Pennsylvania State University, University Park, PA 16802-2320.
A methodology centered around Passive Linear Stability Measurement (PSLM)concept is being developed to investigate dynamic response of rocket motors. A lin-earized analysis of the acoustic motions inside the rocket chamber is developed whichaccounts for propellant response functions, mean flow effects, effects of the nozzle, andvanishingly small two-dimensional waves at area discontinuities. In addition, experi-mental data are analyzed and contrasted to the proposed one-dimensional model. Itwas found that considering a classical homogeneous formulation the pressure-coupledresponse can be determined for the mode frequencies. If noise sources in the controlvolume are accounted for, then in theory it is possible to determine the pressurecoupled response function at all frequencies simply by measuring the pressure fluc-tuations at the head end of the motor. Nonetheless, this study revealed that theeffect of the pressure coupled response function on the pressure oscillations recordedduring stable operation of the rocket motor can be of the same order of magnitudeas that of the noise. Thus, large uncertainties could be associated with the pressurecoupled response function determined using this procedure.
NOMENCLATURE
A = port areaa = sound speedcv = constant volume specific heate = specific energyf = frequencyk = wave numberM = Mach numberm = mass flow rate per unit lengthN = number of pointsp = pressureP = burning surface perimeterR = gas constantRp = pressure-coupled responseRv = velocity-coupled responser = port radiusT = temperaturet = timeu = velocityV = volume flow velocityx = chamber abscissaZ = discontinuity acoustic impedanceα = growth constantβ = coefficient
∗Professor, Dipartimento di Energetica. AIAA SeniorMember.
†Ph.D., Department of Mechanical Engineering, Propul-sion Engineering Research Center. AIAA Member.
Combustion instability is one of the major fac-tors that strongly affects the design of solid rocketmotors. However, combustion instability is still un-predictable and not well understood. The lack ofunderstanding is a result of both the high costs in-volved in experimentally determining the stabilitymargin of a solid rocket motor and the inaccuracyof the methods available for measuring the pres-sure coupled response function of a solid propellant.
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39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit20-23 July 2003, Huntsville, Alabama
Thus, the inability to suppress undesired oscillatorybehavior of a solid rocket motor results from the un-availability of reliable and inexpensive experimentalmethods that allow the coiche of a particular pro-pellant and grain geometry that minimize pressureoscillations produced by combustion and flow pro-cesses.
Recently, Hessler and collegues1 proposed a newexperimental procedure to determine the stabilitymargin of solid rocket motors, known as PassiveLinear Stability Measurement (PSLM). This exper-imental method allows the determination of the sta-bility margin of a solid rocket motor without intro-ducing external disturbances. The PSLM methodis based on the fact that, due to the presence offorcing terms, small pressure oscillations exist atany mode frequency, although the mode consid-ered is a stable one. These pressure oscillationsare then either damped or augmented depending onthe acoustic characteristics of the particular modeconsidered and the coupling between the acousticsof the combustion chamber and the forcing terms.The relation between the acoustical properties of thechamber, the forcing functions, and the acoustic en-ergy gain/loss mechanisms can be expressed in termsof coupling integrals. Thus, two different approachesto the problem can be envisaged. If the couplingintegrals, the energy gain/loss mechanism, and theforcing functions are known, either theoretically orfrom experimental measurements, it is possible topredict the behavior of the pressure oscillation insidethe rocket motor. On the other hand, by measuringthe pressure oscillations inside the chamber duringnormal operation of a rocket motor and from theknowledge of the coupling integrals it is in principlepossible to determine the forcing functions and thegain/loss mechanisms of acoustic energy.
