00 S-593.pdf

Appl. Math. Mech. -Engl. Ed., 2008, 29(11):14111426DOI 10.1007/s10483-008-1103-zcShanghai University and Springer-Verlag 2008

Applied Mathematicsand Mechanics(English Edition)

Monte Carlo simulation of stage separation dynamics of

a multistage launch vehicle

J. Roshanian, M. Talebi

(Department of Aerospace Engineering, K. N. Toosi University of Technology,

P. O. Box 16765-3381, Tehran, Iran)

(Communicated by CHEN Li-qun)

Abstract This paper provides the formulation used for studing the cold and hotseparating stages of a multistage launch vehicle. Monte Carlo simulation is employedto account for the off nominal design parameters of the bodies undergoing separation toevaluate the risk of failure for the separation event. All disturbances, effect of dynamicunbalance, residual thrust, separation disturbance caused by the separation mechanismand misalignment in cold and hot separation are analyzed to find out nonoccurrence ofcollision between the separation bodies. The results indicate that the current designsatisfies the separation requirements.

Key words ullage-rocket, retro-rocket, launch vehicle, Monte Carlo simulation, hotseparation, cold separation

Chinese Library Classification V421.7, O242.22000 Mathematics Subject Classification 74H15

Nomenclature

a0, acceleration of coordinate system (m/s2);

u, v, w, components of body linear velocity alongx, y and z axes (m/s);

p, q, r, components of body angular velocityabout x, y, z axes (()/s);

C, coning angle ();u, critical velocity (m/s);1, gas density (kg/m

3);T , lateral angular tip-off rate (()/s);m, mass (kg);Ma, match number;Pk, pressure in vessel (Pa);Pa, pressure in nozzle (Pa);P02, pressure between shock and first stage

(Pa);PM , pressure between two stages (Pa);

Ixy, Ixz, Iyz, products of inertia in yaw, roll andpitch planes (kg m2);

r, radius of stage one (m);R1,2, relative distance between CG (center

of gravity) of stages 1,2 and coordi-nate system (m);

Ix, Iy, Iz, roll, pitch and yaw moments of iner-tial, respectively (kg m2);

Sa, surface of nozzle (m2);

Tk, temperature in vessel (K);TM , temperature between two stages (K);t, time (s);u1, velocity (m/s);VM , volume of space between two stages

(m3);, , , yaw, roll and pitch angles ().

Received Jun. 19, 2007 / Revised Aug. 7, 2008Corresponding author Jafar Roshanian, Associate Professor, E-mail: [email protected]

1412 J. Roshanian and M. Talebi

Introduction

The dynamics of separating stages has received attention of several investigators. Chubb[1]

has constructed collision boundaries between two separating stages. Palmer and Mitchell[2]

investigated spring separation of spacecraft. Dwork[3] and Wilke[4] provided valuable insightinto disturbances caused by separation mechanisms in a spinning setup. Puglisi[5] analyzedcontrollability of stage separation. A considerable amount of aerodynamic data includingstability derivatives have been generated for the space shuttle type of configurations involv-ing winged bodies[67]. The data so gathered have been utilized for the separation dynamicsinvestigation[8]. Christensen and Narahara[9] reviewed the spacecraft separation, and Mitchelland Palmer[10] developed a high accuracy spacecraft separation system. Waterfall[11] investi-gated multispring systems for separation and spinning and nonspinning bodies. Bolster andGoogins[12] designed, developed and tested a series of air-launched sounding rockets. Longren[13]

analyzed spin-stabilized rockets with guide shoes and rails constraining the lateral motion. Hur-ley and Carrie[14] reviewed the genesis of a four bar linkage separation system and carried outthe analysis for separation of parallel staged shuttle vehicles. Su and Mullen[15] developed aplume impingement force during tandem stage separation at high altitudes. Subramanyam[16]

developed a general model for spring-assisted stage separation. Kalesnikof[17] wrote a bookabout dynamic separation. Saxena[18] investigated upper stage jet impingement on separatedboosters. Lochan et al.[1922] analyzed the separation dynamics of strap-on boosters from thecore rocket utilizing the wind tunnel simulation data for dynamic forces. Cheng[23] developedan analytical procedure based on a coupled gas/structure model to simulate the fairing sepa-ration events. The procedure was validated by comparing the analysis results with full-scalepayload fairing separation test data of the Titan IV launch vehicle. Reubush et al.[24] simulatedhyper-X stage separation with the Monte Carlo method. Jeyakumar and Biswas[2527] providedthe stage separation system design and dynamic analysis of launch vehicles.

