identification of rotor-bearing systems in the frequency domain part ii: estimation of modal...

Mechanical Systems and Signal Processing (2001) 15(4), 775}788doi10.1006/mssp.2000.1334, available online at http://www.idealibrary.com on

IDENTIFICATION OF ROTOR-BEARING SYSTEMSIN THE FREQUENCY DOMAIN

PART II: ESTIMATION OF MODAL PARAMETERS

FRANK PEETERS, RIK PINTELON, JOHAN SCHOUKENS AND YVES ROLAIN

Department of ELEC, Faculty of Applied Sciences, Vrije Universiteit Brussel (VUB),Pleinlaan 2, B-1050 Brussels, Belgium. E-mail: [email protected]

(Received 23 November 1999, accepted 16 August 2000)

This paper discusses the parametric identi"cation of multiple-input}multiple-output(MIMO) rotor-bearing systems in the frequency domain on the basis of the maximumlikelihood estimator (MLE). An identi"cation procedure, considering noise on both inputsand outputs (errors-in variables model), is developed for real as well as for complex modaltesting. Quantitative measures for the uncertainties (covariances) of the estimated para-meters were to be obtained. It is shown that the MLE solution of real and complex modalanalysis is fully equivalent. It follows that, in case of complex modal testing, reduction of thesystem is not allowed in general when noise information is taken into account. A comparisonbetween random and broadband periodic (multisine) excitations is made. Measurements ona rotor test rig demonstrated that multisines yield more accurate estimates of the modalparameters (as compared to random excitation).

( 2001 Academic Press

1. INTRODUCTION

Modal testing of multiple-input}multiple-output (MIMO) rotor-bearing systems can bedone within the classical &real' framework or &complex' framework [1] (i.e. considering realor complex signals). As demonstrated by Lee and coworkers [2], complex modal testingo!ers some distinct advantages with respect to the directivity of the modes, detection ofanisotropy and asymmetry.

In the modal testing society, random excitations are very popular. When modal para-meters are estimated on the basis of data obtained with this type of excitation, the errorbounds are usually not provided. However, if accurate parametric identi"cation is required,uncertainties are of concern. The maximum likelihood estimator (MLE) [3], which isbased on a statistical framework (i.e. taking noise into account), allows one to obtainthese uncertainties. Broadband periodic (multisine) excitations [4] are very suitablein this context, because they allow independent estimation of non-parametric noisemodels [5].

In this paper, we study the estimation of modal parameters of rotor-bearing systems bothfor real and complex modal testing. This is done using the MLE in the frequency domain.Special attention is paid to a correct implementation of the noise information. Two types ofbroadband excitation are considered in this study: random and periodic (multisine) excita-tions. An identi"cation procedure is developed that yields the modal parameters togetherwith their uncertainties. Experiments are performed on a rotor rig and the identi"cation isdone both for real and complex modal analysis.

0888}3270/01/040775#14 $35.00/0 ( 2001 Academic Press

776 F. PEETERS E¹ A¸.

2. MODAL TESTING OF ROTATING STRUCTURES

2.1. REAL MODAL TESTING

In real modal testing, a multi-degree-of-freedom (mdof ) rotor-bearing system (see Fig. 1)is excited by real n

i]1 force vectors f

y(t), f

z(t) and the real n

o]1 response vectors y (t), z (t)

are measured. Here ni(resp. n

o) represents the number of input (resp. output) stations and

displacement was assumed as response for simplicity. The relationship between input andoutput can be written in the frequency domain as Q(R)(u)"H(R)(u) )F(R)(u) or

Q(R)1

(u)

Q(R)2

(u)"

H (R)11

(u) H (R)12

(u)

H (R)21

(u) H (R)22

(u))

F (R)1

(u)

F (R)2

(u)(1)

with Q(R)1

(u), Q(R)2

(u) and F (R)1

(u), F (R)2

(u) the Fourier transforms of the signals y(t), z (t), andfy(t), f

z(t), respectively. The superscript (R) is used to denote quantities within the &real'

framework and u represents the angular frequency. Because H (R) (!u)"H1 (R) (u) (wherethe overbar stands for complex conjugate), only positive frequencies are considered in realmodal testing.

The 2no]2n

ifrequency response matrix (FRM) H (R) (u) can be written in terms of the

modal parameters of the rotor-bearing system:

H (R)(u)"2N+k/1G

R (R)k

ju!jk

#

R1 (R)k

ju!jMkH (2)

i.e. a sum over all modes (2N), including backward (B) and forward (F) modes. The modalparameters are the poles j and the residue matrices R (R)

k, which can be written in terms of

the modal uy, u

zand adjoint modal vectors v

y, v

z:

R (R)"u(R) ) v(R)H"Cuy

uzD ) [vHy vH

z] (3)

where the subscript k has been dropped to simplify the notation. Note that the modalparameters are in general complex.

