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System identification of linear structures based on Hilbert-Huang spectral analysis 2


EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2003; 32:1533–1554 (DOI: 10.1002/eqe.288)

System identi?cation of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes
Jann N. Yang 1; ?; ? , Ying Lei 1 , Shuwen Pan 1 and Norden Huang 2
1 Department 2 Laboratory

of Civil and Environmental Engineering; University of California; Irvine; CA 92697-2175; U.S.A. for Hydrospheric Processes; NASA Goddard Space Flight Center; Greenbelt; MD 20771; U.S.A.

SUMMARY A method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are ?rst decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identi?ed using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and sti ness matrices of the structure. The e ectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identi?ed e ectively using the proposed method. Copyright ? 2003 John Wiley & Sons, Ltd.
KEY WORDS:

system identi?cation; Hilbert transform; linear structures; complex modes; data analysis; Hilbert–Huang spectral analysis

1. INTRODUCTION The Hilbert transform has been applied directly to identify the modal parameters of SDOF structures with very small damping [1; 2]. Based on the Hilbert–Huang transform [3; 4], different methods have been proposed by the authors to: (i) identify natural frequencies and
?

Correspondence to: Jann N. Yang, Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697-2175, U.S.A. ? E-mail: jnyang@uci.edu Contract=grant sponsor: National Science Foundation; contract=grant number: CMS-98-07855. Contract=grant sponsor: NASA; contract=grant number: NAG5-5149.

Copyright ? 2003 John Wiley & Sons, Ltd.

Received 25 January 2002 Revised 10 August 2002 Accepted 14 November 2002

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J. N. YANG ET AL.

damping ratios of in-situ tall buildings [5], (ii) detect damages in MDOF structures [6], and (iii) identify the complete structural dynamic characteristics of MDOF linear structures in which normal modes are assumed to exist, i.e., real eigenvalues and real eigenvectors [7]. For general MDOF linear structures, all the eigenvalues and eigenvectors (mode shapes) may be complex, see, for example, References [8; 9]. In this paper, the method proposed in Part 1 [7] is further extended to identify the general MDOF linear structures with complex modes, using the measured free vibration data that are polluted by noise. In the proposed approach, measured free vibration signals are ?rst decomposed into modal responses using the Empirical Mode Decomposition (EMD) method with intermittency criteria. A modal response is contributed by a complex conjugated pair of eigenvalues and eigenvectors with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude as functions of time using the Hilbert transform. Based on one appropriate measurement of the impulse response time history at one location, the complex eigenvalues, including the natural frequencies and damping ratios, can be identi?ed using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and sti ness matrices of the general MDOF linear structure. Simulation studies have been conducted using two MDOF systems with large damping coe cients to illustrate the e ectiveness of the proposed method. The accuracy of the proposed method in identifying the complete dynamic characteristics of linear structures with complex modes is demonstrated to be quite reasonable.

2. MODAL RESPONSE OF MDOF STRUCTURES The equation of motion of a general n-DOF structure can be expressed as ˙ MX + CX + KX = F (1)

in which X(t) = [x1 ; x2 ; : : : ; xn ]T = n-displacement vector, F = n-excitation vector, and M; C and K are (n × n) mass, damping and sti ness matrices, respectively. In the state-space, Equation (1) can be expressed as ˙ AY + BY= G
or

˙ Y = DY + A?1 G

(2)

in which Y(t) = [x1 ; x2 ; : : : ; xn ; x1 ; x2 ; : : : ; xn ]T is a 2n state vector and ˙ ˙ ˙ Y= X ˙ X ; A= C M 0
?M?1 K

M 0

; B= I
?M?1 C
j

K 0

0
?M

; G=

F 0

;

D = ?A?1 B = Complex eigenvalues

(3)

j

and eigenvectors D
j

can be obtained from the system matrix D as (4)

=

j

j;

j = 1; 2; : : : ; 2n

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in which eigenvalues and eigenvectors are all in complex conjugate pairs, and
j

=
j

j j

(5)

where j is an n complex vector. Let be a (2n × 2n) complex matrix consisting of j as the j-th column and be a (n × 2n) complex matrix consisting of j as the j-th column. Then, the response of the state vector can be expressed as Y= q =
2n j qj (t) j=1

(6)

in which q is the generalized modal coordinate vector with the j-th complex element qj (t). Substituting Equation (6) into Equation (2) and using the orthogonal properties of complex mode shapes, one can decouple Equation (2) into ˙ a j q j + b j qj = in which
T j A j T j G

(7)

= aj ;

T j B j

= bj ;

j

= ?bj =aj ;

j = 1; 2; : : : ; 2n

(8)

The acceleration response of the structure can be expressed as X= q=
2n j=1 j qj (t)

(9)

in which j is given in Equation (5). By applying an impact loading F0 (t) at a particular location of the structure, e.g., the k-th DOF, the responses qj (t) and qj (t) can be obtained from Equations (7) and (8) as qj (t) = where
kj

F0 kj j t e ; aj
j

qj (t) = Bkj e

jt

(10)

is the k-th element of

and Bkj = F0
2 j kj

aj

(11)

Eigenvalues j and eigenvectors conjugates, e.g.,
j

j

as well as aj , bj , qj (t) and qj (t) are all n pairs of complex
n+j

= ? j !j + i!dj ;

=

? j

= ? j !j ? i!dj

j = 1; 2; : : : ; n

(12)

√ where i = ?1, !j = j-th modal frequency, j = j-th modal damping ratio, and !dj = 2 !j (1 ? j )1=2 . Hence, if eigenvectors are arranged such that the last n columns are complex conjugate pairs of the corresponding ?rst n columns, i.e., = [ 1 ; 2 ; : : : ; n ; 1? ; 2? ; : : : ; n? ]T where
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? j

J. N. YANG ET AL.

is the complex conjugate of j , then the acceleration response xp (t) of the structure at p(p = 1; 2; : : : ; n) DOF follows from Equations (9) and (10) as
2n 2n pj qj (t) = pj Bkj e j=1

xp (t) =
j=1

jt

=

n

xpj (t)

(13)

j=1

where xpj (t) is the j-th modal response given by xpj (t) = Rpj; k e? j !j t cos[!dj t + ?( Rpj; k = 2|
pj | |Bkj | pj )

+ ?(Bkj )]

(14) (15)

In Equations (14) and (15), ?( pj ) and ?(Bkj ) are phase angles of the complex quantities and Bkj , respectively, and | pj | and |Bkj | are their corresponding amplitudes. The measured acceleration response vector Z(t) = [z 1 (t); z 2 (t); : : : ; z n (t)]T is polluted by noise, i.e.,
pj

Z(t) = X(t) + V(t)

(16)

where V(t) = [v1 (t); v2 (t); : : : ; vn (t)]T is a white noise vector in which each element, say vp (t), is a band-limited Gaussian white noise process. The measured acceleration response zp (t) at the p-th DOF is given by zp (t) = xp (t) + vp (t) =
n

xpj (t) + vp (t)

(17)

j=1

in which xpj (t) is given by Equations (14) and (15).