In a multistage vehicle mission, the separation phase is very critical. What one envisagesin a separation phase is a clean separation, which means avoidance of any change in attitudedue to lateral angular rates and elimination of collision between the separation stages. Cleanseparation is difficult to achieve due to separation disturbances which are due to tip-off forcesintroduced by the separation mechanism and dynamic unbalance of the separation stages. An-other disturbance is due to the residual burning characteristics of the lower stage.

The prediction of the separation trajectory must include the impingement force from thecontinuing stage propulsion system, and this force is especially large if the propulsion systemfires in close proximity to the spent stage.

Analysis of separation dynamics is essential to determine separation disturbance, to definecollision boundaries and to generate inputs for the guidance program. This may range incomplexity from the analysis of simple, rigid-body, one-degree-of-freedom models to intricate,nonlinear computer simulations, in which each body has six rigid-body degrees of freedom,which may be spinning, and where elastic effects are considered. Customarily, the analysisis used to predict the nominal performance of the mechanism and to estimate tip-off errorsdue to the standard tolerances on the various design parameters. It has also become commonto perform a failure-modes-and-effects analysis in which failure modes are assumed and theireffects on the performance of the separation mechanism are evaluated, at least in a qualitativesense.

In general, two levels of analysis have been used successfully. In cases where no complexforces act on the bodies, simple planar models have been used to analyze nonspinning separa-tions, and simple transverse-moment models have been used to analyze spinning separations.The separation mechanism is designed to operate successfully when each parameter that affectstip-off or hang-up, such as tolerance effects, assumes its most adverse value. Often, severaliterations of the analysis are necessary to determine the worst combination of parametric val-

Monte Carlo simulation of stage separation dynamics of multistage launch vehicle 1413

ues (equivalent, in statistical terms, to large sigma values, possibly 9 or 10) and to confirmsatisfactory separation of the bodies under this condition. Although the simple analysis doesnot account for the effects of coupling and of complex forces and moments, this shortcoming isconservatively compensated for by the requirement for satisfactory separation under the worstcombination of parametric values. This type of analysis has proven successful for most simpleseparations and has led to the design of separation mechanisms with high intrinsic reliability.

When complex forces or moments act on the bodies, the mission is man rated and theseparation-mechanism weight is critical; or when mission requirements are stringent, then com-plex, nonlinear computer simulations are performed. In this type of analysis, the forces andmoments acting on each body are accounted for in detail, each body is allowed rigid-body mo-tion in six degrees of freedom, gyroscopic coupling is included for spinning cases, and elasticeffects may be considered. In addition, statistical studies are performed to assess the effect onseparation motion when the value of each parameter is allowed to vary throughout its toleranceband. The expense of conducting these studies is often reduced by using Monte Carlo tech-niques instead of computing the separation motion for every possible combination of parametricvalues.

Another technique for complex separation analysis is to determine the partial derivative ofeach error source. The partial derivatives, together with the range of values of each parameter,can be used to determine the possible tip-off error attributable to the variation in the valueof each parameter. The tip-off errors from all parameters are then combined to give the totalpossible error. The favored technique for combining the errors is to add directly all the errorsattributable to correlated parameters, and to add the errors attributable to uncorrelated pa-rameters by the root-sum-square method. This technique gives a close approximation of theresults that can be obtained with a more rigorous mathematical approach and avoids a largenumber of computer runs.

The partial-derivative error analysis also identifies the principal sources of tip-off error andthe parameters which should be closely controlled, and those which can be allowed to varywithout producing excessive errors. This information is especially useful for cost/weight tradeoffstudies. The partial-derivative approach offers the best practical way to analyze the separationmechanism and to evaluate possible errors.

Depending on the time in the flight when separation is programmed to occur, a rigorousseparation analysis considers the effects of the aerodynamic environment, wind shears or gusts,fuel sloshing, engine-nozzle flow separation, sequencing of events, control-system interactions,mass and inertia properties of the separating bodies, gyroscopic coupling, and details of theseparation mechanism itself[28].

