26-J2010-693

Journal of Mechanical Science and Technology 25 (5) (2011) 1307~1315

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0229-y

Identification of modal parameters from nonstationary ambient vibration data

using the channel-expansion technique†

Dar-Yun Chiang and Chang-Sheng Lin* Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

(Manuscript Received November 8, 2010; Revised December 31, 2010; Accepted December 31, 2010)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract The identification of modal parameters from the response data only is studied for structural systems under nonstationary ambient vi-

bration. In a previous paper by the authors, the modal parameters of a system were identified using the correlation method in conjunction with the curve-fitting technique. This was done by working within the assumption that the ambient excitation is a nonstationary white noise in the form of a product model. In the present paper, the Ibrahim time-domain method (ITD) is extended for modal-parameter iden-tification from the nonstationary ambient response data without any additional treatment of converting the original data into the form of free vibration. The ambient responses corresponding to various nonstationary inputs can be approximately expressed as a sum of expo-nential functions. In effect, the ITD method can be used in conjunction with the channel-expansion technique to identify the major modes of a structural system. To distinguish the structural modes from the non-structural modes, the concept of mode -shape coherence and confidence factor is employed. Numerical simulations, including one example of using the practical excitation data, confirm the validity and robustness of the proposed method for identification of modal parameters from the nonstationary ambient response.

Keywords: Ibrahim time-domain method; Nonstationary ambient vibration; Channel-expansion technique; Mode-shape coherence and confidence factor ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Modal-parameter identification from ambient vibration data has gained considerable attention in recent years [1, 2]. The use of ambient vibration survey in determining the dynamic characteristics of engineering structures is a valuable tool for practical structural health monitoring [3, 4]. Ibrahim [5] ap-plied the random-decrement technique coupled with a time-domain parameter identification method [6] to process ambi-ent vibration data. James et al. [7] developed the so-called natural excitation technique using the correlation technique coupled with the time-domain parameter extraction. The cross correlation between two response signals of a linear system with classical normal modes and subjected to stationary white-noise inputs is of the same form as free-vibration decay or impulse response. Chiang and Cheng [8] proposed a correla-tion technique for the modal-parameter identification of a linear, complex-mode system subjected to stationary ambient excitation. Chiang and Lin [9] applied the correlation tech-nique coupled with the Eigensystem Realization Algorithm to identify the modal parameters of a system subjected to realis-

tic ambient excitation. Chiang and Lin [10] further presented a theoretical justification of the cross-correlation technique for general linear systems excited by non-stationary white-noise input in the form of a product model. The practical problem of insufficient data samples available for evaluating nonstation-ary correlation is approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. The correlation functions of the stationary response are then treated as free vibration response. Thus, the Ibrahim time-domain method (ITD) can be applied to identify the modal parameters of a system. However, research shows [10] that if a system has relatively heavy damping or very low natural frequencies compared with the major frequency of the ampli-tude-modulating function of the excitation, then identifying some of the modal parameters of the system can become diffi-cult.

In previous studies on modal -parameter identification, most of the methods used free decay or the approximate response of free decay transformed from forced response. The error in-volved in the approximate free-decay response generally leads to a distortion in the modal parameters of identification. Ibra-him [11] tried to reduce the effects of noise and to improve the accuracy of identified modal parameters by using an aug-mented mathematical model of the measured free-decay re-

† This paper was recommended for publication in revised form by Associate EditorOhseop Song

*Corresponding author. Tel.: +011 886 6 2892417, Fax.: +011 886 6 2389940 E-mail address: [email protected]

© KSME & Springer 2011

1308 D.-Y. Chiang and C.-S. Lin / Journal of Mechanical Science and Technology 25 (5) (2011) 1307~1315

sponses of a system and applying a technique of channel ex-pansion. However, the modes identified by the ITD method generally include some fictitious modes caused by the nu-merical computation. Thus, Ibrahim introduced the modal confidence factor (MCF) [11], which is computed for each identified mode. The structural modes were then separated from the noise (or computational) modes accordingly. Gao and Randall further presented the so-called mode-shape co-herence and confidence factor (MSCCF) [12] in conjunction with the ITD method to distinguish physical modes from the identified modes. The MSCCF values are generally more reli-able than the MCF values in distinguishing the structural modes from the computational modes because the calculation of the MSCCF value uses the overall information of mode shapes, not only the information of a single mode as in the calculation of MCF.