3. DETERMINATION OF MODAL RESPONSE USING THE EMD METHOD As presented in Reference [7], the modal response xpj (t) can be obtained by processing the measured data zp (t) through the Empirical Mode Decomposition (EMD) with appropriate intermittency criteria. Hence, the measured response zp (t) in Equation (17) can be decomposed into n modal response functions and m ? n Intrinsic Mode Functions (IMFs) cpj (t) as follows [7]: zp (t) ≈
n

xpj (t) +

m?n j=1

cpj (t) + rpm (t)

(18)

j=1

where rpm (t) is the residue. The isolation of modal responses using the EMD method has a signi?cant advantage in that the frequency content of the signal at each time instant t has been preserved. However, the numerical computation based on this approach may be quite involved if the modal frequency is high. To simplify the computational e orts for high frequency modal responses, an alternative approach based on the band-pass ?lter and EMD has also been proposed [7], and detailed procedures have been presented in Reference [7].
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4. HILBERT TRANSFORM AND SYSTEM IDENTIFICATION Taking the Hilbert transform of xpj (t) in Equation (14) and forming the analytic signal Ypj (t) of the j-th mode [7] ? Ypj (t) = xpj (t) + ixpj (t) = Apj (t) ei?pj (t) (19)

one obtains the instantaneous amplitude Apj (t) and phase angle ?pj (t) of the signal. The results for Apj (t) and ?pj (t) are identical to Equations (19)–(23) of Reference [7], except that Bpj; k and ’j + ( 2 ) + ’pj; k should be replaced by Rpj; k and ?( pj ) + ?(Bkj ), respectively, in these equations. For a special case in which j is very small and !j is large, one obtains [7]: ? xpj (t) = Rpj; k e? j !j t sin[!dj t + ?( Apj (t) = Rpj; k e? j !j t ; It follows from Equation (21) that ln Apj (t) = ? j !j t + ln Rpj; k ; !j (t) = d ?pj (t)= d t = !dj (22)
pj )

+ ?(Bkj )]
pj )

(20) (21)

?pj (t) = !dj t + ?(

+ ?(Bkj )

For a general case in which j is not small, the solution for Apj (t) and ?pj (t) can be obtained in a similar manner as given by Reference [7]. Consequently, each decomposed modal response xpj (t) can be processed easily through the Hilbert transform to obtain the amplitude Apj (t) and phase angle ?pj (t) numerically as functions of time t. 4.1. System identi?cation With the measured impulse response vector Z(t) = [z 1 (t); z 2 (t); : : : ; zp (t); : : : ; z n (t)]T given by Equation (16), either the EMD method with appropriate intermittency or the combination of the EMD method and band-pass ?lter can be used to decompose each measurement zp (t) into n modal responses, xpj (t) for j = 1; 2; : : : ; n, as shown in Equations (14) and (17). Then, each modal response xpj (t) thus obtained will be processed through the Hilbert transform to determine the instantaneous amplitude Apj (t) and instantaneous phase angle ?pj (t) numerically. For the identi?cation of natural frequencies !j and damping ratios j for j = 1; 2; : : : ; n, only one appropriate measured response (i.e., one sensor), say zp (t), is su cient. The procedures are described in the following. For small j , it follows from Equation (22) that the damped natural frequency !dj can be obtained from the slope of the phase angle ?pj (t) vs. time t plot, whereas ? j !j can be estimated from the slope of the decaying amplitude ln Apj (t) vs. time t plot, Equation (22). Consequently, linear least-square ?t procedures have been proposed in Reference [7] to estimate the mean values of natural frequencies and damping ratios. For the general case in which j is not small, the linear least-square procedures presented above are applicable as described in detail in Reference [7]. After identifying !j and j for j = 1; 2; : : : ; n, the complex eigenvalue j can be computed from Equation (12). To identify the complex mode shapes of a structure, the response time histories at all DOF should be measured. It follows from Equations (11), (13) and (22) that the ratio of
Copyright ? 2003 John Wiley & Sons, Ltd.

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1538 the absolute value of modal elements identi?ed as
|
pj |= |

J. N. YANG ET AL.

pj

and

qj

(p; q = 1; 2; : : : ; n) of the j-th mode can be (23)

qj | =

exp[Apj (t0 ) ? Aqj (t0 )]

in which Apj (t0 ) and Aqj (t0 ) are the magnitudes at time t = t0 of the least-square straight lines of ln Apj (t) and ln Aqj (t), respectively. Further, the di erences between the phase angle of the modal element pj and that of qj follows from Equation (21) as ?(
pj )

? ?(

qj ) = ? (t0 ) pj

? ?qj (t0 )

(24)

in which ?pj (t0 ) and ?qj (t0 ) are the magnitudes of the least-square straight lines of the phase angles ?pj (t) and ?qj (t) at time t = t0 , respectively. In Equations (23) and (24), t0 may be taken as a middle point in the range of time in which data for ?pj (t) and ln Apj (t) are available. Thus, both the absolute values and phase angles of all modal elements relative to an arbitrary element in the complex modal vector j have been determined. After identifying the complex eigenvalues j and the modal vectors j for j = 1; 2; : : : ; n, the complex eigenvectors j can be computed from Equation (5). The physical mass, damping and sti ness matrices of the structure can be determined using the following procedures. The amplitude of complex value aj can be determined from Equations (11), (15) and (22) as
|aj | =
2 2F0 | pj kj j | ; exp[Apj (0)]

j = 1; 2; : : : ; n

(25)

in which Apj (0) is the magnitude of the least-square straight line of the amplitude ln Apj (t) at time t = 0, and F0 is the level of impact loading, that can be measured during the test. The phase angle of aj can be determined again from Equations (11), (15) and (22) as
2 ?(aj ) = ??pj (0) + ?( j ) + ?( pj )