1 Stage separation modeling

A typical launch vehicle may involve several separation events, such as strap-on separation,stage separation, heat shield separation, ullage rocket separation and spacecraft separation. Ina multistage rocket configuration the most important event is staging. Obviously, the processcommences from the detection of the burn-out of the ongoing stage, and continues with theignition of the next stage and the separation of the spent stage.1.1 Cold separation

The problem is modeled under the influence of forces and moments of the individual bodiesundergoing separation, treating the bodies as rigid. A rigid body has six degrees of freedom(three displacements of a point fixed in the body and three orientation angles). Each bodyundergoing separation has twelve state variables (namely, three displacements of a point fixedin the body, three orientation angles, three components of the velocity vector, and three com-ponents of the angular velocity vector). Two basic types of frames of reference are used in theformulation apart from the usual geo-centric or Earth-centered inertial frame of reference (to


compute the gravity effect) and the topo-centric or launch point inertial frame.A local inertial frame in which the dynamics is described and the body coordinate system,

separate for each of the body undergoing separation, is used. The transformation from thebody frame to the local inertial frame and vice versa can be achieved through a transformationmatrix in which a prefixed sequence of rotation of the Euler angles is used.

In principle, equations of motion can be expressed in any coordinate system. It is advanta-geous to express them in the body coordinate system, as the moment of inertia, engine tail-offthrust and aerodynamic computations are very simple in the body coordinate system, whilegravity computation is simpler in the geo-centric inertial coordinate system.

The coordinate system for the analysis includes two body-fixed axis systems (see Fig. 1),O1x1y1z1 and O2x2y2z2, for lower and upper stages, respectively

[17]. The six degrees-of-freedomfor the separating stages are three translational rates (u, v, w) along and three rotational rates(p, q, r) about the x, y, z axes, respectively. Exyz is the inertial coordinate system. At theinitiation of separation (t = 0) all the three coordinate systems are mutually parallel to oneanother and O1 coincides with E.

O1

O2

y1

y2

x2

x1

yP x

P

Fig. 1 Coordinate axis systems

The equations are described below:

mdV

dt+ V = Ft, (1)

where m is the mass of the body; V = Vxi + Vyj + Vzk is the velocity; = pi + qj + rkis the angular velocity, which has components p, q and r along the body axes; t is the time;Ft = Fxi+Fyj +Fzk is the total external force vector. Equations for the translational motionalong body x, y, z axes are

m(u + qw rv) = Fx,m(v + ru pw) = Fy,m(w + pv qu) = Fz.

(2)

The equation for the angular motion is

dL

dt+ L = Mt, (3)

where L = Lxi + Lyj + Lzk is the angular momentum vector; Mt = Mxi +Myj +Mzk isthe total external moment vector; i, j,k are the unit vectors with respect to the center of mass


of the respective body frame. The components of the angular momentum vector are obtainedfrom

LxLyLz

= I

pqr

=

Ixx Ixy IxzIyx Iyy IyzIzx Izy Izz

pqr

. (4)

Equations for the angular motion about body x, y, z axes areIxp Ixy q Ixz r p(Ixzq Ixyr) qr(Iy Iz) Iyz(q2 r2) =Mx,Iy q Iyz r Ixy p q(Ixyr Iyzp) rp(Iz Ix) Ixz(r2 p2) =My,Iz r Ixz p Iyz q r(Iyzp Ixzq) pq(Ix Iy) Ixy(p2 q2) =Mz.

(5)

Mass, moments of inertia and products of inertia are, in general, functions of time. Fx, Fyand Fz and Mx, My and Mz are sums of external forces and moments, respectively. Thesefunctions can include effects due to thrust, gravity, aerodynamics separation forces and distur-bances. The effect of dynamic unbalance is simulated by taking nonzero values for products ofinertia.

The Euler angles define the body attitude with respect to the inertial coordinate systems.Their rates , , in terms of , , and p, q and r are

= (q sin + r cos )/ cos,

= p+ tan(q sin + r cos ),

= q cos r sin .(6)

The components of the velocity in the inertial coordinate system arex = u(cos cos) + v(cos sin sin sin cos ) + w(cos cos sin+ sin sin ),y = u(cos sin) + v(sin sin sin cos cos ) + w(sin cos sin cos sin ),z = u sin+ v(sin cos) + w(cos cos).

(7)

The above equations are first order nonlinear simultaneous equations. Two sets of suchequations, one for each stage, are programmed. The output of these equations consists of thetime histories of x, y, z, x, y, z, p, q, r, , , , u, v, w for each stage. Relative velocity and motionbetween the separating stages are obtained by differencing the respective parameters. If we usea guide pin in the separation mechanism, the differential equations change into the followingequations:

Equations for translational motion along the body x, y, z axes with a guide pin aremB(aOx (q2 + r2)RB + RB) +mH(aOx (q2 + r2)RH + RH) = Fx,mB(aOy + (r + pq)RB + 2rRB) +mH(aOy + (r + pq)RH + 2r RH) = Fy,

mB(aOz (q pr)RB 2q RB) +mH(aOz (q pr)RH 2qRH) = Fz.(8)

