PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Piecewise Bilinear Characteristics of Acoustic ModeIn Dual Pulse Solid Rocket Motor Combustion Chamber
Hao Le1; Yincheng WANG2; Jun CHEN1; Xiaoming SHI3
1

Shanghai Space Propulsion Technology Research Institute, China
2

3

Beijing Deep Blue Aerospace Technology Co.Ltd, China

Shanghai Electro-Mechanical Engineering Institute, China

ABSTRACT
Based on the telemetry data of flight tests, the axial abnormal oscillation with frequency doubling
characteristics occurred at the end of the first pulse motor’s operation while it was normal during the second
pulse. With detailed analysis, it was concluded that there was a resonance, the spacecraft structure and the
combustion chamber’s acoustic cavity spectra are qualitatively similar, exhibiting a unique peak at theresonance frequency. In order to study the acoustic modal characteristics of the dual pulse solid rocket motor
comprehensively, a standard model for estimating the acoustic mode of the motor was established by meansof simulation and experiment at room temperature. Moreover, based on the law of grain motion and the statefrozen method, 3D model of the acoustic cavity was rebuilt accurately, which was provided to the analysis.
According to the model above, the acoustic mode’s piecewise bilinear characteristics during motor’soperation was proposed completed. With the comparison, it was known that there was a good agreement
between simulation and flight test at the end of first pulse operation, which proved that the model established
was correct. Consequently, it is considered that the results in this paper can characterize the real change
process of the acoustic modes during the motor’s whole operation.
Keywords: Dual pulse solid rocket motor; Combustion chamber; Acoustic mode; Piecewise bilinear

1. INTRODUCTION
Recently, in order to improve the spacecraft’s control performance, viability and propulsive range,
dual pulse solid rocket motor has gradually become the main power device of spacecraft (1). Usually,
the motor consists of two burning chambers, separated by a bulkhead, designated as pulse separation
device which achieves the restarting operation and provides intermittent thrust. Actually, the pulse
separation device protects the propellant grain in the second pulse chamber against high temperatureand pressure impact during the first pulse operation. At initiation of the second pulse, pulse separation
device reliably opens for gas flow through the empty first pulse chamber and the nozzle to atmosphere.
Based on the telemetry data of flight tests, the axial abnormal high frequency oscillation occurredat the end of first pulse operation while it was normal during the second pul se. One of remarkablefeature of the data was the large number of resonant peaks observed. As many as 20 peaks can be seen
in one configuration over the 200Hz to 5kHz frequency range. In fact, the structural mode of spacecraft
remained basically unchanged at the above moments, but the acoustic cavity of the motor combustion
chamber changed greatly due to the complex structure of motor. Although several possible causes forthe observed vibrations were suggested, it seemed that the most likely cause was combustion -driven
acoustic wave in the motor. As shown in Figure1, the resonance between the spacecraft structure andthe combustion chamber’s acoustic cavity has occurred at a frequency of 225Hz. To date, thevibrations are not known to have prevented completion of the spacecraft’s mission but there is concernthat minor changes in one or more parameters which occasionally occur during a moto r’s production
history may increase the vibration level and consequently interfere with some vital function such as1

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guidance or thrust termination. Compared with traditional one pulse motor, dual pulse motor has more
complex acoustic structure, which makes it difficult to study its dynamic characteristic. However,
because of the complex situation, little insight had been gained concerning the mechanisms involved.

(a)Time evolution of the Axial overload(b) Frequency domain of the response signal

Figure-1 Vibration data of flight test
Therefore, a series of investigations was initiated to the acoustic characteristi cs of the interior of
the motor by some scholars. Browning carried out acoustic tests on the Minuteman II Stage III motor
and obtained the sound pressure distribution in the combustion chamber(2). Mathes carried out the
acoustic modal analysis of the 1/4 full scale model of the Poseidon second stage motor, and theexperimental data were in good agreement with the numerical simulation (3). Francois determined theacoustic mode of the test motor and estimated the stability range of the motor (4). Anthoine obtained
the first three order acoustic modes of the 1/15 scale model of Ariane-5 MPS P230 booster combustion
chamber through experiments(6).French took the standard stability prediction program SSP toestimate the tangential acoustic modes of motor(6). Nicoud proposed a numerical method for
determining the thermoacoustic modes of combustors, which was verified by two specific cases (7).
Zhang Xiangyu grained the acoustic modes of solid rocket motor by simulation and made somecomparison between the simulation and the flight test (8). Zhou Xinxin used acoustic resonancesimulation method to investigate the acoustic response of combustion chamber acoustic cavity (9).
Most attention was pay on the acoustic mode of traditional solid rocket motors under subscale orcold-flow condition, but not including the acoustic modal characteristics during the motor’s operation.
So far, there were few research on the acoustic mode of dual pulse solid rocket motor during working
process.
In this paper, numerical simulation under the state of the end of first pulse and second pulse
operation were carried out. The experiment was designed to determine the frequencies and structuralcharacteristic of standing acoustic wave in the motor. The first three axial acoustic natural frequencies
and mode shapes were obtained by the test under room temperature, which proved the finite element
model established was correct. Therefore, it was employed as standard procedure for analysis. Then,
combined with the law of grain motion and the "state frozen" method, the evolution law of the motor
acoustic cavity with time was presented, which leading to the accurate 3D model of the motor at
specific moments. By taking advantage of the model established above, the piecewise bilinear
characteristics of acoustic mode in dual pulse solid rocket motor was proposed, which has greatsignificance on the cooperative design of motor and spacecraft.

