On the Two-Stage Time Domain Modal Identification - CiteSeerX

ON THE TWO-STAGE TIME DOMAIN MODAL IDENTIFICATION

Lingmi Zhang1 Institute of Vibration Engineering Dept. of Mechanics & Structural Engineering Nanjing University of Aeronautics and Astronautics Nanjing, 210016, China, P.R. Email: [email protected] 1

ABSTRACT A generalized Time Response Function (TRF) is proposed to represent Impulse Response Function (IRF), cross-correlation of input/output (I/O) measurements, free decay response, auto-correlation and Random Decrement (RDD) signature of random response data in output only (O/O) case, or their data correlation and combination. A unified framework is developed for two(2)-stage time domain (TD) modal identification, based on a formula of modal decomposition or system matrix decomposition of the TRF. Possible implementations for the TD 2-stage modal identification, which cover a variety of well-known techniques, are summarized to provide better understanding of the different techniques, and guidelines for effective applications. Relations and features and major issues of different implementations of the 2-stage TD modal identification are further discussed. 1.

INTRODUCTION

System identification is of great importance in many engineering area. Modal identification, developed in vibration engineering, is to build modal model, described via modal parameters, i.e. modal frequencies, damping rations and mode shapes. One of the major features of modal identification is that it can make use of Frequency Response Functions (FRFs) in frequency domain (FD), or Free Decay Response (FDR), Impulse Response Function (IRF), Correlation Function (COR), as well as Random Decrement (RDD) in time domain (TD). Free Decay Response (FDR) can be measured either from transient excitation or sudden termination of board band random excitation; IRF is the counterpart of FRF in TD, and can be calculated via inverse FFT; Correlation Functions (COR) can either be estimated directly from stochastic response, or calculated from Power Spectrum Density (PSD) via inverse FFT in the output only case; Random Decrement (RDD) signature was explained as free decay response in the beginning [1], [2], and then proved to be correlation function of the response, and can be computed through many ways from random response of the system [3], [4].

It has been proved that FDR, IRF, COR and RDD can all be expressed as summation of exponentially decayed sinusoids. Each one of these decaying sinusoids has a damped natural frequency and decay rate, which are identical to the one of the corresponding structural mode. Therefore, Time Response Function (TRF) is defined in this paper to represent all these TD features. It should be noted that in frequency domain FRF can only be estimated from both input and output (I/O) data. However, TRF can be obtained with either input/output (I/O) measurements or output only (O/O). Two-stage TD modal identification is to estimate time response function (TRF) as first stage, and then modal parameters are identified based on TRF data. Historically, Complex Exponential algorithm was one of the earliest multiple degree-of-freedom (M-DOF) 2-stage TD modal identification techniques(1974). The technique was improved via Least Squares solution and expended to Single Input Multi-Output (SIMO) case as Least Squares Complex Exponential (LSCE) algorithm in 1977 [5]. In the same year, well-known modal identification procedure Ibrahim Time Domain (ITD) was developed. ITD was formulated as SIMO technique [6]. However, it can easily be extended for MultiInput Multi-Output (MIMO) application [7]. The first MIMO version of modal identification algorithm, which was an important breakthrough in experimental modal analysis, was the technique known as Polyreference Complex Exponential (PRCE), as an extension of LSCE, developed in 1982 [8]. Eigensystem Realization Algorithm (ERA), based on system realization theory in linear system analysis, was derived in 1984 [9]. To reduce the influence of noise contamination in the TRF data, an improved PRCE called Improved Polyreference technique (IPCE), which makes use of correlation of the TRF data, was developed in 1987[10]. Data correlation version of ERA, as ERA/DC [11], followed in 1988. Due to data correlation, which acts as a correlation filter, the order of the model to be identified can be reduced and identification accuracy is increased. All aforementioned 2-stage TD modal identification techniques are normally making use of IRF obtained via inverse FFT of FRF data. Therefore, input and output (I/O)