A joint research program to develop the PSLMwas started in the year 2000 at the Politecnico di Mi-lano and at the Politecnico di Torino. The first stageof this study consisted in experimentally measuringthe frequency response of a subscale rocket motorwithout either mean flow or combustion.2 The testmotor considered was a constant-area cylinder fittedwith an outward burning cylindrical grain and ter-minated with a conical nozzle. In addition, in orderto explore the influence of the chamber geometryon its acoustic characteristics, piecewise constant-area mock-ups were tested as well. At the sametime, an improved one dimensional acoustic modelof the same rocket chamber was developed at the Po-litecnico di Torino. Results from this study clearlyshowed that a purely one-dimensional model was un-able to accurately predict the acoustic response of achamber, even when a extremely simple piecewiseconstant-area geometries were considered.3 Also,it was shown that by accounting for vanishing two
Fig. 1 Schematic of the sub-scale rocket model(not to scale).
dimensional acoustic waves at area discontinuities,excellent agreement with the experimental resultswas obtained.4
This paper is a follow-up to the previous one4 andis aimed at providing a more thorough descriptionof the acoustic field that is typically encounteredin solid rocket chambers. To this end, the one di-mensional approach is retained but the conservationequations are rewritten to account for the contribu-tions of a non-zero axial mean flow and injection ofmass from the wall of the motor due to the com-bustion of the solid propellant. In addition effectsof non-planar vanishing waves at the area disconti-nuities along with the nozzle admittance have beenconsidered. Experimental data have been analyzedand contrasted to the results obtained using the pro-posed one-dimensional model.
EXPERIMENTAL RESULTS
Experimental data were provided to the authorsby Fiat Avio.5 These data included both low fre-quency and high frequency head-end pressure tracesobtained during actual firings of sub-scale rocket mo-tors. The geometry of these motors at 0% burn isshown schematically in Fig. 1. The motor hardwareis a constant area cylinder fitted with an outward-burning, unrestricted-end cylindrical grain charac-terized by a 3:1 length to diameter ratio at ignitionand a 2:1 length to diameter ratio at burnout. Thedesign of the grain is such that the burning sur-face propellant is roughly constant during a firing,so that the operating pressure of the motor shouldremain constant for the duration of a test. Lowfrequency pressure data were measured by a Taber206/SA pressure transducer and were acquired at afrequency of 1000 Hz. Conversely, high frequency
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0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4
-0.02
-0.01
0
0.01
0.02
High-frequencypressure trace
Low-frequencypressure trace
Time t (s)
Nor
mal
ized
pres
sure
Nor
mal
ized
pres
sure
Fig. 2 Low-frequency and high-frequency pres-sure traces recorded during a firing of the sub-scale rocket motor. For both traces the pressureis divided by the average chamber pressure.
pressure fluctuations were obtained by means of aKistler model 601A pressure transducer placed atthe head-end of the motor. The output from thepressure transducer was fed into an Endevco 2721Bcharge amplifier and sampled at 48 kHz by a 16 bitdata acquisition card.6 For a given propellant for-mulation three sets of tests at low, medium and highchamber pressure were performed. In the presentstudy we focused our attention on three data recordscorresponding to different propellant formulationsobtained for the same chamber pressure. Each ofthe propellant formulations delivered the same aver-age burning rate, but had slightly different dynamiccombustion characteristics.
An example of the pressure traces recorded bythe low-frequency and high frequency pressure trans-ducers are given in Fig. 2. Both the low and highfrequency pressure traces were divided by the av-erage operating chamber pressure of the subscalerocket motor. As seen from the low-frequency trace,after ignition the mean pressure rapidly increasedfrom atmospheric to the target chamber pressure,and remained roughly constant until burnout. Slightchanges in the mean pressure were related to changesin the propellant burning surface and/or propellantcomposition encountered during the test. All of thehigh frequency pressure data traces available to theauthors were plagued by two problems, namely 50Hz line noise and saturation of the charge amplifierduring ignition and burnout transients, which madea good portion of the data unusable for the analysis.