Equations for angular motion about the body x, y, z axes are

p Ixx + (r2 q2) Iyz + (pr q)Ixy (pq + r)Ixz + rq(Izz Iyy) =Mx,

q Iyy + (p2 r2) Ixz + (pq r)Izy (qr + p)Ixy + pr(Ixx Izz)

(mBRB +mHRH)aOz + (2mBRB RB + 2mHRHRH)q =My,

r Izz + (q2 p2) Ixy + (rq p)Ixz (pr + q)Iyz + pq(Iyy Ixx)

+ (mBRB +mHRH)aOy + (2mBRBRB + 2mHRHRH)r =Mz.

(9)


Additional equations are

FBx = mB

(aOx (q2 + r2)RB + RB

),

FHx = mH

(aOx (q2 + r2)RH + RH

),

mB RB +mH RH = 0.

(10)

1.2 Hot separation

In hot separation, the reversed flow effects are significant when the tandem stages are rel-atively close together (X/D 2). These effects result from the flow to the spent stage cavitybeing directed back toward the continuing stage. In this part, we describe an analytical tech-nique to predict the plume impingement force during the tandem stage separation, and considerthe reversed flow effects.

The flow model of a plume impinging on the lower stage at high altitudes is shown in Fig. 2.The exhaust gas expands rapidly once it leaves the nozzle and passes through a bow shock wavebefore entering the lower stage cavity. The flow then reverses direction and leaves the cavity atthe nearly sonic speed.

321

Lower stage

(Spent)

Upper stage

(Continuing)

Bow shock

Task dome

Plume boundary

R

x

Fig. 2 Flow model and control surface of plume impingement during tandem stage separation

By applying the momentum theorem, one obtains the impingement force on the lower stagedome at separation distance x. If one assumes that the flow behind the shock wave is nearlyparallel to the axis of symmetry and the flow leaves the cavity at the sonic speed, then

F1x = 2pi

r10

P02rdr + 2piu

r10

1u1rdr,

F1y =

s1y

PMds, F1z =

s1z

PMds.(11)

The above equation can be integrated numerically with the characteristic solution of the ex-haust plume. However, the characteristic calculation is time-consuming, and at large distancesfrom the nozzle, computational difficulties occur.

The calculation can be greatly simplified if one integrates the above equation analytically


with the approximate analytical expression for the exhaust plume flow field by[15]P02 = PaSa1(Ma, X)(1 + kM

2a ),

u1 = ua[1 +2

(k 1)M2a]1/2,

1 = a2(Ma, X)k(k 1)M2a ,

(12)

1(Ma, X) =

[k(k 1)M2a + 2][cos(arctan(r/X))]k(k1)M2a+4

2piX2,

2(Ma, X) =1

2

(raX

)2[cos(arctan(r/X))]k(k1)M

2a .

(13)

Critical velocity u according to the gas dynamic computation is

u =

2k

k + 1gRTk. (14)

Inserting Eq. (12) into Eq. (11) and carrying out the integration, and after some manipulationsone obtains

F1x = kM2aPaSa{(1 +

1

kM2a)[1 fa1 (X)] +

k(k 1)M2a[k(k 1)M2a 2]

(2

k + 1

)12

[k 12

+2

M2a+

2

(k 1)M4a]12 [1 f b1(X)]}, (15)

F1y = PMS1y, F1z = PMS1z, (16)

f1(X) =1

1 + ( r1X)2, a =

k(k 1)M2a2

+ 1, b =k(k 1)M2a

2 1, (17)

where

SkPSa

=Ma(

k+12 )

k+12(k1)

(1 + k12 M2a)

k+12(k1)

,PkPa

= (1 +k 12

M2a )k

k1 , (18)

which define the axial plume impingement force on the lower stage at the nozzle exit. TM , PMis calculated by the following equations:

dTMdt

=RTMPMVM

[(kTk TM )kOPk

g

RTk

ni=1

SikP

c(k 1)TMkO

g

RTM pi(d1 + d2)

2xPM (k 1)CTSCT (TM TCT )

R

], (19)

dPMdt

=0.3kO

gRTk

VMPk

n1

SikP 1

VM[ckO

0.3gRTk

pi(d1 + d2)

2x+ SM x]PM . (20)

The axial plume impingement force on the upper stage is

F2x = PM (S2 n

i=1

Sia). (21)


1.3 Separation mechanisms force and momentum

Stage separation by the force of the ullage rocket and retro rocket thrust is considered.The separation impulse is provided by release of the energy stored in the relatively low thrustshort burn duration solid motor. Its location and support hardware design is dictated by thegeometry requirements, as the jet impingement from the retro motor should not affect thehighly sensitive payload interface.