In a previous paper by the authors [10], the modal parame-ters of a system were identified through the correlation method in conjunction with the curve-fitting technique. How-ever, this is done by assuming the ambient excitation to be nonstationary white noise in the form of a product model. In the present paper, we propose a technique to extend the ITD method for modal-parameter identification from the nonsta-tionary ambient vibration data without using input data or any additional treatment of converting the original forced -vibration data into the form of free vibration. In doing so, we can avoid a distortion in the modal parameters of the identifi-cation induced by the error involved in the approximate free-decay response obtained by computing the correlation func-tions [10]. The ambient response corresponding to the nonsta-tionary input of various types can be approximately expressed as the sum of exponential functions. Thus, we can use the ITD method to identify directly the major modes of a structural system. Hence, we get rid of the restriction of the product model, which was employed in the previous paper [10] for modeling ambient excitation assumed to be nonstationary white noise. For the purpose of improving the accuracy of modal-parameter identification, the channel-expansion tech-nique is also employed. Numerical simulations, including one example of using practical excitation data, are performed to confirm the validity and robustness of the proposed method in identifying the modal parameters from the general nonstation-ary ambient vibration data.

2. Extension of Ibrahim time-domain method for mo-

dal-parameter identification from ambient vibra-tion data

The conventional ITD method uses free-decay responses of a structure to identify its modal parameters in complex form [6]. In the present paper, we extend the ITD method to iden-tify the modal parameters of a structure solely and directly from forced response data, i.e. without using input data or any additional treatment of converting responses into the form of free vibration. Remarkably, the essence of the ITD method is

in utilizing the free-decay responses, which can be expressed as linear combinations of the exponential functions. With this method, the extraction of the modal parameters can be per-formed using different time-delayed sampled responses. In light of this concept, we came up with a new idea that if the ambient vibration data can be approximated directly as a lin-ear combination of the exponential functions, then the ITD method can be used for modal-parameter identification di-rectly from ambient response data. In the following, we show that, under appropriate conditions, nonstationary ambient re-sponses can be approximately expressed as a linear combina-tion of exponential functions.

2.1 Modeling of ambient excitation

In this paper, we consider that the ambient excitation can be modeled as a nonstationary process, which is represented by a proper composition of a stationary process and a deterministic time-varying function. We start by considering that a station-ary process ( )W t can be approximately expressed as [13]

1

( ) Re e k

Ni t

kk

W tω

ωψ=

⎧ ⎫≈ ⎨ ⎬

⎩ ⎭∑ , (1)

where 4 ( ) ki

k WW kS e ϕψ ω ω −= ∆ , k kω ω= ⋅ ∆ , ( )WWS ω is the power spectral density function of ( )W t (if ( )W t is white noise, WWS is a constant), and kϕ , 1 ~k Nω= , are a set of independent random variables (phases) uniformly dis-tributed over ( )0,2π . Note that for each k , e ki t

kωψ has a

form similar to the vibration behavior of a mode; thus, herein we call it a “mode” of excitation. The corresponding kψ can then be considered a component of the “mode shape” of exci-tation (corresponding to a certain degree of freedom). Eq. (1) is originally used to simulate stationary Gaussian processes and to generate sample functions with prescribed power spec-tra. Noted that a stationary sample function ( )W t , as de-scribed in Eq. (1), can be approximately expressed as a linear combination of e ki tω , which is of a similar form to the un-damped free vibration of a discrete linear system.

In general, a nonstationary process can be expressed, on the basis of a stationary process ( )W t , as follows:

1. Product model

( ) ( ) ( )p pf t Γ t W t= (2)

where ( )p Γ t is a deterministic envelope function (or ampli-tude-modulating function) used to describe the time-varying amplitude (variance) of a nonstationary process. 2. Additive model

( ) ( ) ( )a af t Γ t W t= + , (3)

where ( )a Γ t is a deterministic trend function used to de-

D.-Y. Chiang and C.-S. Lin / Journal of Mechanical Science and Technology 25 (5) (2011) 1307~1315 1309

scribe the time-varying mean of a nonstationary process. The time-varying functions ( )p Γ t and ( )a Γ t mentioned

above can be further approximated as

( )1

kn

tk

kΓ t a eα

=

≈ ∑ . (4)