In order to be able to extract pressure amplitudeand phase information from these traces, data pre-processing aimed at eliminating unusable parts ofthe data, low frequency trends and 50 Hz line noisewere necessary. The first step in data preprocessing
consisted of clipping the ignition and burnout tran-sients. Next, the pressure traces were divided intosegments consisting of N = 8192 points, and cor-responding to 0.1707 s of data. Upon inspection itwas found that the records presented low frequencycomponents with a characteristic period significantlylarger than their lengths. As suggested by Ben-dat and Piersol7 these components were filtered outby fitting a polynomial to the experimental dataand subtracting it from the original pressure trace.The records thus obtained still presented a strong50 Hz line noise component that was eliminated bymeans of an “ignore A/C” filter.6 This filteringtechnique consists of performing an FFT on eachdata record and substituting the magnitude of the50 Hz component of the spectrum with the averageof the magnitudes of nearby points while leaving thephase unchanged. Since the 50 Hz peak was indeedcomposed of three points, the magnitude of the twopoints adjacent to the 50 Hz peak was first calculatedby averaging the magnitudes of the two points imme-diately preceding and following them, and then themagnitude on the 50 Hz peak itself was calculatedby averaging the magnitudes of four adjacent points.This completely eliminated the 50 Hz line noise fromthe power spectrum of the pressure. An inverse FFTwas then used to reconstruct the signal in the timedomain. Then, an estimate of the power spectrum ofthe signal was computed by zero padding the recordsin the time domain with N zeroes, by performinga 2N point FFT, and by frequency averaging theFourier transform using a Hanning window so thatfrequency leakage could be minimized. From thepressure power spectrum we were then able to ex-tract amplitude and phase of pressure oscillations asa function of frequency. Figure 3 shows a waterfallplot for the pressure power spectral density around50% burn for one of the propellant formulations con-sidered. It is seen that the pressure fluctuation
Time t (s)
Frequency f (Hz)
Normalized spectral density
Fig. 3 Waterfall plot of the head-end pressurepower spectral density around 50% burn.
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spectrum exhibits a peak around 1600 Hz, whichcorresponds to the first longitudinal mode of thechamber. Additional peaks are present at higherfrequencies. As expected in the case of an acous-tic enclosure in which stochastic noise sources arepresent, the pressure power spectrum is composedof several broad peaks centered around the modalfrequencies of the combustion chamber.
IMPROVED 1-D MODEL
Homogeneous One-Dimensional FormulationThe purely one-dimensional model presented in
previous papers was improved to account for gas in-jection from the burning surface of the solid propel-lant and for the effect of the mean flow Mach num-ber. The governing equations for one dimensionallongitudinal waves were obtained starting from theconservation equations of mass, momentum and en-ergy written for a thermodynamically and caloricallyperfect gas, and a constant cross-section area cylin-drical chamber with mass injection from the walls:
1A
∂
∂t(ρA) +
∂ρu
∂x=
mb
A
ρ∂u
∂t+ ρu
∂u
∂x+
∂p
∂x= −u
mb
A
ρ∂e
∂t+ ρu
∂e
∂x=
mb
A
(hb − e +
u2b
2+
u2
2
)− p
∂u
∂x
The energy equation can be further elaborated byremembering that e = cvT, by assuming isothermalinjection, and by multiplying the equation thus ob-tained by R/cv and subtracting RT times continuityfrom it. This yields an equation for pressure and ve-locity:
∂p
∂t+ u
∂p
∂x+ γp
∂u
∂x=
R
cv
(u2
b
2+
u2
2
)mb
A+ a2 mb
A
Using a linear approach the variables are written assum of an average part plus a fluctuation which isassumed to be small compared to the average quan-tities. By linearizing the equations and subtractingthe conservation equation for average quantities, thefollowing equations for pressure and velocity fluctu-ations are obtained:
ρ∂u′
∂t+
∂p′
∂x= −ρ′u
∂u
∂x− ρ
∂uu′
∂x− u′mb
A− m′
bu
A
∂p′
∂t+ u
∂p′
∂x+ γp
∂u′
∂x= −u′ ∂p
∂x− γp′
∂u
∂x+ a2 m′
b
A+
+R
cvA
[(u2
b
2+
u2
2
)m′
b + mbubu′b + mbuu′
]
It is now assumed that separation of variables can beapplied so that pressure, velocity and injected mass
fluctuations can be written as:
p′ = p (x) eiakt
u′ = u(x)eiakt
m′b = ˆmb(x)eiakt
By substituting these definitions into the previousequations, assuming that the pressure fluctuationsare isentropic, i.e.:
ρ′ =p′
a2
and considering that
u′b =
m′b
ρP− ρ′
ρub
the following equations for the variables p (x) andu(x) are obtained:
Q(x)d
dx
[up
]+ B(x)
[up
]= C (x) ˆmb (1)
where the coefficient matrices Q(x), B(x), and C(x)have the following form:
Q(x)=[
ρu 1γp u
]
B(x)=
[ρdu
dx + mb
A + iakρ uc2
dudx
dpdxu − γ−1
A mbu γ dudx + γ−1
A
mbu2b
ρc2 + iak
]
C (x)=
[− u
Aγ−1A
[(ub
2
2 + u2
2
)+ mbub
ρP
]+ a2
A
]
Assuming that
ˆm = D(x)[
up
]
where
D(x) = m
[Rv/aRp/p
]
Eq. (1) can be rewritten as:
d
dx
[up
]+ E(x)
[up
]= 0 (2)
with
E(x) = Q−1(x)B(x) − Q−1(x)C(x)D(x)
Although these equations cannot be solved analyt-ically, an approach similar to that used by the au-thors in a previous paper is employed in this case aswell. This approach is detailed in the next section.