measurements are required. In early 1990s, Natural Excitation Technique (NExT) was proposed for modal identification from output data only in the case of natural excitation [12]. NExT suggests to use correlation function (COR) of the random response of the structure subjected to natural excitation, which often is broadband stochastic process. All the MIMO versions 2-stage TD modal identification procedures mentioned above can be used as NExT. Based on this idea, the two-stage modal identification techniques can be adopted not only for traditional modal analysis, but also for ambient or operational modal analysis. A q-Markov Covariance Equivalent Realization (QMCover, or QMC) approaches was developed, which makes use of both IRF and Auto-correlation (A-COR) of the random response [13]. So called Observability Range Space Extraction approach employs both A-COR and Cross-correlation (CCOR) of both input and output (I/O) measurements [14] as one of covariance driven subspace state-space system identification (4SID) techniques, which make use of covariance, i.e. correlation function with zero-mean stochastic response, developed in 1990’s. All aforementioned 2-stage TD modal identification techniques are based on all kinds of mathematical formulations, range from continuous-time second and first order matrix differential equations, discrete-time state-space equations to innovation (or Kalman filter) state-space formulation. A unified 2-stage TD modal identification framework has been developed during a re-visit to modal identification developed in the last three decades. The unified approach is based on the formula of modal decomposition or system matrices decomposition of TRF, and can cover all aforementioned TD modal identification algorithms. Under the unified framework, we may have better understanding of all different implementation, the relation and features of different techniques, as well as major issues for the two-stage TD modal identification will further be discussed in the paper. 2.

UUFIED FORMULATION FUNCTION (TRF)

OF

TIME

RESPONSE

Dynamic behavior of a linear time-invariant (LTI) structural system is traditionally described via second order matrix differential equations,

Mξ(t ) + Dξ(t ) + Kξ (t ) = bf (t ) y (t ) = cξ (t )

(2-1)

Where f(t), y(t) and ξ(t), are input, output and displacement state vector with dimensions of l, m and N, respectively. M, K and D are N×N mass, stiffness and damping matrices, respectively. b, c is N×l input influence matrix and m×N output influence matrix, respectively. The dynamic equation of the LTI structural system can also be expressed as first order form, or state-space equations,

x (t ) = Ac x(t ) + Bc u (t ) y (t ) = Cx (t )

(2-2)

Where

ξ  x =   , ξ 

 0 A= −1 − M K

I  , − M −1C 

 0  Bc =  −1 , C = [c 0]  M b The free decay response (FDR), impulse response function (IRF), cross and auto correlation matrices can be derived from first order equation as follows

y (t ) x = Ce Act x0

(2-3a)

H (t ) = Ce Act Bc

(2-3b)

0

R yu (t ) = Ce Bc Ruu (0) Ac t

(2-3c) ∞

R yy (t ) = Ce Act B R , B R = α ∫ e Acσ Bc BcT e Acσ dσ ⋅ C T 0

(2-3d) The state-space equation of the LTI structural system can be written in modal space making use of the modal matrix as a linear transformation matrix. Therefore, the above time domain response can be expressed as modal decomposition,

y (t ) x = Φe Λt γ 0 , γ 0 = Ψ −1 x0

(2-4a)

H (t ) = Φe Λt Γ T

(2-4b)

0

Λt

R yu (t ) = Φe Γ Ruu (0) T

(2-4c) ∞

R yy (t ) = Φe Λt ΓRT , ΓRT = α ∫ e Λσ Γ T Γe Λσ dσ ⋅ Φ T 0

(2-4d) Eigenvalue and eigenvector matrices obtained from corresponding homogeneous state equation can be written as

Λ 0  Ψ  2 N ×2 N Λ =  *  ∈ C 2 N ×2 N , Ψ =   ∈C Ψ Λ  0 Λ  Φ = cΨ and Γ = bT Ψ are mode shape matrix and modal participation factor matrix corresponding to m measurements and l inputs respectively.

