Performance model and

in-flight fault detection for Solid Rocket Motor

V.N. Smelyanskiy*, D.G. Luchinsky, V. Osipov,

D.A. Timucin, C. Kiris

NASA Ames Research Center

• Serdar Uckun (NASA Ames Research Center )

• Fred Culick (California Institute of Technology)

Contents

• Our goal is to predict the catastrophic events in SRMduring the flight

• High-fidelity model is introduced to simulate known faults

• Low-dimensional models are derived to infer SRMparameters with redundancy and to calculate theprobability of the catastrophe to occur at a given time

• The information contents of two sensors is evaluated usingBayesian model inference algorithm to ensure that we havemultiple evidence of the fault. Prognostics algorithm isdeveloped to predict the redline time.

• We present an analysis of an example fault: anoverpressure due to various causes (i.e., bore chocking)

• Conclusions are drawn and the future work is discussed

)t()c;x(fx !rrrr&r +=

Measurement modelMeasurement model

{ }K:0k);i,t(I kk ==r

Low-dimensional dynamical modelLow-dimensional dynamical model

),0(N~)t(xˆi kkkk !""+#=rrrr

measurement error

Diagnostics and Prognostics:Identification of nonlinear stochastic models from data

Algorithm for reconstruction of system state, parameters of dynamical and measurement models from time series

Data recordData record

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,)t(bxxxx

,)t(xxxrxx

, )t()xx(x

ijjii

z3213

y31212

x321

!=""="

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Stochastic dynamical system outputStochastic dynamical system output

Noise intensity is greaterthan deterministic forces

by ~ 1000 times

Lorenz system IILorenz system II

1522.1-0.020.0020.0130.06510.77-10.55

16000-100128

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1713.4-0.0160.01760.9951-2.7590.569-0.4294

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1621.5-0.012-0.9850.01050.0011-0.194

27.53

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(t), + ÷ xxb + xa x imk

mkikm

j

jiji !!"

=&

We use 50 blocks with 20000 points each to infer parameters shownin the table. Convergence can be improved by increasing the numberof points.

Inferential framework:

Inference of noiseintensities andcorrelations

t e

0 0 0 0, , , v 0T p ! = , , ,p pH r T S

L

Geomet ry of an Idealized Solid Rocket

Pressuresensor

T* p* ρ* u*TL pL ρL uL

Temperaturesensor

0

1. Bore choking: Bore choking occurs when the propellant deforms (bulge) radially inward and disruptsthe exhaust gas flow, causing a choked flow condition inside the motor. Bore choking can be mostlikely realized near radial slots and segment joints between two sections with a smaller radius of the aftsection. This critical effect is typically caused by localized areas of low pressure arising near suchinhomogeneous. Bore choking has the potential of causing booster over-pressure and catastrophicfailure.

2. Nozzle Failure: Nozzle failure will reduce the thrust being generated. Failures down stream of thethroat will have no impact on the chamber pressure. A non uniform failure of the nozzle (such asloosing a chunk of the aft exit cone, or partially failing a joint) will result in a non-axial component ofthrust. A failure would also result in the plume moving closer to the aft skirt causing increased heatingthe could adversely affect the TVC system.

3. Debonding: Potentially large parts of the propellant debond from the liner and got loose. They canbend and stick inside the bore. In the large rocket with the large aspect ratio of the bore volume thedepleted propellant can significantly obscure the bore volume leading to chocking.

4. Propellant structural failure: Critical defects are cracks and voids in solid propellant and slots ofbooster joint segments. These defects can stimulate the increase of local burning rate that can result inabruption of lager enough piece of the propellant. This piece can stick to a narrow place of the burningpropellant or choke minimum cross section of the nozzle. This can cause a sharp catastrophic jump ofthe booster trust and overpressure in the chamber head.