To compute the actual thrust from a propulsion unit in the body frame, a transformationmatrix is generated between the thrust frame (coordinate frame assumed at the nozzle) andthe body frame. This matrix is defined by the following sequence of anti-clockwise rotation ofthe body frame: about the x-axis through (measured from the y-axis) is the azimuth angledefining the propulsion unit location; about the z-axis through the nozzle is the cant angleas shown in Fig. 3.

z

yO

x

z1

y1

x2 x1y2y1O2

O1 y

L'

CG

z

d

Fig. 3 Transformation of retro rocket thrust frame to body frame

The thrust vector in the body frame[27] is

Fb = T (t)(cosi sin cos j + sin sin k). (22)

The propulsion unit thrust location is given by

r = Li + d cos j + d sin k. (23)

The moment due to thrust is given by

M = r Fb. (24)

The mathematical models for thrust F (t) are shown below, the inputs for the mathematicalmodel for the ullage rocket and retro rocket are I, , t0, t1, t2, t3, where ti is time, I is the motorimpulse, and is the angle (see Figs. 4 and 5).

F(t)

t0O

t1 t2 t3 t

Fig. 4 Mathematical model for retro rocket

F(t)

t1O

t2 t3 t

Fig. 5 Mathematical model for pyrotechnic bolt


The surface under the F (t) curve is equal to I:

I =

t3t0

F (t)dt. (25)

The thrust vector F (t) is given by

T =

t t0t1 t0h, t0 < t < t1;(t1 t2)(t t1)

t2 t1 tan+ h, t1 < t < t2;t t3t2 t3 (h (t2 t1) tan), t2 < t < t3,

(26)

where

h =I + 0.5(t2 t1)2 tan+ 0.5(t2 t1)(t3 t2)

0.5(t1 t0) + 0.5(t3 t2) + (t2 t1) . (27)

Inputs for the pyrotechnic bolt are I, t0, t1, t2. Similar to the above equations the force of thepyrotechnic bolt is given by

T =

t t0t1 t0

2I

t2 t0 , t0 < t < t1;t t2t1 t2

2I

t2 t0 , t1 < t < t2.(28)

2 Monte Carlo method and disturbance

Analysis of separation dynamics is essential to determine separation disturbances and todefine collision boundaries. The effects of dynamic unbalance, residual thrust and separationdisturbance occurr due to the separation mechanism.2.1 Monte Carlo method

An important aspect of separation analysis is the uncertainty of model parameters (massproperties, initial conditions, separation mechanism parameters, etc.). In order to account forthe randomness associated with such uncertainty, the Monte Carlo technique was incorporatedin the stage separation analysis. The Monte Carlo simulation provides a unified framework forthe quantitative analysis of model uncertainty and assessment of associated risk, as well as inthe formulation of trade-off studies relative to design parameters. The use of high-speed work-stations has made the Monte Carlo simulation more practical as a design and verification tool.

Enhanced by the Monte Carlo technique, the separation analysis predicts the statisticalbounds of separation parameters. These statistical boundes are used to assess the performanceof the stage separation hardware design under the worst case conditions[34].2.2 Effect of dynamic unbalance

The product of inertia causes dynamic unbalance in each stage. Dynamic unbalance producestip-off rates and in turn coning motion.2.3 Effect of residual thrust

Residual thrust accelerates the lower stage forward and it may catch up with and hit theupper stage.2.4 Effect of separation disturbance which occurs due to the separation mecha-

nism

Due to manufacturing tolerances, it is practically impossible to get identically acting sepa-ration mechanisms. The possibility of one or more separation mechanisms failing to cause force


can not be completely ruled out. Such disturbance creates lateral moments on the separationstages inducing tip-off rates.

Manufacturing tolerances also introduce variation in separation velocity and time. In thispart we introduce some parameters. Lateral angular tip-off rate T and coning angle C are[34]:

T =q2 + r2, (29)

C = arctan

(IyIx

q2 + r2

p

). (30)

In the hot separation the point must be attentioned in distance between two stages. Becauselittle volume and distance cause pressure to rise up and cause cutoff flow in the nozzle, to avoidthis phenomenon, the following conditions must be applied[17]:

dadkP

1 + 0.33k(PkPM

1fkP

)0.6,

fkP =

(2

k + 1

) kk1

.