By utilizing the above formulation as well as choosing appro-priately the values of n , ka , and kα , we can describe a wide variety of nonstationary behaviors in reality. Substituting both Eqs. (1) and (4) into Eqs. (2) and (3), respectively, the models can be expressed as follows:

( ) ( ) ( )( )11

1

Re k n k

Ni t i t

p k nkk

f t e eω

α ω α ωψ ψ+ +

=

⎧ ⎫′ ′= ⋅ + + ⋅⎨ ⎬

⎩ ⎭∑ , (5)

( )1 1

Rek k

Nnt i t

a k kk k

f t a e eω

α ωψ= =

⎧ ⎫= + ⎨ ⎬

⎩ ⎭∑ ∑ (6)

where nk n kaψ ψ′ = . From Eqs. (5) and (6), a nonstationary process in the form of either a product model or an additive model can also be approximately expressed as a linear combi-nation of exponential functions. In the following, we analyze the responses of a system subjected to nonstationary inputs discussed above.

2.2 Analysis of ambient response

Consider an N-degree-of-freedom (DOF), discrete-time, linear system subjected to ambient excitation resulting from a single source, which is assumed to be a nonstationary process in the form of either Eq. (5) or (6). The displacement response, either ( )p tx or ( )a tx , which is a vector of displacement corresponding to different DOFs of a system can be obtained by combining the complementary solution and the particular solution, as expressing in the following equation:

2

1( ) e k

m n Nt

p p k p pk

tω

λ+ ⋅

=

= ∑x φ = Φ Λ (7)

where m is the number of (real) modes of the system, and pΦ is an (2 )N m n Nω× + ⋅ matrix, which is the complex

“modal” matrix composed of p kφ for 1 ~ 2k m n Nω= + ⋅ , including the 2m mode shapes of the structure as well as the n Nω⋅ “mode shapes” of the nonstationary excitation in the form of Eq. (5). pΛ is a (2 ) 1m n Nω+ ⋅ × matrix composed of e k tλ for various k .

2

1( ) e k

m n Nt

a a k a ak

tω

λ+ +

=

= ∑x φ = Φ Λ (8)

where the roles of a kφ , aΦ , and aΛ are similar to those in Eq. (7). Note that in Eqs. (7) and (8), some kλ ’s are associated with structural modes, whereas the others may be associated

with the excitation. Eqs. (7) and (8) show that both ( )p tx and ( )a tx can be expressed as a linear combination of exponen-

tial functions, indicating that- the forced-response data can be put into the form of free-vibration without any additional treatment. The major modes of a structural system can then be obtained via the time-domain modal-identification method, such as the ITD method, as will be described next.

2.3 ITD Extraction

In the following, we extend the ITD method to identify the modal parameters of a structure from forced-response data. Recall that in the conventional ITD analysis, which uses the free-decay responses of a structure to identify its modal pa-rameters in complex form [6], we define a system matrix A as

A X Y= (9)

where Y is a response matrix obtained from the time-shifting of a data-expansion matrix .X Eq. (9) implies that the system matrix A can be determined from X and Y using the least-squares method when the number of sampling points is larger than the number of measurement channels. The order of the system matrix A is chosen to be at least twice the number m of the (real) modes of interest. If the number of measurement channels does not actually reach the number of modes of interest, we can use the channel expan-sion technique [6], which uses time-delayed responses as addi-tional response channels, such that the total number of meas-urement channels reaches the desired order of the system ma-trix A . In the present paper, the ITD method is extended for modal-parameter identification directly from the force re-sponse data. Thus, the noise caused by the excitation generally leads to distortion in identifying the modal parameters of a system. To take advantage of the property of consistency in the theory of system identification to improve the effective-ness of the extended ITD method, we tend to use more meas-urement data in the analysis, which can be practically achieved by extending the sampling period and/or performing channel expansion. In performing the channel expansion, we usually construct a data-expansion matrix X with more measurement channels to obtain richer information of struc-tural modes, as in the following:

