1

Parametric Uncertainty Quantification of an

Inflatable/Rigidizable Hexapod

Jiann-Shiun Lew*

Tennessee State University, Nashville, TN 37203

Lucas G. Horta†

NASA Langley Research Center, VA 23681

Space exploration has always been constrained by our ability to develop systems that can be flown at reasonable cost within launch-vehicle volume constraints. Ultra-lightweight inflatable/rigidizable structures offer many advantages over conventional structures in this area. Recently, a 3m-diameter hexapod structure was designed and built, with these new materials and fabrication techniques, to conduct research on modeling and vibration control. For the space community to embrace this technology, systems like the hexapodmust be studied to understand their performance. This paper presents a study of parametric uncertainty quantification of the dynamic model of this hexapod structure. Our goal is to develop a model with parametric uncertainty for robust control design and analysis. To obtain parametric uncertainty, experiments with various kinds of excitation, such as random input and sine-sweep input, at different levels of force are conducted. Time-domain and frequency-domain system identification techniques are applied to analyze the experimentaldata collected from the hexapod. A singular value decomposition technique is used to model the parametric uncertainty via an interval transfer function, where each interval represents one bounded uncertainty parameter.

I. INTRODUCTION

There is an increasing interest in large ultra-lightweight space structures for space exploration. The applications include solar sails, large solar arrays, large aperture telescopes, and communication antennas. However, new technologies to reduce construction cost, weight, and pre-deployment size of space structures are needed in the development of ultra-lightweight space structures. Research related to these applications has been conducted under the auspices of NASA’s Spacecraft initiative.1-4 Recently a 3m-diameter hexapod structure5 was designed and built to conduct research on modeling and vibration control of this type of systems. A hexapod arrangement, often found in antennas and telescopes, is used because it incorporates design features and materials that can be used on a variety of future missions.5,6

Nearly all the missions envisioned for inflatable and rigidizable structural systems will benefit from vibration suppression, line of sight control, and shape control systems. The design of these control systems requires accurateanalytical models. To date, only a few experimental studies have been conducted to examine the vibration of membrane structures.7,8 Recently, new experimental methods for shape and dynamic characterization of future

* Research professor, Center of Excellence in Information System, Senior Member AIAA.†Assistant Branch Head, Structural Dynamics Branch, Associate Fellow AIAA.

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1826

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

space structures have been developed and applied at the NASA Langley Research Center.7,8 The goal of our research is to develop mathematical models and associated parametric uncertainties from experimental data for the design of vibration suppression algorithms.

Modeling this hexapod structure is a challenging task because of the structure's flexibility and the substantial number of modes in the low frequency range. A series of experiments with different kinds of excitation, such as random input and sine-sweep input, are conducted to investigate the structural dynamics of this hexapod. For vibration testing, conventional instrumentation, such as accelerometers, can greatly increase the mass of the structure, thus altering its structural vibration. Instead, laser vibrometry, a non-contact measurement technique, is used for the vibration testing. To obtain the identified parameters of each mode from Frequency Response Function (FRF) data, a sine-sweep signal in a specified narrow frequency range is used to excite the structure. Following the FRF estimation, data are curve fitted with a least-squares technique to obtain the identified parameters. An alternate approach with time-domain data uses random white noise to excite the structure and state space models are synthesized with the Eigensystem Realization Algorithm9 (ERA).

Robust control design of large flexible structures has received considerable attention in the last decade. The first step in robust control design or analysis is to model the system with uncertainty. Accurate models of the system “uncertainty” are crucial in obtaining a nonconservative robust control design. Various approaches for the quantification of model uncertainty have been proposed recently.10-12 In this paper a singular value decomposition (SVD) technique10 is used to model a dynamic system with parametric uncertainty via an interval transfer function, i.e., the parameter uncertainty is represented by intervals of transfer function coefficients. This SVD technique can also be applied to quantify the parametric uncertainty via an interval state-space model for a multi-input/multi-output system.13

II. DESCRIPTION OF THE TESTBEDFig. 1 shows a photograph of the assembled test-bed suspended in a vertical orientation. Fig. 2 shows a view in

the direction perpendicular to the torus plane. As shown, the torus segmented construction has twelve 0.181-meter diameter tubes arranged to form a circle. Six tapered struts, with diameters ranging from 0.0795 to 0.130 meters, connect the torus to a triangular aluminum frame. Tubes for the torus and struts are fabricated using a proprietary thermoplastic graphite epoxy composite, developed by ILC-Dover, Inc. When this material is heated above its glass-transition temperature, the stiffness decreases enough to allow it to be folded or rolled, and thus packaged into a small volume. Rigid joints connecting the sections of the torus are cast from glass filled urethane. A 25.4µm thickKapton membrane, coated with vapor deposited aluminum, is stretched and held within the torus inner diameter. The structure was rigidized prior to its assembly.