+ ?(

kj )

(26)

in which ?pj (0) is the magnitude of the least-square straight line of the phase angles ?pj (t) at 2 2 t = 0, ?( j ) is the phase angle of the complex value j , and ?( pj ) and ?( kj ) are the phase angles of the complex modal elements pj and kj , respectively. Then, the complex value of bj can be obtained from Equation (8) as bj = ? j aj (27) The mass, damping and sti ness matrices (M; C; K) can be determined using the orthogonal properties of complex modes in Equations (3) and (8). These equations can be expressed in the following matrix forms
T

K 0

0
?M

= diag[bj ];

T

C M

M 0

= diag[aj ]; for j = 1; 2; : : : ; 2n

(28)

We determine the mass matrix M and the sti ness matrix K from the inversion of the ?rst equation in Equation (28) K 0 0
?M

=

?T

diag[bj ]

?1

(29)

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Although the damping matrix C can be determined from the second equation of Equation (28) by a straightforward inversion, the identi?ed matrices may not satisfy the equation of motion, due to estimation error. Here, we shall estimate C from the equation of motion in Equation (1). Substituting Equations (9)–(12) into Equation (1) for the free vibration F = 0, one obtains
2

M[

?

]

?2

+ C[

?

]

?

+ K[

?

]=0
j

(30) and
2 j,

in which and 2 are (n × n) diagonal matrices with the diagonal elements respectively. From Equation (30), one obtains M
2

+C

+ K =0
?1 ?1

(31)

Thus, the damping matrix C is estimated from Equation (31) as C = ?{M
2

+K }

(32)

Owing to estimation errors for the mode shapes and j , the sti ness and damping matrices K and C estimated from Equations (29) and (32) may not be symmetric. Consequently, the symmetry of K and C is imposed by using the averaging process K= K + KT ; 2 C= C + CT 2 (33)

In many circumstances, the masses of the system can be estimated more accurately. Hence, the mass matrix M can be assumed to be known, and only K and C should be identi?ed. In this situation, it follows from Equation (4) that D j = j j where D is given by Equation (3). Using Equations (3) and (4), K and C can be identi?ed as follows: [M?1 K M?1 C]
j

=?

2 j

j;

j = 1; 2; : : : ; 2n

(34)

An alternative approach to evaluate the mass and sti ness matrices M and K more accurately was presented by Liu [10]. The second and third quadrants of the ?rst equation of Equation (28) can be expressed as follows
T ?T

K
?

?
?

T ?T

M M

= diag[bj ]; =0

j = 1; 2; : : : ; n

(35a) (35b)

K

in which Equation (5) has been used. These two complex equations can be expressed by (4n × n) real equations using the following substitutions =
R

+i

I;

=

R

+i

I

(36)

and in which ( R ; R ) and ( I ; I ) and are the real and imaginary parts of matrices , respectively. With the resulting 4n × n equations, the 2n × n unknown elements of the sti ness and mass matrices M and K can be estimated based on the least-square method using MATLAB built-in function ‘MLDIVIDE’.
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J. N. YANG ET AL.

5. SIMULATION RESULTS To demonstrate the e ectiveness and accuracy of the system identi?cation methodology presented above, two numerical examples with complex modes are considered in the following. 5.1. System identi?cation of a 3-DOF building model A 3-story shear-beam type building model, as shown in Figure 4(a) of part 1 [7], with the following parameters is considered. The mass and sti ness of each story unit are identical with mj = 1000 kg and kj = 9:8 kN= m, for j = 1; 2; 3. The damping coe cients of each story unit are c1 = 7:035 kN · s= m, c2 = 2:814 kN · s= m, and c3 = 0:704 kN · s= m, respectively. In this case, the structure has complex modes. Suppose an impact loading is applied to the 2nd oor and the acceleration responses of all oors are measured. Based on the proposed identi?cation method, all natural frequencies !j and damping ratios j (j = 1; 2; 3) can be identi?ed using any one of the measured signals. The simulated time history x3 (t) of the third oor acceleration response without noises is shown in Figure 1(a). The noise level associated with each measurement z j (t) is expressed by Rpj = j = max |xj (t)| in which j is the rms value of the noise vj (t) associated with the measurement z j (t). In other
25 acc. m/s2 0 -25 (a) 25 acc. m/s2

0

1

2

3

4 5 6 7 time (sec)

8

9 10

0

-25 (b) 25 acc. m/s2

0

1

2

3

4 5 6 7 time (sec)

8

9 10

0

-25 (c)

0

1

2

3

4 5 6 7 time (sec)

8

9 10

Figure 1. Acceleration impulse response of 3-DOF structural model with complex modes: (a) response of 3rd oor x3 (t), (b) noise v3 (t), and (c) measured response of 3rd oor z 3 (t).
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8 amplitude (m/s) amplitude (m/s) 4 8 12 frequency (Hz) 16 20 (b) 6 4 2 0 0 (a)

8 6 4 2 0 0 4 8 12 16 frequency (Hz) 20

Figure 2. Fourier Transforms: (a) Fourier Transform of z 3 (t) without noise, and (b) Fourier Transform of z 3 (t) with 5% noise to peak signal ratio.