(31)

With these equations we can arrive at the minimum allowable initial volume.

3 Simulation results

The separation between two stages of a rocket utilizing 4 springs for separation is chosen.The initial conditions for separation are[16]

x1 = y1 = z1 = 0, 1 = 1 = 1 = 0,

u1 = v1 = w1 = 0, p1 = 6pi rad, q1 = r1 = 0.

In the nominal case where separation disturbances are absent, the relative velocity andrelative distance are shown in Figs. 6 and 7.

0 0.02 0.04 0.06 0.08 0.10t/s

5

4

3

2

1

0

x rel

/cm

Fig. 6 Relative distance in separation

0.8

0.6

0.4

0.2

00 0.02 0.04 0.06 0.08 0.10

t/s

ure

l/(m

s

1)

Fig. 7 Relative velocity in separation

The separation distance between the stages increases monotonically and hence there is nopossibility of collision. Now, effect of each separation disturbance is discussed.

In the present example, dynamic unbalance is assumed to be in the pitch plane only. Dy-namic unbalance produces tip-off rates and in turn coning motion. The coning angle for thelower stage behaves as a periodic function of time, and the maximum value of the coning angleis 2.5; for the upper stage this parameter increases continuously with time, see Fig. 8.

Now, the effect of spring force variation is discussed. Due to manufacturing tolerances,it is practically impossible to get springs of identical action. The possibility of one or more


springs failing to impart ejection force can not be completely ruled out. In the present example,maximum variation in stiffness of spring is assumed to be 5% from the nominal value, see Fig. 9.

Separation between two stages of a launch vehicle with 6 pyrotechnic bolts, 2 retrorocketsand rolling rockets is simulated. Separation occurs in vacuum and hence aerodynamic force isabsent. External forces and moments are due to the thrust, gravity and separation mechanism.In cold separation, separation occurs after lower stage burnout and hence mass, moments andproducts of inertia are constant. In the hot separation, separation occurs before the lower stageburnout. In the nominal case where separation disturbance is absent, the relative parametersare shown in Figs. 16 and 17.

Lower stage

Upper stage

Separation

0 0.2 0.4 0.6 0.8 1.0t/s

4

3

2

1

0

Co

nin

g a

ng

le C

/()

Fig. 8 Effect of dynamic unbalance in con-ing angle

Lower stage

Upper stage

1.0

0.8

0.6

0.4

0.2

00 0.02 0.04 0.06 0.08

t/sA

ng

ula

r ti

p-o

ff r

ate T

/((

)

s1)

Fig. 9 Effect of 5% variation in spring stiff-ness in tip-off rate

0 0.5 1.0 1.5t/s

2.0

2.0104

1.5104

1.0104

0.5104

0

Pyrotechnic bolt

Retro rocket

Rolling rocket

Fo

rce F

/N

Fig. 10 Separation mechanism forces

800

600

400

200

0

200

a2x /(m

s

2)

0 1104 2104 3104

t/s4104 5104

Fig. 11 Component x of acceleration instage 2

0 0.5 1.0 1.5t/s

10

5

0

5

10

a2

y /(m

s

2)

Fig. 12 Component y of accelerationin stage 2

25

20

15

10

5

00 0.5 1.0 1.5

t/s

p2

/(r

ad

s

1)

Fig. 13 Component x of angle veloc-ity in stage 2


0 0.5 1.0 1.5t/s

45.4

45.3

45.2

45.1

45.0

2/(

)

Fig. 14 angle in stage 2

0 0.5 1.0 1.5t/s

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

q2

Fig. 15 Mathematical model for py-rotechnic bolt

60

55

50

45

40

35

300 0.5 1.0 1.5 2.0

t/s

x rel

/m

Fig. 16 Relative distance in hot sep-aration

0 0.5 1.0 1.5 2.0t/s

40

30

20

10

0

10

ure

l/(m

s

1)

Fig. 17 Relative velocity in hot sep-aration

The plume impingement force and PM are shown in Figs. 18 and 19.Table 1 provides the dynamic parameters considered for the separating bodies at the time

of separation. Table 2 provides the dispersion parameters considered in the studies of theseparation mechanism. Table 3 provides the expected value and the time variance of the 12degrees of freedom of the separated bodies.

Note the 1000 simulation performed and the results of separation parameters which arepresented in terms of their statistical bounds: minimum, maximum, mean 3.

Figures 2027 summarize the statistical bounds of the separation parameters.