2. Theoretical Background
2.1 Physical hypothesis
In order to solve the wave equation in the combustion chamber of solid rocket motor conveniently,
the mass conservation, momentum conservation, energy conservation and equation of state in the
combustion chamber are linearized. The simple harmonic sound field in the combustion chamber canbe described by Helmholtz equation as follows (10):4661 2 P ( 2 / a0 2 ) P  0(1)Here,  2 is Laplace operator; P is sound pressure; ω is angular frequency; a0 is sound velocity.
Assuming that the acoustic space consists of a finite number of elements, the gas vibration equation
in a closed cavity can be obtained based on Equation (1).

 K     M P  0
2Here,  K  

m

e

m

(2)

e

  K  ；  M     M  ； m is the total number of gas elements, [K]e、[M]e
e1

e1

are the stiffness and mass matrices of the system respectively, they are all real symmetric matrices of n×n.
The necessary and sufficient conditions for the solution of Equation (1) are as follows, and it gives thenatural frequencies of each order of the enclosed cavity.

K  2 M   0(3)

The actual pressure oscillation in solid rocket motor combustor may be the result of combinationof one or more modes, or the superposition effect of high-order harmonic oscillation. These factors
lead to complex waveforms, and the main acoustic mode often controls the development trend of
oscillating combustion in combustor. For a typical cylindrical combust or chamber with radius R andlength L, the axial acoustic frequency is:na n   RgT
f a c o u s ti c 
2L
2L

（4）

Here, n is the acoustic mode number; γ is the specific heat ratio; Rg is the average gas constant; T
is the gas temperature; L is the length of the acoustic cavity.

3. Numerical simulation
3.1 Physical hypothesis
Considering the limitation of the test facility, the simulation analysis was carried out under thestatic conditions. The physical hypothesis was shown as follow:
1) Grain is a surface dense material without considering its damping effect.
2) The structural stiffness of motor case is relatively large, and the acoustic-solid coupling problem
is neglected.
3) The influence of gas flow in combustion chamber is not considered.
3.2 Finite element model
The fluid property density and sound velocity in the acoustic chamber are defined as follows:  1.225 kg / m3 , a  340 m / s .The three-dimensional model of the motor under first pulse andsecond pulse condition was established by using tetrahedral mesh, as shown in Figure 1 and 2. The
mesh size is controlled within 50 mm, which ensures the calculation accuracy. The three-dimensionalmodel of the chamber of one pulse combustor consists of 93 012 units while the model of the chamberof second pulse consists of 121 404 units (11).

Figure 1–The chamber acoustic field model of first pulse motor

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Figure 2–The chamber acoustic field model of second pulse motor
3.3

Simulation results

Based on LMS.virtullab , the estimated natural frequencies and mode shapes of the motor under
different conditions were obtained, as shown in Table1 and Figure 3 and 4.
Table 1–Results of the stimulation
StateFirst pulse

Second pulse(a)first order

Order

Result, Hz

1

69.13

2

138.173

206.99

1

47.49

2

92.13

3

145.20(b)second order(c)third orderFigure 3–Mode shapes of the first pulse motor(a)first order(b)second order(c)third orderFigure 4–Mode shapes of the second pulse motor

4. Test4.1 Test instrument
Figure 5 shows the test instrument which consists of horn, microphone, electric horn and conduit.
The electric loudspeaker horn is first extended from the nozzle into the combustion chamber as an
excitation source to generate sound field. The signal generator generates random white noise electric
signals. After power amplifier treatment, the electric signal is input into the electric loudspeakerhorn to produce sound field, which stimulates the acoustic oscillation in the combustion chamber.
Axial wave structural characteristics were determined by moving a microphone along the model axis.
The microphone could be used to measure on the model centerline to measure the pressure
distribution in a line parallel to the axis of symmetry, give a longitudinal pressure distribution near
the model wall. The instrumentation system provided means for detecting the pos ition of the moving
probe and for continuously recording the acoustic pressure as a function of position in the motor. At
the same time, the sound field level in the chamber is controlled by the power amplifier regulator.
The fluctuating pressure signal received by the microphone is transformed and amplified by the4663

microphone power supply and then input into the data acquisition system. After spectrum analysis of
the power spectral density of the data acquisition system, the combustion chamber acoustic moda lfrequency can be obtained.