words, Rpj is the ratio of the noise rms to the peak signal. A sample function of the noise process v3 (t) associated with the measured acceleration z 3 (t) of the third oor is shown in Figure 1(b), in which the noise level is Rp3 = 5% with a bandwidth of 500 Hz. The measured acceleration z 3 (t), that is the sum of Figures 1(a) and (b), is shown in Figure 1(c). Note that x3 (t) in Figure 1(a) will also be considered as the measured data z 3 (t) where Rp3 = 0% (without noise pollution). The Fourier Transforms of z 3 (t) in Figures 1(a) and (c), i.e., without and with noise, are presented in Figures 2(a) and (b). As observed from Figure 2, there are three dominant frequencies around 2.2, 6.2 and 9 Hz, respectively. The empirical mode decomposition (EMD) method is capable of extracting all of the three vibration modes from any single measurement of the acceleration response time history. The following frequency ranges are used for intermittency criteria: (i) 7:7 Hz = !3L ?!3 ?!3H = 16:7 Hz for the 3rd mode, (ii) 4:3 Hz = !2L ?!2 ?!2H = 7:7 Hz for the 2nd mode, and (iii) 1:8 Hz = !1L ?!1 ?!1H = 4:3 Hz for the 1st mode. The EMD procedures used to obtain the modal responses have been described in detail in Reference [7]. The modal responses thus obtained are presented in Figures 3 and 4 as solid curves. Also shown in Figures 3 and 4 as dotted curves are the theoretical modal responses for comparison. It is observed that the correlation between the identi?ed modal responses and the theoretical modal responses is quite good except in the region near t = 0 due to the end boundary e ect (see Huang et al. [3; 4]). After removing a small segment near t = 0, the Hilbert transform is applied to the extracted modal responses in Figures 3 and 4 to obtain the corresponding instantaneous amplitudes A3j (t) in the natural logarithm scale and phase angles ?3j (t) for j = 1; 2 and 3. Linear leastsquare ?t procedures are used to ?t the phase angles ?3j (t) and amplitudes ln A3j (t). The plots of the phase angle ?33 (t) and amplitude ln A33 (t) as functions of time t for the third mode in Figures 3 and 4 are presented in Figure 5 as dotted curves for the cases with and without noise pollution, i.e., Rp3 = 0 and 5%. Also shown in Figure 5 as solid straight lines are the linear least-square ?ts. As expected, both ?33 (t) and ln A33 (t) oscillate around the straight lines. From the slopes of these straight lines, the natural frequency !3 and damping ratio 3 have been estimated. Repeating the same procedures for the other two modes, we obtain !j and damping ratios j for j = 1; 2 and 3. The results are presented in Columns (4)–(7) of Table I for both cases Rp3 = 0 and 5%. Also presented in Columns (2) and (3) of Table I are the theoretical frequencies and damping ratios of the building for comparison.
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15 acc. (m/s2) 5 -5 -15 0.0 (a) 10 acc. (m/s2) 5 0 -5 -10 0.0 (b) 10 acc. (m/s2) 5 0 -5 -10 (c) 0

J. N. YANG ET AL.

0.2

0.4

0.6 0.8 time (sec)

1.0

1.2

0.5

1.0 time (sec)

1.5

2.0

1

2

3

4 5 6 time (sec)

7

8

9 10

Figure 3. Impulse modal response of z 3 (t) with noise: (a) 3rd mode, (b) 2nd mode, and (c) 1st mode.

It is observed from Table I that the correlation between the theoretical values (!j and j ) and the identi?ed results is quite good. With the aid of Equation (12), the complex eigenvalues j for j = 1; 2; 3 have been computed. It should be noted that only one measurement of the acceleration response is su cient to identify all the natural frequencies and damping ratios, and the variation of these results by using di erent measurements is very small. The procedures described above have been applied to the other measurements z 1 (t) and z 2 (t) again with Rp1 = Rp2 = 0% and 5%, respectively, to obtain the corresponding three modal responses. With all the oor acceleration responses measured, all the mode shapes can be identi?ed using Equations (23) and (24). The time instant t0 on the linear least-square ?t line as indicated in Figure 5 is t0 = 0:35 sec. Note that t0 is used in Equations (23) and (24) to determine Apj (t0 ) and ?pj (t0 ). The amplitude and phase angle of the identi?ed complex modal (normalized) matrix of the structure is presented in Table II for Rpi = 0% and Rpi = 5%. The physical mass, sti ness and damping matrices (M; K; C) have been identi?ed using Equations (25)–(29), (32) and (33) where the level of the impact force F0 is measured to be 1000 N. The identi?ed results are presented in Table II, along with the theoretical values for comparison. When the mass matrix is known, the physical sti ness and damping matrices can be identi?ed using Equations (33) and (34), and the identi?ed results are shown in Table III.
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SYSTEM IDENTIFICATION OF LINEAR STRUCTURES
15 acc. (m/s2) 5 -5 -15 (a) 10 acc. (m/s2) 5 0 -5 -10 0.0 (b) 10 acc. (m/s2) 5 0 -5 -10 (c) 0 1 2 3 4 5 6 7 time (sec) 8 9 10 0.5 1.0 1.5 time (sec) 2.0 2.5 0.0 0.2 0.4 0.6 0.8 time (sec) 1.0 1.2

1543

Figure 4. Impulse modal response of z 3 (t) without Noise: (a) 3rd mode, (b) 2nd mode, and (c) 1st mode.

A comparison of the results in Tables II and III indicates that the accuracy of identifying K and C improves slightly when the mass matrix is known. Further, the modi?ed equations proposed by Liu [10] in Equations (35a) and (35b) are used to identify the M and K matrices, and Equation (32) is used to estimate the C matrix. The results are presented in Table IV, referred to as the modi?ed approach. A comparison of the results presented in Tables II and IV indicates that the accuracy of identifying the mass matrix M has been improved using the modi?ed approach. With known mass matrix and the shear-beam model, the sti ness and damping coe cients for all stories are determined from Table III using the least square estimation as follows: k1 = 942 kN= m, k2 = 969 kN= m, k3 = 993 kN= m, c1 = 6:362 kN · s= m, c2 = 3:178 kN · s= m, and c3 = 0:734 kN · s= m for the case without noise pollution (Rpi = 0%). The results for the case Rpi = 5% are: k1 = 1000 kN= m, k2 = 935 kN= m, k3 = 1002 kN= m, c1 = 6:884 kN · s= m, c2 = 3:406 kN · s= m, and c3 = 0:338 kN · s= m. Finally, a careful evaluation of the system parameters identi?ed above and in Tables II– IV demonstrates that all the identi?ed results, including the mode shapes, mass, damping and sti ness matrices, are reasonable as compared with the theoretical values, indicating the validity of the system identi?cation technique proposed in this paper.
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50 30 10 -10 0.0

J. N. YANG ET AL.

phase angle (rad)

phase angle (rad) 0.2 0.4 0.6 0.8 (c)

50 30 10 -10 0.0

t0

0.2

t00.4 time

0.6

0.8

(a) 3 ln amplitude 2 1 0 -1 0.0 (b) 0.2

time

3 ln amplitude 0.4 t0 time 0.6 0.8 (d) 2 1 0 -1 0.0 0.2 t0 0.4 time 0.6 0.8

Figure 5. Plots of phase angle and ln amplitude for 3rd mode of z 3 (t): (a) – (b) with noise, and (c) – (d) without noise.