0.50 1.0 1.5 2.0t/s

10105

8105

6105

4105

2105

0

F1

x /N

Fig. 18 Axial force caused by gasdynamic in hot separation

7105

6105

5105

4105

3105

2105

1105

00 0.1 0.2 0.3 0.4 0.5

t/s

PM

/P

a

Fig. 19 PM in hot separation


Table 1 Dynamic parameters of the separating bodies

Parameters m/kg Ix/(kg m2) Iy/(kg m2) Iz/(kg m2) Ixy/(kg m2) Ixz/(kg m2) Iyz/(kg m2)

Stage 1 2000 2 500 5 25000 20 25000 20 0 10% 0 10% 0 10%

Stage 2 500 2 50 2 200 5 200 5 0 10% 0 10% 0 10%

Table 2 Sensitivity of the separation mechanism

ParametersDispersion level

Retro rocket Rolling rocket Pyrotechnic bolt

Jettisioning rocket thrust dispersion & py-rotechnic bolt force dispersion p/% 4 4 4

Jettisioning rocket ignition delay & pyrotechnicbolt ignition delay t/ms 10 10 1

Jettisioning rocket cant angle /() 0.5

Differential impulse between the core engines &between pyrotechnic bolts Pe/(N s) 10 5 2

Thrust line offset between the core engines &between pyrotechnic bolts e/mm 0.3 0 0.1

Table 3 Variation in 12 degrees of freedom for the separating bodies

Physical Stage 1 Stage 2

quantities 3 value Expected value +3 value 3 value Expected value +3 value

vx/(m s1) 2973 2976 2979 3000.4 3004 3007.6

vy/(m s1) 79.1 88 96.9 280 231 182

vz/(m s1) 239.2 249 258.8 4.5 5.3 6.1

x/m 2706 3378 4050 2716 3397 4078

y/m 233.6 194 154.4 233 195 157

z/m 3559 2957 2355 3527 2968 2409

p/(rad s1) 0 0 0 21.2 24.2 27.2

q/(rad s1) 0.0087 0.0087 0.0087 0.0075 0.0065 0.00055

r/(rad s1) 0.0097 0.0087 0.0077 0.007 0.01 0.013

/rad 0.11 0.107 0.102 0.095 0.091 0.086

/rad 0.78 0.79 0.792 0.787 0.79 0.792

/rad 0.105 0.102 0.098 22 29.5 37.2

Fig. 20 Component z of angle acceler-ation in stage 1

Fig. 21 Component x of angle ac-celeration in stage 2


Fig. 22 Angular tip-off rate in stage 2 Fig. 23 Component z of angle velocityin stage 1

Fig. 24 Relative angle of stages Fig. 25 Relative angle of stages

Fig. 26 Relative angle of stages Fig. 27 Component y of accelerationin stage 2

4 Conclusions

Limits on various disturbances can be specified with the knowledge of allowable tip-off rateswhich depend on vehicle performance and control power plant limitation. Separation dynamicsanalysis generates some of the inputs for designing on-board control power plants and also forchoosing and evaluating a separation mechanism. A statistical method is followed to examinethe influence of the design variables on the separating bodies and detect the statistical bounds


of separation parameters.

References

[1] Chubb W. The collision boundary between the two separating stages of the SA-4 saturn vehicle[R].NASA-TND-598, August 1961.

[2] Palmer G D, Mitchell D H. Spring separation of spacecraft[R]. NASA-CR-64009, 1963.

[3] Dwork M. Coning effects caused by separation of spin stabilized stages[J]. AIAA Journal, 1963,1(11):26392640.

[4] Wilke R O. Comments on coning effects caused by separation of spin stabilized stages[J]. AIAAJournal, 1964, 2(7):1358.

[5] Puglisi A G. Saturn IB/S-IVB stage separation controllability report[R]. Douglas Report SM-46758, 1964.

[6] Decker J P, Pierpont P K. Aerodynamic separation characteristics of conceptual parallel-stagedreusable launch vehicle at Mach 3 to 6[R]. NASA-TMX-1051, January 1965.

[7] Decker J P. Aerodynamic abort-separation characteristics of a parallel staged reusable launchvehicle from Mach 0.60 to 1.20[R]. NASA-TMX-1174, November 1965.

[8] Decker J P, Gera J. An exploratory study of parallel-stage separation of reusable launch vehi-cles[R]. NASA-TND-4765, 1968.

[9] Christensen K L, Narahara R M. Spacecraft separation[J]. Space Aeronautics, 1966, 46(7):7482.