2.2. COMPLEX MODAL TESTING

Complex signals are introduced in complex modal testing according tog(t)"f

y(t)#jf

z(t), g6 (t) and p (t)"y(t)#jz(t), p6 (t). Now, the relationship between input and

output in the frequency domain can be written by a similar equation as equation (1):Q (C) (u)"H (C) (u) )F (C)(u). The superscript (C) now denotes quantities in the &complex'

Figure 1. A mdof rotor-bearing system excited at station k (k"1,2, ni) by forces ( f

yk(t), f

zk(t)) and responses

(yl(t), z

l(t)) measured at station l (l"1,2, n

o). X represents the angular rotation speed.

777ESTIMATION OF MODAL PARAMETERS

formalism: Q(C)1

(u)"P(u), Q(C)2

(u)"P) (u), F (C)1

(u)"G(u) and F(C)2

(u)"GK (u) where P (u),P) (u), G (u), G) (u) denote the Fourier transforms of p(t), p6 (t), g (t), g6 (t), respectively.

Writing the FRM H (C)(u) in terms of the modal parameters, one obtains

H (C) (u)"2N+k/1G

R(C)k

ju!jk

#

R3 (C)k

ju!jMkH (4)

with

R (C)"u(C) ) v(C)H"Cuc

u(cD ) [vHc v( H

c] (5a)

R3 (C)"u8 (C) ) v8 (C)H"Cu(c

ucD ) [v( Hc vH

c]. (5b)

The circular modal and adjoint modal vectors uc, u(

c, v

c, v(

care related to u

y, u

z, v

y, v

zaccording to

uc"

1

J2(u

y#ju

z) (6a)

u(c"

1

J2(u

y!ju

z) (6b)

vc"

1

J2(v

y#jv

z) (6c)

v(c"

1

J2(v

y!jv

z) . (6d)

In real modal testing, all four block FRMs in equation (1) are needed to fully characterisethe system. This will be also the case in complex modal testing if only positive frequenciesare considered. On the contrary, if negative frequencies are considered too (i.e. usingtwo-sided FRMs H (C) (u)), only two block FRMs are needed.

3. PARAMETRIC IDENTIFICATION IN THE FREQUENCY DOMAIN

3.1. THE MAXIMUM LIKELIHOOD ESTIMATOR

Here, we consider a multiple-input}multiple-output (MIMO) system with measurementsnoise both on inputs and outputs (errors-in-variables model) as shown in Fig. 2. The FRMdepends on the frequency u (in practice a discrete set u

k) and a parameter vector h to be

estimated: H(h, uk).

A maximum likelihood estimator (MLE) can be derived by minimising a quadratic costfunction ¸ with respect to h:

¸"

Nf

+k/1

l (h, uk)H ) l (h, u

k) (7a)

"

Nf

+k/1

e (h, uk)H )C~1

e(h, u

k) ) e(h, u

k) (7b)

Figure 2. The errors-in-variables (EV) model. A system with impulse response h(t) has &true' inputs f0(t) and

outputs q0(t). Due to measurement noise, the measured inputs and outputs are f (t)"f

0(t)#n

f(t) and

q(t)"q0(t)#n

q(t). The corresponding quantities in the frequency domain are also shown in the "gure.


where Nf

is the number of frequencies considered. The equation error and covariancematrices are given by

e (h, uk)"B (h, u

k) )U(u

k) (8a)

Ce(h, u

k)"B (h, u

k) )C(u

k) )BH(h, u

k) (8b)

with

B (h, uk)"[H(h, u

k)!I] (9a)

U(uk)"C

F (uk)

Q(uk)D (9b)

(I is a identity matrix). Here, we have assumed that the covariance matrix C"cov(U) of themeasurement noise is a priori known. The uncertainty on the estimated parameters h

%45is

given by the Cramer}Rao inequality

C (%45)h "cov(h

%45)*Fi~1"E G

L2¸

Lh H~1

(10)

where Fi is the so-called Fisher information matrix and EM2N denotes the expectationvalue. If the covariance matrix C is estimated using a small number of data sets, a correctionhas to be applied to equation (10) as explained in Appendix A. More details about the MLEcan be found in [3].