To minimize gravity effects on the tensioned membrane, the torus is mounted on a pair of steel rods rigidly attached to an opposing pair of torus joints. For dynamic testing, Fig. 3 shows a permanent magnet shaker, Ling Dynamic Systems V203, connected to the torus perpendicular to the membrane plane. Forces were imparted to the torus at a hard point on the polyurethane collar. Displacement measurements were taken with a Keyence LK-503 single point laser displacement sensor as shown in Fig. 4. The effective distance between the laser sensor and the measurement point is 350mm with a measurement resolution of 10µm

III. LEAST-SQUARES APPROACHIn this paper, a least-squares technique is developed for curve-fitting frequency domain response data. The

transfer function of a single input and single output system can be written as

)()(1

sfcsfn

ii∑

=

+= (1)

where fi is the ith component of the system transfer function. For example, fi can be chosen as the transfer function corresponding to the ith mode. The ith component fi is expressed as

3

0 0, 1 ,2 1 0

, 11 2,1 ,2 , 1 ,

00, 1

1, 1 ,2

1 2,1 ,2 , 1 ,

( )

( )

( )

( ).

( )

i

i i

ii i i

i i

i

i

i i

i i i

i i

mi m i m

i i mm m mi i i m i m

ii m

i

mi m i m

m m mi i i m i m

i

i

f s ff s f

s f s f s f s f

n sf

d s

f s f

s f s f s f s f

n s

d s

+ ++− −

−

+

−+

− −−

+ += −

+ + + + +

= −

+ +=

+ + + + +

=

L

L

L

L

(2)

To solve for the parameters of the transfer function, the following procedure is used. First, the experimental data y(jωl ) in the ith chosen frequency range are used to curvefit the ith transfer function component and a cost function is computed as

∑+

+=

−=1

1

20)0( |)()()(|i

i

k

kllilili jnjdjyV ωωω . (3)

There are ki+1– ki samples in the ith chosen frequency range. This is a linear least-squares problem of the variables fi,l, and a unique solution can be obtained by minimizing this cost function. The corresponding optimal cost function

for each component is computed as (0)iV . The initial value of constant c in Eq. (1) is computed as

∑=

+=n

imi i

fn

c1

01,

1. (4)

Then the experimental data in the ith frequency range are used to form the cost function as

∑+

+=

−−=1

1

2)1( |)()())((|i

i

k

kllilili jnjdcjyV ωωω . (5)

(1)iV is a function of constant c. For each constant c, we can find a unique solution of variables fi,l to minimize (1)

iV ,

where the optimal cost function is computed as (1) ( )iV c . The system cost function is defined as

(1)

(0)1

( )ni

i i

V cV

V=

=∑ . (6)

Here (0)iV is chosen as the weighting of the cost function V. This is a nonlinear optimization problem with one

variable c. The Matlab program fmins, which uses the Nelder-Mead simplex (direct search) method, is applied to get a solution for c and the initial identified parameters of fi. To get the optimal solution, the cost function of the ith

component is formed as

∑+

+=

−−−=1

1

20

)2( |)()())()((|i

i

k

kllililli jnjdjycjyV ωωωω (7)

4

with

)()(,1

0 l

n

ikkkl jfjy ωω ∑

≠=

= (8)

where y0 is the effect of the transfer function from other modes, and it is computed from the updated identified parameters. For each constant c, we can find a unique solution of variables fi,l to minimize Vi

(2) , where the optimal cost function is computed as Vi

(2) (c). Then the system cost function is computed as

(2)

(0)1

( )ni

i i

V cV

V=

=∑ . (9)

This is a nonlinear optimization problem with one variable c. The Matlab program fmins is used to get the solution of c and the identified optimal parameters of fi.