0

τ

⎡ ⎤= ⎢ ⎥⎣ ⎦

XX

X (10)

where τ X is obtained from the τ -time shifting of the origi-nal response data matrix o X . It uses almost the same proce-dure as that used in ITD to solve the eigenvalue problem asso-ciated with the corresponding system matrix A determined using the least-squares method. Afterwards, the major modes of a structural system can be identified. Note that the well-known least-squares method is a classical strategy to find so-


lutions for over-determined systems, usually interpreted as a method of fitting data. Estimates obtained by the least-squares method also have optimal statistical properties of being unbi-ased, efficient, and consistent [14]. Based on the concept men-tioned above, the higher-order A leads to the more complete and accurate modal parameters of a system. Note that the ei-genvalues of A are directly related to the natural frequencies and the damping ratios of the original vibrating system and that the eigenvectors of A correspond to the mode shapes of the system [6].

On the contrary, the modes identified by the extended ITD method directly from the forced response data generally in-clude the vibrating modes of a structural system, the “modes” of excitation (as mentioned above), and possibly some ficti-tious modes caused by numerical noise. In the following, we show how to extract the structural modes from the identified modes. Recall in Eqs. (7) and (8), the nonstationary response in the form of either a product model ( )p tx or an additive model ( )a tx can be approximately expressed as a linear combination of exponential functions. The time-delayed re-sponses obtained from the τ -time shifting of ( )p tx and

( )a tx , denoted respectively as ( )p t +τx and ( )a t +τx , can be derived as follows:

2

1( ) e k

m n Nt

p p kk

t +ω

λττ+ ⋅

=

= ∑x φ, (11)

2

1( ) e k

m n Nt

a a kk

t +ω

λττ+ +

=

= ∑x φ

(12)

where e k

p k p kλ ττ =φ φ

and e k

a k a kλ ττ =φ φ . According to Eqs.

(11) and (12), as well as the aforementioned relationship be-tween o X and τ X in Eq. (10), a typical eigenvector iφ of A corresponding to the i th eigenvalue iλ can be ex-pressed as

0 i

iiτ

⎡ ⎤= ⎢ ⎥⎣ ⎦

φφ

φ (13)

where

e i

i 0 iλ τ

τ =φ φ (14)

provided that o iφ is the mode-shape vector of a structure mode or excitation mode [cf. Eqs. (11) and (12)]. The reason is that the vibration in each “mode” demonstrates exponen-tially decayed harmonic behavior. Based on the concept men-tioned above, to extract further the structural modes from the identified modes, we can employ the MSCCF [12], which is defined as

*

*

eMSCCF( , )

iTo i i

0 i i Ti i

λ ττ

ττ τ

=φ φ

φ φφ φ

(15)

where the superscripts * and T denote the complex conjugate

and matrix transpose, respectively. When the MSCCF value of an identified mode shape is approximately equal to 1, it implies that the mode shape represents structure or the excita-tion. Identification results are then sorted as either structural parameters or input-force characteristics by computing the MSCCF values.

Furthermore, to distinguish the vibrating modes of the struc-tural system from the excitation, we can use the modal assur-ance criterion (MAC) [15], which is originally used to check for agreement between the identified mode shapes as well as exact shapes, and has been extensively used to confirm the vibrating modes of a structural system in experimental modal analysis. MAC is defined as

2

MAC( , )T

iA jXiA jX T T

iA iA jX jX

∗

∗ ∗=φ φ

φ φφ φ φ φ

(16)

where iAφ and jXφ denote the i th theoretical and the j th identified mode shape, respectively. The value of MAC varies between 0 and 1. When the MAC value is equal to 1, the two vectors iAφ and jXφ represent exactly the same mode shape. However, when two mode shapes are orthogonal with each other, the MAC value is zero. Thus, the verification of the vibrating modes of a structural system can be performed by computing the MAC values. If the mass matrix or the stiffness matrix of a structural system is available, we can distinguish the structural modes from the excitation or fictitious modes according to the orthogonality of the mode shapes with respect to the stiffness matrix or to the mass matrix. If neither the exact information of the structural modes nor the mass matrix (or the stiffness matrix) of the structural system is available, the structural modes can usually be identified by observing the identified damping ratios, due to that the “excitation modes” are related to the steady-state responses corresponding to per-sistent (in contrast to shock or impulse) excitation and the corresponding damping ratios are generally close to zero (i.e., showing no decay), whereas the damping ratios corresponding to the vibrating modes of damped structures are normally non-zero.

