Physics Based Modeling for Stage Separation ...

Physics Based Modeling for Stage Separation Recontact 1 and Vasyl Haychuk2 , Vadim N. Smelyanskiy3 , and Igor C. Kulikov4

Dmitry G. Luchinsky

Applied Physics Group, Intelligent Systems Division,

NASA Ames Research Center, MS 269-1, Moett Field, CA, 94035, USA 5

John M. Hanson

NASA Marshall Space Flight Center/EV40, Huntsville, AL, 35812 6

Ashley D. Hill

Dynamic Concepts, Inc, Huntsville, AL, 35806 7

8

9

Donovan Mathias , Scott Lawrence , and Mary Werkheiser

Supercomputing Division, NASA Ames Research Center, MS 258-1, Moett Field, CA, 94035, USA

Physics based modeling for a stage separation recontact fault is presented. Numerical models and analytical estimations are applied to analyze the physics of the failure and reconstruct the following sequence of events: structural dynamics of the nozzle extension during impact; yielding and melting of the damaged nozzle under plume loadings during engine start up; reduction of the actual thrust and side loads in the steady burning regime; response of the thrust vector control (TVC) system to the fault-induced torque; and rocket trajectory variations due to the fault. The obtained results provide a foundation for engineering risk assessment and for development of onboard diagnostic and prognostic system for stage separation failure. Analysis using 1 2 3 4 5 6 7 8 9

Research Scientist, MCT Inc, AIAA Member. Research Scientist, SGT Inc Research Scientist, Applied Physics Group, Intelligent Systems Division. Research Scientist, Intelligent Systems Division, AIAA Member. Lead for Flight Mechanics Analysis and Design AIAA Member. Senior Engineer. Aerospace Engineer, Supercomputing Division, AIAA Member. Aerospace Engineer, AUS, AIAA Member. Aerospace Engineer, Supercomputing Division, AIAA Member.

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the models developed as part of this research shows that the damage results in three possible outcomes: actuator failure with resulting loss of control, loss of performance resulting in an inability to reach orbit, and eects that are suciently minor that orbit is still attainable. In the case of crewed missions, abort triggers based on navigation and ight control data (as described in this paper) may be used to determine the need to abort immediately or to estimate the likelihood that orbit will be reachable.

2

Nomenclature p

= gas pressure, Pa

ρ

3 = gas density, kg/m

m ˙

= mass ow rate, kg/sec

V

= gas velocity, m/sec

t

= time,

τ

= characteristic time scale,

T

= temperature,

F

= thrust, N

K

= force, N

f

= torque, N·m

M

= Mach number

ρn

3 = density of the nozzle material, kg/m

∆L

= characteristic length of the impact, m

cp

= specic heat for constant pressure, J/kg/K

cv

= specic heat for constant volume, J/kg/K

γ

= ration of specic heats

R

= radius vector, m

l

= actuator level arm, m

u, v, w

= displacements of the nozzle surface, m

Y

= Airy stress function

h

= nozzle shell thickness, m

s, θ

= coordinates of conical shell, m, rad

α

= bending angle, rad

S

2 = fault area, m

η

2 = heat transfer coecient, J/sec/m /K

κ

= thermal conductivity, J/sec/m/K

Γ

= dimensionless parameter

sec sec

K

3

Subscripts w

= the nozzle wall

ex

= the nozzle exit

t

= the nozzle throat

a

= the actuator force

0

= stagnation values and asymptotic quasi-steady thrust

main

= the main thrust

nom

= the nominal thrust

side

= the side thrust

bend

= the characteristic time scale of bending

I. Introduction UNDERSTANDING critical engineering issues of the next-generation Crew and Heavy-Lift Launch Vehicles is a prerequisite for the safe exploration of space in this century. One of such issues is related to the separation failure. The stage separation failure may have various origins including e.g. failure of the accelerating/decelerating motors during the separation. The combined expected failure rate is relatively high and is the second most common cause of launch failures [1]. Despite this fact data related to the analysis of the recontact caused by separation failure are sparse and largely unavailable. Such an analysis is a challenging [2] engineering problem. The main diculties stem from the fact that the phenomena underlying fault dynamics are highly non-linear in structural, thermal, and hydrodynamical domains. Accordingly, the solution requires 3D analysis of the thermal/uid/structure interaction in the supersonic ow. In addition the number of sensors available onboard is severely limited and the safe time window between the detectable onset of the fault and possible catastrophic failure is typically a few seconds. Therefore, reconstruction of the time-line for a sequence of events during the fault separation failure may provide a valuable information for the risk management of the next generation vehicles. In the context terms of assessment of risk to crew of launch vehicle failures two elements have to be considered: the reliability of the launch vehicle and the consequences in terms of hazards

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to the crew should a vehicle failure occur. In particular, characterization of the vehicle reliability requires detailing the potential failure modes of the vehicle, including a relative likelihood of the various modes. Evaluation of the failure consequences involves analysis of the propagation of the failure within a vehicle subsystem and, critically, from one subsystem to another. The present paper focuses on one such failure mode: the failure of the upper stage to cleanly separate during the staging process. In order to develop an understanding of the impact of the failure on crew safety, the following questions are considered. What type of damage is incurred by recontact between the stages? How does the engine nozzle ow act to mitigate or exacerbate the initial damage? What are the consequences of the resulting damage in terms of engine performance and/or o-nominal forces and moments? Can the resulting o-nominal forces and moments be mitigated by the vehicle control system? If not, are the existing abort triggers capable of providing sucient warning for successful abort? Answers to these questions can then be used to develop loss-of-crew probabilities conditional on occurrence of a recontact event during staging. The present paper describes analysis that have been performed to answer these questions with respect to an Ares I-like launch vehicle. The approach adopted for the current analysis combines explicit modeling of the fault evolution with analytical estimations based on simplied physicsof-failure models. High-delity models and analytical estimations are applied to an analysis of the following sequence of events: structural dynamics of the nozzle extension during the impact; yielding and burning through the damaged nozzle under the plume loading during engine start up; and reduction of the actual thrust and side load in the quasi-steady burning regime. The results of the calculations are compared with the results of analytical estimations based on the quasi-twodimensional model of the pressure distribution in the model. Explicit modeling of the fault evolution is applied to predict response of the TVC system to the fault induced loads and to determine variations in the rocket trajectory. The results of the assessment are used to suggest diagnostics and prognostics of the stage separation fault. The diagnostic is based on the measurements of the impact torque. It is shown that the time delay, the strength of the rst peak of the impact signal, and the orientation of the torque provide a reliable basis for the detecting and identifying the relevant fault mode. It is further shown that prognostics can be based on the analysis of the correlation between

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the actuator signal, the fault-induced damage on the nozzle, and changes in the main and side thrust of the damaged nozzle. The paper is organized as follows. After a brief introduction the impact dynamics is discussed in Sec. II. The plume loading on the damaged nozzle during ignition transient is considered in Sec. III. The thrust reduction, side load in the notched nozzle, and the time line of the dynamics of the separation failure are analyzed in Sec. IV. The diagnostic of the impact based on the analysis of the TVC response is discuss in Sec. V. The analysis of the fault dynamics using simulation package MAVERlC 3.7 is presented ion Sec. VI. Finally, the results are summarized and conclusions are drawn in Conclusions.

