On the Operational Modal Analysis of Solid Rocket Motors

Sebastiaan Fransen1, Daniel Rixen2, Torben Henriksen1, Michel Bonnet3 1European Space Agency, ESTEC, P.O. Box 299, 2200 AG Noordwijk, The Netherlands, EU

2Delft University of Technology, 3mE, Mekelweg 2, 2628 CD Delft, The Netherlands, EU 3European Space Agency, ESRIN, P.O. Box 64, 00044 Frascati, Italy, EU

ABSTRACT: ESA’s new small launcher – VEGA – has been designed as a single body launcher with three solid rocket motor stages and an additional liquid propulsion upper module used for attitude and orbit control, and satellite release. In order to verify the performance of the solid rocket motors, all of the motors are tested in static firing tests on a test bench. In the frame of the correlation of the solid rocket motor mathematical models, an operational modal analysis tool was developed that is based on the Least Squares Complex Exponential method. The tool allows the computation of experimental poles and modeshapes from the accelerometer data recorded during a firing test. Convergence can be verified by means of the classical stabilization diagram and by the reconstruction of the correlation functions on the basis of the stable poles. Introduction In order to verify the thrust performance of the solid rocket motors of ESA’s new small launcher VEGA, static firing tests are conducted. In such test the motor is suspended on a test bench and ignited. Besides the measurement of the thrust by a loadcell, also other performance parameters are recorded such as temperatures, pressures and vibratory accelerations of the motor case. In figure 1 the firing test of VEGA’s third stage is depicted, for which the test bench in Sardinia (Italy) is used.

Figure 1: Firing test of VEGA’s third stage solid rocket motor (Z9) in Sardinia

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

As the finite element models of the motors shall be dynamically correlated, before using them in a launcher-satellite coupled dynamic analysis, it is essential to extract the modal information from the test measurements as accurately as possible. For this purpose an operational modal analysis tool was developed in the MATLAB environment. The implemented methodology is based on the Least Squares Complex Exponential (LSCE) method [1] and enables the generation of stabilization diagrams and the recovery of wireframe modes. Besides these standard features, a solution was implemented to perform operational modal analysis in the presence of harmonic excitations [2]. Harmonic excitations in combustions chambers of solid rocket motors are a well-known phenomenon that could jeopardize the convergence of the poles in the vicinity of those harmonic frequencies. Finally, the tool was completed with a feature that enables the detection of the most dominant modes and the reconstruction of the original correlation functions on the basis of those modes. In order to demonstrate the capabilities of the tool, the operational modal analysis of one of the solid rocket motor firing tests is discussed. LSCE Method In the LSCE method [1], the correlation functions between the various accelerometer outputs are used as an input for the computation of the modal characteristics of the structure. The correlation function between the

response signals i and )(tRij

j at an instant of time t is equal to the response of the structure at i due to an impulse at j , see figure 2.

Figure 2: Correlation function between signals i and j (impulse response) Assuming the damping to be small, the correlation function is given by a summation of N decaying sinusoids:

)sin()(1

rdr

tN

rdrr

rjriij tem

AtR rr

(1)

Each modal contribution is characterized by the multiplication of several modal parameters indexed by the modal subscript r: Modal parameter ri is the modeshape displacement at DOF i , is the input intensity at DOF rjA j ,

r is the modal damping ratio, r is the natural frequency, r is the phase angle, is the generalized mass, and is the damped eigenfrequency given by:

rmdr

21 rr

dr (2)

It is evident that the modal parameters can be computed from eq.(1), once experimental correlation functions

between various accelerometer outputs on the structure are available. In order to explain how the modal parameters are exactly solved, we will first write eq.(1) in terms of complex modes:

)(tRij

white noise input

correlation function or impulse response ijR

ii i

ij

ii

N

rrij

tksN

rrij

tksij CeCetkR rr

1

*

1

*

)( (3)

where the complex eigenvalue is given by, rs

21 rrrrr is (4) Expressing eq.(3) in terms of complex conjugate forms we obtain:

N

rrij

tksij CetkR r

2

1

)( (5)

A polynomial of the order ( ) – known as Prony’s equation – exists of which N2 Nk 21 tsre are roots (number of timesteps equals the number of roots):