3. Numerical simulation

To demonstrate the effectiveness of the proposed method, we first consider a linear 6-DOF chain model with viscous damping. A schematic representation of this model is shown in Fig. 1. The mass matrix M , stiffness matrix K , and the damping matrix C of the system are given as follows:

2

2 0 0 0 0 00 2 0 0 0 00 0 2 0 0 0

= N s /m,0 0 0 2 0 00 0 0 0 3 00 0 0 0 0 4

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

⋅⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M


1 1 0 0 0 01 2 1 0 0 0

0 1 2 1 0 0600 / ,

0 0 1 2 1 00 0 0 1 3 20 0 0 0 2 5

N m

−⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥− −

= ⋅ ⎢ ⎥− −⎢ ⎥

⎢ ⎥− −⎢ ⎥

−⎢ ⎥⎣ ⎦

K

6 6

1 10.05 0.001 0.2 s/ .

1 1N m

×

⎡ ⎤⎢ ⎥= + + ⋅⎢ ⎥⎢ ⎥⎣ ⎦

C M K…

…

Note that because the damping matrix C cannot be ex-pressed as a linear combination of M and K , the system has non-proportional damping (and complex modes in gen-eral). Consider that the ambient vibration input is modeled as nonstationary white noise, as represented by the product model or the additive model, given by Eqs. (2) and (3), respec-tively. Using Eqs. (2) and (3), the simulated stationary white noise is multiplied and added, respectively, by an amplitude-modulating function 0.002 0.004( ) 4 (e e )t t

p Γ t − −= ⋅ − and a trend function 0.002 0.004( ) 50 (e e )t t

a Γ t − −= ⋅ − to obtain the nonstation-ary white noise inputs, which act on the sixth mass of the sys-tem. Note that the stationary white noise, whose bandwidth extends well past all the frequencies of interest, is generated using the spectrum approximation method [13] as a zero-mean band-pass noise, for which the power spectral density constant is 2 30.02m /(s rad)⋅ with a frequency range from 0-50 Hz. The sampling interval is chosen as ∆t = 0.01s , and the sam-pling period is T = 1310.72s. The time signal of a simulated sample of the nonstationary white noise and the power spec-trum of the corresponding stationary part are shown in Figs. 2 and 3, respectively.

The displacement responses of the system are obtained us-

ing Newmark’s method. The ITD method is then applied to identify the modal parameters of the system. As the nonsta-tionary white-noise input can be expressed as the sum of ex-ponential functions, according to the theory presented in the previous sections, we can use the ITD method in conjunction with the channel-expansion technique to identify the major modes of a structural system solely and directly from the am-bient response data. The results of the modal-parameter identi-

1x

2x

3x

4x

5x

6x

Fig. 1. Schematic plot of the 6-DOF chain system.

(a)

(b)

Fig. 2. Sample function of non-stationary white noise with a slowly-varying amplitude-modulating function or trend function (a) Productmodel; (b) Additive model.

Fig. 3. Power spectrum associated with the stationary part of the simu-lated nonstationary white noise.


fication of the 6-DOF system subjected to nonstationary white-noise input in the form of a product model are summa-rized in Table 1, which shows that the errors in natural fre-quencies identified are less than 1 %, and the errors in damp-ing ratios are less than 10%. Note that the “exact” modal damping ratios listed in Table 1, as well as the exact mode shapes, are actually the equivalent values obtained using ITD from the simulated free-vibration data of the nonproportion-ally damped structure. The concepts of MSCCF and MAC can be used to confirm further the vibrating modes of the struc-tural system. The results are also listed in Table 1. In general, when the MAC values are larger than 0.9 and the MSCCF values are closer to 1, the identified results can be considered as “accurate.” The identified mode shapes are also compared with the exact values in Fig. 4, where good agreement is ob-

served. Note that if the system has non-proportional damping, then it has complex modes in general. However, in the present case, the imaginary parts of the mode shapes (both identified and exact) are small relative to the real parts. Hence, in Fig. 4, the mode shapes are presented only in the form of real parts of the modal-shape vectors. The errors of identified damping ratios and mode shapes are somewhat higher, which may be due to the fact that the system response generally has lower sensitivity to these modal parameters than to the modal fre-quencies.