[10] Mitchell D H, Palmer G D. Analysis and simulation of a high accuracy spacecraft separationsystem[J]. Journal of Spacecraft and Rockets, 1966, 3(4):458463.

[11] Waterfall A P. A theoretical study of the multi-spring stage separation system of the black ar-row satellite launcher[R]. Royal Aerospace Establishment, TR-682016, Farnborough Hants, UK,August 1968.

[12] Bolster W J, Googins G C. Design, development and testing of a series of air-launched soundingrockets[J]. Journal of Spacecraft and Rockets, 1969, 6(4):460465.

[13] Longren D R. Stage separation dynamics of spin stabilized rockets[J]. Journal of Spacecraft andRockets, 1970, 7(4):434439.

[14] Hurley M J, Jr, Carrie G W. Stage separation of parallel-staged shuttle vehicles: a capabilityassessment[J]. Journal of Spacecraft and Rockets, 1972, 9(10):764771.

[15] Su M W, Mullen C R, Jr. Plume impingement force during tandem stage separation at highaltitudes[J]. Journal of Spacecraft and Rockets, 1972, 9(9):715717.

[16] Subramanyam J D A. Separation dynamics analysis for a multistage rocket[C]. In: Kobayashi S(ed). Proceedings of the International Symposium of Space Science and Technology, Tokyo: AGNEPublishing, 1973, 383390.

[17] Kalesnikof K S. Dynamic separation[M]. 1977 (in Russian).

[18] Saxena S K. Upper stage jet impingement on separated booster[J]. Aeronautical Journal, 1979,616:7174.

[19] Lochan R, Adimurthy V, Kumar K. Separation dynamics of strap-on boosters[J]. Journal ofGuidance, Control and Dynamics, 1992, 15(1):137143.

[20] Lochan R. Dynamics of bodies separating from launch vehicles[D]. Ph D Dissertation. Departmentof Aerospace Engineering, Indian Institute of Technology, Kanpur, May 1993.

[21] Lochan R, Adimurthy V, Kumar K. Separation dynamics of ullage rockets[J]. Journal of Guidance,Control and Dynamics, 1994, 17(3):426434.

[22] Lochan R, Adimurthy V. Separation dynamics of strap-on boosters in the atmosphere[J]. Journalof Guidance, Control and Dynamics, 1997, 20(4):633639.

[23] Cheng S C. Payload fairing separation dynamics[J]. Journal of Spacecraft and Rockets, 1999,36(4):511515.

[24] Reubush D E, Martin J G, Robinson J S, et al. Hyper-X stage separation-simulation developmentand results[C]. In: 10th International Space Planes and Hypersonic Systems and TechnologiesConference, Kyoto, Japan, April 2001.


[25] Jeyakumar D, Biswas K K. Design and analysis of the stage separation system of a massive liquidrocket stage[C]. In: Proceedings of the International Conference on Modelling Simulation andOptimization for Design of Multidisciplinary Engineering Systems, Goa, India, 2003.

[26] Jeyakumar D, Biswas K K. Stage separation system design and dynamic analysis of ISRO launchvehicles[J]. Journal of Aerospace Sciences and Technologies, 2003, 55(3):211222.

[27] Jeyakumar D, Biswas K K. Stage separation dynamic analysis of upper stage of a multistage launchvehicle using retro rocket[J]. Mathematical and Computer Modelling, 2005, 41(8/9): 849866.

[28] Mitchell D H. Flight separation mechanism[R]. NASA-SP-8056, 1970.

[29] Logan J W. DSV-3E first-second stage separation analysis[R]. Rept SM-46446, Douglas Aircraftco, April 1965.

[30] Ball K J, Osborne G F. Space vehicle dynamics[M]. Oxford: Oxford University Press, 1967.

[31] Orlik-Rukemann K J, Iyengar S. Example of dynamic interference effects between two oscillatingvehicles[J]. Journal of Spacecraft and Rockets, 1973, 10(9):617619.

[32] Xue Yu. Separation between stages of multistage carrier rocket[R]. FTD-ID (RS) T-1143-83, Sept1983.

[33] Naftel J C, Wilhite A W, Cruz C I. Analysis of separation of a two-stage winged launch vehicle[C].In: 24th AIAA Aerospace Sciences Meeting, Reno, NV, Jan 1986.

[34] Papoulis A. Probability, random variables, and stochastic processes[M]. McGraw-Hill, 1991.

Date post:	06-Mar-2016
Category:	Documents
Upload:	anonymous-iqcujjoi
View:	40 times
Download:	5 times

00 S-593.pdf

Documents