0)2()12(122

210

tNstNs

Ntsts rrrr eeee (6)

or,

0212

122

210

NtsNts

Ntsts rrrr eeee (7)

or,

0212122

21

10

N

rN

rNrr VVVV (8) Before we can solve the roots we first need to determine the coefficients rV k (note that 12 N ). For this purpose we multiply the correlation functions at time instant with the coefficient k k and superimpose these values for (a summation over the time co-ordinate): Nk 20

N

r

N

k

krkrij

N

k

N

rrij

krk

N

k

N

rrij

tkskij

N

kk VCCVCetkR r

2

1

2

0

2

0

2

1

2

0

2

1

2

00)( (9)

Note that at least equations shall be written to solveN2 120 N . By the application of a time shift strategy, i.e. by starting at successive time samples, a linear system of equations can be build to solve the coefficients k :

Nnij

NnijN

nij

nij RRRR

21212

110

(10) where and . The above equations can also be written as: kijij RtkR )( Ln 1 ijij RR ' (11) Where is known as the Hankel matrix. Assuming we have R p response stations of which are reference stations, the number of successive time samples required to solve the coefficients

qL k follows from:

NqqqpL 22

)1(

(12)

Having only one reference station, i.e. , the required number of successive time samples is given by: 1q L

pNL 2 (13)

In order to stay within the time window T with sample time , the number of successive time samples shall be limited to:

dT L

NdTTL 2 (14)

Figure 3: Stabilization Diagram

Analysis Settings Value

Time span of correlation functions T =2s Sample time dT =0.0005 s Max Polynomial Order Prony’s Equation N =130 Number of time steps in timeshift window L =3400 (1.7 s) Number of sensors p =18 Number of reference sensors q =1

Table 1: Analysis settings

The system of equations takes the following form for p response stations of which are reference stations: q

qpqp R

RR

R

RR

'

''

12

11

12

11

(15)

Having solved the coefficients from the system of equations given by eq.(14) in a least square sense using pseudo-inverse techniques, we can now solve the roots from the polynomial (Prony’s equation) given by eq.(8). By variation of the polynomial order of Prony’s equation, a so-called stabilization diagram can be constructed, which helps to identify the stable modes.

rV

As an example case throughout this paper we will discuss the operational modal analysis conducted in the frame of the qualification firing test of the 1st stage of the VEGA launcher. This test was conducted at Kourou Spaceport in French Guiana in December 2007. The settings for this operational modal analysis are listed in table 1. The resulting stabilization diagram is shown in figure 3. Harmonics In the LSCE method the excitation is assumed to be random white noise. In case the excitation also includes harmonics, then those harmonics will be identified as modes with negligible damping. As those harmonic modes potentially could disturb the identification of the structural modes, especially when the harmonic and structural mode are close in frequency, one could include them as predefined poles in the solution sequence. This will improve the quality of the true poles of the structural modes [2]. From eq.(4) we can see that for pure harmonics, i.e. zero damping, the complex eigenvalue equals rr is . As such we have two extra roots of Prony’s polynomial, namely , which are roots of eq.(8). This means we can write two extra time signals (correlation functions) in accordance with eq.(10):

)sin( ti r)cos( teeV rtits

rrr

)2cos()2sin(

)12(cos)cos(1)12(sin)sin(0

12

1

0

tNtN

tNttNt

r

r

N

rr

rr

(16)

These two independent linear equations for the coefficients must be satisfied in order to represent the harmonics in the time signals. Let us assume that m harmonic frequencies in the frequency range of interest exist. Adding the linear equations (16) to the linear system defined by eq.(15), and assuming only one reference station (i.e. ), we get: 1q

12

2

2

12

1

0

111

111

2212221

121

21

121

01

)12(cos)2(cos)12(cos1)12(sin)2(sin)12(sin0

)12(cos)2(cos)12(cos1)12(sin)2(sin)12(sin0

N

m

m

mmm

mmm

NLp

mLp

mLp

Lp

Nmm

b

b

tNtmtmtNtmtm

DBtNtmtmtNtmtm

RRRRCA

RRRR

)2(cos)2(sin

)2(cos)2(sin

1

1

12

21

tNtN

FtNtN

RE

R

m

m

NLp

N

(17) In symbolic form we can write :