Table I. Natural frequencies and damping ratios of 3-DOF building model. Mode Theoretical values Frequency (Hz) (2) 2.22 6.24 8.94 Damping ratio (%) (3) 3.47 6.26 7.30 Identi?ed values (Rp3 = 0%) Frequency (Hz) (4) 2.21 6.22 8.97 Damping ratio (%) (5) 3.53 6.27 7.57 Identi?ed values (Rp3 = 5%) Frequency (Hz) (6) 2.21 6.28 8.95 Damping ratio (%) (7) 3.66 6.50 7.09

(1) 1 2 3

5.2. System identi?cation of a 4-DOF mechanical system A 4-DOF mechanical system, as shown in Figure 4(b) of Reference [7], with the following properties: m1 = m2 = m3 = m4 = 1 kg, k1 = k3 = k5 = 7000 N= m, k2 = k4 = 8000 N= m, c1 = c3 = c5 = 4:2 N · s= m, c2 = c4 = 3:2 N · s= m, is considered. This system has complex modes and the damping coe cients are much larger than for those studied by Ruzzene et al. [11] and in part 1 [7]. The acceleration impulse responses of all masses xj (t) for j = 1; 2; 3; 4 due to an impact loading applied to the second mass are measured. The acceleration impulse responses x1 (t) and x3 (t) of the 1st and 3rd masses are shown in Figures 6(a) and 7(a), respectively. Since the impulse response functions are highly transient, the noise level of the measurement
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

Table II. Mode shapes and mass, sti ness and damping matrices of 3-DOF structural model.
Theoretical results

| |

Phase angle of 0 0.03 0.05
?980

(rad) 1000 0 0 0 1960 ?980 Identi?ed results (Rpi = 0%) 0 1000 0 0 0 1000 1960
?980 ?980

M (kg) 0

K (kN/m)
?2:81

C (kN · s= m) 9.85 0
?2:81 ?0:70

Copyright ? 2003 John Wiley & Sons, Ltd.

1.00 1.80 2.25

1.00 0.46 0.80

1.00 1.23 0.55

0 3.52 ?0:70

0 0.33 ?3:08

0 2.90 ?0:39

980

0.70

| |

Phase angle of 0 0.02 0.04
?959

(rad) 901 8 64 91 Identi?ed results (Rpi = 5%) 8 981 12 64 12 978 1717

M (kg)

K (kN/m)
?959 ?978

C (kN · s= m) 91 1900 ?978 971
?2:98

1.00 1.79 2.14

1.00 0.42 0.80

1.00 1.26 0.56

8.57 0.01

?2:98

?0:66

0.01 3.85 ?0:06 0.74

0 0.39 ?3:05

0 2.95 ?0:34

| |

Phase angle of 0 0.04 0.05
?87 ?65

(rad) 884 951 126
?87

M (kg)
?65

K (kN/m) 126 1004
?1071

C (kN · s= m) 1916
?1071 ?822

SYSTEM IDENTIFICATION OF LINEAR STRUCTURES

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

1.00 1.75 2.07

1.00 0.50 0.76

1.00 1.31 0.65

62 62 1785 ?822 884

9.87
?3:62 ?0:28

?3:62

?0:28

0 0.46 ?2:99

0 2.99 ?0:36

3.88 0.04

0.04 0.10

1545

1546

J. N. YANG ET AL.

Table III. Sti ness and damping matrices of 3-DOF structural model with known mass matrix. Theoretical results K (kN/m)
?980

C (kN · s= m)
?980

1960 0

?980

0

1960 ?980

?2:81

9.85

?2:81

980

0

3.52 ?0:70

?0:70

0

0.70

Identi?ed results (Rpi = 0%) K (kN/m)
?980

C (kN · s= m)
?987

1911 5

?980

5

1952 ?987

1010

?3:14 ?0:33

9.54

?3:14

3.95 ?0:62

?0:33 ?0:62

0.81

Identi?ed results (Rpi = 5%) K (kN/m) ?943 1929 ?1006 C (kN · s= m) ?3:21 3.94 ?0:24
?0:35 ?0:24

1935 ?943 10

10 ?1006 1006

10.29 ?3:21 ?0:35

0.24

Table IV. Mass, sti ness and damping matrices of 3-DOF structural model based on modi?ed approach. Theoretical results M (kg) 1000 0 0 0 1000 0 0 0 1000
?980

K (kN/m) 1960 0
?980

C (kN · s= m)
?980

0

1960 ?980

?2.81

9.85 0

?2.81

980

3.52 ?0.70

?0.70

0

0.70

Identi?ed results (Rpi = 0%) M (kg) 946 ?3 42
?3 957 26

K (kN/m) 42 26 981
?938

C (kN · s= m)
?939

1783 44

?938

44

1841 ?939

962

?3.00 ?0:13

8.97

?3.00

3.76 ?0:60

?0.13 ?0.60

0.76

Identi?ed results (Rpi = 5%) M (kg) ?20 962 81 K (kN/m) ?959 1779 ?867 C (kN · s= m) ?3.26 3.83 ?0.04
?0.45 ?0.04

?20

989 7

7 81 1013

?959

1787 37

?867

37

927

?3.26 ?0.45

9.88

0.18

Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

SYSTEM IDENTIFICATION OF LINEAR STRUCTURES

1547

80 acc. (m/s2) 40 0 -40 -80 0.0 (a) 80 acc. (m/s2) 40 0 -40 -80 0.0 (b) 80 acc. (m/s2) 40 0 -40 -80 0.0 (c) 0.5 1.0 1.5 2.0 time (sec) 2.5 3.0 0.5 1.0 1.5 2.0 time (sec) 2.5 3.0 0.5 1.0 1.5 2.0 time (sec) 2.5 3.0

Figure 6. Acceleration impulse response of 4-DOF system with complex modes: (a) response of 1st mass, (b) noise (R1 = 20%), and (c) measured response of 1st mass.