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0
0.5
1.0
0 0.5 1.0
f = 1500 Hzα = 0
Rv
= 10 + i0
5 + i0
0 + i0
Normalized chamber abscissa ξ
Nor
mal
ized
pres
sure
ampl
itude
δ
Fig. 4 Changes in standing wave amplitude dueto changes in real part of Rv (Rp = 0 + i0).
Boundary ConditionsThe motor is composed of several constant cross-
section area elements and the nozzle. In the constantelements the equations can be solved for each partof the motor separately. The interaction betweenthe different parts is then accounted for by enforc-ing appropriate boundary conditions at the junctionbetween two adjacent elements. Since non-planarvanishing waves have been shown to be importantfor accurately predicting the frequency of a mockupmodel of the current chamber, their effects are ac-counted for as well.4 This is done by introducing anappropriate impedance, Zj , at the j-th area changeand by writing the boundary conditions for acousticpressure and acoustic velocity at the area disconti-nuity as:
pj (Lj) − pj+1 (0) = ZjVj j = 1, m − 1
uj (Lj) Aj = uj+1 (0) Aj+1 j = 1, m − 1
where the volume flow velocity Vj is:
Vj = Ajuj (Lj) = Aj+1uj+1 (0)
and the area discontinuity acoustic impedances fora flow expansion (rj+1 > rj) is:
Zj = ρaMj
πr2j
(λ4 − 1
)+
iωρ
rj
83π2
HT (λ) ; λ =rj+1
rj
(as suggested by Peat8 a contraction can be studiedas a “reversed” expansion ). In order to evaluateHT (λ) an infinite system of linear algebraic equa-tions needs to be solved to determine the infinitenumber of coefficients that appear in its definition.A detailed description of the method for calculat-ing the function HT (λ) in absence of mean flow was
given in a previous paper and is used here as well.4
Early calculations8 showed that, for Mach numberssuch as those encountered in this study, the real partof HT is hardly affected by Mach number, and thatits imaginary part is negligible compared to the realpart. Therefore, HT (λ) is evaluated assuming thatM = 0 and its imaginary part is assumed equal tozero.
In addition to specifying lumped impedances atcross section area discontinuities, boundary condi-tions at the motor head-end and at the nozzle in-let section must be given as well. At the motorhead-end, a zero acoustic velocity is enforced. Theimpedance of the nozzle can be estimated assumingthat the short choked nozzle approximation holdstrue. In this approximation the nozzle has onlya damping effect on the acoustic oscillations insidethe chamber, i.e., the imaginary part of the nozzleimpedance is equal to zero. The magnitude of thedamping provided by the nozzle is a function of theflow Mach number at the inlet of the nozzle, and isgiven by:
ASN =u′
a
γp
p′= Mi
γ − 12
In certain cases this impedance has been shownnot to be sufficiently rigorous to accurately predictmode frequencies and mode shapes of a combustionchamber.9 In these cases a fully three-dimensionalanalysis of the acoustic motions inside the nozzle isnecessary to evaluate the nozzle impedance. In orderto explore the effects of the nozzle on the acous-tic motions inside the rocket chamber, in the cur-rent study nozzle impedances obtained using boththe short nozzle approximation and the fully three-dimensional approach proposed by Bussi et al.9 areused and contrasted.