3.2. IDENTIFICATION PROCEDURE

In this section, we will describe the procedure we have followed to estimate the modalparameters. After input of the data (excitation and response signals in the time domain), thespectra Q (u

k), F (u

k) are calculated by means of the FFT and the covariance matrix C(u

k) of

the noise is calculated.Because the MLE cost function is a non-quadratic function of the parameters h, initial

values are needed to start the minimisation. Valuable start values can be obtained byreplacing the equation error matrix in equation (7a), (7b) and (8a) by

e8 (h, uk)"N(h, u

k) )F (u

k)!D(h, u

k) )Q(u

k) (11)

where N (h, uk) represents the numerator matrix and D(h, u

k) the common denominator of

the FRM. Moreover, if Ce(h, u

k) is replaced by the identity matrix and N(h, u

k), D(h, u

k) are


written in their polynomial form, a linear least-squares problem is obtained leading to h(1)*/*5

.Improved start values [6] h(2)

*/*5are generated with the aid of writing e (h, u

k) as

e(h, uk)"

N (h, uk) )F (u

k)!D(h, u

k) )Q (u

k)

D(h, uk)

"

e8 (h, uk)

D(h, uk)

(12)

leading to

¸"

Nf+k/1

eJ H (h, uk) )W~1(u

k) ) e8 (h, u

k) (13)

with W(uk)"DDI (u

k) D2C3

e(u

k). Here, DI (u

k) is obtained by substituting h(1)

*/*5into D(h, u

k) or

simply setting all coe$cients equal to one. C3e(u

k) is obtained by replacing H (h, u

k) by

non-parametric estimates [see equations (8b) and (9a)].The core of the procedure is the minimisation of the non-linear cost function ¸ in

equation (7). This done in the residue form of the FRM [see equations (2) and (4)] usingLevenberg}Marquardt algorithm. To speed up convergence, a change to the Newton}Gauss method is made after a few successful iteration steps. Depending on the quality of theinitial estimates, the minimisation in residue form can be preceeded by a similar one inpolynomial form. The output of the procedure is the "nal parameter vector h

&*/!-[i.e. poles

and (adjoint)modal vectors] and their uncertainties [see equation (10)]. Hereto, the Fishermatrix is approximated by Fi:2Re(JH ) J) where J represents the Jacobian of l (h), evalu-ated during the last iteration.

3.3. EXCITATION

3.3.1. Random excitation

The classical excitation signal in modal testing is random excitation. Unfortunately,random excitation has some major disadvantages: long measurement times are needed inorder to obtain a reasonable signal-to-noise ratio SNR (averaging), the covariance matrix ofthe measurement noise is not available in general and spectral leakage cannot be avoided.

Leakage can be reduced with a Hanning window for example. However, "ltering willalways lead to a loss in frequency resolution. An alternative method to take leakage intoaccount is by estimating the coe$cients h

Lof the extra leakage terms in the cost function

[7]. Leakage e!ects can be represented as

e(h, hL, u

k)"H(h, u

k) )F (u

k)#

T (hL, u

k)

D(h, uk)!Q (u

k) (14)

where T (hL, u

k) is the leakage polynomial matrix, depending on the leakage coe$cients h

L,

and D (h, uk) is again the common denominator of the FRM H(h, u

k). However, note that

without "ltering, the non-parametric FRM will still be contaminated by leakage. Moreover,if the classical H

1-estimator is used, systematic errors cannot be avoided if input noise is

present, even if "ltering is performed.As discussed in [8], consistent estimates of the FRM are possible by using the &instrumen-

tal variables' estimator Hiv

(i.e. no bias due to input noise). There it is also explained that,under certain circumstances, the covariance matrix C of the measurement noise can beobtained for random excitations too.

3.3.2. Periodic excitation

A better way to excite the system is using periodic signals. Broadband periodic signalsgive rise to a better SNR within the same measurement time, do not exhibit spectral leakage


problems and the covariance matrix of the measurement noise can be retrieved by measur-ing a few periods of the steady-state response. We opted for multisines (i.e. sum of sineswhere the amplitudes and phases can be adjusted to optimise the signal) as the excitationsignal. Details about the estimation of the non-parametric FRM (using the estimator H

1%3)

and the covariance matrix C can be found in [8].

3.5. REAL VS COMPLEX MODAL ANALYSIS

As stated before, in real modal testing only positive frequencies are used and the full FRMis needed to characterise the system. As a result, the MLE has to estimate the parametervector h for a system with N

i"2n

iinputs and N

o"2n

ooutputs.

If in complex modal testing only positive frequencies are used, one arrives again at theidenti"cation of a 2n

i]2n

osystem. From a numerical point of view, both strategies are fully

equivalent. This can be easily understood by considering the condition number c (A) ofa matrix A. Because

H(C)(u)"T~1 )H(R)(u) )T (16a)

with

T"

1

2 C1 1

!j jD (16b)

where T is the transformation matrix to convert real signals into complex ones, one obtainsc(H (C))"c (H (R)) (see also Appendix B). As a result, no numerical advantages should beexpected by using complex modal analysis.