is de?ned by Ri = i = (xi ), where i is the rms of the noise, and of xi (t) over the time period T = 1:5 sec., i.e., 1 (xi ) = T
T 0 1=2

(xi ) is the temporal rms

x2 (t) d t i

(37)

Sample functions of the noise processes v1 (t) and v3 (t) associated with the measured accelerations z 1 (t) and z 3 (t) of the ?rst and third masses are shown in Figures 6(b) and 7(b), respectively, in which the noise level is R1 = R3 = 20% with a bandwidth of 500 Hz. The measured accelerations z 1 (t) and z 3 (t), which are the sum of (a) and (b) in Figures 6 and 7, respectively, are shown in Figures 6(c) and 7(c). Note that x1 (t) and x3 (t) in Figures 6(a) and 7(a) can be considered as the measured responses z 1 (t) and z 3 (t) where the noise level is zero (no noise). The Fourier Transforms of x1 (t), z 1 (t), x3 (t) and z 3 (t) are shown in Figures 8. As observed from Figures 8(a) and (b), each mode falls in the following frequency ranges: (i) 7Hz = !1L ?!1 ?!1H = 10Hz for the ?rst mode, (ii) 13Hz = !2L ?!2 ?!2H = 19Hz for the second mode, (iii) 19 Hz = !3L ?!3 ?!3H = 24 Hz for the third mode, and (iv) 24 Hz = !4L ?!4 ?!4H = 35 Hz for the fourth mode. Similar observations can be found in the Fourier Transform of x2 (t) and z 2 (t). From the Fourier Transform of x3 (t) and z 3 (t) in
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

1548
80 acc. (m/s2) 40 0 -40 -80 0.0 (a) 80 acc. (m/s2) 40 0 -40 -80 0.0 (b) 80 acc. (m/s2) 40 0 -40 -80 0.0 (c)

J. N. YANG ET AL.

0.5

1.0

1.5 2.0 time (sec)

2.5

3.0

0.5

1.0

1.5 2.0 time (sec)

2.5

3.0

0.5

1.0

1.5 2.0 time (sec)

2.5

3.0

Figure 7. Acceleration impulse response of 4-DOF system with complex models: (a) response of 3rd mass, (b) noise (R3 = 20%), and (c) measured response of 3rd mass.

Figures 8(c) and (d), it is noted that the third and fourth modes of the system are closely spaced. Such a situation also exists in the Fourier Transforms of x4 (t) and z 4 (t). Since the frequencies of this problem are quite high, in order to obtain a reasonable resolution of the high frequency modal response by the EMD procedures, it is necessary to increase the number of sifting. As a result, the computational e ort is quite involved. Consequently, we shall use the alternative approach proposed in Reference [7], i.e., the use of a 4th order band-pass ?lter twice and EMD. The procedures are brie y described as follows. We process the signal z 1 (t) and z 3 (t) in Figures 6(c) and 7(c) through the 4th order band-pass ?lters twice, each with a frequency band !jL ?!j ?!jH (j = 1; 2; 3; 4) given above. The resulting time histories thus obtained are denoted by x? (t) and x? (t) (j = 1; 2; 3; 4). Then, we process 1j 3j each x? (t) and x? (t) through EMD and the ?rst IMF is the modal response x1j (t) and x3j (t). 1j 3j The resulting modal responses x1j (t) and x3j (t) for j = 1; 2; 3; 4 are presented in Figures 9 and 10. Note that the band-pass ?lter used should have as small a phase shift as possible. After removing a segment near t = 0, in which the response is not a decaying function, the modal responses shown in Figures 9 and 10 have been processed through the Hilbert transform to obtain the instantaneous phase angles, ?1j (t) and ?3j (t), and instantaneous amplitudes, A1j (t) and A3j (t), as functions of time t. The plots of these quantities for the 4th mode, i.e., ?14 (t), ln A14 (t), ?34 (t) and ln A34 (t), are presented as dotted curves in Figure 11. Also shown in
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

SYSTEM IDENTIFICATION OF LINEAR STRUCTURES

1549

10 amplitude (m/s) 8 6 4 2 0 0 (a) 10 amplitude (m/s) 8 6 4 2 0 (c) 0 10 20 30 40 frequency (Hz) 50 (d) amplitude (m/s) 10 20 30 40 50 (b) frequency (Hz) amplitude (m/s)

10 8 6 4 2 0 0 10 20 30 40 50

frequency (Hz) 10 8 6 4 2 0 0 10 20 30 40 frequency (Hz) 50

Figure 8. Fourier Transforms of: (a) z 1 (t) (without noise), (b) z 1 (t) (with 20% noise), (c) z 3 (t) (without noise), and (d) z 3 (t) (with 20% noises).

Figure 11 as solid straight lines are the linear least-square ?ts. As expected, the dotted curves oscillate around the solid straight lines. From the slopes of these two solid straight lines, the natural frequency !4 and the damping ratio 4 have been estimated. Repeating the same procedures for all modal responses, one obtains all the natural frequencies and damping ratios. Based on a single measurement of the 1st mass z 1 (t), the results are presented in Columns (6) and (7) of Table V. Also shown in Columns (2) and (3) of Table V are the theoretical values for comparison. Further, the identi?ed results when the noise pollution is zero, i.e., Ri = 0% are presented in Columns (4) and (5) of Table V. It is mentioned that the identi?ed results based on a single measurement of the 3rd mass are very close to the results presented in Table V. Compared with the theoretical values, the identi?ed results presented in Table V are excellent. The procedures above have been repeated for the responses at all DOFs with Ri = 0% and 20%, respectively. Then, all the complex mode shapes have been identi?ed using Equations (23) and (24). The amplitude and phase angle of the identi?ed complex modal (normalized) matrix of the system is presented in Table VI for Ri = 0% and Ri = 20%. The time instant t0 on the linear least-square ?t line as indicated in Figure 11 is t0 = 0:285 sec. Note that t0 is used in Equations (23) and (24) to determine Apj (t0 ) and ?pj (t0 ). The physical mass, sti ness and damping matrices have been identi?ed using Equations (25)–(29), (32) and (33) when the level of the impact force F0 is measured to be 1 N. The identi?ed results for M, C, and K are presented in Table VI, along with the theoretical values for comparison. When the mass matrix is known. The physical sti ness and damping matrices can be determined using Equations (33) and (34), and the identi?ed results are shown in Table VII. As expected, the identi?ed physical sti ness and damping matrices in Table VII are slightly
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