IV. MODEL UNCERTAINTY QUANTIFICATIONThe transfer function of a single input and single output system is represented by Eq. (1) with fi as the ith

component for parametric uncertainty quantification. By selecting a particular model structure, fi can be chosen as the transfer function corresponding to the ith mode. With this selection, now the parameter vector of the ith

component is defined as

,1 ,2 ,2[ ]Ti i i i mp f f f= L . (10)

The elements of parameter vector need not to be selected as the transfer function coefficients of each component. For instance, they can also be physical parameters, such as damping ratios and natural frequencies. However, the identified parameter vector changes with test conditions, such as input force levels. To distinguish the identified parameter vector let the ith component under the jth test condition be denoted by pij. To quantify the model uncertainty for the ith component, first compute the average parameter vector for h tests as

∑=

=h

jiji p

hp

10

1. (11)

The deviation of the jth parameter vector about the average is defined as

0iijij ppp −=∆ . (12)

Then, an uncertainty matrix is defined as

][ 21 ihii pppP ∆∆∆=∆ L (13)

To reflect the degree of variation in each coefficient, define the weighted uncertainty matrix as

PWPW ∆=∆ −1 (14)

where W is a diagonal matrix with its jth diagonal element being the standard deviation of the jth row of P∆ . The dimension of the parameter vector pi is 2m, so the maximum dimension of the space used to describe the uncertainty is 2m.

5

In this paper, an interval modeling technique10 is used to generate a linear interval model, which represents the ith

component with uncertainty, as

],[|{2

1,0

+−

=

∈+== ∑ ijijij

m

jijijii qpppP αααα } (15)

where pi0 is the nominal model, and αij are the identified bounded uncertainty parameters corresponding to the basis vectors qij. The corresponding interval transfer function is expressed as

∈+

+= +−

=

=

∑

∑],[,

)()(

)()(

:)(),(2

10

2

10

ijijijm

jijiji

m

jijiji

sdsd

snsn

sgsG αααα

αα . (16)

The polynomials ni0(s) and di0(s) are the numerator and denominator, respectively, of the nominal model, whereas nij(s) and dij(s) are polynomials corresponding to qij. To determine the nominal polynomials, the base polynomials, and parameter bounds, a singular value decomposition (SVD) technique is used. The SVD process involves four computational steps:10

1)SVD14 is used to compute the basis matrix UW for WP∆

][, 21 mTWW ssdiagSSVUP L==∆ (17)

and to compute the basis matrix U for P

][, )2(1 miiW qqUWUU L== . (18)

The singular values sj are in descending order, this leads to a descending order of perturbation distribution in qij.2) Compute the base polynomials corresponding to qij.

∑

∑

=

−

=

−

=

+=

m

k

kmijij

m

k

kmijij

skqsd

smkqsn

1

1

,)()(

,)()(

(19)

where qij(k) is the kth element of qij.

3) Compute the coordinate vector14 of ijp∆ corresponding to the basis vectors qij

ijij pU ∆= −1β . (20)

4) Compute the parameter bounds as

)}.(,),(),(min{

)}(,),(),(max{

21

21

jjj

jjj

ihiiij

ihiiij

βββα

βββα

L

L

=

=−

+

(21)

6

This representation yields an optimal linear interval system for capturing the parameter uncertainties.10 Also, the interval length for each identified parameter is defined as

−+ −=∆ ijijij ααα . (22)

This interval length represents the distribution of the uncertainty in the direction of qij.

V. EXPERIMENTAL RESULTSTo collect the data required to implement the uncertainty quantification procedure, the first set of experiments

are conducted using sine-sweep excitation to compute the FRF. A sample set of results is shown in Fig. 5 for the frequency range between 1 and 100 Hz. To improve the accuracy of identified parameters for each mode, blocks of data are collected with the sine-sweep frequency range restricted to a narrow frequency range. Fig. 6 shows the FRF data in the frequency range between 6 and 9 Hz for 3 input levels, and Fig. 7 shows the FRF data in the frequency range between 11 and 13 Hz. The FRF data of other modes shows no significant change due to increased input levels. After applying the least-squares identification technique, results for two identified models are presented in Table 1. Model 1 corresponds to data in the 6 to 13 Hz range with an input gain of 1. Similarly, model 2 uses a gain of 2. The identified natural frequency and damping ratio of mode 9 change significantly with the input gain. Figs. 8-10 show the identified results of some components of model 1. The component in Fig. 8 has two modes with natural frequencies close to each other. From Figs. 8-10 and other observations, the model error is around one order of magnitude lower than the magnitude of the experimental data.