A unified Time Response Function (TRF) is defined in this paper to represent free decay response (FDR), Impulse response function (IRF), as well as cross and auto correlation function (COR) in general. Sampling the time response series and letting t=tk=k∆t, where fs=1/∆t is the sampling frequency, the Time Response Function (TRF) can be written in generalized discrete form in physical coordinate or modal space,

Yk ≡ Y ( k ) = CA k B g

(2-5a)

Yk ≡ Y ( k ) = ΦZ k ΓgY , Z = e Λ∆t

(2-5b)

3. UNIFIED 2-STAGE TD MODAL ID BASED ON MODAL DECOMPOSITION OF TRF

The procedure of the unified 2-stage TD modal Identification based on Modal Decomposition of TRF as follows. Making q block row shift of the basic Eq. (2-5b), when the number of measurements is smaller than modes to be identified, yields

Y0   Φ  Y     1  =  Φ Z ΓT  #   #     q   Y q  Φ Z 

(3-1)

A1 " Aq −1

]

can be estimated via Least

parameters contained in Z and Φ via block matrix form of Eq. (3-3)

− A1 " − Aq −1

 Φ   ΦZ    = ΦZ q  #   q −1  Φ Z 

]

(3-2)

(3-8)

A standard eigenvalue problem can be formulated by constructing a companion matrix of the AR coefficient matrices

Assuming there are N-pair of complex modes, and mp=2N, ~ therefore the matrix Φ has full row rank of mq. Since the ~ m(1+q) × mq matrix Φ has m more rows than columns, ~ there must exist a m × m(1+q) matrix A so that

~~ AΦ = 0

0

Having the estimates of the system characteristics matrix [A0 A1 " Aq −1 ] , we can build its relation with modal

0

Y 0   Φ  Y   ΦZ  ~ ~ 1  where H =   , Φ =  #   #     q Φ Z  Yq 

[A

Squares (LS) technique, when p is selected to satisfy the relation of lp> mq.

[− A

Or in compact form

~ ~ H = ΦΓT

The matrix

(3-3)

 0  0   0   # − A0 

I 0

0 I

0

0

# # − A1 − A2

0  Φ   Φ  0   ΦZ   ΦZ      " 0   ΦZ 2  =  ΦZ 2  Z     % #  #   #  " − Aq −1  ΦZ q −1  ΦZ q −1  " "

(3-9)

From Eq. (3-1) we can obtain the following equation ~~ AH = 0

(3-4)

~

Partitioning matrix A into (1+q) blocks m × m matrices yields

[A

0

A1

"

Aq

Y 0 Y  1  #   Y q

]

   = 0   

(3-5)

[A

A1 " Aq −1

0

(3-6)

]

In order to solve AR coefficient matrix m×m matrix Ai (i=0,1,", q-1), an over determined equation could be

~ arranged via block column shift of matrix H as follows

[A

A1 " Aq −1

0

[

 Y0 Y1 Y Y 2  1  # #  Yq −1 Yq

]

= − Yq Yq +1 " Yp + q −1

]

Yp −1  " Yp  % #   " Yp + q − 2 

βr =

([

])

 ℜ( z r )  2 2 1 1 ln ℜ( zr ) + ℑ( zr ) tan −1  , α r = ∆t ℑ ( z ) 2 ∆ t r  

] [

where λr = −α r + jβ r , zr = ℜ( zr ) + jℑ( zr ), r = 1,2," N

An multivariable AutoRegression (AR) equation is therefore obtained, which can be normalized as Aq=I and written as

 Y0  Y   1  = −Y q  #    Yq −1 

From the eigenvalue matrix Z of the above eigensolution, the complex modal frequency matrix Λ=diag[λr] can be calculated by the following formulas

"

(3-7)

Modal frequencies and damping ratios can then be computed from damped natural frequencies βr and decay rates αr, r=1, 2, ",N. The modal matrix can be obtained from the first m lines of the eigenvector matrix. To obtain full sets of modal parameters, the MPF matrix Γ should be calculated. Again, over determined equation could be formulated to compute MPF matrix from measured TRF ~ data and estimated Φ by LS solution