1550
15 acc. (m/s2) 5 -5 -15 0.0 (a) 25 15 5 -5 -15 -25 0.0

J. N. YANG ET AL.

0.6

1.2 1.8 time (sec)

2.4

3.0

acc. (m/s2)

0.3

(b) 30 acc. (m/s2) 10 -10 -30 0.0 (c) 40 acc. (m/s2) 20 0 -20 -40 0.0 (d) 0.2 0.3

0.6 0.9 time (sec)

1.2

1.5

0.6 time (sec)

0.9

1.2

0.4 0.6 time (sec)

0.8

1.0

Figure 9. Modal response of 1st mass of 4-DOF system with R1 = 20%: (a) 1st modal response, (b) 2nd modal response, (c) 3rd modal response, and (d) 4th modal response.

more accurate when the mass matrix is known. The modi?ed equations proposed by Liu [10] in Equations (35a) and (35b) are also used to identify the M and K matrices, and Equation (32) is used to determine the C matrix. The accuracy of the results for M, C, and K thus obtained improves only slightly over that given in Table VI. Hence, these results are not presented. From Table VII, the model parameters (sti ness and damping coe cients) of the mechanical system, in which the mass matrix is known, are identi?ed using the least square estimation as follows: k1 = 6886 N= m, k2 = 8054 N= m, k3 = 7123 N= m, k4 = 7969 N= m, k5 = 6751 N= m, c1 = 4:43 N · s= m, c2 = 3:03 N · s= m, c3 = 4:09 N · s= m, c4 = 3:55 N · s= m, and c5 = 4:67 N · s= m for the case without noise pollution (Ri = 0%). The results for the case Ri = 20% are: k1 = 6970 N= m, k2 = 7680 N= m, k3 = 8050 N= m, k4 = 7550 N= m, k5 = 6180 N= m, c1 = 3:34 N · s= m, c2 = 2:94 N · s= m, c3 = 4:70 N · s= m, c4 = 4:05 N · s= m, and c5 = 2:83 N · s= m.
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

SYSTEM IDENTIFICATION OF LINEAR STRUCTURES

1551

30 acc. (m/s2) 15 0 -15 -30 0.0 (a) 20 acc. (m/s2) 10 0 -10 -20 0.0 (b) 15 acc. (m/s2) 5 -5 -15 0.0 (c) 50 acc. (m/s2) 0 -50 0.0 (d) 0.3 0.6 time (sec) 0.9 1.2 0.3 0.6 0.9 time (sec) 1.2 1.5 0.6 1.2 1.8 2.4 3.0

time (sec)

0.2

0.4 0.6 time (sec)

0.8

1.0

Figure 10. Modal Response of 3rd mass of 4-DOF system with R3 = 20%: (a) 1st modal response, (b) 2nd modal response, (c) 3rd modal response, and (d) 4th modal response.

It is observed from the system parameters identi?ed above and in Tables V–VII that the accuracy of the system identi?cation technique presented herein is quite reasonable.

6. CONCLUSIONS Based on the Hilbert–Huang spectral analysis, a method has been proposed for the identi?cation of the dynamic characteristics of general linear structures with complex modes. The method uses measured free vibration time histories, which are polluted by noise. Natural frequencies and damping ratios have been identi?ed based on a single measurement of the free vibration time history. When the response time histories are measured at all DOFs, the
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

1552
phase angle (rad) 90 65 40 15 -10 0.0 0.1 0.2 0.3

J. N. YANG ET AL.

phase angle (rad)

90 65 40 15 -10 0.0 0.1 0.2 0.3 t0 0.4 0.5

t0

0.4

0.5

(a)
4 3 2 1 0 -1 0.0 ln amplitude

time (sec)

(c)
4 ln amplitude 3 2 1 0 0.0 0.1

time (sec)

0.1

0.2

t0

0.3

0.4

0.5

0.2

0.3 t0

0.4

0.5

(b)

time (sec)

(d)

time (sec)

Figure 11. Plots of phase angle and ln amplitude for the 4th mode of 1st and 3rd masses: (a) ?14 (t), (b) ln A14 (t), (c) ?34 (t), and (d) ln A34 (t). Table V. Natural frequencies and damping ratios of 4-DOF system using the measured acceleration of 1st mass. Mode Theoretical values Frequency (Hz) (2) 8.37 15.73 22.64 26.26 Damping ratio (%) (3) 1.45 2.90 3.19 4.02 Identi?ed values (Ri = 0%) Frequency (Hz) (4) 8.37 15.71 22.62 26.28 Damping ratio (%) (5) 1.47 2.88 3.33 4.08 Identi?ed values (Ri = 20%) Frequency (Hz) (6) 8.36 15.71 22.63 26.20 Damping ratio (%) (7) 1.46 3.04 3.25 3.95

(1) 1 2 3 4

proposed methodology is capable of identifying the complex mode shapes as well as the mass, sti ness and damping matrices of the structure. Thus, the complete dynamic characteristics of general linear structures can be identi?ed. Simulation results demonstrate that natural frequencies and damping ratios can be identi?ed quite accurately using the proposed approach, in particular, for the 4-DOF mechanical system in which the 3rd and 4th modes are not well separated, see Figures 8(c) and (d). The accuracy for the identi?cation of complex modes and system matrices is also reasonable. The proposed method o ers an e ective tool for the identi?cation of complete dynamic characteristics of general linear MDOF structures with complex modes.

ACKNOWLEDGEMENT

This research is supported by the National Science Foundation through Grant No. CMS-98-07855 and NASA Grant No. NAG5-5149.
Copyright ? 2003 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

Table VI. Mode shapes and mass, sti ness and damping matrices of 4-DOF system.
Theoretical results

| |

Phase angle of
?8

(rad) 15
?7 ?8 ?8

M (kg) 0

K (kN/m)
?3:2

C (kN · s= m)

Copyright ? 2003 John Wiley & Sons, Ltd.