II. Impact dynamics of the nozzle extension Consider two-stage, vertically stacked space vehicle. The vehicle is propelled by the rst stage booster at the rst phase of the ight. After the rocket booster completes its mission, the stages separate and the second stage engine provides the propulsion during the second phase of the ight. Due to separation faults, the rst stage can collide with the second stage engine nozzle and damage it. A typical geometry of a separation failure is sketched in the Fig. 1. Fault induced lateral velocity of the inter-stage results in a separation trajectory that crosses the nozzle at the distance

∆L

from

the nozzle exit plane. The larger is the lateral velocity the larger is the characteristic impact length

∆L. In turn the area S length

∆L

of the dent caused by the impact is proportional to the characteristic impact

and, therefore, more severe is the consequence of the impact.

Fig. 1 A sketch of the stage separation fault geometry. ∆L is the impact length. Two separation trajectories are shown by arrows. 6

The goal of the present analysis is to reconstruct the dynamics of the fault, predict the nozzle damage, estimate fault induced reduction of the main thrust and side load, and determine the response of the system to the fault induced torque. As the rst task we perform an analytical description of nozzle damage during the impact using shell theory.

A. Analytical model Note that the nozzle extension, a composite material, is a very thin wall cooled by lm injection (see also [4]) rather then by heat exchangers. The wall nozzle extension has complex weakly parabolic dense orthogrid structure of variable thickness. However, the averaged thickness of the nozzle wall is very small as compared the the nozzle diameter. The ratio of the averaged thickness to the diameter is of the order

10−3 .

Because this ratio is so small while orthogrid structure is dense the

main characteristic features of the nozzle dynamics can be captured using a simplied model in the form of truncated conic shell shown in the Fig. 2. The bending and shear stiness can be written as

D = Eh3 /12(1 − ν 2 )

and

G = Eh

correspondingly, where

Poisson's ration, with the eective thickness

h

E

and

ν

are Young's modulus and

being the main tting parameter.

Because of the very small thickness of the nozzle extension walls the dynamics of the second stage engine nozzle extension under the impact can be well approximated by Donnell's equation of shallow shell theory [3]

D∆2 w − ∇2R Y + ch

∂2w ∂w + ρh 2 = f (t, s, θ), ∂t ∂t

1 2 ∆ Y + ∇2R w = 0, G where

w

transverse displacement,

∆= D

and

G

Y

(1)

(2)

- an Airy type of stress function

∂2 1 ∂ 1 ∂2 + + 2 2 , 2 ∂s s ∂s s sin α ∂θ2

are bending and shear stiness

c

∇2R =

1 ∂2 , s tan α ∂s2

- damping coecient,

h

(3)

- thickness of the plate. A

simplied geometry of the nozzle extension corresponding to this model is shown in the Fig. 2 For the clamped short end and free large end the boundary conditions can be written as follows

u=v=w=

∂w = 0, ∂s

Ns = Ssθ = Vs = Ms = 0, 7

at

at

s = s1

(4)

s = s2 .

(5)

Fig. 2 Sketch of the nozzle extension as a conical shell dened by the coordinates s and θ. Here

u, v ,

and

respectively,

w

are the orthogonal components of displacement in the

Ns , Ssθ

s, θ,

and normal directions,

are the membrane forces related to the Airy stress function,

Kirchho shear stress,

Ms

Vs

is the Kelvin-

is the meridional moment resultant [3].

To nd the eigenfunctions and eigenfrequencies of the problem let us introduce the following separation of variables

w(t, s, θ) = eiωt w(s) sin(nθ),

(6)

Y (t, s, θ) = eiωt Y (s) sin(nθ).

(7)

We obtain for the system (1), (2)



 −bb

(

where

b a=

[

d4 ds4

+

2 d3 s ds3

The eigenfunctions

−

1+2n2 d2 s ds2

+

1+2n2 d s2 ds

wmn (s), Ymn (s)





( )  hρn ω  w(s)   =  Y (s) 0 ] 2 2 ) 1 d2 − n (4−n , bb = s tan s4 α ds2 .

a(n)  Db  w(s)     Y (s) −1 bb G b a(n)

)

2

satisfying (8), (4),(5) and natural frequencies

(8)

ωmn

of the

cone can be nd by Galerkin's or RayleighRitz methods. Expanding the radial displacement and Airy function

Y (t, s, θ)

w,

in the series of the shallow shell eignemodes

w(t, s, θ) =

∑

cmn (t)wmn (s)sin(nθ),

(9)

mn

Y (t, s, θ) =

∑

dmn (t)Ymn (s)sin(nθ)

(10)

mn Using that normal modes with dierent indexes are orthogonal we obtain equations governing the amplitudes

cmn

as

∂cmn ∂ 2 cmn + 2δ + (ω 2 mn )cmn = fmn (t) 2 ∂t ∂t 8

(11)

where

δ = c/2ρn

and

ωnm

are frequencies of the eigenmodes.

fmn (t) =

1 hρ

∫ ∫ f (t, s, θ)wmn (s)sin(nθ)dsdθ

To analyze the nozzle damage dynamics one can use a solution of Donnells equations (1) and (2) to simulate the impact-induced torque applied to the nozzle and perform computer simulation of the upper stage thrust vector control response to the impact (see Sec. VII). Solution of (11)

1 √ cmn (t) = ωmn 1 − δ 2

∫t fmn (τ )eδωmn (t−τ ) sin[ωmn

√ 1 − δ 2 (t − τ )]dτ

(12)

0

is known as Duhamel's integral. The response of the nozzle extension shell during cold impact is mainly determined by the low-frequency eigenfunctions of (12). The corresponding eigenfrequencies are calculated using Ritz method similar to one proposed in [10]. For the parameter values corresponding material properties of the nozzle ex tunica with eective thickness of eigenfrequencies are found in the range

4

to

20

h ∼ 1mm

a number

Hz, which are mainly determined by the exural

modes of the nozzle bending. It will be shown in the next section that the response of the nozzle extension to the cold impact can be characterized by the low-frequency eigenmodes in the same frequency range.

B. Finite Element Model Further insight into the nozzle response to the impact can be obtained by simulating the nite element model (FEM) of the nozzle extension using ABAQUS software package. In our simulations the FEM reproduces accurately the geometry (including dense orthogrid structure shown in the inset of Fig. 3(a)) and material properties of the nozzle extension. Because the nozzle extension shell is very thin as compared to the thickness of the interstage walls the interstage is modeled as a rigid body. We x nozzle extension at the narrow end and use explicit dynamics with general contact properties to simulate the impact. This boundary condition overconstrains boundary and excludes from the analysis the situation when nozzle extension is ripped o the nozzle during cold impact. Typical results of the simulations are shown in Fig. 3(a). The dent caused by the impact can be clearly seen in the gure. The dent can be characterized by two key factors: the area bending angle

β.