 Y0   Φ   Y   ΦZ   1 = ΓT  #   #     s −1  Y s −1  Φ Z 

(3-11)

Where ms>2N is required to form LS solution. Another way of manipulating the modal decomposition of the TRF is to make block column shift instead of row shift

[Y

0

[

Y1 " Y p ] = Φ Γ T

ZΓ T

" Z p ΓT

]

(3-12)

Where lp=2N is assumed. In light of the exact same argument, and following the same procedures, we can derive the equation for AR coefficient matrix

 h0 h  1  #  hq −1

h1 h2 # hq

h p −1   A0   hp     h  " hp A   1  = −  p +1   #  % #  #      " h p + q − 2   Ap −1  h p + q −1  "

(3-13)

We can also have the standard eigenvalue problem for Modal frequencies, damping ratios and, instead of modal matrix, MPF matrix

 0  0   0   #  − A0 T

I

0

"

0

I

"

0

0

"

# − A1

# T

− A2

% T

"

 Γ   Γ  0   ΓZ   ΓZ      0   ΓZ 2  =  ΓZ 2  Z  #  #   #      T − Ap −1   ΓZ p −1   ΓZ p −1  0

(3-14) The modal matrix in this case can be obtained from the following linear equations,

[Y0

[

Y1 " Ys −1 ] = Φ Γ T

ZΓ T

" Z s −1 Γ T

]

(3-15)

 0 I 0 " " 0   Y0 Y1  0 I "" 0   Y 0 Y2 1  0 0 0 "" 0    #  # # # #  #  Y Y  − A0 − A1 − A2 " − Aq −1   q −1 q Y Y " Y  1  2 p Y Y " Y p +1  2 3 =  #  Y q Y q +1 " Y p + q −1 

Y p −1  Yp   % #   " Y p+ q−2  " "

Or in compact form

AI H 0 = H 1

(4-1) (3-14)

where

 Y0 Y1 Y Y2 1 H0 =  #  #  Y q −1 Y q

Y p −1  " Yp  Y1 Y 2 Y Y Yp  " Y p +1  2 3 , H =   1 % #  #   Y q Y q +1 " Y p + q −1  " Y p+ q−2 

" "

System matrix AI can then be solved by normal equation

[

][

]

−1

[

][

−1

AI = H 1 H 0T H 0 H 0T

In summary, the procedure for the unified 2-stage TD modal identification based on modal decomposition of the TRF can be implemented in two approaches. The first one actually is an MIMO extension of Ibrahim Time Domain (ITD) technique, and the other is exactly the same as Polyreference Complex Exponential (PRCE) technique, which is an extension of SIMO Least Squares Complex Exponential (LSCE) derived from applying Prony’s algorithm.

This is exactly the ITD implementation [6]. A double LS solution (DLS) was suggested as the average of the above two solutions [15].

The two approaches, EITD and PRCE-types, consist of three steps. (1) Solving an over determined linear equations by LS technique to estimate AR coefficient block matrices with dimensions of m×m, or l×l, where m and l are number of outputs and inputs, respectively; (2) To calculate modal frequencies and damping ratios, and mode shapes, or MPFs, by solving standard eigenvalue problems; (3) To obtain MPFs or mode shapes, another linear equation should be solved via LS estimation (LSE). 4. NUMERICAL IMPLEMENTATIONS FOR TD MODAL ID BASED ON MODAL DECOMPOSITION OF TRF

(4-2)

It should be noted that when the number of response locations is less than the number of modes, i.e. m2N is required.

0 0 0 # 0

I 0 0 # 0

0 I 0 # 0

" " − A0  "" − A1  "" − A2  #  "" − Aq −1 

The similar LSE techniques can be applied.

(4-3)

4.2 LSE via RQ Decomposition

are referred to Observability and Controlability matrix, respectively.