1.00 1.53 1.53 1.00 0 0 15 ?7 0 15 Identi?ed results (Ri = 0%) Phase angle of
?6.91 ?0.87

1.00 0.65 0.65 1.00

1.00 0.65 0.65 1.00

1.00 1.53 1.53 1.00

0 0 0 0

0 0 ?3.15 ?3.14

0 0 3.13 ?3.13 3.13 0 0 ?3.14

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 ?8 15

7.4 ?3.2 0 0 7.4 ?4:2 0 0 ?4.2 7.4 ?3.2 0 0 ?3.2 7.4

| |

(rad) 13.27

M (kg)

K (kN/m)
?6.91 ?6.89

C (kN · s= m) 0.50 14.11 ?6.89 0.16
?0.87 ?5.88

1.00 1.51 1.51 0.99 0.50 Identi?ed results (Ri = 20%) Phase angle of (rad) M (kg)
?8.02 ?2.96

1.00 0.63 0.63 1.01

1.00 0.64 0.73 1.05

1.00 1.52 1.54 0.98

0 0 0.02 ?0.01 0.01 ?3.15 0.01 ?3.15

0 0 0.92 3.17 ?3.12 0.05 3.14 0.03 0.03 0.04 ?3:09 ?0:04

0.05 0.95 0.00 0.02

0.03 ?0.04 0.00 0.02 0.90 0.09 0.09 0.89

13.10

6.6 ?2.8 1.0 ?0.4 0.16 ?2.8 6.5 ?4:1 ?0:5 ?5.88 1.0 ?4:1 6.6 ?2:5 12.38 ?0:4 ?0:5 ?2:5 7.0

| |

K (kN/m) 14.50
?8.02 ?8.48

C (kN · s= m) 2.02 2.02 14.62 ?8.48 3.50
?2.96 ?9.8

SYSTEM IDENTIFICATION OF LINEAR STRUCTURES

Earthquake Engng Struct. Dyn. 2003; 32:1533–1554

1.00 1.51 1.52 0.99

1.00 0.63 0.71 1.05

1.00 0.66 0.71 1.18

?0.01 0.08 ?0.19 1.00 0 0 0 0 1 1.69 0.01 0.01 3.22 ?3:07 ?0:01 0.99 ?0.02 0.24 1.61 ?0.01 ?3.20 3.05 0.04 0.08 ?0.02 0.93 ?0.14 1.19 0.01 ?3.20 ?0.01 ?3.06 ?0.19 0.24 ?0.14 1.03

15.92

6.5 ?2.6 1.8 ?2.4 3.50 ?2.6 7.3 ?4.5 0.9 ?9.80 1.8 ?4.5 9.1 ?5.6 15.43 ?2.4 0.9 ?5.6 7.7

1553

1554

J. N. YANG ET AL.

Table VII. Sti ness and damping matrices of 4-DOF system with known mass matrix. Theoretical results K (kN/m)
?8

C (kN · s= m)
?7 ?8

15 0 0

?8

0

15 ?7 0

15

0 0 ?8 15

?3.2

7.4

?3:2

0 0

7.4 ?4.2 0

?4.2 ?3:2

0

7.4

0 0 ?3.2 7.4

Identi?ed results (Ri = 0%) K (kN/m)
?8:07

C (kN · s= m) 0.01
?0.08 ?7.90

14.94 0.01 0.02

?8.07

15.16 ?7.07 ?0.08

?7.07 ?7.90

0.02

?3.26

7.46

?3:26

15.16

14.72

0.91 0.04

6.89 ?4:25 ?0:60

?4:25 ?3:49

0.91 7.70

?0:60 ?3:49

0.04 8.22

Identi?ed results (Ri = 20%) K (kN/m)
?8:06

C (kN · s= m) 0.45
?1:32 ?7:02

14.65 0.45 1.08

?8:06

15.35 ?7:90 ?1:32

?7:90

1.08

?2:69 ?0:30

6.28 0.89

?2:69

?7.02

16.13

13.73

7.89 ?4:10 ?1:09

?4:10 ?3:71

0.89 9.09

?0:30 ?1:09 ?3:71

6.88

REFERENCES 1. Feldman M. Non-linear system vibration analysis using Hilbert Transform—I: Free vibration analysis method FREEVIB. Mechanical Systems and Signal Processing 1994; 8(2):119 –127. 2. Feldman M. Nonlinear free-vibration identi?cation via the Hilbert Transform. Journal of Sound and Vibration 1997; 208(3):475– 489. 3. Huang NE, Shen A, Long SR, Wu MC, Shih HH, Zheng Q, Yen NC, Tung CC, Liu HH. The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society London—Series A 1998; A454:903 – 995. 4. Huang NE, Shen A, Long SR. A new view of nonlinear water waves: the Hilbert spectrum. Annual Review of Fluid Mechanics 1999; 31:417– 457. 5. Yang JN, Lei Y. Identi?cation of tall building using noisy wind vibration data. Advances in Structural Dynamics, Vol. 2, 1093 –1100, 2000, Elsevier (Proceedings of the International Conference on Advances in Structural Dynamics, Hong Kong). 6. Yang JN, Lei Y, Huang N. Damage identi?cation of civil engineering structures using Hilbert–Huang transform. Proceedings of the 3rd International Workshop on Structural Health Monitoring, 544 – 553, 2001, Stanford University, CA, CRC Press: New York. 7. Yang JN, Lei Y, Pan S, Huang N. System identi?cation of linear structures based on Hilbert–Huang spectral analysis. Part I: Normal modes. Earthquake Engineering and Structural Dynamics 2003; 32:1443–1467. 8. Naylor S, Cooper JE, Wright JR. On the estimation of model matrices with non-proportional damping, Proceedings of the 13th International Modal Analysis Conference 1371–1378, 1995. 9. Caughey TK, Ma F. Complex modes and solvability of nonclassical linear systems. Journal of Applied Mechanics (ASME) 1993; 60(1):26 – 32. 10. Liu S. Improvement of analytical dynamic models using complex modal test data. International Journal of Mechanical Sciences 1992; 34:817– 829. 11. Ruzzene M, Fasana A, Garibaldi L, Piombo B. Natural frequencies and dampings identi?cation using wavelet transform: application to real data. Mechanical Systems and Signal Processing 1997; 11(2):207– 218.
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Earthquake Engng Struct. Dyn. 2003; 32:1533–1554


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