failure probability.

According to Fig. 3, \[4\] the probability of failure as defined by the
damage model is defined as,

P~f~ = P (d ≥ 1.0) (7)

![](media/image4.jpeg){width="3.252083333333333in"
height="1.7236111111111112in"}

Figure 3: Failure probability distribution due to cumulative damage.

The probabilistic approach suggests that the damage model be developed
as a failure distribution using statistical information on the input
parameters such as temperature, humidity, and shock as measured by the
health monitoring systems. The variability in the input parameters could
provide distribution of the damage function that can be evaluated for
failure probability and reliability of the component.

WEIBULL SERVICE LIFE PREDICTION METHOD

Weibull analysis is widely used in reliability work to predict component
service life and long-term aging degradation due to induced stress
\[9\]. Weibull probability analysis provides a useful tool to predict
time-dependent reliability and component service life \[8\].

The analysis methodology has a great advantage in stress and strength
approach where non-linearity in the strength or stress function is
involved. The application of Weibull distribution in the prognostic
analysis is illustrated with an example of monitoring and predicting
reliability of missile component like gyros in the guidance section.

PROBABILITY ANALYSIS OF GYROS:

Gyros are used in missile guidance and navigation systems and their
precision is impacted due to long term exposure to humidity and
temperature in storage environment. Failure mode analysis of gyros has
identified several failure modes \[16\] in non-operating storage
environment. Some of them are listed below:

-   Separation in Spin Bearing lubrication due to temperature and
    humidity

-   Creep and change in dimensional tolerances due to temperature
    variations

-   Adhesive failure in Gimbals gas bearings due to humidity.

Data on gyros reliability as a function of storage time have been
reported in reference \[16\]. To illustrate the application of
probabilistic analysis for predicting failures and improve shelf life,
some of the data from \[16\] are used. Weibull probability analysis has
been performed to predict the reliability of gyros as a function of
storage time.

According to Weibull analysis, the cumulative failure distribution is
defined by function F(t) as follows:

$F(t) = 1 - e^{- (\frac{t}{\eta})^{\beta}}$ (8)

where

t = time in storage (hours, years, etc.)

= scale parameter

= shape parameter

Component reliability at a given storage time t is therefore,

$R(t) = 1 - F(t) = e^{- (\frac{t}{\eta})^{\beta}}$ (9)

Weibull parameters and can be estimated by using field reliability data
and then used to predict the component reliability over time. Some of
the data on gyros described in \[16\] are used to illustrate the
methodology.

Taking the national log of both sides and negating both sides, (9), we
get

ln \[1/R(t)\] = (t/n) (10)

And, by taking the natural log again, we arrive at

ln {ln \[1/R(t)\]} = ln(t) - ln() (11)

Given component reliability versus storage time data, Weibull parameters
(, ) can be estimated using linear regression analysis of equation (11)
An example of gyros field reliability versus storage time are taken from
\[16\] and are shown in Table 2 below

######  Table 2: Reliability Data on Gyros

+-----------------+-----------------+-----------------+-----------------+
| Storage Time    | Reliability     | ln ln           | Ln (t)          |
|                 |                 | \[(1/R(t)\])    |                 |
| (years)         | Success/Total   |                 |                 |
|                 | tested          |                 |                 |
| t               |                 |                 |                 |
|                 | R(t)            |                 |                 |
+-----------------+-----------------+-----------------+-----------------+
| 1.14            | 0.9988          | -6.718409       | 0.127061        |
+-----------------+-----------------+-----------------+-----------------+
| 1.50            | 0.9942          | -5.154005       | 0.405592        |
+-----------------+-----------------+-----------------+-----------------+
| 3.27            | 0.9824          | -4.030075       | 1.185539        |
+-----------------+-----------------+-----------------+-----------------+
| 4.35            | 0.9844          | -4.150703       | 1.46978         |
+-----------------+-----------------+-----------------+-----------------+
| 5.00            | 0.9548          | -3.074563       | 1.609438        |
+-----------------+-----------------+-----------------+-----------------+

The data are plotted in Figure 4. A least square fit of data in Table 2
gives the following values.

Slope () = 1.432

Intercept (- ln()) = -5.774

, the scale parameter is equal to exp (5.774/1.432) = exp (4.032) = 56.4
years. Thus 63.21 % of gyros failures are expected to occur in 56.4
years.

Using the values of and , Figure 5 is constructed to show the predicted
reliability of gyros as a function of time.

![](media/image5.wmf){width="5.594444444444444in"
height="3.091666666666667in"}

Figure 4: Weibull plot of gyro data.

![](media/image6.wmf){width="5.594444444444444in"
height="3.091666666666667in"}

Figure 5: Pct failures versus storage (time).