1)122(2

)22()12(1

)2( LpxmxNmNLpxmxmLpxEbCbA

(18)

and

12)122(2

)222()12(1

)22( mxmxNmNmxmxmmxEbDbB

(19)

From eq.(18) we can solve : 1b 211 bDFBb (20) Substituting in eq.(17) yields : FBAEbDBAC 121 (21)

From eq.(21) can be found as a least square solution. The coefficients 2b 1b can then be solved from eq.(20). Together and provide the coefficients of Prony’s polynomial. The roots of the polynomial, solved again from eq.(8), will include the harmonic frequencies since the procedure presented here enforces them exactly.

1b 2b

Figure 4: Stabilization Diagram with predefined poles at harmonic frequencies

In figure 4 the stabilization diagram is shown when predefined harmonics are included as poles of Prony’s polynomial. Those predefined poles are located at the 50, 100, 150 and 200 Hz and coincide with the known acoustic harmonic frequencies of the combustion chamber. In the stabilization diagram, the harmonics are indicated with a + sign. If we compare figures 3 and 4, we can see that the stabilization of the low frequency modes below the first harmonic at 50Hz is significantly improved when using predefined harmonics. The same applies to the modes found around the second harmonic at 100Hz. Recovery of Mode Shapes Once the roots are known from Prony’s polynomial, we can find the residues rV rjririj AC from eq.(5), which are the complex mode shapes times the modal participation factors : rjA

N

rrij

kr

N

rrij

ktsN

rrij

tkskijij CVCeCeRtkR rr

2

1

2

1

2

1)( (22)

where,

1011

Lkqjpi

(23)

Assuming that the number of reference stations 1q , eq.(22) can be written as follows in matrix form:

Lxpp

L

p

p

p

MxpMpM

p

p

p

MLx

LM

LM

LL

MM

MM

RR

RRRRRR

CC

CCCCCC

VVVV

VVVVVVVV

01

111

01

211

01

111

01

011

)2(12112

13311

12211

11111

)2(

12

112

12

11

22

212

22

21

21221

1111

(24)

Or,

Lxp

TL

T

T

T

MxpM

TM

T

T

T

MLx

LM

LM

LL

MM

MM

R

RRR

A

AAA

VVVV

VVVVVVVV

1

2

1

0

)2(22

33

22

11

)2(

12

112

12

11

22

212

22

21

21221

1111

(25)

Figure 5: Wireframe mode of 1st stage at 57.8 Hz – Forward dome mode

Figure 6: Accelerometer positions (red) and wireframe model with respect to FEM

The modeshapes in eq.(25) can be solved in a least squares sense. Eq.(25) presumes M poles were found from Prony’s polynomial. In practice only those poles will be used that are in the frequency range of interest, i.e. M

Threshold for mode selection

Figure 8: Maximal normalized modal gains per mode over all sensors

Figure 9: Correlation function sensor 17 – original versus recovered

Conclusions The Least Squares Complex Exponential method has been used in the frame of modal characterization of solid rocket motors. In order to avoid the detection of strong harmonics that are associated to the forcing function rather than the structure of the solid rocket motor, the method of predefined poles was used. It was shown that stable modes in the frequency range of interest can be identified easily from a stabilization diagram. In this diagram the predefined poles can be highlighted as well. In order to find the dominant modes amongst the identified stable modes, the modal gains can be used which are computed anyway as part of the modeshape recovery procedure. On the basis of the dominant modes, one should be able to reconstruct the correlation functions without significant loss of accuracy. References [1] Brown, D.L. et al., Parameter Estimation Techniques for Modal Analysis. SAE Technical Paper Series,

(790221), 1979. [2] Mohanty, P. and Rixen, D.J., Operational Modal Analysis in the Presence of Harmonic Excitations,

Journal of Sound and Vibration, 270(Issues1-2):93-109, Feb. 2004. [3] Fransen S. et al., Damping Methodology for Condensed Solid Rocket Motor Structural Models, IMAC

2010, Jacksonville, USA, 2010

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