S

and the

In addition, for strong impacts nozzle surface can wrinkle (see e.g. [5]). Note that

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the dent area is primarily determined by the impact length

∆L, which in turn is well dened by the

impact trajectory. It is therefore reasonable well estimated in the simulations. The uncertainty in the coecients of structural and material damping result, however, in the uncertainty of the bending angle

β,

which can vary in simulations between 30 to 90 degrees. For this reason the estimations of

the response of the damaged nozzle to the hot gas ow during ignition transient was performed in the next section for three dierent angles.

buckling bending

Fig. 3 (a) Finite element model of the nozzle extension. The inset shows the detailed view of modeling rectangular grid. (b) The results of simulation of the nozzle extension damage caused by the impact. The buckling of the nozzle observed in simulations is in agreement with the analytical predictions. In fact, according to eq.(5) in [11] buckling moment for a small end of the cone is

√ M ∼ πEh2 R cos2 α/ 3(1 − ν 2 ). Taking parameters values for the nozzle extension shell estimation for the buckling moment of the order of

E ∼ 1011 P a, h ∼ 10−3 m

M ∼ 105 N m

one obtains an

which is in with the results of

numerical simulations. The impact can be characterized quantitatively by the time-traces of the reaction forces and of the impact-induced torque. Examples of the torque time-traces are shown in the Fig. 4(a) for three impact trajectories. The following features can be noticed from the gure. The amplitude of the rst peak of the impact torque is strongly correlated with the impact trajectory and with the

10

1

0

−10

2

−20

3

−30 0.35

0.55

0.75

1

10

10

K × 10−4 , N · m

K × 10−4 , N · m

20

0.95

1.15

1.35

2

0

3

−10 −20 −30 0.35

0.55

time, sec

0.75

0.95

1.15

1.35

time, sec

Fig. 4 (a) The reaction torque obtained in the simulations for three dierent impact lengths ∆L:

(1) 11 in (dotted line); (2) 18 in (solid line); (3) 24 in (dashed line). (b) (1) The reaction

torque is compared with a (2) response of damped oscillator (open circles) induced by (3) a short rectangular excitation pulse (solid line, not in scale). initial time of the impact signal. These correlations can be used to dierentiate between dierent failure modes of the stage separation fault. The corresponding analysis can be simplied by taking into account the fact that response dynamics is governed mainly by a few low-frequency eigenmodes of the nozzle extension. The frequency analysis of the nite element model reveals a number of eigenfrequencies in the range

5

to

20

Hz in agreement with estimations obtained using analytical

model in the previous section. Note also that the time traces of the impact-induced torque can be t by a response of a damped oscillator with frequency

∼

4 Hz to a short rectangular excitation

pulse as shown in the Fig. 4(b). This fact will be used in Sec. VII to model the TVC response by an oscillator dynamics. The simulations also reveal correlations between the amplitude of the impact torque and the dent area. Therefore, it becomes possible to predict the fault-induced changes in the thrust of the damaged nozzle based on the measurements of the time-traces of the impact torque. The predictions of the thrust dynamics can be performed in a few steps. First, one can analyze the gas ow through the damaged nozzle during ignition transient and estimate temperature and pressure loads on the nozzle walls induced by the ow. Next, simulations of the structural dynamics of the nozzle in the

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presence of the temperature and pressure loads during ignition transient can be performed. Finally, the fault induced-changes in the main and side thrust of the damaged nozzle in the quasi-steady regime can be estimated. The results of the corresponding analysis are presented below.

III. Ignition Transient Loads on the Damage Nozzle In this section we analyze the gas ow through the damaged nozzle. The transient response of the nozzle to the hot gas ow in the presence of thermal-structural-uid interaction and structural instability is very complicated and dicult to simulate. The approach adopted throughout this project for the order of magnitude estimations of the eect of transient fault dynamics was as follows. At the rst step, the results of the cold impact analysis discussed in the previous section were used to estimate analytically the transient aerodynamic loads on the dent including aftershock loads. At the second step, the deformed shape of the nozzle obtained in cold impact analysis was substituted into Fluent nite volume model to verify analytical estimations of aerodynamic loads on the dent. At the third step, transient aerodynamic loads were substituted back into the Abaqus nite element model to simulated coupled thermal-structural response of the damaged nozzle to transient plume loading on the nozzle wall. It was shown that the most likely scenario of the fault dynamics during ignition transient is reversed buckling of the dent under plume loading followed by its yielding and melting away leaving notch in the nozzle. The temperature rise above melting temperature in the dent was further veried using analytical model. At the next step, the order of magnitude of the aerodynamic loads and the fault induced torque during reversed buckling, yielding, and melting away of the dent were roughly estimated analytically. Finally, the shape of the notched nozzle was substituted back into Fluent to estimate the fault-induced side loads and losses of the main thrust by simulating nozzle ow in the notched nozzle in quasi-steady regime. The results of the Fluent simulations of the side loads were veried using simplied analytical model with point forces approximation as will be discussed in the next section. In what follows it will be shown that these approach allows one to reconstruct a complex sequence of events during ignition transient and to provide order of magnitude estimations of the transient thermal-structural response of the damaged nozzle to the hot gas ow.

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Following this approach a quasi-two-dimensional approximation for the distribution of the pressure and temperature loads on the walls of the nozzle extension during ignition transient will be now introduced.

A.

Quasi-two-dimensional Approximation To estimate the side and axial loads on the dent in quasi-two-dimensional approximation let

us recall rst one dimensional isentropic model of the nozzle ow for a given transient time traces of the pressure

p0 (t)

and temperature

and temperature

Tx (t)

T0 (t)

at any axial location

in the main combustion chamber. The pressure

x

px (t)

along the nozzle axis (including nozzle exit) can be

written in the form

( ) ( ) γ (γ − 1) 2 (γ − 1) 2 γ−1 Tx = T0 1 − Mx , px = p0 1 − Mx . 2 2 We note that the Mach number at a given location geometry and the specic heat ratio

Mx = Vx /c0

(13)

is determined entirely by the nozzle

γ = cp /cv

Fig. 5 Quasi-2D approximation for the pressure distribution. Radial pressure distribution is shown for four dierent locations along the nozzle axis. Insert shows radial pressure distribution (circles) as compared with the results of high-delity simulations (dashed line). ( ) 1 (γ−1) 2 γ−1 St Mx 1− Mx = , 2 ΓSx where

St

is the nozzle throat area and

Sx

( Γ=

γ+1 2

γ+1 ) 2(γ−1)

is the nozzle cross-section area at

,

(14)

x-location. In particular,

time variation of the thrust is completely determined by the time variation of the chamber pressure

13

and the exit Mach number according to the following expression

Fmain (t) =

γ p0 (t)St Mex . Γ

(15)

Alternatively the thrust can be calculated as the pressure on the nozzle walls that takes into account radial distribution of the ow parameters. Note that contribution to the side loads due to ow separation during ignition transient considered in earlier research (see e.g. [4]) is small at high attitudes as will be discussed in more details in Sec. V. To calculate thrust as an integral of pressure load on the nozzle wall one can build a quasi-twodimensional approximation for the pressure distribution in the nozzle by satisfying the following conditions: for every location along the nozzle axis the mean value of the radial pressure distribution coincides with the values obtained using one-dimensional model (13); the wall pressure coincides with the results of the numerical simulations; and the values of the vacuum thrust calculated using Eq. (15) correspond to the vacuum thrust obtained by integrating pressure on the nozzle walls. These conditions can be satised using polynomial t to the radial pressure distribution as shown in the insert of Fig. 5. Similarly we obtain quasi-two-dimensional distribution of the ow temperature. The transient nozzle ow parameters in this approximation are obtained by using nominal timetraces for the chamber pressure

p0 (t)

and temperature

T0 (t).