Failure modes leading to overpressure

Direct detection

High-fidelity model of SRM (Parameters and variables)

Conservation of mass

Momentum conservation

Energy conservation

• ρ [kg·m-3] is the gas density, ρp [kg·m-3] is the propellant density

• u [m·s-1] is the gas velocity

• p [N·m-2] is the gas pressure

• S [m2] is the cross-sectional area of the propellant; S* is the minimum cross section of the nozzle

• L [m] is the length of the burning propellant

• l(x)=2πR(x) [m] is the perimeter of the internal cross-section of the burning propellant

• F(x) [m2] is the burning surface of the grain up to the point x along the axis of the SRB

• R(x) is the internal radius of the propellant

• e = cVT+u2/2 [J·kg-1] is the specific energy of the gas

• h = cPT+u2/2 [J·kg-1] is the specific enthalpy of the gas

• H [J·kg-1] is the heat released by the burning propellant

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),()(

),()()(

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xtxlufpSSuuS

xtxlpmuSS

xxt

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xt

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0

2

1

!++"#$#"=$"

!+$#"#$#"=$"

!++$#"=$"

( ) ( / )n nb c c br p r p p a p= !Burning rate model

1D time-accurate PDE model with dynamical noise sources: stochastic integration

Fault model: nozzle blocking

uL ~ 200 m/sec

Nozzle blocking

grain boundary after chocking

grain boundary before chocking

c0 ~ 1000 m/sec

0 20 40 60 800

20

40

60

80

t, sec

p, atm

0 20 40 60 800

3

6

9

12

T/T

0

t, sec

(b)

Chamber head temperature and pressure

Patm = 101.23 kPa Tatm = 288.2 K

Noise is enhanced during the chocking event

Averaging over the combustion volume1. The mass dynamics

2. The energy dynamics

3. The dynamics of the burning area

Noise sources1. Gas flow density fluctuation in the throat

2. Gas flow density fluctuation in the burning surface

( )

( )

( )!!!

"

!!!

#

$

%+=

%+&'(+(

'&=

%+(&(+(&=(

)(

)()()(

)()()(

tpprdt

dR

tpptBp

ptAdt

dp

tptBptAdt

d

nb

np

np

30

2

1

)(

)()()( *

tV

tSctA

!"= 0

)(/)( 00ptpp! ( )

)(

)()(

tV

tFprtB b 0=

A(t) describes fast nozzlearea change in chocking

B(t) is a slow variable

Low-dimensional dynamic model

0 0. 5 1.0 1.5 TIME (S EC)

A FT G AP - EX CESSIVE RAD I A L DE F O RM ATI ON C AUSES HIGHE R PRE S SU R E DROP - HI GHER PRESSU RE D R O P C AU SES M OR E RADIA L DEFORM A TI O N

M EA SU RED DELTA G AP( Pf -P a)

P (A FT )

2 00 0

1 50 0

1 00 0

50 0

0

P (F OR WARD)

,

Fault model: Titan IV overpressure fault

The model includes state driving noises, which are assumed to have unknown parameters.Noise characterization is important for designing the fault estimation algorithm.

Low-dimensional dynamic model:System view

Combustion Model (3 states) • gas pressure• gas density• mean radius of the bore

Bore volume

Burning areaof the grain

Throat area change(fault parameters)

Pressure

Temperature

We used an algorithm for reconstruction of system state, parameters of dynamical andmeasurement models from time series, see V. N. Smelyanskiy, D. G. Luchinsky, D. A.Timucin, and A. Bandrivskyy, Physical Review E 72, 026202 (2005).

Fault estimationbased on low-dimensional dynamic model

Optimal Estimation Algorithm

Bore volume

Burning area& radius of the

grain

Throat area change(fault parameter)

Pressure

Combustion Model

Case Burst Prediction Expectedredline time

Temperature

NOTE. That the algorithm of dynamical inference will be embedded into inferential learningframework that will continuously update the models and parameters beginning from the

stage of development and firing tests.

0 2 4 6 8 10 12 14 16 18 200

50

100

p, atm measurements

actrual

inference

0 2 4 6 8 10 12 14 16 18 200

5

10

15

!, kg/m

3

measurements

actrual

inference

0 2 4 6 8 10 12 14 16 18

0.8

1

R, m

t,sec

actrual

inference

Inference of the hidden dynamical variables

Measurement noise

In this figure we show preliminary results of the inference of the gas flow parameters inthe case when the measurements of the pressure, p (a) and density, ρ (b) are corruptedby noise and the propellant radius, R (c) is not measured. The actual values of thepressure, density, and radius are shown by the red, blue and green solid linescorrespondingly. The measured values of the pressure and density are shown by blackjuggling lines.