Gyros failures in storage are generally caused by exposure to
temperature and humidity cycling over a long period of time. Humidity
and temperature cycling causes material degradation and induce stress
which accumulates over time resulting in component failure. The Weibull
analysis can be further extended by evaluating a relationship between
stress and temperature/humidity cycling and integrating it over the
storage time. Temperature, and humidity data from RRAPDS could then be
directly input into the model to predict component reliability and shelf
life.

In situations where there is a potential of crack initiation and/or
crack propagation in materials due to environmental exposures, Weibull
probability analysis provides a useful tool to predict time-dependent
reliability and component service life \[11\]. In the case of tactical
missiles, the potential for crack initiation and propagation exists in
propellant, liners, solder joints and metallic hardware because of
temperature and humidity cycling and exposure to corrosive environments.

The Weibull probability analysis for failure modes related to
time-dependent crack propagation can be developed by defining the
Weibull statistics in terms of parameters used in the fracture mechanics
analysis. According to Weibull distribution, the reliability of a
component under stress ( ) is given by:

$R(\sigma) = \text{exp}\left\lbrack - \frac{\sigma}{\sigma_{o}} \right\rbrack^{\beta}$
(12)

From the fracture mechanics model, the material fracture toughness at a
given stress level and critical flaw size is given in reference \[14\]

$K_{\text{IC}} = Y\sigma\sqrt{a_{c}}$ (13)

In equations (12) and (13),

a~c~ = critical crack length (mm)

= applied stress (psi)

y = is a geometrical factor (assumed = 1.0)

-   = Weibull modulus or shape parameters

> ~o~ = Weibull scale factor (also characteristic value of ultimate
> material strength)
>
> K~i\ c~ material fracture toughness properties to for a critical flaw
> size (determined by testing or handbooks)

The Weibull shape parameter () and the scale parameter ~o~ are
determined from material test data.

Fracture toughness (K~i\ c~ ) is a measure of material toughness to
withstand cracks \[13\] \[14\]. When both sides of equation (13) are
equal, a material balance is maintained and the material is tough enough
to withstand the effect of stress concentration about the crack. If
stress () or crack length (a) increases due to environmental thermal
cycling or corrosive atmosphere, the balance is broken and the crack
propagates and ultimately cause a failure.

The propagation of crack due to cyclic stress is defined by power law
\[15\]

$\frac{\text{da}}{\text{dt}} = A(K_{1})^{n}$ (14)

where K~1~ is as defined as in equation (13) for any crack length (a)
below the critical length (A ~c~) and stress level () is equal to a^0.5^
for y =1. The variables a and n are two constants that are determined
experimentally or from historical data.

The life time, t~f~ , can be determined by integrating equation (14)
from initial crack length a~i~ to the critical length a ~c~ as shown in
equation (15) below:

$t_{f} = \int_{a_{1}}^{{}^{a}c}\frac{\text{da}}{A(K_{1})^{n}}$ (15)

The integral steps are given in \[15\]. The final Weibull reliability
model for crack propagation as a function of time is shown in equations
(12) and (13).

$R(t) = \text{exp}\left\lbrack \frac{\text{Ct}^{\frac{1}{(}n - 2)}\sigma^{\frac{n}{(}n - 2)}}{\sigma_{o}} \right\rbrack^{\beta}$
(16)

where C is given by

$C = \left\lbrack \frac{A(n - 2)}{2K_{\text{IC}}^{2 - n}} \right\rbrack^{\frac{1}{(}n - 2)}$
(17)

Equation (16) provides a measure of component service life for given
environmental conditions. If the initial defect or flaw size is known
from manufacturing or historical data, the service life of the component
can be predicted with equation (16). In the case of tactical missiles,
the solder joint failures and or the failures of propellant, liners or
other similar structures are caused by propagation of cracks when
subjected to thermal cycling, shock, vibration or corrosive environment.
Reliability models based on crack propagation theory are good prognostic
tools for improving the reliability of the tactical missiles.

**CONCLUSION**

This paper addresses the application of an integrated health monitoring
system to monitor health and perform prognostics and diagnostics
analysis of army missile systems in storage and field deployment. The US
Army has been field testing an integrated health monitoring system
called RRAPDS that include diagnostic and prognostic models to assess
the reliability of the weapons.

The application of probabilistic engineering methods to analyze RRAPDS
data and predict component reliability during the life cycle of the
weapon systems was discussed. Probabilistic methods in the diagnostic
and prognostic analysis provide a realistic reliability assessment for
decision making purposes.

The limitations of deterministic methods to predict component
survivability were discussed. Probabilistic methodologies based on
stress and strength approach, cumulative damage functions, and Weibull
analysis were presented for use with data from health monitoring
systems.

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