The power of complex modal testing lies in the fact that negative frequencies can beconsidered too. In this case, only two block FRMs have to be considered, i.e. only a 2n

i]n

osystem has to be identi"ed. However, if noise information is used (the covariance matrix C),one has to be very careful to apply the MLE in a correct manner. This will be illustrated bycomparing a full 2]2 complex system with the reduced 2]1 complex system. Let usconsider a 2]2 real system with noise variances p2

1, p2

2for the inputs and p2

3, p2

4for the

outputs. For uncorrelated input/input, output/output and input/output noise, thecovariance matrix will be C (R)"diag(p2

1, p2

2, p2

3, p2

4). For the full 2]2 complex system,

the covariance matrix becomes

C (C)"

a b 0 0

b a 0 0

0 0 c d

0 0 d c

(17)

with a"p21#p2

2, b"p2

1!p2

2, c"p2

3#p2

4and d"p2

3!p2

4. Thus now, input/input and

output/output noise becomes correlated. The cost function ¸(C)"eHC~1e

e is then given by

¸(C)"R11

De1D2#R

22De2D2#2Re(R

12eN1e2) (18)

with equation error matrix

e"e1

e2

"

Q1!(H

11F1#H

12F2)

Q2!(H

21F1#H

22F2)

(19)


and Re(2) means real part. For simplicity, the superscript (C) and the summation over thepositive frequencies have been omitted. The matrix R"C~1

ecan be written as

R"

1

c11

c22!Dc

12D2

c22

!c12

!cN12

c11

(20)

with ckl

the elements of Ce. These are given by

c11"( DH

11D2#DH

12D2)a#2Re(HM

12H

11)b#c

c22"( DH

21D2#DH

22D2)a#2Re(HM

21H

22)b#c

c12"(H

11H

21#H

12HM

22)a#(H

11HM

22#H

12HM

21)b#d

c21"cN

12. (21)

For the reduced 2]1 complex system with positive and negative frequencies considered,the covariance matrix of the noise is

C (C)3%$

"

a b 0

b a 0

0 0 c

(22)

and the cost function is now given by

¸(C)3%$

"R (104) De(104)1

D2#R (/%') De(/%')1

D2 . (23)

Using the relationships between quantities at positive and negative frequencies, it is easy toshow that R (104)"1/c

11and R (/%')"1/c

22. As a result, the cost function for the reduced

system:

¸(C)3%$

"

De1D2

c11

#

De2D2

c22

(24)

does not equal ¸(C). Equations (18) and (20) show that ¸(C)"¸(C)3%$

when c12

can be neglected.According to equation (21), this will be the case if:

(1) H(C)12

"H(C)21

"0 and b"d"0, thus p1"p

2and p

3"p

4. This means that the system

should be isotropic (bearing) and the noise variances along the y- and z-directions should beequal (for the real signals).

(2) a"b"d"0, i.e. p1"p

2"0 and p

3"p

4. Now, the real input signals have to be

noise free and the real responses along the y- and z-directions have again the same noisevariances.

So in general, ¸(C) and ¸(C)3%$

will lead to di!erent estimates. Therefore, the full complexsystem must be considered if noise information is taken into account. As a result, from anidenti"cation point of view, complex modal testing does not o!er any advantages ascompared to real modal testing when the MLE is used. However, if condition (1) or (2) isful"lled, a reduction is allowed and complex modal testing can be preferred. It is interestingto note that condition (2) is usually assumed when random excitations are used.

4. VALIDATION OF THE IDENTIFICATION METHOD

The identi"cation procedure was validated on simulated data. Simulation and evaluationof the data were performed in MATLAB on a Power Macintosh 8100/80. A 4-dof(two stations) anisotropic rotor-bearing system was studied (see [9, Section 5]). The exact

TABLE 1

Comparison of exact and simulated modal parameters

Mode 2B 1B 1F 2F Remark

j !165.182!1922.088 j !38.8686!987.9864 j !36.1314#1010.4945 j !168.151#2297.122 j Exactj (!165.182$0.010)# !38.8702$0.0039)# (!36.1328$0.0038)# (!168.185$0.010)# Simulated