Parametric studyThe knowledge of the mean quantities and their
derivatives, along with the values of the regressionrate, combustion gas properties, and velocity andpressure coupled response functions is needed tosolve the system of equations Eq. (2). Mean pres-sure, mean Mach number and their derivatives withrespect to the axial coordinate are calculated accord-ing to the formulas given in the Appendix.
A parametric study was performed by changingthe real and/or imaginary part of each of the re-sponse functions while maintaining the other fixed.If the boundary condition imposed by the nozzleis not taken into account, and the values of ω andα are assigned, a simple forward integration of theequations, starting from the head-end, yields pres-sure and velocity wave amplitudes as a function ofthe non-dimensional axial coordinate ξ. Figure 4is an example of the results obtained during this
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1600
1650
1700
1750
0 2 4 6 8 10
-200
-100
0
100
200
αL1
fL1
Coefficient β
Mod
efr
eque
ncy
f L1(H
z)
Gro
wth
cons
tant
α L1(s
-1)
Fig. 5 Influence of Rv = β + iβ on the frequencyand growth constant αL1 of the first longitudinalmode(Rp = 0 + i0).
parametric study. These results were obtained as-suming that ω = 2πf , where f = 1500Hz, α = 0,and Rp = 0 + i0. The presence of vanishing two di-mensional waves at area discontinuities was includedas well. Similarly to results obtained by other au-thors,11 it was seen that the shape of pressure wavewas affected by Rv more than it was affected by Rp.In order to ascertain the effects of two-dimensionalvanishing waves at area discontinuities, these stud-ies were performed both neglecting and includinglumped impedances that account for the presenceof two-dimensional acoustic motions at area discon-tinuities. In contrast to results previously observedfor cold flow simulations,4 the results show that aminimum change in mode growth constant and modefrequency is obtained.
1600
1650
1700
1750
0 2 4 6 8 10
Rp
= 0 - iβ
Rp
= 0 + iβ
Rp
= β + i0
Coefficient β
Mod
efr
eque
ncy
f L1(H
z)
Fig. 6 Effect of the real and imaginary partof the pressure-coupled response function onthe frequency of the first longitudinal mode(Rv = 0 + i0).
On the other hand, when imposing the nozzle ad-mittance as a boundary condition, the values of themodal frequencies, i.e., fn, and the growth constantof each mode, i.e., αn can be found solving for thecorresponding boundary value problem (BVP). TheBVP is solved by the means of a numerical procedurebased on Newton’s method.12 Tentative values areassumed for the unknown head-end pressure (imag-inary and real part) and are progressively correctedto satisfy the prescribed boundary conditions. Ac-cording to the present formulation, the magnitudeof the acoustic pressure at head-end does not affectresults and can be arbitrarily assumed.
Figure 5 shows the effect of the velocity coupledresponse function Rv = β + iβ on the dampingand frequency of the first longitudinal (1L) acousticmode of the combustion chamber obtained assumingthat Rp = 0+ i0. One can see that, in clear contrastwith the results obtained neglecting the presence ofthe nozzle, Rv has a negligible effect both on fL1
and on αL1. Therefore, in the remainder of thisstudy, the velocity-coupled response function Rv isassumed equal to zero.
Figures 6 and 7 show the effect of the pressurecoupled response function on the growth constantand frequency of the 1L mode of the chamber as-suming that Rv = 0 + i0. One can see that the realpart of the pressure coupled response function hasa larger effect on the growth constant αL1, and anegligible effect on the mode frequency. Conversely,the imaginary part of Rp affects the mode frequencywhile leaving almost unchanged the mode growthconstant.
Simulations were also performed to establish theeffect of a more rigorous three-dimensional treat-
-200
-100
0
100
200
0 2 4 6 8 10
Rp
= 0 - iβ
Rp
= 0 + iβ
Rp
= β + i0
Coefficient β
Gro
wth
cons
tant
α L1(H
z)
Fig. 7 Effect of the real and imaginary partof the pressure-coupled response function on thegrowth constant αL1 of the first longitudinalmode (Rv = 0 + i0).