The results of the modal-parameter identification of the 6-DOF system subjected to nonstationary white-noise input in the form of an additive model are summarized in Table 2. The result of the identification is still satisfactory. In the present method, it is significant to have sufficiently long measurement data to reduce the error caused by noise effect in the process of modal-parameter identification, especially for the identifi-cation of damping ratios. In addition, to improve the effec-tiveness of the extended ITD method, we can use a data -expansion matrix X with more measurement channels. The corresponding system matrix A , which is determined using the least-squares method, leads to more complete and accurate modal parameters of a system. Based on the numerical studies of this particular paper, the results of modal-parameter identi-fication using the extended ITD are satisfactory with the selec-tion of the number of measurement channels of X as 16m .

In the first part of this paper, we considered the nonstation-ary excitation to be modeled as a composition of stationary white noise and an amplitude-modulating or trend function.

Table 1. Results of the modal-parameter identification of a 6-DOFsystem subjected to nonstationary white-noise input in the form of a product model.

Natural Frequency (rad/s) Damping Ratio (%) Mode

Exact ITD Error (%) Exact ITD Error

(%) MSCCF MAC

1 5.03 5.06 0.66 5.24 5.32 1.56 1.01 0.99

2 13.45 13.42 0.23 1.07 1.14 6.64 1.01 0.99

3 19.80 19.77 0.16 1.13 1.11 1.49 1.01 0.99

4 26.69 26.51 0.69 1.43 1.32 8.04 1.01 0.99

5 31.66 31.43 0.73 1.66 1.72 3.55 1.02 0.99

6 33.73 33.40 0.99 1.74 1.81 3.88 1.02 0.99

(a) (b) (c)

(d) (e) (f) Fig. 4. Comparison between the identified mode shapes and the exact mode shapes of the 6-DOF system subjected to nonstationary white-noise input in the form of a product model.


However, in practice, the ambient excitation may be nonsta-tionary in a more complex form. To address this issue, we consider a more complicated nonstationary input given by 0.4 0.2 2 0.1 2( ) 0.2 cos (3 ) 0.1 cos (10 )f t t t t t t= + ⋅ ⋅ − ⋅ ⋅ , which serves as the excitation acting on the sixth mass of the 6-DOF chain model as shown in Fig. 5. Note that the 2t in the co-sine terms in this case implies that the frequency content may change with time. Thus, this nonstationary input generally contains the time-varying mean, variance, and frequency con-tent. The displacement responses of the system are obtained using Newmark’s method. The modal-parameter identification is performed using the simulation responses. Table 3 summa-

rizes the results of the modal identification in this case, and presents the well-defined modal parameters of the system. Such results are obtained in this case because the simulated non-stationary input can also be approximately expressed as the composition of exponential functions.

To examine the effectiveness of the present method for more complex structural systems, we consider a 20-DOF chain model with nonproportional damping. Assume that each mass is 1 kg and that all spring constants are 600 N / m . The damping matrix of the system is assumed as

20 20

1 10.05 0.001 0.02 s/

1 1N m

×

⎡ ⎤⎢ ⎥= + + ⋅⎢ ⎥⎢ ⎥⎣ ⎦

C M K…

…

In this example, we still use the previous nonstationary white-noise in the form of a product model as input acting on the twentieth mass of the chain model. The corresponding dis-placement responses obtained by Newmark’s method are then used for modal-parameter identification. The results of the identification are summarized in Table 4, which shows that the modal identification in this case is satisfactory. Note that the higher modes are not identified as accurately as the lower ones, because the contribution of higher modes to the system response is less than that of the lower modes. Observing the MAC values, which signify the consistency between the iden-tified and the theoretical mode shapes, we found that 17 out of the 20 modes are identified accurately (MAC ≥ 0.9). Thus, the

Table 4. Results of the modal-parameter identification of a 20-DOF system subjected to nonstationary white -noise input in the form of a product model.