(!1922.088$0.010) j (!987.9864$0.0040) j (1010.4948$0.0040) j (2297.112$0.010) j

uy

1.000000 1.000000 0.25229$0.12105 j !0.000915#0.878182 j Exactuy

(1.000000$0.000030)# (1.000000$0.000046)# (0.25235$0.00015)# (!0.000898$0.000031)# Simulated(0.000000$0.000030) j (0.000000$0.000045) j (0.12104$0.00014) j (0.878244$0.000030) j

uz

!0.001066!0.856361 j 0.25244!0.11842 j 1.000000 1.000000 Exactuz

(!0.000996$0.000036)# (0.25232$0.00017)# (1.000000$0.000041)# (1.000000$0.000027)# Simulated(!0.856353$0.000035) j (!0.11837$0.00016) j (0.000000$0.000040) j (0.000000$0.000027) j

vy

(0.0278#76.1115 j )10~6 (4.1631!66.2592 j )10~6 (!8.898!15.848 j )10~6 (65.1789#0.0363 j )10~6 Exactvy

(0.0316$0.0038)10~6# (4.1662$0.0074)10~6# (!8.903$0.012)10~6# (65.1775$0.0037)10~6# Simulated(76.1045$0.0039)10~6 j (!66.2626$0.0074)10~6 j (!15.839$0.013)10~6 j (0.0383$0.0037)10~6 j

vz

(65.1789#0.0573 j )10~6 (!8.898#16.233 j )10~6 (4.1685#64.8160 j )10~6 (0.1186!74.2202 j )10~6 Exactvz

(65.1785$0.0039)10~6# (!8.895$0.011)10~6# (4.1654$0.0078)10~6# (0.1178$0.0036)10~6# Simulated(0.0548$0.0039)10~6 j (16.239$0.011)10~6 j (64.8165$0.0075)10~6 j (!74.2271$0037)10~6 j

uc

1.000000 1.00000 1.00000 1.000000 Exactuc

(1.0000000$0.0000027)# (1.00000$0.00013)# (1.00000$0.00011)# (1.000000$0.0000020)# Simulated(0.0000000$0.0000028) j (0.00000$0.00013) j (0.00000$0.00012) j (0.0000000$0.0000020) j

uLc

0.077376#0.000619 j 0.70155!0.38405 j !0.69804!0.38214 j !0.064860#0.000519 j ExactuLc

(0.077381$0.000035)# (0.70170$0.00016)# (!0.69802$0.00014)# (!0.064824$0.000030)# Simulated(0.000578$0.000036 j) (!0.38393$0.00016) j (!0.38222$0.00015) j (0.000509$0.000031) j

vc

(!0.1027#131.1430 j )10~6 (!16.2360!40.5051 j )10~6 (!15.8449#39.8451 j )10~6 (0.0817!130.9085 j )10~6 Exactvc

(!0.0918$0.0049)10~6# (!16.2326$0.0076)10~6# (!15.8446$0.0075)10~6# (0.0840$0.0045)10~6# Simulated(131.1355$0.0048)10~6 j (!40.5037$0.0077)10~6 j (39.8482$0.0074)10~6 j (!130.9179$0.0046)10~6 j

vLc

(0.0732#10.1474 j )10~6 (4.1658!34.6517 j )10~6 (!4.1658!33.8686 j )10~6 (!0.0732#8.4906 j )10~6 ExactvLc

(0.0748$0.0053)10~6# (4.1726$0.0080)10~6# (!4.1580$0.0096)10~6# (!0.0706$0.0052)10~6# Simulated(10.1413$0.0053)10~6 j (!34.6536$0.0078)10~6 j (!33.8649$0.0092)10~6 j (8.4987$0.0051)10~6 j

782F

.PE

ET

ER

SE¹

A¸

.


modal parameters for excitation at station 2 and response (displacement) at station 1 aregiven in Table 1. The system is characterised by four modes (2B, 1B, 1F, 2F) together withtheir conjugate modes (2FM , 1FM , 1BM , 2BM ). Data were generated for multisine excitation withall amplitudes equal and phases uniform random distributed. Gaussian measurement noisewas added in the time domain on both inputs and outputs. In order to estimate thenon-parametric FRFs, two experiments have to be generated [8]. Here, we have chosen forthe optimal combination, i.e. the force signals for experiment 1 are equal and oppositefor experiment 2. Table 1 shows the exact modal parameters and the estimated parametersfor a simulation with 128 samples per period, 40 periods and a signal-to-noise ratioSNR"104. The data were evaluated using both real modal analysis (leading toj, u

y, u

z, v

y, v

z) and complex modal analysis (leading to j, u

c, uL

c, v

c, vL

c). The identi"cation

procedure was already described in Section 3.2. The displayed results show that the exactand estimated parameters agree very well within the uncertainty (as represented by thestandard deviation). As expected, real and complex modal analysis yielded exactly the samevalues for the poles j. The modal vectors were normalised, per mode, with respect to thelargest component. The equivalence of real and complex modal analysis was also revealedby inspection of the cost function and condition number of the Jacobian J during the lastiteration. Both analyses yielded identical values: ¸(C)"¸(R)"232.6 (as compared to theoptimum value ¸