Next, this approximation is applied to

determine aftershock ow parameters and to estimate the pressure and temperature loads on the dent.

B. Temperature and Pressure Loads on the Dent To estimate temperature and pressure loads on the wall we recall that whenever a supersonic ow is turned into itself  as shown in Fig. 6 an oblique shock wave will occur [6, 7]. The analysis

◦ shows that for the deection angles as large as 40 and relevant gas parameters the ow at the corner will correspond to the detached shock wave as shown in Fig. 6. Accordingly the estimation from above of the pressure and temperature loads on the dent can be performed using equations for

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Fig. 6 Oblique shock wave with deection angle

α

near the corner of the dent is shown by

the dashed bended lines. Parameters of the incident and aftershock aws are indicated by sub-indexes 1 and 2 respectively. the normal shock wave in the form

1 + [(γ + 1) /2] M12 , γM12 − (γ − 1) /2 ) 2γ ( 2 M1 − 1 , =1+ (γ + 1) 1 + [(γ − 1) /2] M12 . = 1 + [(γ − 1) /2] M22

M22 = p2 p1 T2 T1

(16)

To estimate temperature and pressure loads from below the Eqs. (16) are solved for the the smallest deection angle

α

= 30

◦

of the dent obtained in structural dynamics simulations. In this

case eq. (16) are solved for the Mach numbers the value of the shock angle

β

M1,n

and

M2,n

of the ow normal to the shock, while

is found by solving the equation (17)

tan α = 2 cot β

M12 sin2 β − 1 . M12 (γ + cos 2β) + 2

(17)

The resulting pressure loads on the damaged wall range from approximately 1 atm (weak oblique shock with deection angle

α

◦ = 30 ) to a few standard atmospheres (normal detached shock). The

corresponding temperature loads on the wall varies from 2200 K to 3700 K exceeding the melting temperature for the nozzle material. These analytical estimation were conrmed by the numerical simulations of the gas ow in the damaged nozzle using Fluent. An example of the velocity distribution obtained in these simulations

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is shown in Fig. 7. The formation of the detached oblique shock wave can be clearly seen near the bending corner of the nozzle. Note that the obtained pressure and temperature loads may result in yielding and melting away the nozzle dent as will be discussed in the following section. We now estimate the time scales required of yielding and burning through the damaged area of the nozzle.

Fig. 7 (left) Distribution of the velocity of the gas ow in the damaged nozzle with the dent area S ≈ 0.4m2 and bending angle α = 45◦ . The formation of the detached oblique shock wave can be seen in the lower right corner of the nozzle. (right) Distribution of the velocity of the gas ow in the notched nozzle after the dent was melted away.

C. Coupled Thermal-Structural Dynamics During Ignition Transient It was shown in the previous section that the temperature loads on the damaged wall exceed the melting temperature. Accordingly, the transient dynamics of the dent in the presence of the ow will be mainly determined by the time required to heat the wall to the melting temperature. However, before the melting will occur the rise of the pressure load during ignition transient will result in yielding of the dent. To estimate corresponding characteristic time scales transient temperature and pressure loads obtained in the previous section were applied to the nite element model of the nozzle as follows. Nominal values of pressure

p1 , temperature T1 , and Mach number M1

near the wall are found using

quasi-2D distribution for the nominal values of the transient chamber pressure

p0 (t) and temperature

T0 (t) (see e.g. [4]). The initial pressure and temperature jump in the nozzle ow occurs 1.7 sec after 16

the ignition (see

[4]). It is followed by a subsequent rise of the ow parameters to the nominal

values with characteristic time scale

≈

0.5 sec for the temperature and

≈

1.0 sec for the pressure.

Aftershock pressure, temperature, and Mach number are found using Eqs. (16). The shock angle is found using Eq. (17). In the nite element model the nominal pressure and temperature loads are applied everywhere to the nozzle wall except the dent. At the dent the aftershock values of the pressure and temperature loads are applied. The coupled structural-thermal simulations are performed in ABAQUS.

Fig. 8 Dynamics of the temperature of the internal (blue solid line) and external (black dashed line) surfaces of the damaged nozzle. Red horizontal line shows melting temperature on the nozzle material.

The simulations reveal the following characteristic features of the fault dynamics during ignition transient. The temperature of the thin nozzle wall is rising towards the melting temperature with characteristic time scale approximately tT = 0.2 sec. The temperature rise promotes yielding of the nozzle wall material. Simultaneously pressure load on the dent is rising to 1 atm on the same time scale. This interplay of the nonlinear thermal and structural dynamics initiates a reversed buckling of the dent at around 0.1 sec after pressure jump. Finally, the reversed bucking is followed by melting and burning through the nozzle wall at

≈

0.2 sec after pressure jump.

The temperature dynamics is shown in the Fig. 8. It can be seen from the gure that the

17

temperature of the external side of the thin nozzle walls approaches melting temperature

Tmel

in approximately 0.23 sec. However, even before the melting begins the pressure load on the wall exceeds the critical value for reversed buckling. To see this one can notice that the dent of the nozzle can be approximately considered as a plate one side of which is free and all others are clamped. As a result maximum stress for the dent uniformly loaded with pressure

σ ∼ pS/h2 ,

where

S

is the surface of the dent and

p

can be estimated as

p ∼ 105 P a, S ∼ 0.2m2 , h2 ∼ 10−6 m.

It can

be shown using these estimates that stress is greater than Yield strength of the nozzle material (less than

2 × 108 P a).

The gas pressure

p≈

7

×

4 10 Pa is reached in approximately 0.16 sec after

the pressure jump (i.e. approximately 1.85 sec after engine ignition). At this moment the reversed buckling of the damaged area of the nozzle is initiated by the pressure loads induced by the nozzle ow. After reversed buckling was initiates the nonlinear dynamics of the dent should be considered. It will be shown now that the dent will begin to melt down approximately 40 msec later. So in

≈

2 sec after the ignition the dent will be melted away by plume loading leaving notch in the nozzle wall. To estimate the characteristic time of the heating of the nozzle wall analytically one can use use Bartz' approximation [8, 9] for the heat transfer from the gas ow to the nozzle wall. Since the nozzle wall is very thin the analysis is performed in 1D using the following equation for the

T (x, t)     T,t = Cmetκρmet T,xx x1 > x > x0 , t > t0     ∂T (x,t) −κ = η (Tinf − T )|x=x0 ∂x  x=x0       −κ ∂T (x,t) = 0, T (x, t0 ) = T0 , ∂x

dynamics of the nozzle temperature

(18)

x=x1

x

=

x0 )

x

=

x1 )

where boundary conditions correspond to the heat ux from the gas ow to the internal ( surface of the nozzle with the heat transfer coecient

η

and zero ux on the external (

surface of the nozzle. The solution of this equation we seek in the form

∞ ∑ 2 T (x, t) − T∞ sin(λn L) cos(λn x) =2 e−λn α t , Ti − T∞ λ L n + sin(λn L) cos(λn L) n=1 where

cot λn L =

λn κ h and

α=

this solution are of the order

(19)

κ Cp ρ . The estimations of the characteristic melting time obtained with

Tmel ≈ 0.2

sec, which conrms the numerical estimations discussed

above.