Figure 5–Schematic diagram of the test facility
4.2 Test results
By fitting the response amplitudes of each measuring point, the first thre e order axial acoustic
modes were obtained, as shown in Figures 8 and 9. Comparison of simulation-based andexperimentally determined frequencies was made in Table 3.

(a)first order(b)second order(c)third orderFigure 8–Mode shapes of the first pulse motor(a)first order(b)second order(c)third orderFigure 9–Mode shapes of the second pulse motor
Table 3–Summary of the above results
StateFirst pulse

Second pulse

Order

Test, Hz

Simulation, Hz

Error,%

1

70.0

69.13

1.24

2

140.0

138.17

1.31

3

209.5

206.99

1.20

1

48.5

47.49

2.08

2

93.0

92.13

0.94

3

147.0

145.20

1.23

From table 2,it was clear that the test results showed good agreement with the corresponding ones
obtained from the FEM, the error between them was only 0.94%~2.08%, which proved the FEM
model established was reliable.

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5. Modal analysis of motor’s operation
5.1 Acoustic Mode Determination
According to the classical acoustics theory (12), it’s well known that the acoustics mode of
combustion chamber is only related to motor’s geometric variables and gas property. Based on the
acoustic modal test of motor at room temperature, the simulation model adopted in this paper is soreliable that it is capable of estimating the acoustic modal characteristics during the motor’s operation.
On the basis of test results, it was obvious that the acoustic modal frequency was generally large.
That means, its response period was far shorter than the solid propellant burning’s. Based on the above
analysis, some characteristic moments during working process were chosen by the "state frozen"
method which was generally considered from the time-varying system point of view as an operational
method, so the 3D model of the motor at specific time were rebuilt respectively (13).
Due to the steady cavity volume variation during the working process, 5 characteristic time points,
each interval of 2 seconds, were selected respectively. Then, instantaneous propellant grain shapes ofthe motor were obtained by the specialized software which could accurately calculate the scale changeof grain during the working process, so the 3D model of the motor at specific time were rebuilt, as
depicted in Figure 9.Finally, the complete 3D model of the motor during working process wasestablished, regarded as the basement for analysis.

(2s)

(6s)

(10s)

First pulse’s operation

(2s)

(4s)

(6s)

Second pulse’s operation
Figure 9–Time evolution of the motor’s grain scale
For the sake of simplicity, combined with the frequency doubling characteristics of the acoustic
modes, it is more effectively to study the change of the fundamental frequency of the acoustic modes
to illustrate the change of the acoustic modes during motor’s operation. Therefore, according to theabove 3D model at specific time, combined with the important attributes such as gas property, thestandard model is used to analysis the acoustic frequencies. However, the classical method neglects thenatural frequency variation caused by the structure change during its operation so that there is a certain
error for the actual frequency prediction of the motor. In order to emphasize the accuracy of thismethod, it was given in Figure 10, compared with the classical case.

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300

First pulse

Second pulse

Simulation

Frequency(Hz)Classcial
250

200

150
0

10

20

30

Time(s)Figure 10–Time evolution of the frequency
5.2 Results and discussions
As indicated by figure 10, the axial acoustic modes of the motor shows piecewise linearity during
the motor’s operation. The main reason is that the opening of the pulse separation device makes theacoustic cavity of first pulse chamber and second pulse chamber interconnected so that the axial
internal geometry of the combustor increase significantly, resulting in the frequency drop at the
beginning of the second pulse motor’s operation. Besides, it can be seen that the error between
simulation and test is less than 10% whereas the error between classical method and test is about 20%.
This might be explained by the fact that the simulation model is an accurate configuration of motor’s
acoustic cavity based on the law of grain motion. As a result, there is a great difference between themethod presented in this paper and classical one, whose change in the acoustic frequency almost keeps
a constant trend.
It is well known that when the solid rocket motor is working, all the burning surface moves along its
normal direction at the same speed, leading to the increase of inner diameter of the grain. At the
beginning of the first pulse motor’s operation, the initial inner diameter of the grain is close to that of
the nozzle. During the motor’s operation, the configuration of acoustic cavity changed from theoriginal slender tube to the cylindrical cavity, so the equivalent length of acoustic cavity decreased
gradually, which results in the slow increase of the fundamental frequency of acoustic mode.
During the period of second pulse, this is no longer the case. Actually, it was interesting to note thatthe fundamental frequency decreases monotonously with time while the degree of variation is greaterthan that of first pulse. These results suggest a link between the pulse separation device and acoustic
cavity. In fact, at the beginning of the second pulse motor’s operation, the sudden change section effectof the pulse separation device produces a large acoustic load affecting the refraction and reflection of
the acoustic wave in the combustion chamber, which reduces the efficiency of acoustic wave
transmission and increases the path. It can be considered that the equivalent length of th e acoustic
cavity increases gradually, so the fundamental frequency of the acoustic mode decreases
monotonously.
According to the standard model, the results of acoustic frequencies during operation are obtained.
The error between simulation and flight test at the end of first pulse is less than 10%. According to
the existing physical parameter errors and the influence of distributed combustion on the gas
temperature, it is considered that the present model is actuate and reliable to estimate the acoustic
modal characteristics during dual pulse motor’s operation.