015"(number of outputs)](number of points)!(number of estimated

parameters)/2"232) and c(J(C))"c(J (R))"1.5]105. The FRFs H11

( f ) and H12

( f ) forcomplex modal analysis are displayed in Fig. 3. Shown are the exact, non-parametric and

Figure 3. Comparison on the non-parametric FRFs (sss), estimated (*) and exact (} }}) parametric FRFsfor the simulation: amplitude in dB, phase in rad and frequency in Hz.

TABLE 2

<alidation for random excitation (M¸E equations (7) and (14) vs [10])

Mode 2B 1B 1F 2F Method

f0

(Hz) !61.54 !27.55 29.01 64.43 MLEf0

(Hz) !61.53 !27.51 29.00 64.39 [10]d 0.026 0.046 0.036 0.032 MLEd 0.032 0.054 0.041 0.033 [10]


parametric FRFs. Note that negative frequencies are used for visualisation purposes; duringthe identi"cation only positive frequencies were used (2]2 real and complex modalanalysis).

The identi"cation procedure was also validated on experimental data obtained usingrandom excitation. These data were provided by Lee et al. The experimental set-up andmeasurement procedure are described in detail in [2, 10]. The data originate again froma two-station situation and were analysed in terms of four modes (2B, 1B, 1F, 2F). Theprovided data represent the signals f

y(t), f

z(t) and y(t), z (t) during one record (i.e. no

averaging), each containing 8192 points. The identi"cation was performed using the methoddescribed in Section 3.3.1: leakage was taken into account by estimating the coe$cients h

Lof the extra leakage terms. On the basis of the non-parametric FRMs, only the frequencybands of interest were selected and parasitic peaks at multiples of the rotational frequency(due to unbalances, etc.) were cut away. As a result, 557 points were used for identi"cationusing real modal analysis. The results are given in Table 2. Shown are the naturalfrequencies f

0and logarithmic decrements d. Also shown are the results obtained by Lee

et al. [10] using an anisotropic dARMAX time domain method. Unfortunately, because thecovariance matrix of the data C(u) could not be retrieved here, uncertainties of theparameters are not provided. The modal vectors are not shown because they were notidenti"ed by Lee et al. [10]. Inspection of the table reveals that there is an excellentagreement in the natural frequencies. Comparison of the logarithmic decrements show thatour values are systematically smaller, this can be easily explained on the basis of theHanning "lter used by Lee et al. [10] to suppress leakage: the resulting broadening of thepeaks leads to higher d's. Finally, it is interesting to note that their results were obtainedusing averaging.

5. IDENTIFICATION OF A ROTOR TEST RIG

Experiments were performed on a rotor test rig (rotation speed X"600 rpm"11 Hz)using a multisine and a random excitation. In order to excite the "rst and second #exuralmode of the system, power was put in the frequency band 10}120 Hz. The experiments weredesigned to obtain the same measurement time for multisine and random measurements.For all measurements, the sampling frequency was f

S"400 Hz. For multisine measure-

ments, two experiments were required: each one consisting of 40 periods with 2048 pointsper period. For random measurements, 80]2048 points were measured. Identical peakvalues in the time domain were used for the random and periodic excitation. Details aboutthe experimental set-up, measurement and estimation of non-parametric FRFs can befound in [8].

The identi"cation was performed on the complete frequency band (10}120 Hz) usingN"5 #exural modes. This was necessary in order to take into account the extra modes

TABLE 3

Results real modal analysis for multisine and random excitation

Mode 1F 1B1 2F 2BM Excitation

j (!1.0053$0.0043)# (!2.7698$0.0048)# (!15.772$0.041)# (!12.324$0.029)# Multisine(168.2805$0.0042) j (173.5342$0.0049) j (421.549$0.041) j (444.364$0.029) j

j (!0.66$0.11)# (!2.55$0.12)# (!17.2$1.2)# (!11.55$0.30)# Random(168.85$0.11) j (174.18$0.12) j (424.3$1.2) j (444.81$0.30) j

uy

(1.00000$0.00046)# (1.00000$0.00065)# (1.0000$0.0028)# (0.1436$0.0014)# Mutisine(!0.00000$0.00044) j (0.00000$0.00064) j (0.0000$0.0028) j (!0.0525$0.0014) j