To obtain time-domain data, a Dspace system is used to generate random input signals with a sampling rate of 200 Hz. Models for the system are synthesized using the Eigensystem Realization Algorithm (ERA) curve fittingimpulse response functions synthesized from time-domain input/output data. There are 18 sets of data, each setwith 6000 samples. Four input levels are used to excite the hexapod structure. Table 2 lists the identification results for the cases when the model order in ERA is chosen as 32. Fig. 11 shows the impulse response for the input excitation with a gain of 6. The mean-squared-root of error is about 10% that of the experimental data. The model uncertainty quantification technique in the preceding section is used to identify the parametric uncertainty from seven identified models of the systems with input gains ranging from 1.2 to 6. In this example, we choose each component with one mode. The uncertainty quantification results for 3 modes are listed as follows:

−−−

+−−

+

−−−−+−−−

+

−−−+−−

+

−−+−−−

+

+−−

+−

=

136.6

423.9

171.1

204.6

123.1

307.4

022.8

276.9

159.6

398.1

096.7

111.1

126.5

365.2

126.1

281.8

006.2

265.2

380.2

111.7

43216

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

P αααα

with

[ ] [ ] [ ] ]312.0239.0[,494.0602.0,882.0956.0,.261.2260.2 4321 −∈−∈−∈−∈ αααα

+−−+−−

+

−−−+−−−

+

+−−−+−−

+

+−−−+−−

+

+−−

++

=

079.1

219.1

197.1

188.1

142.7

385.3

201.1

239.1

074.1

241.2

026.8

371.7

043.1

223.1

166.6

153.1

118.3

201.2

346.6

058.1

43219

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

P αααα

with

[ ] [ ] [ ] ]195.0205.0[,247.0363.0,387.1112.1,.431.2822.1 4321 −∈−∈−∈−∈ αααα

7

+−−+−−−

+

+−−

+−

+

−−−−

+−−

+

+−+−−

+

+−++

=

049.4

310.1

123.2

382.2

024.2

349.2

140.1

232.3

149.7

399.4

092.4

236.2

074.3

381.1

194.1

213.2

146.7

204.5

477.3

014.2

432112

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

P αααα

with

[ ] [ ] [ ] ]406.0448.0[,212.1749.0,376.1104.1,.127.2060.2 4321 −∈−∈−∈−∈ αααα

where Pi represents the interval model (Eq. (15)) corresponding to the ith mode in Table 2. The first component of Pi

is iiωζ2 and it corresponds to the damping term, the second component is 2iω , the third component is close to zero

for displacement measurement, and the fourth component is the amplitude of the displacement. In general, the identified natural frequency is much less sensitive to noise than the identified damping and amplitude. The interpretation of the natural frequency uncertainty is based on its shift. The interpretation of the damping uncertainty or amplitude uncertainty is based on its percentage change. The third term is close to zero, so the effect from this term is not significant. For mode 9, the uncertainty corresponding to α1 is dominated by the second component related to natural frequency. This coincides with the FRF data in Fig. 7, which shows that the natural frequency of this mode shifts significantly as the gain of input force changes. The damping of mode 9 also shows some change, but the change of the displacement amplitude is relatively small. Comparing the interval lengths of uncertainty parameters of this mode, the uncertainty in the direction of the third vector drops to one order lower than the uncertainty in the direction of the first vector. The model uncertainty of mode 9 is dominated by the first two uncertainty parameters α1 and α2. For mode 6, the uncertainty is dominated by the fourth component corresponding to the amplitude of displacement. Also the damping and natural frequency of mode 6 have some changes. For mode 12, the identified parameters have insignificant uncertainty uncertainty, and this mode can be considered as a mode without uncertainty for control design. In real applications, each component used for model uncertainty quantification can include more than one mode with close natural frequencies. Also the elements of uncertainty vectors can be physical parameters, such as natural frequencies and damping ratios. The high frequency modes in Table 1 can also be identified when the model order in ERA is chosen as 100 (50 modes). Here we demonstrate the results of model uncertainty quantification from the time-domain identified models and the parametric uncertainty is quantified as an interval transfer function. The SVD technique can also be applied to quantify the parametric uncertainty via an interval state-space model for a multi-input/multi-output system.