The direct utilization of normal equation is sometimes unstable due to the condition number of the data matrix. The more accurate way for LSE technique is making use of orthogonal decomposition

Q1  H 0 = [R 0][Q ] = [R 0]  = RQ1 Q2 

(4-4)

Where H0 is an mp×lq data matrix, Q is an lp×lp orthogonal matrix, and R is of dimension mp×mp, or 2N×2N, lowertriangular matrix with positive numbers in its diagonal. Orthogonal decomposition can numerically be implemented in one of the following methods: (1) Householder transformation, (2) Given method, or (3) improved GramSchmitt orthogonal method. 4.3 LSE via SVD

(1) One of the implementations is based on Eq. (5-1). The data matrix H0 is decomposed via SVD at first as Eq. (4-5). The system matrix AE can then be estimated via LSE as follows,

A = AE = Σ −s 1 / 2U sT H 1V s S s−1 / 2

(5-5)

From Eq. (5-2), the input and output inference matrices Bg, C of the state-space equation can also be obtained from the SVD of the Hankel matrix H0. (2) The other implementation is to decompose an extended Hankel matrix with q+1 block rows. Then Pq and

Pq are defined as

Another possible implementation of LS solution with illconditioned coefficient matrix, in the first step of the 2-stage TD modal identification based on modal decomposition of TRF is to adopt Singular Value Decomposition (SVD) of the TRF data matrix H0 (4-5)

H 0 = UΣV T = U s Σ sVsT

Where Σ is diagonal singular value matrix U, V are orthogonal matrices consists of corresponding left and right singular vectors. Σs is the sub-matrix with first s dominate singular values, Us and Vs are the partitions with corresponding singular vectors. The system matrix for EITD technique AI can then be estimated as

AI = H1H 0+ = H1Vns S ns−1U nsT

(4-6)

5. UNIFIED 2-STAGE TD MODAL ID BASED ON SYSTEM DECOMPOSITION OF TRF 5.1 System Realization with Two Implementations The 2-stage TD modal identification can also be implemented based on system matrix decomposition of the time response function (TRF), or TRF in physical coordinate as Eq. (2-5a). The data matrix H is called as Hankel matrix in system realization theory. It can be shown that the two Hankel matrices can be written as

H 0 = PQ ,

There are two possible implementations for calculation of system matrix:

H 1 = PAE Q

(5-1)

[

(5-3)

The system matrix AE can then be calculated via general inverse +

AE = P q Pq

(5-4)

From system realization point of view, there could be infinite solution for the system matrices, i.e. matrix triple [A, Bg, C], due to non-uniqueness of the inverse problem. Between different realizations, there exists a similarity transformation which corresponding to different coordinates of the statespace equation. Since the matrix triple [A, Bg, C] can be identified in the first step, all the modal parameters, i.e. modal frequencies, damping rations, mode shapes and MPF matrix can be calculate from matrix triple [A, Bg, C]. Therefore, the 2-stage procedures based on system matrices decomposition of TRF need only two-step, instead of 3-step procedures base on modal decomposition of TRF. 5.2 The Relation between EITD, PRCE and ERA The first system identification algorithm based on system realization for modal identification, known as Eigensystem Realization Algorithm (ERA) [9], makes use of the first implementation. It is interesting to observe that there is a similarity transformation between system matrices estimated from ERA and SVD implementation of EITD

Where

 C  AC  p −1 P =  , Q = B g AB g " A B g #  q −1  A C

CA  C  CA2  CA  Pq =  , P =   q # #  q  q −1  CA  CA 

]

(5-2)

AE = T −1 AI T , where T = U s Σ −s 1

(5-6)

The eigenvalues calculated for two system matrices are exactly the same; the two eigenvector matrices are related by the linear transformation.