18

Fig. 9 Torque induced by the transient loads on the damaged nozzle area

S ≈

0.15

m2 .

The

inset shows the shape (colored contour) of the damaged area 0.18 sec after the pressure jump and the initial shape (gray contour).

The corresponding sequence of events obtained numerically by solving coupled temperaturedisplacement problem in ABAQUS for the relevant parameters of the ow

[12] is shown in Fig. 9.

The analysis reveals that the reversed buckling occurs at approximately 0.16 sec after the pressure jump and the dent is completely buckled back 0.25 sec after the pressure jump. By this time the dent will be melting away. Following the line of reasoning outlined in the beginning of this section the order of magnitude of the aerodynamic loads and the fault induced torque during reversed buckling, yielding, and melting away of the dent will be estimated analytically in the following subsections.

D. Ignition Transient The results obtained in the previous section allow one to estimate the dynamics of the pressure and temperature loads on the dent during its yielding and melting away. To do so the dynamics of the bending angle found in coupled-thermal-structural simulations in ABAQUS explicit is substituted back into Eqs. (16) and(17) to nd updated values of the aftershock pressure and temperature loads on the dent. The results of the calculation are shown in the Fig. 10 for three dierent initial bending angles

α

corresponding to the results of simulations of the impact (discussed in Sec. II) with three

dierent values of the damping parameters of the material damping. It can be seen from the gure that for small bending angles (weak shock solution) the variation of pressure and temperature is

19

smooth. For larger angles corresponding to a detached shock the variation of the temperature and pressure in time is abrupt indicating transition from a detached to a weak shock during yielding of the dent.

15

4000

exit wall o α1 = 90

3000

α = 90o

10

1

α = 30o

T, K

P´ 10 − 4, Pa

exit wall

2

o

α3 = 50

5

o

α2 = 30

2000

α = 50o 3

1000

0

1.8

2

2.2

0

2.4

t, sec

1.8

2

2.2

2.4

t, sec

Fig. 10 The time variation of the pressure (left) and temperature (right) loads on the dent for three dierent bending angles. The nominal values of the pressure and temperature at the axis and at the wall are shown by blue and black lines respectively. Dashed lines correspond to the predictions starting from initialization of inverse buckling.

The worst case scenario corresponds to large bending angles and reverse buckling as shown by the green (uppermost) curves in the gures. Analysis of the reversed bending results (see Sec. III C and Fig. 9) shows that bending dynamics can be tted by the following equation

(

(t − ti ) θ(t) ∝ exp − 2 τbend where

ti

is the bending initiation time, and

τbend

2

) ,

(20)

is the characteristic bending time determined

in Abaqus simulations. For the worst case scenario corresponding to the longest bending time, the highest pressure rise, and the largest uctuations of the thrust and torque the upper limit estimation for

τbend

is

≈

0.23 sec and initiation time

ti

approximately 1.85 sec after engine ignition.

The time traces corresponding to the ignition transient are determined by an interplay of the rise of

p0 (t)

and

T0 (t)

and decay of the shock strength due to reversed buckling. The analysis indicates

that on a small timescale

t < τbend

the angle of bending is changing slowly while the pressure and

temperature in the main combustion chamber continue to rise and as a consequence the aftershock pressure and temperature loads will also continue to rise. When bending angle becomes suciently

20

small the shock becomes weak and continues to decay with characteristic time scale

τbend .

As a

consequence both loads will quickly decay. Note that these estimations do not take into account the lm cooling of the nozzle extension (which will also be impaired in the damaged area) and possible burning of the nozzle material. In the following sub-section similar order of magnitude estimations of the transient loads and fault-induced torque are provided during melting away of the dent.

E. Burning through the nozzle It was shown in the Sec. III C that the nozzle wall temperature

Tw

will reach the melting

temperature in approximately 0.2 sec after the transient pressure jump. Once the nozzle

Tw

reaches

the melting temperature the dent will be melt away. In this section we provide simple estimations of the transient loads during burning through the nozzle dent for the bending area

S = 0.43m2

and

three dierent bending angels: 30, 50, and 90 degrees.

3000

exit wall o α1 = 90

8 6

α2 = 30

o

α = 50

α = 30o 2

α = 50o

1500

3

1000

3

2 0

2000

o

4

exit wall o α1 = 90

2500

T, K

P´ 10 − 4, Pa

10

500

2

2.5

0

3

t, sec

2

2.5

3

t, sec

Fig. 11 The pressure (left) and temperature (right) loads on the damaged nozzle wall are shown for three dierent dent angles. The nominal values of the pressure and temperature at the axis and at the wall are shown by blue and black lines respectively. Dashed lines correspond to the predictions starting from initialization of inverse buckling.

Note that high-delity modeling of the nozzle ow during burning through the nozzle was not attempted in this project. Instead, for the purpose of engineering risk assessment a simplied dynamical model of this process was introduced. This model is based on the analysis of the results of the ground ring tests that reveals a short time scale for this process. To take this time scale into

21

4

K × 10−4 , N · m

α1 = 90o α = 30o

2

2

α3 = 50o

0 −2 −4 −6

2

2.5

3

time, sec

Fig. 12 (left) A sketch illustrating the point force approximation of the fault-induced side load. (right) Predictions of the transient fault-induced torque for three dierent bending angles. Dashed lines correspond to the predictions starting from initialization of inverse buckling. account in the analytical estimations an exponential relaxation factor for the wall temperature and pressure was introduced as follows

( ) (t − tburn ) Pw (t), Tw (t) ∝ exp − . τburn

(21)

Because of this factor both the temperature and pressure on the dent are reduced to zero with characteristic burning through time

τburn

and initialization time

tburn .

The resulting time traces

are shown in the Fig. 11. It can be seen from the gure (cf. with Fig. 10) that in the presence of the relaxation term (21) the pressure and temperature loads on the damaged part of the wall are reduced to zero corresponding to the fact that the dent melts away leaving a notch in the wall. The notch causes a reduction in actual thrust and non-symmetric loading on the nozzle, resulting in a side force and reaction torque on the nozzle. Recall that the burning through the nozzle wall analyzed in this section is similar to the fault observed during the rst high-speed test of X-15 hypersonic airplane [7]. In that test the shock wave from the engine impinged on the bottom surface of the X-15, and because of the locally high aerodynamic heating in the impingement region, a hole was burned in the X-15 fuselage. The asymmetry of the load on the damaged nozzle induces signicant torque. To estimate this torque one can apply the results of the calculation of the wall pressure obtained above within 2D

22

approximation. The mean fault-induced force aftershock wall pressure over the fault area nominal force

fn

applied to the equal area

The torque is then estimated as

ff

Sf .