VI. CONCLUDING REMARKSDynamic testing of an inflatable/rigidizable structure presents many challenges in terms of high modal densities,

model uncertainty, and structural nonlinearities. Our goal was to characterize and develop analytical models with parametric uncertainty for use in efforts to control the structure. Results from both frequency-domain and time-domain analysis showed that there are significant variations in the response due to variations in the input level. For example, the identified natural frequency of the mode around 13 Hz reduces significantly from 13 Hz to 12.6 Hz when the input force increases. This model uncertainty brings a challenge for model uncertainty quantification, and robust control design and analysis because there are three modes between 11.4 Hz and 13 Hz. Both precise model identification and model uncertainty quantification are needed for robust control design and analysis. To precisely investigate and identify each mode from FRF data, a sine-sweep signal in a specified narrow frequency range is used to excite the structure. The developed least-squares technique fits the experimental FRF data well. The magnitude of the model error is about one order lower than the magnitude of the experimental data. Also the identified model based on time-domain data fits experimental impulse response well with model error one order of magnitude lower than that of the experimental data. The time-domain and frequency-domain identification results are coincident. In real applications, we should combine the advantages of both time-domain and frequency-domain identification results.

A singular value decomposition technique is used to quantify the parametric uncertainty via an interval transfer function with its associated uncertainty basis vectors and parameter bounds. The uncertainty vectors represent the

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direction of uncertainty and they can be used to indicate the dominance of parameter uncertainty and the magnitude of uncertainty. Foe example, the uncertainty vectors for mode 12 in Table 2 indicate that the uncertainty of this mode is insignificant and this model uncertainty can be excluded in robust control design. The associated interval lengths indicate the distribution and dimension of uncertainty. This SVD technique can precisely quantify parametric uncertainty in the form of the specified model structure from the identified models. The elements of uncertainty parameter vector are not necessarily the transfer function coefficients of each component. They can be the physical parameters, such as damping ratios and natural frequencies. Also the model structure can be the state space model in modal form. Further research on model uncertainty quantification and validation of this hexapod structure will be conducted.

ACKNOWLEDGMENTSThis research is supported in part by NASA Grant NCC5-228 and NSF Grant HRD-9706268. The support is

greatly appreciated.

REFERENCES1Jenkins, C.H.M., (editor), Gossamer Spacecraft: Membrane and Inflatable Structures Technology for Space Applications,

Volume 191, Progress in Astronautics and Aeronautics Series, AIAA, Reston VA, 2001.2Chmielewski, A.B., “Overview of Gossamer Structures,” Gossamer Spacecraft: Membrane and Inflatable Structures

Technology for Space Applications, Jenkins, C.H.M., Editor, Volume 191, Progress in Astronautics and Aeronautics Series, AIAA, Reston VA, 2001, pp. 1-20.

3Cadogan, D., Grahne, M., and Mikulas, M., “Inflatable Space Structures: A new Paradigm for Space Structure Design,” IAF-98-I.1.02, 49th International Astronautical Congress, Melbourne, Australia, Sept 28-Oct 2, 1998

4Adetona O, Keel, L.H., Horta L.G., Cadogan, D.P., Sapna, G.H., and Scarborough, S.E., “Description of new inflatable/rigidizable hexapod structure testbed for shape and vibration control,” Paper # AIAA-2002-1451, Proc.43rd AIAA structures, structural dynamics and materials conference, Denver, Co, April 22-25, 2002.

5Adetona O, Horta L.G., Taleghani, B.K., Blandino, J.R., and Woods, K.J., “Vibration Studies of an Inflatable/Rigidizable Hexapod Structure with Tensioned Membrane,” Paper # AIAA-2003-1737.

6Wada, B.K., Lou, M., “Pre Flight Validation of Gossamer Structures,” AIAA-2002-1373, Proceedings of the 3rd Gossamer Spacecraft Forum, Denver, CO, April 2002.

7Blandino, J.R., Pappa, R.S., Black, J.T., “Modal Identification of Membrane Structures with Videogrammetry and Laser Vibrometry,” AIAA 2003-1745, Proceedings of the 4th Gossamer Spacecraft Forum, Norfolk, VA, April 2003.

8Pappa, R.S., Jones, T.W., Black, J.T., Walford, A., Robson, S., and Shortis, M.R., “Photogrammetry Methodology Development for Gossamer Spacecraft Structures,” NASA/TM-2002-211739, June 2002.

9Juang, J.N., Horta, L.G., Phan, M. “System/Observer/Controller Identification Toolbox (SOCIT),” NASA Technical Memorandum 107566, Feb. 1992.

10Lew, J.-S., and Lim, K. B., “Robust Control of Identified Reduced-interval Transfer Function,” IEEE Transactions on Control Systems Technology, Vol. 8, No. 5, 2000.