uy

(1.000$0.013)# (1.000$0.020)# (1.00$0.23)# (0.176$0.031)# Random(0.000$0.013) j (0.000$0.020) j (0.00$0.24) j (!0.039$0.032) j

uz

(!0.43790$0.00034)# (0.56824$0.00068)# (0.2412$0.0018)# (1.0000$0.0038)# Multisine(!0.23152$0.00034) j (0.27798$0.00067) j (!0.0722$0.0018) j (0.0000$0.0038) j

uz

(!0.4427$0.0087)# (0.476$0.019)# (0.32$0.10)# (1.000$0.047)# Random(!0.2473$0.0087) j (0.165$0.019) j (!0.02$0.11) j (0.000$0.048) j

vy

(!0.07006$0.00060)# (!0.04717$0.00058)# (!0.2167$0.0010)# (0.7343$0.0077)# Multisine(0.79308$0.00061) j (0.36578$0.00058) j (3.8832$0.0010) j (!0.5999$0.0077) j

vy

(!0.123$0.019)# (0.122#0.015)# (1.213$0.035)# (0.901$0.061)# Random(0.725$0.019) j (0.428$0.015) j (4.967$0.034) j (!1.048$0.062) j

vz

(0.36309$0.00071)# (!0.46032$0.00053)# (0.3650$0.0049)# (!0.7333$0.0013)# Multisine(!1.04274$0.00072) j (0.70845$0.00050) j (!0.4133$0.0049) j (3.3149$0.0012) j

vz

(0.332$0.019)# (!0.534$0.016)# (0.42$0.29)# (!0.743$0.015)# Random(!1.177$0.019) j (0.725$0.016) j (0.05$0.30) j (3.631$0.015) j

785E

ST

IMA

TIO

NO

FM

OD

AL

PA

RA

ME

TE

RS

Figure 4. Comparison on the non-parametric FRFs ())))) and parametric (*) FRFs for real modal testing withmultisine excitation: amplitude in dB vs frequency in Hz.


generated by the shakers [8]. As explained in [8], a Hanning window was applied both forrandom and multisine excitation due to the unbalance of the rotor. For random excitation,the covariance matrix C (u

k) was estimated using the instrumental variables method [8] and

the modal parameters were estimated on the basis of one record. The estimated modalparameters and their uncertainties for real modal testing are given in Table 3 (only the "rstand second #exural modes of the rotor are shown). A correction was made because only10 blocks could be used to estimate C(u

k) for multisine excitation (see also Appendix A).

A comparison of the non-parametric and parametric FRFs is shown for multisine excitationin Fig. 4.

It can be observed that more accurate results are obtained with multisine excitation.However, this statement should somewhat be relativated. The results for multisine exci-tation were obtained on the basis of two experiments, each consisting of 40 periods(as explained in [8], only 10 blocks could be used for identi"cation). On the contrary, for therandom excitation only one record was used for the identi"cation [C (u

k) was estimated

using 80 records]. Therefore, if all records were used, an improvement of the accuracy with

a factor J80:9 can be expected. Clearly, this argument does not explain the di!erence inaccuracy. Both types of excitation yielded similar poles but discrepancies can be found inthe (adjoint) modal vectors. It should be noted that the modal vectors were normalised withrespect to the largest component. An identi"cation on the basis of equations (7) and (14),thus estimating the leakage coe$cients instead of using a Hanning "lter, did not lead tosigni"cant better results and agreements.


Usually, in case of random excitation, noise is not taken into account. As a result, C(uk) is

not available. In this case, disturbances (e.g. unbalances) will deteriorate dramatically thequality of the estimated parameters if no precautions are taken. Therefore, the disturbancepeaks should be cut away, reducing the amount of points. In our case, this was not necessary(and not done).

The modal parameters for complex modal testing can be easily found using equation (6).On the basis of the circular modal vectors, the directivity of the modes was determined(see Table 3): for the backward modes themselves, one should take the complex conjugate ofthe displayed parameters.

6. CONCLUSIONS

In this paper, we have shown that when noise information is correctly implemented forMLE, real and complex modal testing are fully equivalent. An identi"cation procedure inthe frequency domain was developed yielding the modal parameters and their uncertainties.Experiments on a rotor test rig demonstrated that multisine excitation allows moreaccurate identi"cation as compared to random excitation.

ACKNOWLEDGEMENTS

We thank Prof. D. J. Ewins (Imperial College, London) for giving us the opportunity toperform the measurements in his laboratory. We thank Prof. C. W. Lee (KAIST, Taejon,Korea) for providing us experimental data.