Exact ITD Error(%) Exact ITD Error

(%) MSCCF MAC

1 3.66 3.65 0.27 5.49 4.81 12.54 1.01 0.992 7.30 7.31 0.14 0.71 0.67 5.63 1.00 0.993 10.90 10.88 0.18 0.94 0.97 3.19 1.01 0.994 14.44 14.41 0.21 0.89 0.82 8.89 1.01 0.995 17.90 17.86 0.22 1.06 1.17 9.35 1.01 0.996 21.26 21.16 0.47 1.17 1.29 9.32 1.01 0.997 24.49 24.38 0.45 1.33 1.37 2.24 1.01 0.998 27.60 27.37 0.83 1.45 1.68 14.29 1.02 0.999 30.54 30.30 0.79 1.59 2.02 25.46 1.02 0.9910 33.32 32.99 0.99 1.71 2.11 21.26 1.02 0.9911 35.91 35.47 1.29 1.83 2.09 11.76 1.03 0.9912 38.30 37.75 1.44 1.94 1.92 3.03 1.02 0.9913 40.48 39.89 1.46 2.03 1.80 13.87 1.02 0.9914 42.43 41.84 1.39 2.12 1.84 15.60 1.03 0.9915 44.14 43.45 1.56 2.20 2.08 7.96 1.03 0.9916 45.60 44.68 2.02 2.26 2.01 14.10 1.03 0.9517 46.81 45.88 1.99 2.26 1.72 28.03 1.03 0.9618 47.76 46.95 1.70 2.20 1.49 38.93 1.02 0.3919 48.44 47.79 1.34 2.51 1.70 31.17 1.02 0.3920 48.85 48.41 0.90 2.41 3.50 40.56 1.05 0.07

Table 2. Results of the modal-parameter identification of a 6-DOFsystem subjected to nonstationary white-noise input in the form of an additive model.


Exact ITD Error (%) Exact ITD Error

(%) MSCCF MAC

1 5.03 5.12 1.79 5.24 5.29 0.95 1.03 1.00

2 13.45 13.44 0.07 1.07 0.90 15.89 1.02 1.00

3 19.80 19.77 0.15 1.13 1.06 6.19 1.03 1.00

4 26.69 26.49 0.75 1.43 1.27 11.19 1.04 1.00

5 31.66 31.45 0.66 1.66 1.59 4.22 1.05 1.00

6 33.73 33.24 1.45 1.74 2.06 18.39 1.06 1.00

Table 3. Results of the modal-parameter identification of a 6-DOFsystem subjected to nonstationary input containing time-varying mean, variance, and frequency content.

Natural Frequency (rad/s) Damping Ratio (%)

Mode Exact ITD Error

(%) Exact ITD Error (%) MSCCF MAC

1 5.03 5.06 0.60 5.24 4.95 5.53 1.03 1.00

2 13.45 13.39 0.45 1.07 1.23 14.95 1.03 1.00

3 19.80 19.70 0.51 1.13 0.98 13.27 1.03 1.00

4 26.69 26.48 0.79 1.43 1.24 13.29 1.04 1.00

5 31.66 31.46 0.63 1.66 1.76 6.02 1.05 1.00

6 33.73 33.40 0.98 1.74 1.42 19.32 1.04 1.00

Fig. 5. Sample function of non-stationary input containing time-varying mean, variance, and frequency content.


proposed method is valid in identifying modes with close (or even identical) frequencies, as long as the modal damping ratios of each mode of a system are different and small. When a sys-tem has light damping, the interference between modes gener-ally does not lead to large errors in separating the closely spaced modes of a system. The errors of identified damping ratios and mode shapes are larger than those of modal frequencies. The reason is that the system response generally has lower sensitiv-ity to these modal parameters than to the frequencies.

To examine further the effectiveness of the proposed method, we consider the previous 6-DOF chain model excited by a vibration practically recorded at Sun-Moon Lake on Sep-tember 21, 1999 when the Chi-Chi Earthquake with a moment magnitude of 7.6 occurred in central Taiwan. The sampling interval and period of this seismic record are ∆t = 0.005 s and T = 59.995s, respectively. A sample of the seismic re-cord, which serves as the excitation input acting on the sixth mass of the model, is shown in Fig. 6. The displacement re-sponses of the system are obtained using Newmark’s method. The modal-parameter identification is then performed using the simulation responses. The excellent results obtained are summarized in Table 5. When the practical ambient excitation, such as earthquakes, can be approximately expressed as the composition of the exponential functions, the proposed method is applicable to identify the modal parameters of a structural system subjected to realistic excitation.