~ ~ ~ Φ E = T −1Φ I = Σ s−1 / 2U sT Φ I

(5-7)

It should be noted that PRCE can be classified as observability or controllability system realization, and the EITD is linear regression implementation from system realization point of view. 5.3 QMC and HQC Approaches as 2-stage TD Modal ID Impulse response function (IRF) and cross-correlation (CCOR) estimated from I/O measurements, or auto-correlation (A-COR) calculated from O/O data can be employed for system realization algorithms. A q-Markov covariance equivalent approach (QMC) was developed using both IRF and COR [13 ]. With I/O measurements, ERA/DC and Hankel QMC (HQMC) were proposed to use data correlation to further reduce noise contamination in the IRF data [11], [16]. Another stochastic realization approach, in termed of Observability Range Space Extraction (ORSE) [14] makes use of the following matrix as time response function (TRF):

R ( k ) = R yy ( k ) − R yu ( k ) Ruu−1 ( k ) Ruy ( k )

(5-8)

6. FUTHER DISCUSSION In the above three sections, most TD modal identification approaches using FDR, IRF, C-COR or A-COR, RDD or their data correlation or combinations, have been put in a unified framework. The relationship between some algorithms and implementations has been discussed. 1. With Input/Output measurement (I/O case), normally IRF data is estimated in the first stage, and employed for modal identification via different implementations based on either modal decomposition or system decomposition of TRF. It is interesting to notice that C-COR of I/O data with A-COR of input data can also be used for modal identification in the second stage based on Eqs. (2-3c) and (2-4c). Adopting formula in Eq. (5-8) as generalized TRF yields ORSE technique. 2. We can use free decay response (FDR) measurement for the first stage as one of the output only (O/O) cases. It should be noted that multi-output measurements with respect to one set of initial condition is equivalent to SIMO system. However, In this case, γ 0 = Ψ −1 x0 is no longer MPF. Moreover, in order to handle closely spaced or even repeated modes, multiple sets of initial conditions are required during the first stage. 3. In the O/O measurements of a structural system subjected to stochastic, ambient or operational, excitation, A-COR matrix is employed in second stage. It is important to aware that the input is not only temporary white noise,

 R (0), k = 0 Ruu ( k ) =  uu k ≠0 0,

(5-9)

but also white spatially, i.e.

Ruu (0) = αI

(5-10)

where α is a constant scalar. 4. In the O/O case, the Γg is no longer the MPF matrix. When making use of EITD procedure, only first two steps are needed. While using PRCE procedure, the third step is still required to calculate mode shapes. 5. The unified formula of Eq (2-5a) developed in this paper are based on continuous-time (C-T) domain state-space formulation. The ERA, however, is based on discrete-time (D-T) domain counterpart. It should be aware the difference between the two. In ERA, the input is supposed to be unit pulse function in D-T domain, which is different from the unit impulse function in C-T domain state-space formulation. The A-COR function in this case is written as

R yy ( k ) = CA k −1 B

(5-11)

Therefore the D-T input inference matrix B obtained from ERA is different from C-T one, and cannot be used to calculate MPF matrix directly! 6. In the O/O system subjected to random excitation, stochastic state space system identification (4SID) techniques have been proposed which make use of A-COR data estimated at the first stage. The all 4SIT techniques are developed upon so called innovation D-T state space formulation with Kalman Filter state zk instead of xk as state variable.

R yy ( k ) = CAk −1G zy (1)

(5-12)

where

Gzy (1) = E {zk +1 ykT } Application of Eqs. (2-5a) and (2-5b), derived in C-T statespace equation makes the modal identification in the O/O case much easier. 7. As TD modal identification techniques, structural mode sorting is still a serious issue remain to be solved for the 2stage TD modal identification approach. The number of computational modes, predefined in TD modal identification, should be much larger then the number of structural modes. Not only because the number of structural modes is normally unknown, but due to the need of accommodating all unwanted effects in the measured TRF data, e.g. input and response noise, leakage, residuals, non-linearity, etc. Therefore, after the entire modes have been identified, the true structural modes should then be distinguished from other computational modes, or fictitious modes, used for compensation of these unwanted effects. . There are three relevant approaches for modal sorting, i.e. to distinguish structural modes from computation ones. First, the total number of modes should be defined before