6. CONCLUSIONSIn this paper, the acoustic investigation is a follow-on to phenomena seen in the motor, which
provide a valuable guide. In the present case, the acoustic modes of the dual pulse solid rocket motor
were investigated by numerical simulation and experiment. Comparison of experimental results withsimulation indicates that the simulation does a reasonably good job of predicting the acoustic

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characteristics of a cavity with complicated geometry under the end of first pulse and second pulse
motor’s operation. Furthermore, a procedure capable of tracking the actual modal characteristics
variation during motor’s operation was presented. The procedure took advantage of the law of grain
motion and “state frozen” method, then the 3D model of the motor during its operation was rebuilt
accurately, which was provided to the modal analysis, consequently, the standard model wasestablished.
Based on the standard model, the piecewise linear characteristics of motor’s acoustic modes were
grained and in deep analysis. Combined with the flight test, it is certainly that the method presented
shows a greatly improved prediction accurate about the axial acoustic modes determination in dual
pulse solid rocket motor combustion chamber. The result of such analysis provide a useful guide forstudy of motor instability as well as the cooperative design of motor and spacecraft.

REFERENCES
1. HOU Xiao,FU Peng,WU Yuan. Energy management technology of SRM and its development. Journalof Solid Rocket.2017; 40(1):1-6.
2. Browing S C,Krashin M,Thacher J H. Application of combustion instability technology to solid
propellant rocket motor problem. Journal of Spacecraft and Rockets.1972, 9(5): 293-294.
3. MATHES H B, PRICE E W. Methods for determining characteristics of acoustic waves in rocket motor.
Journal of Spacecraft and Rockets.1975;12(1): 39-43.
4. Francois Vuillot. Acoustic Mode Determination in Solid Rocket Motor Stability Analysis[J].Journal of
Propulsion and Power.1987;3(4):381-384.
5. Anthoine J,Planquart P,Olivari D. Cold flow investigation of the flow acoustic coupling in solid
propellant boosters. AIAA1998-0475.
6. French J C.Tangential mode instability of SRMs with even and odd numbers of slot. AIAA 2002-3612.
7. Nicoud F,Benoit L,Sensiau C,et al. Acoustic modes in combustors with complex impedance andmultidimensional active flames. AIAA 2007-426.
8. ZHANG Xiangyu,GAn Xiaosong,Gao Bo,et al.Research on the influence of the overload of the missileon the unstable combustion of SRM.The 33th session of the CSA;25-28 August 2016;Dan
Dong,China.2016.p.527-533.
9.ZHOU Xinxin,GAN Xiaosong,ZHANG Xiangyu,et al. Research on acoustic instability trend of solid
rocket motors based on acoustic resonance simulation. Journal of Solid Rocket.2019;42(2):143-148.
10.ZHANG Qi,ZHANG Wei-hong. Reserch about enclosure cavum acoustic characteristics. Structure &Environment Engineering.2008;35(6):36-39.
11.LE Hao,WANG Yingcheng,XUE Muyao,et al. Simulation Analysis and Experimental Study On
Acoustic Property of Dual Pulse Solid Rocket Motor’ Combustion Chamber. Aerospace
Shanghai.2019;36:35-42.
12.MA Da You. The Theoretical Basis of Modern Acoustics. Beijing, China; Science Press;2004
13.LE hao,WANG Lei,CHEN Jun,et al. Modal Simulation Analysis and Experimental Study on the
Working Process of a Dual Pulse Solid Rocket Motor. Fifteenth International Space Conference ofPacific-basin Societies;10-13July 2018;Montreal, Canada 2018.p.317-327.

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