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ment of the acoustic motions inside the choked exitnozzle.9 The results obtained showed that, whena three-dimensional approach is employed insteadof the short nozzle approximation, a slight increasein the damping effect of the nozzle and a shift ofthe mode frequencies of the combustion chamberis observed. Since mode frequencies obtained us-ing the short nozzle approximation are very closeto those determined experimentally, the effect of athree-dimensional treatment of the exit nozzle wouldbe to increase the imaginary part of the pressurecoupled response function.
These results suggest that one can determine thepressure coupled response function of a solid pro-pellant by measuring the growth constant and thefrequency of the acoustic modes of the combustionchamber during stable operation. Conversely, if thereal part of the pressure coupled response functionis known, the stability margin of the motor can bedetermined.
are not forced motions.13 Nonetheless, pressuremeasurements performed during stable operation ofa rocket motor reveal the presence of stochastic forc-ing noise sources which appear in the pressure au-tospectrum as non-null pressure oscillations at anyfrequency. Thus, as far as the combustor dynamicresponse is concerned, the rocket chamber can bethought as composed by an infinite series of dampedoscillators driven by broad-band forcing terms. Ifthese forcing terms are given and the combustor re-sponse is measured, one can in principle derive thepropellant response function at any frequency. Inorder to determine the feasibility of this approach,one should measure the non-acoustic forcing terms,and then quantify the combustor dynamic sensitivityto the propellant response function. The presence offorcing functions is taken into account assuming thatthe right-hand term in Eq. (2) is not null, i.e.
d
dx
[up
]+ E(x)
[up
]= F (x) (3)
where
F (x) =[
νu 00 νp
]Q−1(x)C(x)
If the non-acoustic terms are either measured orestimated, and Rp and Rv are given, then one candetermine the head-end pressure for any given valueof the angular frequency ω. Figure 8 shows a com-parison of the results obtained using such an ap-proach, assuming that νu = νp = 0.02 and thatRv = Rp = 0+i0, and compares them to experimen-tally measured pressure oscillations. It is seen from
this diagram that a good qualitative agreement is ob-tained between experimental and calculated pressureamplitudes at the head-end. In order to establishwhether this approach allows determining the pres-sure coupled response function of the propellant, asensitivity study was performed. The results of thisstudy showed that the effect of the pressure coupledresponse function can be of the same order of magni-tude as that of the noise present in the system. Un-fortunately, this represents a major drawback of thePSLM procedure since it implies that large uncer-tainties might be associated with the inferred valuesof Rp.
CONCLUSIONS
An improved one dimensional analysis for de-termining the dynamic combustion characteristicsof subscale rocket motors has been developed andcontrasted to experimental pressure measurements.The theoretical model accounts for mass injectionfrom the side walls of the rocket chamber due to thecombustion of the solid propellant, two-dimensionalnon-planar vanishing waves at area discontinuities,and the presence of a choked nozzle. If the presenceof noise sources in the control volume is neglected,then a classical acoustic problem is obtained whichallows the determination of the growth constantsand frequencies of each acoustic mode. Conversely, ifthe growth constants and frequencies of the acousticmodes are known the propellant response functionscan be estimated at these same frequencies. Thefirst part of this study suggested that the effect oftwo dimensional vanishing non-planar waves at areadiscontinuities was negligible. This was a majordifference with the results obtained for a cold flowtheoretical model of the same combustion chamber.
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Moreover, it was shown that when the presence ofthe nozzle was accounted for by imposing the ap-propriate boundary condition, the pressure coupledresponse function was solely responsible for changesof the growth constants and of the mode frequen-cies. A three-dimensional treatment of the acousticmotions in the choked nozzle affected the mode fre-quencies and, to a lesser extent, the mode growthconstants.
When the presence of forcing terms is accountedfor in the equations, and the propellant responsefunctions are known, the pressure fluctuations atthe head-end of the motor can be calculated. Con-versely, if the noise level is experimentally deter-mined and the amplitude of pressure oscillationsversus frequency are known, then the response func-tions can be estimated at any frequency. Unfortu-nately, the results obtained in the second part ofthis study suggested that the effect of the pressurecoupled response function on pressure oscillationsrecorded during stable operation of the rocket motorwas of the same order of magnitude as that of thenoise. This in turn suggested that large uncertain-ties could be associated with the pressure coupledresponse function determined using this procedure.