Sn

(see Fig. 12 (left)) is found by integrating the This force is only partially compensated by the

at the apposite side of the wall (see Fig. 12 (left)).

K ∝ ff × Rf + fn × Rn ,

where

Rf

and

the origin at the gimbal center pointing to the geometrical centers of

Rn

Sf

are radius vectors with

and

Sn

respectively. The

results of the torque calculations for three dierent bending angles are shown in the Fig. 12 (right).

◦ Note strong dependence of the torque on the bending angle. For bending angle larger then 45 the torque changes sign. The results discussed in the last two subsections provide only an order of magnitude estimations of the transient loads and torque during yielding and melting away of the nozzle dent and are indicate' by the dashed lines in the Figs. 9  12. These estimations depend on a number of parameters as will be briey discussed in the following subsection.

F. Error Estimation The estimations of the transient thrust and torque depend on the a number of parameters. The two key parameters  dent area and the bending angle  were discussed above. Other parameters were introduce in the previous section and are related to the characteristic time scales:

τburn ,

and

tburn .

An additional key parameter is the value of the quasi-steady side load

ti , τbend , F0

which

will be discussed in the following section. The variation of the actual thrust as a function of the model parameters is show in Fig. 13. It can be seen from the gure that the main contribution into uncertainty of thrust estimations is related to the uncertainty of the bending angle. The later in turn is determined by the uncertainty of the structural and material damping of the nozzle extension. For example a change of the damping coecient in the ABAQUS FEM from

βm = 1 × 10−6

to

βm = 2 × 10−6

(where

βm

is material

damping coecient proportional to stiness and to the frequency of oscillations) corresponds to the change of the bending angle of the dent from

30◦

(weak shock) to

45◦

(detached normal shock). This

result in more then threefold variation of the losses of the actual thrust from 0.5% to 1.8%. The large uncertainty of the response is also related to the initialization time of yielding. The later is

23

Fig. 13 The change of the main thrust (in percentage of the main thrust) as a function of: (i) its asymptotic quasi-steady value

F0 ;

(ii)

τbend

= 0.345 and 0.23 sec; (iii)

τburn

0.1 and 0.075

sec; and (iv) initial bending time ti 0.11 and 0.09 sec.

primarily determined by the uncertainty in the heat transfer coecient from the ow to the nozzle wall, which may vary more then twice for the complex ow near the dent corner, resulting in more then twofold variation of the thrust losses. It has to be emphasized however that the changes of the thrust during ignition transient are very short. They develop on the time scale 0.2 sec after the pressure jump when the actual thrust is less then one third of the nominal value. Therefore the described variations do not aect signicantly the ight conditions or TVC response, but they can be used to detect and isolate the fault mode of the stage separation failure. From the point of view of the overall changes of the ight conditions the losses of the actual thrust in the quasi-steady regime are more important. Note that dent area and corresponding side load and the loss of the main thrust depends mainly on the impact length

∆L and the fact the dent

is melted away shortly after pressure jump during ignition. The impact length

∆L

is well dened

by the impact trajectory. For that reason the accuracy of the estimations of the bending area and fault-induced changes in the quasi-steady thrust are relatively higher than accuracy of estimations of the transient loads during ignition. The corresponding estimations of the fault-induced changes in the main and side loading and reaction torque as a function of the notched area in the quasi-steady regime will be provided in the following section.

24

IV. Reduction of the Thrust and the Side Load. To analyze fault-induced changes of the thrust in the quasi-steady regime simulations of a nite volume model of the 3D nozzle ow were performed in FLUENT and the results of the simulations were compared with the results of analytical estimations. Examples of the velocity distribution obtained in the simulations are shown in Fig. 7. Because of the symmetry of the problem only half of the nozzle with the dent in the low-right corner is shown in the gure. To evaluate thrust losses the simulations were performed for undamaged nozzle, for the nozzle with damaged area bended at various angles, and with a notch in the nozzle wall. In the left part of the Fig. 7 the velocity distribution of the ow is shown for the damaged area

Sn

= 0.387

m2

and initial bending angle

θ

o = 45 . The velocity distribution for the ow in the nozzle with a notch in the wall is shown in the right picture.

5

Fside, %

,%

95

4

main

97

F

99

2

1

93

0

3

0.5

1

1.5

2

2.5

0 0

3

2

Fault area [m ]

0.5

1

1.5

Fault area [m2]

2

2.5

3

Fig. 14 (left) The reduction of the actual thrust in the notched nozzle as function of the notch area Sn . (right) The side load developed in the notched nozzle as a function of the Sn . Dierent colors correspond to dierent meshes of the model. The eye guiding lines approximate the mean value (dashed lines) and the error of estimation (solid lines).

The results of the numerical simulations for dierent values of the

Sn

and three dierent meshes

are shown in Fig. 14. The red square in the gure correspond to the coarse mesh, while yellow circles to the nest mesh. The thrust was calculated according to the equation and

m ˙

Fmain = mV ˙ ex ,

where

Vex

are the velocity and the mass ow rate the at the nozzle exit surface. For the notched nozzle

the exit surface was obtained by sweeping a line connecting two points at the nozzle edge alone

25

a vector normal to the nozzle axis and lying in its symmetry plane. The analysis shows that the reduction of the actual thrust and the side load caused by the notch are both linearly proportional to the notch area

Sn .

This result can be readily understood by noticing that the thrust can be calculated as a pressure integrated over the nozzle wall. Therefore in the rst approximation the reduction of the thrust in the damaged nozzle is proportional to the area

∆F ∝ Sn pw .

Sn

of the notch projected onto the nozzle exit:

Similar approximation can be used to estimate the side load except that in this

case the notch area has to be projected in the direction normal to the nozzle axis. According to these estimations both the loss of the exit thrust and the side thrust due to the notch with area

Sn = 0.5m2 ,

◦ bending angle 45 , and wall pressure

pw ≈ 1.75

psi are

(Sn · pw )/Fnom ≈ 0.5%,

which

agrees well with the results of numerical estimations shown in the Fig. 15.

Fig. 15 (left) A sketch explaining analytical estimations estimations of the fault-induced torque. (right) The results of the numerical calculations of the side load as a function of Sn

(magenta rhombus) as compared to the results of analytical estimations (yellow circles).

The eye guiding lines approximate the mean value (dashed lines) and the error of estimation (solid lines).

To calculate the fault-induced torque numerically the cross-product over the nozzle surface area. Here

R

(mV ˙ ex ) × R

was integrated

is a vector radius with the origin at the gimbal center location

pointing to an element at the nozzle exit surface. The results of the numerical calculation are shown in Fig. 15 together with the results of analytical estimations. The later estimations were obtained

26

in the point force approximation. It was assumed that because of the symmetry of the problem the force on the wall in radial direction is compensated everywhere except at the notch area and symmetrically situated undamaged area of the nozzle

Sn .