Recall that in the correlation technique presented in a previ-ous paper by the authors [10], to improve the accuracy of the identified modal parameters, choosing a proper reference channel for computing the correlation functions is important. The richer frequency content the reference channel has, the better the results of the modal-parameter identification. Using the extended ITD method proposed in the present paper, we can prevent computing the correlation functions to convert forced responses to free vibration [10]. Furthermore, we show that the present method is valid for general nonstaionary input. The restriction of the product model can also be avoided, which was employed in the previous paper [10] to model the ambient excitation assumed to be nonstationary white noise.

4. Conclusions

The ITD method is extended in the present paper for direct modal-parameter identification solely from nonstationary ambi-ent vibration data, i.e., without using input data or any additional treatment of converting the original forced-vibration data into free vibration. Under appropriate conditions, the ambient vibra-tion responses corresponding to the various nonstationary inputs can be expressed as the sum of the exponential functions. Hence, we can use the ITD method in conjunction with the channel-expansion technique to identify the major modes of a structural system. To confirm the validity of the proposed method, nu-merical simulations were performed for various cases where the systems were subjected to nonstationary white-noise input in the forms of a product and an additive model, respectively. The

theory proposed can be extended to cover general nonstaionary input which contains time-varying mean, variance, and fre-quency content. Furthermore, to improve the effectiveness of the extended ITD method, we used a data-expansion matrix with more measurement channels such that the corresponding system matrix determined via least-squares method could lead to more complete and accurate modal parameters of a system. Using the extended ITD method proposed in this paper, we can avoid computing the correlation functions to convert forced responses into free vibration. Through numerical examples, the present method has been shown to be valid for general non-staionary input, and the restriction of the product model can be avoided to model ambient excitation assumed to be nonstation-ary white noise.

Acknowledgement

This research was supported in part by the National Science Council of the Republic of China under grant number NSC-96-2221-E-006-188. The authors would like to thank the anonymous reviewers for their valuable comments and sug-gestions in revising this paper.

Nomenclature------------------------------------------------------------------------

A : System matrix ( )a f t : Nonstationary process in the form of an additive

Table 5. Results of the modal-parameter identification of a 6-DOFsystem subjected to a recorded sample of the Chi-Chi Earthquake.

Natural Frequency (rad/s) Damping Ratio (%)

ModeExact ITD Error

(%) Exact ITD Error (%) MSCCF MAC

1 5.03 5.16 2.66 5.24 6.68 27.54 1.07 0.98

2 13.45 14.47 7.61 1.07 1.22 13.90 1.00 1.00

3 19.80 19.95 0.76 1.13 1.25 10.98 1.03 1.00

4 26.69 26.74 0.19 1.43 1.02 28.64 1.03 0.98

5 31.66 32.73 3.38 1.66 1.40 15.52 1.05 0.99

6 33.73 33.01 2.15 1.74 1.87 7.32 1.08 1.00

Fig. 6. A recorded sample of the Chi-Chi earthquake.


model ( )p f t : Nonstationary process in the form of a product

model ( )WWS ω : Power spectral density function of ( )W t

( )W t : Stationary process X : Data-expansion matrix o X : Original response data matrix τ X : Response data matrix obtained from τ -time shift-

ing of o X ( )a tx : Displacement vector corresponding to nonstation-

ary process in the form of an additive model ( )p tx : Displacement vector corresponding to nonstation-

ary process in the form of a product model Y : Response matrix obtained from time-shifting of X

( )a Γ t : Deterministic trend function ( )p Γ t : Deterministic envelope function (or amplitude-modulating function)

iλ : Typical eigenvalue of A iφ : Typical eigenvector of A

o iφ : Mode-shape vector of a structure mode or

: Excitation mode iAφ : i th theoretical mode shape jXφ : j th identified mode shape

kϕ : Set of independent random variables (phases) References

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D. Y. Chiang received his Ph.D. in Applied Mechanics from the California Institute of Technology, USA in 1992. He joined the faculty of National Cheng Kung University, Taiwan in 1993 where he is currently a professor at the Department of Aeronautics and Astronautics. His research interests are

system identification and plasticity.

C. S. Lin received his B.S. and M.S. degrees from the Department of Aero-nautics and Astronautics of National Cheng Kung University, Taiwan in 2002 and 2004, respectively. He is cur-rently a Ph.D. candidate with research interests in random vibration and modal analysis.

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