ACKNOWLEDGEMENTS
We would like to acknowledge support for thiswork from the MIUR (Ministero dell’Istruzione, Uni-versita e Ricerca). The authors also thank FiatAvio-BPD Comprensorio di Colleferro, Rome, formaking available the experimental data presented inthis paper.
APPENDIX: MEAN
PRESSURE AND VELOCITY
For the special case of a constant port area rocketmotor, if friction, erosive burning and effect of pres-sure variations on the regression rate are neglected,mean pressure and velocity along the axis are readilyevaluated.14 If the perfect-gas law is used along withthe momentum equation applied to an incrementalcontrol volume along the grain axis one obtains
p(X) = 0.5p1
[1 +
√1 − KpX2
](A-1)
where
Kp = 4RTc
(PρbrX
p1A
)2
(A-2)
being p1 the mean pressure level at the beginning ofthe grain where X = 0. If it is assumed that thetemperature along the chamber axis is uniform andequal to Tc, one can write:
dp
dx= −0.5p1
KpX√1 − KpX2
(A-3)
and
du
dx= RTc
PρbrX
p2A
(p − X
dp
dx
)(A-4)
By integrating these equations the mean pressureand the mean Mach number versus the axial coordi-nate x can be evaluated.
REFERENCES1Hessler, R. O., “Passive Linear Stability Measurements,”
JANNAF Combustion Meeting, West Palm Beach, FL, Octo-ber 1997.
2Hessler R. O. et alii, “Frequency Response of a SubscaleModel Rocket Motor,” 5th International Symposium on Spe-cial Topics in Chemical Propulsion, Stresa, Italy, June 2000.
3Hessler R. O. et alii, “Acoustic Response of a ModelRocket Chamber,” Second European Conference on LauncherTechnology: Space Solid Propulsion, Rome, Italy, November2000.
4Pastrone, D., Saretto, S.R., and Vassallo, E., “AcousticAnalysis of a Subscale Rocket Motor,” AIAA Paper 2001-3867, July 2001.
5Annovazzi A., private communications, 2002.6Tion, C., Stramezzi, F., “ Sviluppo e applicazione di
un’analisi spettrale di stabilita per motori a propellentesolido”, Thesis, Politecnico di Milano, Jul. 2002.
7Bendat, J., S., Piersol, A., G.,“ Random data : analysisand measurement procedures”, Wiley, 2000.
8Peat, K. S., “The Acoustical Impedance at Discontinu-ities of Ducts in presence of a Mean Flow,” Journal of Soundand Vibration, Vol. 127, No. 1, 1988, pp. 123-132.
9Bussi, G., Colasurdo, G. and Pastrone, D., “Nozzle Ef-fects on Linear Stability Behaviour of Combustors,” PaperISABE 93-7044, September 1993.
10Culick, F. E. C. and Yang, V., “Stability Prediction inRocket Motors,” Nonsteady Burning and Combustion Stabil-ity of Solid Propellants, Progress in Astronautics and Aero-nautics, Vol. 143, AIAA, 1992, pp. 759-761.
11Micci, M. M., Caveny, L. H., and Sirignano, W. A.,“Linear Analysis of Forced Longitudinal Waves in Rocket Mo-tor Chambers,” AIAA Journal, Progress in Astronautics andAeronautics, Vol. 19, No. 2, 1981, pp. 198-204.
12Colasurdo, G., and Pastrone, D., “Indirect Oprimiza-tion Method for Impulsive Transfer,” AIAA Paper 94-3762,Aug. 1994.
13Culick, F. E. C., “Combustion Instabilities: MatingDance of Chemical, Combustion, and Combustor Dynamics,”Paper AIAA 2000-3178, July 2000.
14Hill, P., Peterson, C.,Mechanics and Thermodynamics ofPropulsion, Addison Wesley, 1992, 2nd edition, pp. 604-605.
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American Institute of Aeronautics and Astronautics
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