The force on the notch was equal to zero

and the force on the undamaged area was calculated as a point force applied at the geometrical center of the

Sn

and equal to

pw · Sn

with nominal value of the wall pressure

pw .

The sketch

explaining these calculations is shown in Fig. 15(left) where the disturbed ow is shown by the vectors. The analytical estimations taking into account the dierence between the

fZL

and

fZR

are shown in Fig. 15 (right) by the yellow circles and are in a good agreement with the results of numerical simulations shown by magenta rhombs.

V. Time Line of the Fault-Induced Torque The analysis of the fault separation failure discussed above allows one to reconstruct sequence of events resulting in the nozzle damage and fault-induced torque. The time line of the torque reconstructed from this analysis is shown in Fig. 16. The time is counted from the initial moment of the impact.

40

K × 10−4 , N · m

impact torque ignition transient torqueflow torque 20 0

−20 −40 0

1

2

3

4

time, sec

Fig. 16 Time line of the torque induced by the stage separation failure, impact, and the ow.

Three dierent regions can be clearly seen in the gure. The rst region corresponds to the impact torque. The signal is characterized by the initial delay (determined by the propagation of the impact signal through the nozzle) followed by the high amplitude vibration of the nozzle caused by the impact. During this stage nozzle buckles. And if the impact is strong enough the surface of the

27

nozzle can also wrinkle. The oscillations of the torque at this stage are very informative. Although our analysis provides only order of magnitude estimation for the value of these oscillations, in practice nozzle response can be always calibrated and used to infer the strength of the impact, identify fault mode, estimate nozzle damage, and predict fault dynamics ahead of time. The second region corresponds to the engine start-up transient. The nominal side loads induced at this stage by the ow separation were investigated in details in [4]. According to the results of [4] during the rst 1.7 sec after ignition the side load is less then

104

N·m. Because this value is at

the noise level in our simulations this part of the response was modeled as a random signal with an amplitude less than

104

N·m.

The next stage begins when pressure and temperature jumps occur in the system 1.7 sec after engine start-up. A highly non-linear response at this stage is determined by the formation of the detached shock wave near the dent corner giving rise to strong local temperature and pressure loads. Under this loading the damaged area yields and melts away, leaving a notch in the nozzle wall. In the end of this stage the ow approaches a quasi-steady regime with asymmetric loading on the nozzle wall and reduced actual thrust. In the following sections the results of this analysis will be applied to predict rocket trajectory and response of the thrust vector control system in the presence of the stage separation fault.

VI. Diagnostics of stage separation faults In this section response of the TVC system to the impact is analyzed.

A. Second stage thrust vector control system operation simulation The second stage thrust vector control system has two hydraulic supply strings supporting two actuators, which change the nozzle altitude. At nominal regime of stage separation the actuators do not operate and the nozzle altitude does not change. When the rst stage collides with the nozzle, the nozzle changes its angular orientation and the control system responds to the impact. The impact torque obtained above for a given stage separation fault is implemented into the second stage thrust vector control system simulation software. As the result of the simulation nozzle altitude changes and the actuator forces as functions of time are obtained. The nozzle altitude changes during the

28

0.01 0 −0.01 2

fa × 10−4 , N

Engine position, rad

0.02

1 0 −1 −2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

time, sec

Fig. 17 (a) Nozzle angular positions tilt (the red curve) and rock (the blue curve) during the impact. (b) Induced by the impact actuators' forces components tilt (the red curve) and rock (the blue curve) as functions of time.

impact are shown in Fig. 17 (a) and the actuators’ forces are shown in Fig. 17 (b). The impact torque is simulated for several separation trajectories, and for each case thrust vector control response is calculated. Dierent separation trajectories correspond to dierent amplitudes of torque oscillations and, as the result, dierent nozzle angles and actuators response forces

B. Method of Diagnostics The objective of this section is to describe how one can use accelerometry data to dierentiate between various separation trajectories and to estimate corresponding nozzle damage fault-induced changes of the thrust. The idea of the method is to build an extensive library of the actuator responses for various separation trajectories correlated with nozzle damage and the damage-induced changes of the thrust obtained in simulations of the impact of the gas ow in the notched nozzle using nite element and nite volume models. The real-time telemetry data of nozzle angular oscillations can then be correlated with the library of simulated telemetry data to perform diagnostics of the separation trajectory and to predict nozzle damage and thrust changes in the damaged nozzle approximately 2 sec prior to the pressure and temperature jump. To facilitate diagnostics and prognostics of the impact-induced damage based on the analysis of

29

15

K × 10−4, N · m

10

5

0

−5

−10 0

0.2

0.4

0.6

0.8

1

time, sec

Fig. 18 Actuators tilt (the red curve) and rock (the blue curve) torques components as functions of time. the actuator response a low dimensional model of the nozzle angular changes in the presence of the external torque is introduced. Within this model the second stage engine nozzle altitude changes are given by the equation

J φ¨ + B φ˙ + Cφ = K

where

J , B, C

are the coecients,

φ

(22)

is (rock/tilt) nozzle angle. The equation (22) allows one to

dene nozzle angular changes with respect to the change of the torque

K.

of the thrust vector control operation the torque is given by the equation

fa

is the actuator force and

l

Kc .

K = Ka = fa · l,

where

is the actuator lever arm. If the impact induced torque is applied

to the nozzle, then the total torque impact torque

In the nominal regime

K = Ka + Kc

is the sum of the actuator torque

Ka

and the

Therefore, the knowledge of the actuator force and the nozzle angle allows one

to compute the impact torque with the equation

Kc = J φ¨ + B φ˙ + Cφ − fa · l Measuring the nozzle altitude angles

φ and actuator forces fa

(23)

by the second stage engine sensors

one can use the equation (23) to compute the impact torque applied to the nozzle extension in real time. Actuator torques components obtained by the simulation of thrust vector control response to the impact are shown in Fig.

18. These torque components coincide with the torque components

obtained in the ABAQUS simulations.

30

The order of magnitude estimations of the nozzle extension and TVC response to the impact demonstrates that the information about impact torque dynamics can be used to estimate the nozzle extension damage and to predict the consequences of the impact. As follows from the discussion above, once the impact induced torque is detected, the prediction of the damage is reduced to the analysis of the buckling eigenmodes of the thin conical shell. Which in its turn can be reduced to an analysis of a set of forced oscillators as was shown in Sec. II. The in-ight analysis of these timetraces, at least in principle, can be performed within Bayesian inferential framework similar to the one introduced in earlier work [13, 14] for the in-ight diagnostic and prognostic of the case breach fault in solid rocket motor. The results obtained within this project can be used as basic guiding lines for the future analysis, which should include model validation using experimental data.