11Boulet B., Francis, B. A., Hughes, P. C., and Hong, T., “Uncertainty Modeling and Experiments in H∞ Control of Large Flexible Space Structures,” IEEE Transactions on Control Systems Technology, vol. 5, no. 5, pp. 504-519, 1997.

12Lew, J.-S., Link, T., Garcia E., and Keel L. H., “Interval Model Identification for Flexible Structures with uncertainty parameters,” Proceedings of the AIAA/ASME Adaptive Structures Forum, Hilton Head, SC, 1994.

13Lew, J.-S., Ahmad, S.S., and Keel, L.H., ``Robust Control of Identified Reduced-interval State Space Model,’’ Proceedings of AIAA Guidance, Navigation and Control Conference, Boston, MA, Aug. 10-12, 1998.

14Kailath T., Linear Systems, Englewood Cliffs, NJ: Prentice-Hall, 1980.

9

Figure 1: Hexapod structure for testing.

Figure 2: Hexapod with dimensions in Meters.

.0884

Diameter = .1811

3.8117

2.838

Radius = .4657

10

Figure 3: Shaker set-up for testing.

Figure 4: Laser displacement sensor for testing.

11

Figure 5: FRF data for sine-sweep input.

Figure 6: FRF data between 6 and 9 Hz with various gains: gain=1, ⋅⋅⋅ gain=1.5, - - gain=2.

12

Figure 7: FRF data between 11 and 13Hz with various gains: gain=1, ⋅⋅⋅ gain=1.5, , - - gain=2.

Table1: Identification results from FRF data.

Mode Model 1Fre. Dam.(Hz) (%)

Model 2Fre. Dam.(Hz) (%)

1 1.7500 0.3584 1.7500 0.35832 6.2767 0.6180 6.2646 0.76843 6.9309 0.5543 6.9204 0.53784 7.4166 0.6521 7.3654 0.84105 8.0797 0.4825 8.0414 0.62136 8.3363 0.4654 8.3215 0.85937 11.4719 0.5681 11.4423 0.57058 11.9061 0.2628 11.8990 0.34029 12.6362 0.7184 12.3979 1.398910 19.2538 0.3035 19.2570 0.286011 19.4260 0.1045 19.4297 0.142112 19.7924 0.2666 19.7924 0.270113 30.8226 0.4439 30.8231 0.445914 41.9801 0.2653 41.9748 0.273415 48.8583 0.7155 48.8540 0.723316 68.0061 0.1771 68.0036 0.177517 69.2875 0.0667 69.2878 0.066318 78.5076 0.4693 78.5066 0.4689

13

Figure 8: Identification results of model 1 from FRF data: experimental data, ⋅⋅⋅ model, - - model error.

Figure 9: Identification results of model 1 from FRF data: experimental data, ⋅⋅⋅ model, - - model error.

14

Figure 10: Identification results of model 1 from FRF data: experimental data, ⋅⋅⋅ model, - - model error.

Figure 11: Impulse response for random excitation with gain of 6: experimental data, ⋅⋅⋅ model, - - model error.

15

Table2: Identification results from time-domain response data with various excitation gains.

Mode Gain=1.2 Fre. Dam.

(Hz) (%)

Gain=2 Fre. Dam.

(Hz) (%)

Gain=4 Fre. Dam.

(Hz) (%)

Gain=6 Fre. Dam.

(Hz) (%)1 1.7598 6.3049 1.7224 3.9707 1.7447 6.7519 1.7584 4.63122 6.2410 0.3244 6.2461 0.4142 6.2335 0.4716 6.2235 0.46723 6.9143 0.1871 6.9277 0.2056 6.9285 0.2336 6.9272 0.25754 7.5166 0.6777 7.5126 0.5365 7.4869 0.7795 7.4553 0.73685 8.1091 0.3980 8.1364 0.3226 8.1119 0.3631 8.0934 0.45366 8.3116 0.4882 8.3348 0.4076 8.3262 0.5242 8.2998 0.45047 11.5051 0.2918 11.4978 0.2977 11.4862 0.4677 11.4816 0.26018 11.7962 0.2909 11.8580 0.2051 11.8324 0.1625 11.8394 0.14399 12.9671 0.6141 12.9000 0.8478 12.7420 1.1090 12.6563 1.227310 19.2701 0.2554 19.2629 0.2203 19.2682 0.2372 19.2674 0.228111 19.7587 0.1572 19.7825 0.0954 19.7679 0.1022 19.7698 0.104612 30.9311 0.5435 30.9265 0.5441 30.9132 0.5446 30.9000 0.5641