The inferred values of p, ρ, and R are shown by the red dotted lines.

(a)

(b)

(c)

0 2 4 6 8 10 12 14 16 180

50

100

150

200

p, atm

time, sec

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

St,

m2

time, sec

Prognostics for the neutral thrust curve

Consider fault which time evolution is arbitrary function of time with characteristic timescale of a few seconds.

As an example consider slowly varying area of the nozzle throat, which is described bythe following function f(τ)= f(t-t0)= a(t-t0) +b(t-t0)2+c(t-t0)3: a=-0.6, b=0.4, c=-0.1. Faultoccurs at 9 sec. It leads to the case burst 7 sec later. The problem is how soon we candetect dangerous situation and predict the time of the case burst.

( )2 3Fault as arbitrary function of time

Fault occurs at 9 sec

t tS S S !" #" $"= %& + +

Case burst occurs at 16 sec

We infer parameters of the gas dynamics and use these parameters to predict time when pressure will cross thedangerous level. To estimate the accuracy of the prediction we also build the distribution of predicted pressure valueswhich will occur 5 sec after the fault

Here presents the result of the prediction of the time of the case burst after

(i) 1 sec after the fault (greed lines); (ii) 1.5 sec after the fault (cyan lines); (iii) 2.1 sec after the fault (blue lines)

It is clear that prediction captures highly nonlinear behavior of the fault.

6 10 14 18 22

40

80

120

160

p, atm

time, sec

Predictions

( )2 3

Fault occurs at 9 sec

t tS S S t t t! " #= $% + +

st1 prediction

1 sec after the fault

nd2 prediction

1.5 sec after the fault

rd3 prediction

2.1 sec after the fault

danger

We build distribution

of predicted pressure

5 sec after the fault

We build distribution

of predicted times of the

case burst

10 15 20 250

0.02

0.04

0.06

!(t)

t, sec

1 sec after the fault

1.5 sec after the fault2.1 sec after the fault

0 50 100 150 200 250 3000

0.02

0.04

!(p)

p, atm

1 sec after the fault1.5 sec after the fault

2.1 sec after the fault

Distributions

We build distribution

of predicted pressure

5 sec after the fault

We build distribution

of predicted times of the

case burst

Here presents the resulting distributions of predicted time of the case burst (top figure) and the head pressure,which will occur 5 sec after the fault.

(i) 1 sec after the fault (red lines); (ii) 1.5 sec after the fault (black lines); (iii) 2.1 sec after the fault (blue lines)

It is clear that prediction becomes reasonable 1.5 sec after the fault. The accuracy of prediction can be fatherimproved by adding measurements of the thrust.

actual time

critical pressure

Nonlinear dynamics of the bore clogging fault

z

w

L

2

L

2!

-w

Gas chamber

Moving obstacle of ellipsoidal shape

v

Consider an obstacle passing through the nozzle throat. E.g. it can represent a cloud of particles in theexhaust gases or a piece of the propellant. The effect of this nonlinear dynamical fault is an effectivereduction of the nozzle area. Assume the fault have time evolution corresponding to an obstacle ofelliptical shape passing through the nozzle throat. Then in the first approximation we can write

2

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0

0

1 1 , for / 2 / 2(0) ( )/ 2( )

(0)0, for / 2

tt t tS S t

tf tS

t t

!" # $# $% & '( ( ( ) )( & '% & '= = * + ,+ ,%

>%-

0 2 4 6 8 10 120

100

p, atm

0 2 4 6 8 10 1210

11

12

T, K

0 2 4 6 8 10 120

50

!, kg

"m-3

0 2 4 6 8 10 12

0.75

0.8

0.85

R, m

time, sec

Averaging over the combustion volume

1. The mass dynamics

2. The energy dynamics

3. The dynamics of the burning area

Noise sources

1. Gas flow density fluctuation in the throat

2. Gas flow density fluctuation in the burning surface

( )

( )

( )!!!

"

!!!