VII. Contact Eects on TVC Response Due to the fact that the recontact may impart high loads on the nozzle, the recontact could also aect the TVC response of the US engine. Based on the magnitude of the loads applied, the eect on the TVC response can vary from a small bias to a quick hard over condition of the aected actuator. These eects are imparted in the vehicle ascent simulation as a function of the recontact loads. Following the initial impact load, there is a continuing load from the nozzle damage. Since the TVC system has not been tested for this type of failure, certain assumptions in modeling the TVC response are in order. First, the initial impact loads presented in Section II are quite high and may damage the TVC so badly that it causes complete failure. Hardware testing of the TVC would be the only way to determine if this is the case. Since no hardware test of this sort has been performed, for this analysis the assumption is made that the initial impact loads will not aect the TVC response at all. The duration of the initial impact is short. The impact loads will be applied to the vehicle dynamics, but no reduction in TVC response due to the impact is included. Another assumption involves the actuator response over time. The sustained load from the nozzle damage leads to an actuator bias, or to a hard over condition, depending on the size of the load (as calculated using techniques discussed earlier) and the ability of the actuator to withstand the load. For the cases where the recontact

31

load results in a TVC hard over condition, the internal damage to the TVC system in terms of overpressure and other failures to hardware may lead to other failures not included in the TVC modeling. As such, the hard over may occur more quickly or the actuator may break in such a way that no TVC load is applied and the nozzle is free to move with respect to the thrust oset alone. Once again, hardware tests would be required to truly know how the TVC system would respond to these failures that are out of the scope of normal TVC testing. For this reason, the assumption is made herein that the hard over condition proceeds as provided by the current TVC modeling and no other failures are applied to alter the hard over. For cases where the recontact loads result in a TVC bias, no assumption is necessary since the current TVC modeling is sucient for this condition. In terms of the TVC modeling, the eects on the rock and tilt actuator responses are treated separately. That is, based on the location of the damage to the nozzle, and thus the direction of the applied loads, the components of the loads are computed along each actuator. The actuators have a certain stall load, such that if the load applied to the actuator exceeds this stall load, the actuator is shown to go to a hard over condition. For loads less than the stall load, a small bias to the actuator command results as indicated by the TVC model within MAVERIC the vehicle simulation. For this reason the resulting load on each actuator is compared to the actuator stall load, and if the load is less than the stall load that load is sent as a bias torque to the actuator model in MAVERIC. If the computed load is above the stall load, then a table lookup is performed to determine the hard over angle and rate that the actuator should experience for that load. The table employed here was constructed from several runs of a high-delity quad-valve model of the TVC system that is able to simulate this situation with the assumptions mentioned previously. The higher the load on the actuator, the higherlarger the hard over angle and the higherfaster the rate the actuator moves to that angle.

VIII. Vehicle Response and Abort With the impact loads, nozzle and thrust eects from the recontact, and the resulting actuator eects computed, simulation of the vehicle response to these events can proceed. A model of the

32

stage separation recontact scenario has been added to release 3.7 of MAVERIC (Marshall Aerospace Vehicle Representation in C) for inclusion in failure and abort analyses. A sample analysis was performed including uncertainties in the impact, nozzle and thrust eects, and contact location, along with representative uncertainties for other ascent parameters. A single scale factor was uniformly dispersed with a 50% uncertainty for each run with this scale factor applied proportionally to the impact load, nozzle side load, reaction torque, and thrust reduction. The large uncertainty was chosen because the desire is to see the eect of recontacts that may be from dierent causes, not just the specic impact scenarios analyzed in this paper. Therefore, the larger initial impact leads to a larger nozzle deformation, the larger side load, and a higher reduction in nozzle thrust. The contact location, and therefore, the direction of the resulting side force, was uniformly dispersed about the nozzle circumference to dene the direction of the contact torque, side force, and reaction torque. For the cases where the resulting load causes an actuator to go to hard over, the resulting attitude rates will climb quickly following the onset of the hard over actuator rates. Once the attitude rates signicantly exceed those expected from normal dispersed ight, the mission should be aborted. As the attitude rates rise quickly and since the structural loads during this phase of ight should not be that high, a successful abort should be feasible. For the many cases that continue without a hard over condition, where a bias is applied to the actuator command, the vehicle's ight control system senses the bias and adjusts the actuator commands appropriately to y the vehicle to the desired orbit. This is feasible as long as there is sucient gimbal margin to accommodate the bias. With the reduction in thrust, in combination with other dispersed ascent parameters, there is a possibility that the vehicle may run out of propellant prior to reaching the desired orbit insertion condition. However, most should reach orbit satisfactorily. Failure analysis of this type of failure in MAVERIC revealed that 61.7% of the cases ran had a hard over actuator followed by abort, 38.2% reached the desired orbit, and 0.1% did not have a hard over but failed to reach orbit. All cases that had a hard over condition were sensed by the abort logic that compares the attitude rates with those from normal dispersed with margin applied. These results are representative only and depend on the particular models, actuator capabilities, propellant margins, and dispersions applied. A higher percentage may not reach orbit in

33

cases where there is not extra propellant margin beyond the normal ight performance reserve.

IX. Conclusion In this paper an analysis of the stage separation failure for two-stage vertically stacked space vehicle was reported. It was shown that the recontact between the stages results in the nozzle buckling and wrinkling of the nozzle surface. The amplitude of the rst peak of the impact torque is strongly correlated with the impact trajectory and with the initial time of the impact signal. These correlations can be used to diagnose the damage and to predict fault-induced changes of the thrust ahead of time. It was shown further that fault induced damage of the nozzle will result in signicant ignition transient torque imposed by the ow on the dent in the plane of separation in the direction opposite to the impact induced torque. It was next shown that the heat and pressure loads generated by the oblique shock on the dent will result in its reverse buckling and subsequent melting away on the time scale approximately 0.23 sec. As a result the torque induced by the ow will change its sign one more time and will approach quasi-equilibrium value proportional to the damaged area. The quasi-equilibrium value of the fault-induced torque was estimated numerically and analytically and was shown to be directly proportional to the damage area. The value and sign of this torque are strongly correlated with the strength and the direction of the impact. The estimated fault-induced changes of the main thrust were shown to be less then 2.5% of the nominal value. A model of the TVC actuator response to the fault-induced torque was introduced and it was shown that parameters of the model can be inferred using actuator signal. The estimated values of the fault-induced changes of the main thrust and side load were used to analyze vehicle response to the stage separation failure using 3.7 release of the MAVERIC simulation code. Analysis shows that a successful abort is feasible in many cases of failure to actuator hard over, and that most other cases may be able to reach orbit if sucient propellant remains onboard. The obtained results should be considered as an order of magnitude estimations of the nozzle and TVC response to the impact and as basic guiding lines for the future analysis, which should include model validation using experimental data. Such an analysis may pave a way for development

34

of the failure diagnostics and prognostic system for the stage separation failure and mitigation of its possibly catastrophic consequences. Use of the understanding of the longer-term damage eects enables determination of the overall eects on the ascent ight.

Acknowledgments The authors thank C. Kiris for extended and valuable discussion of the results and Engineering Group at MFSC, including in particular P. R. Gradl, M. Shadoan, and M. I. Johnston, for providing the discussions and information that helped to conduct simulations and developed analytical estimations.

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[5] Lindberg, Herbert E. and Florence, Alexander L.,

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Introduction to Heat Transfer ,

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Free vibration analysis of rocket nozzles using energy methods , Master Thesis, Luleå Tekniska

Universitet, Trollhåttan, Sweden, 1998.

35

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