identification. Two indication techniques are available, which are based on Error Chart and Rank Chart, respectively. The second approach for modal sorting is making use of Stability Diagram, a plot of modal frequency points, versus the model order. The structural modes are supposed to converge with the increasing of the model order. The convergence criterion can be the difference of subsequent two modal frequencies, or/and modal damping, and or/and mode shapes, represented by Modal Assurance Criterion (MAC). The third method of distinguish structural modes from computational modes is to adopt specific modal indicator. Several modal indicators were proposed for different modal identification algorithms, e.g. Modal Confidence Factor (MCF) for ITD [17], extended latter for PRCE [18], Modal Amplitude Criterion (MAC) and Modal Phase Collinearity (MPC) [19], among others. An effort was made to extend all modal indicators mentioned above plus a Modal Partition Indicator (MPI) for unified two-stage modal identification [20]. Unfortunately all these approaches can not be used to distinguish the structural modes robustly and reliably. 8. Accuracy is another important issue for 2-stage TD modal identification. The essence of the 2-stage modal identification approach is actually a Least Squares Estimation (LSE), no mater employ normal equation, QR decomposition or SVD. The advantages of LSE are noniterative and fast in computation. However, there is a serious drawback in LSE, i.e. it could possibly cause bias error in the estimates. In the LSE, a prediction error, or residual, ε k = Yk − Yˆk ( k = 1, 2,") is assumed when IRF, FDR and COR are directly used as TRF to form the data matrix. It is well know that LSE would be unbiased only if the prediction error is white noise. However, in reality, it would never be such a white noise; even the system corrupted only by output measurement noise and it is white! Therefore, bias error caused by color prediction error becomes one major problem in the LSE-based 2-stage modal identification. There are two possible ways to overcome bias problem caused by LSE: one is to properly model the noise (noise modeling methods); the other is to eliminate bias error without noise modeling.

ERA, ERA/DC, QMC, and ORSE, etc. Much better understanding for different techniques can be obtained under the unified framework. Estimation accuracy and/or efficiency can be increased when numerical considerations are taken care of in the implementation. 4.

The unified 2-stage TD modal identification with many different implementations can be applied to real world complex structures to identify full sets of modal parameters, including modal frequencies, damping ratios, scaled mode shapes and modal participation factors with MIMO I/O measurements. It can also be utilized for ambient or operational modal analysis using free decay response, or correlation functions from output measurements only to obtain modal frequencies, damping ratios and un-scaled mode shapes.

5.

The two-stage TD modal identification, after almost three decades development, seems matured. However, several issues are still remaining to be resolved. For examples, the influence of data selection to the estimation uncertainty (with both variance and bias errors); the modal sorting, i.e. distinguishing true structural modes from spurious ones; the accuracy and further reduction of bias and variance error in the estimates while dealing with array of measurements from large complex structures, or very noisy data from operational or ambient measurements.

6.

This paper summaries the two-stage modal identification in time domain. It’s counterpart in frequency domain and one-stage modal identification based on state-space model will be followed as the outcomes of re-visiting to the modal identification developed in the last three decades. ACKNOWLEGEMENT

The author of the paper would like to express his appreciation for the support of the National Natural Science Foundation of China (NSFC) and Aeronautical Science Research Foundation (ASRF).

7. CONCLUDING REMARKS 1.

2.

3.

A generalized Time Response Function (TRF) is defied to represent the impulse response function (IRF), crosscorrelation function (C-COR) with the Input/Output (I/O) measurements, as well as free decay function (FDR), auto-correlation function (A-COR) of the stochastic response and random decrement (RDD) signature, based on the observation that all these time response function can be expressed as summation of exponentially decayed sinusoids; A unified framework for two-stage time domain modal identification has been developed based on unified formulas of modal decomposition or system matrix decomposition of the TRF; There are variety of possible implementations for the 2stage TD modal identification, including well-known techniques, such as ITD, EITD, LSCE, PRCE, IPCE,

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