This work is sponsored by the Fund for Scienti"c Research (FWO-Vlaanderen), theFlemish Government (GOV-IMMI), the Belgian Program on Interuniversity Poles ofAttraction initiated by the Belgian State, Prime Minister's O$ce, Science Policy program-ming (IUAP 4/2), and the European Project SALOME (BE96-3443).

REFERENCES

1. C. W. LEE 1993 <ibration Analysis of Rotors. Dordrecht: Kluwer Academic.2. C. Y. JOH and C. W. LEE 1996 Journal of <ibration and Acoustics 118, 64}69. Use of dFRFs for

diagnosis of asymmetric/anisotropic properties in rotor-bearing system.3. J. SCHOUKENS and R. PINTELOM 1991 Identi,cation of ¸inear Systems: A Practical Guideline to

Accurate Modeling. Oxford: Pergamon Press.4. J. SCHOUKENS, P. GUILLAUME and R. PINTELOM 1993 In Perturbation Signals for System

Identi,cation, pp. 126}159. Englewood Cli!s, NJ: Prentice-Hall. K. Godfrey (ed.). Design ofbroadband excitation signals.

5. J. SCHOUKENS, R. PINTELON, G. VANDERSTEEN and P. GUILLAUME 1997 Automatica 33,1073}1086. Frequency-domain system identi"cation using non parametric noise models esti-mated from a small number of data sets.

6. Y. ROLAIN and R. PINTELON 1999 Automatica 35, 965}972. Generating robust starting values forfrequency domain transfer function estimation.

7. R. PINTELON and J. SCHOUKENS 1997 Automatica 33, 991}994. Identi"cation of continuous-timesystems using arbitrary signals.

8. F. PEETERS, R. PINTELON, J. SCHOUKENS, Y. ROLAIN, E. GUTIERREZ and P. GUILLAUME 2001Mechanical Systems and Signal Processing 15, 759}773. Identi"cation of rotor-bearing systems inthe frequency domain Part I: estimation of frequency response functions.

9. C. W. LEE and Y. D. JOH 1993 Mechanical Systems and Signal Processing 7, 57}74. Theory ofexcitation methods and estimation of frequency response functions in complex modal testing ofrotating machinery.

10. C. W. LEE, J. P. PARK, J. S. YUN and C. Y. JOH 1997 Mechanical Systems and Signal Processing 11,827}842. Complex time series analysis for rotor dynamics identi"cation.


APPENDIX A

In [5] it is shown that when non-parametric noise models are estimated from the smallnumber of data sets, due to the fact that the exact covariances C(u

k) are replaced by their

sample estimates C) (uk), a loss in e$ciency will result (the covariances of the estimated

parameters increases). One can write for the &real' covariances of the estimated parameters:

Ch,CK "bCh,C (A1)

with Ch,A the covariance matrix of the estimated parameters h using the covariancesA"C, C) . For the estimation of complex parameters in MIMO systems, the loss ine$ciency is given by

b"M!N

o!1

M!No!2

. (A2)

Here, M represents the number of blocks (periods, records) and Nothe number of outputs.

However, not the &real' covariance is determined in practice, but an estimated one on thebasis of equation (10). Linearisation of the noise contributions leads to

C(%45)h,CK "

M!No!1

M!1C(%45)

h,C (A3)

Combining equations (A1)}(A3) yields

Ch,CK "M!1

M!No!2

C(%45)h,CK (A4)

which allows to determine the &real' covariance of the estimated parameters in terms ofthe estimated covariance when the sample covariance C) (u

k) is used. If su$cient blocks

are used, the correction can be neglected. However, for M"10 and No"2 one obtains

Ch,CK "1.5C(%45)h,CK .

APPENDIX B

In this appendix, we will show that c (H(R))"c (H (C)). The relationship between the FRMsin real and complex modal testing can be written as [see also equation (16)]

H (C)"T3 ~1 )H (R) )T3 (B1)

with T3 "J2T an unitary matrix (i.e. T3 H"T3 ~1). The condition number of a matrix can befound using the singular value decomposition. For H (R) one obtains

H (R)"U )S )VH (B2)

with U and V unitary and S a diagonal matrix consisting of the singular values (diagonalelements di!erent from zero). The condition number c(H (R)) is then given as the ratio of thelargest and smallest singular value: c (H (R))"p

.!9/p

.*/. Left multiplication of equation (B2)

with T3 ~1 and right multiplication with T3 gives

H (C)"U3 ) S )V3 H (B3)

with U3 "T3 ~1 )U and V3 "T3 H )V. It is easy to show that U3 and V3 are unitary. As a result,H (C) has identical singular values as H (R), yielding the same condition number.

identification of rotor-bearing systems in the frequency domain part ii: estimation of modal...

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