#

$

%+=

%+&'(+(

'&=

%+(&(+(&=(

)(

)()()(

)()()(

tpprdt

dR

tpptBp

ptAdt

dp

tptBptAdt

d

nb

np

np

30

2

1

)(

)()()( *

tV

tSctA

!"= 0

)(/)( 00ptpp!

( ))(

)()(

tV

tFprtB b 0=

A(t) describes fast nozzle areachange in chocking

B(t) is a slow variable

Low-dimensional dynamic model

Nonlinear fault dynamics

Inference of the model parameters

Convergence of the model parameters: 7 lines from the top to the bottom correspond to the results ofthe inference 1.5, 1.3, 1.1, 0.9, 0.7, 0.5, 0.3 sec after the fault. Red lines show actual values of theparameters.

3.1 3.3 3.5 3.7 3.80

20

40

c1

!(c 1)

-110 -100 -90 -80 -700

0.2

0.4

0.6

c2

!(c 2)

600 700 800 900 10000

0.02

0.04

0.06

c3

!(c 3)

Model space for nonlinear dynamical faults of the nozzle clogging

1

2 2/ / / / / / ; ; ; ; ; ; ; ; ;

n n nn p p p p p p p p pp p pp t t t t

R R R R R R R R R

! ! ! ! ! ! ! ! !!+" #$ %$ %& '

2. Model parameters

These base functions span part of the model spacecorresponding to the dynamics of model clogging fault

1. Base functions

1 1 0 1 1 1 2

1 1 0 1 1 1 2

0 2 - 2 0 0 - - - 0 0

0 2 0 - 2 - 0 0 0 - -

1 0

a p a p a p

a p a p a p

!" ! ! !

!"

0 0 0 0 0 0 0 0

# $% &% &% &' (

These model parameters correspond to the dynamics ofmodel clogging fault

0

0

1

0

*

th

b

c Sa

Lr R!

"=

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

p, atm

!(p)

0 5 100

20

40

60

80

100

120

140

160

t, sec

p, atm

x 5

Continuous prognostic of the nonlinear dynamical fault

Standard threshold formission cancellation due to

the overpressure

Nonlinear dynamical inference predicts,however, that the fault is not critical and

that pressure will not exceed 90 atm maximand then will relax to the normal value

Critical level ofcase burst

Nominaldistribution

Prognostics of the off-nominal pressure

distribution at its maximum

67.35 atm

Continuous prognostic of the fault evolution (we predictthe value of the maximum pressure for the fault)

Pred

ictio

n tim

e lin

e

0.5 sec

0.9 sec

1.3 sec

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

p, atm

!(p)

0 5 100

20

40

60

80

100

120

140

160

t, sec

p, atm

x 5

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

p, atm

!(p)

0 5 100

20

40

60

80

100

120

140

160

t, sec

p, atm

x 5

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

p, atm

!(p)

0 5 100

20

40

60

80

100

120

140

160

t, sec

p, atm

x 10

Low-dimensional model gives accurate representation ofthe transient overpressure phenomena in a number ofimportant fault modesAccordingly a Bayesian framework is introduced forprognostics and diagnostics of these failure modesCase breachBore chockingGas leak

Conclusions and Further Work

DIRECT DETECTION of faultsChanging of an efficient cross section of the nozzle S*(t) can be found from measurement

of the averaged head pressure p0(t), thrust Fth(t) or pressure jump at a rapid chocking:dS*/d(t)0 at the burning case

1 0 0*1

*0 0

( ) ( )(3) 1 pressure jump due to the shock wave

( )

!= ! !

L

p t p t cS

S p t u"

( )

( )

1 1

2 1 22 2 2

0

1

2 12*

*0

( ) / 1 1(2 ) 1 1 1

( ) 2 2

( ) 1(2 ) 1 variations of thust and pressure

2

th ex aF t S pa M M M

p t

S t Mb M

S

!

!

!

!

! !!

!

+" "

"

+"

"

# $+ " "% & % &' (= + + +) * ) *' (+ , + ,

- .

"% &= + ") */ + ,

(1) pressure variation

Time-resolved picture of fast nozzle blockingP/

P atm

T/T a

tm

P/P a

tm

T/T a

tm

PRESSU

RE

TEMPERATURE