Assessment of the Frequency Domain Decomposition Method: Comparison of Operational and Classical Modal Analysis Results

Aleks KUYUMCUOGLU Arcelik A. S., Research & Development Center, Istanbul, Turkey

Prof. Dr. Kenan Yuce SANLITURK Istanbul Technical University, Mechanical Engineering Department, Istanbul, Turkey

ABSTRACT: Operational Modal Analysis (OMA) is originally used for extracting modal parameters (natural frequency, mode shape and damping) of mainly civil engineering structures. However, in recent years it is also becoming popular for modal analysis of mechanical structures. The main advantage of the method is that neither artificial excitation needs to be applied to the structure nor force signal(s) is/are to be measured. The parameter estimation is based upon the response signals only, thereby minimizing the cost for modal testing. One of the techniques that is often used for OMA is called Frequency Domain Decomposition (FDD) method which is a non-parametric technique. This study addresses identifying modal parameters of some relatively simple mechanical systems under operating conditions using FDD method. The FDD procedure requires auto- and cross-spectrums between measured response data. The success of the FDD technique is examined using both numerical simulations as well as experimental studies. Various types of excitations have been applied in order to assess the advantages and limitations of the FDD method. The results of OMA are compared to those of classical modal testing. It is shown that OMA can yield satisfactory results if the underlying assumptions behind this approach, the most significant one being that the excitation is stationary white noise, are satisfied. However, it is also seen that the accuracy of the method starts to decline when a structure is subjected to complicated excitations as in rotating machinery.

1 INTRODUCTION

Classical modal analysis methods can be applied to those cases where the input forces can be measured. However, if it is not possible or practical to measure the input forces, these methods lose their applicability. In some cases, it is impossible or very difficult to excite structures using shakers or impact hammers. These conditions are usually faced when engineers deal with very large structures such as bridges, buildings and large mechanical systems. Also, in some other cases, the level of excitation forces might be so large that they cannot be measured by available sensors.

The measured responses from linear systems contains: i) Responses due to the input forces ii) Environmental noises iii) Measurement Noises. In classical modal analysis, measured responses are supposed to depend only on the input forces which are generated and applied in a controlled manner. Therefore, to minimize the other effects, input forces must be sufficiently strong. This requirement in classical modal analysis brings additional cost, especially in terms of hardware. However, Operational Modal Analysis (OMA) techniques do not require any artificial excitation to be applied to the structure in order to determine the dynamic properties of systems under actual working conditions. As a result, if OMA approach is applicable, the whole modal analysis process can become relatively easier and more economical.

Large civil engineering structures are often excited by natural loads under operating conditions that cannot easily be controlled, for instance wave loads (offshore structures), wind loads (buildings) or traffic loads (bridges). A similar argument can also be made for most

machines under operating conditions. They are also excited by natural sources such as noise from bearings or vibrations from other machines nearby or internally generated random forces. For example, a typical vehicle on the road in operating conditions is excited by the road, wind and the engine in very complicated manner. Another advantage of OMA is that the dynamic properties of systems are estimated under actual operating conditions. These dynamic properties are the system’s natural frequencies, damping and mode shapes.

In practice, many forces have some randomness and contain some characteristics of noise. It is known that this type of forces can excite all the modes of structures. The most important assumption in most Operational Modal Analysis techniques is that the excitation to the system is stationary white noise. This white noise is able to excite the structure in all frequencies and has a flat spectrum. Therefore, the spectrum of white noise can be used as an appropriate excitation for any system. Based on this fundamental assumption, Operational Modal Analysis methods are able to determine the dynamic properties of systems by using the measured vibration levels under operating conditions without the need for measuring the excitation forces.

For the past 15 years or so, different OMA methods have been proposed by researchers to determine the dynamic characteristics of systems and these methods are applied to many different situations. In 1995, G.H. James [1] used Natural Excitation Techniques (NExT) to determine the physical parameters of the systems under working conditions. R.Brincker [2] introduced Frequency Domain Decomposition Technique and later Enhanced Frequency Domain Decomposition (EFDD) Technique. At the same time, different time domain methods [3] were also developed and published in the literature, for example, Stochastic Subspace Identification (SSI) method [4]. Among all the proposed OMA techniques, FDD and SSI are the most well-known and used methods. FDD uses Complex Mode Indicator Function [5] approach for processing the measured responses and then determines the natural frequencies and mode shapes of the structure with simple Peak Picking technique. Therefore, the main advantage of this method is that it is relatively easy to implement and use as it requires relatively fewer complex calculations. On the other hand, as stated, SSI method is a time-domain method and it is more complex and time consuming when compared to FDD. For detail information interested reader may refer to [3], [4], [6], [7], [8].

The FDD method is used in this paper. This method is based on the formulation of the relationship between the input and the output power spectral density functions, and the modal parameters are estimated via Singular Value Decomposition (SVD) of the cross-spectrum matrix. Singular values contain frequency and damping information and singular vectors contain mode shape information. The aim of this study is to determine the advantages and the limitations of FDD technique and to investigate the applicability of this method in practice. The main reason for choosing this method is its simplicity, requiring less computation and practical applicability being easier than other methods. This paper is organized as follows: First, the theory behind the FDD method is described briefly. Then, some numerical simulations are made in order to assess the performance of the FDD method. In these simulations, dynamic forces are applied to the test structure and structural responses are calculated at all selected co-ordinates. However, the excitation forces in this paper are considered unknown for OMA purposes. In numerical simulations, finite element model of a simple plate structure is developed and used. The modal parameters obtained from simulated OMA are compared with those obtained via traditional approach. After that, experimental studies are presented. In the experimental studies, modal shakers or modal hammers are used to excite the plate structure but excitation forces are not utilized during OMA. Piezoelectric accelerometers are used to measure the system responses. Then, the measured data are processed using a MATLAB code developed for FDD analysis so as to determine the system modal parameters. These modal parameters are then compared with those obtained via classical modal testing approach that required excitation forces to be measured. After the plate structure, a practical applicability of FDD method is assessed by applying this approach to a rotating machinery under operating conditions. The results are again compared with those obtained via traditional approach. Finally, some concluding remarks are given about the accuracy, reliability and applicability of FDD method.

2 THEORY

Frequency Domain Decomposition technique determines the modes by applying the method of Singular Values Decomposition to the spectral density matrices. This decomposition corresponds to a single degree of freedom identification of the system for each singular value. The relationship between the input x(t) and the output y(t) of a linear system can be written as: [9]

[ ] [ ] [ ] [ ] Tmxrrxrxxmxrmxmyy HGHG )()()()( * ωωωω = (1)

Where [ ])(ωxxG is the input power spectrum matrix that is constant in case of stationary zero mean white noise input and this is expressed in terms of constant C. Also r indicates the number of inputs. [ ])(ωyyG is the output power spectrum matrix. m indicates the number of the responses. [ ])(ωH is the Frequency Response Function (FRF) matrix. ‘*’ and ‘T’ superscripts refers to complex conjugate and transpose, respectively. As seen in Equation (1), [ ])(ωyyG is very sensitive to the input constant C. The next equations and single degree of freedom system identification are based on the assumption that the input power spectrum is represented by a constant value.

As shown in Eq. (2) the FRF matrix can be expressed in the form of poles and residues as in classical modal analysis.

[ ] [ ] [ ]∑

= −+

−=

m

k k

k

k

k

j

R

j

RH

1*

*

)(λωλω

ω (2)

dkkk jωσλ +−= (3)

Here, m is the total number of modes, kλ is the pole of the ��� mode, kσ modal damping (decay constant), dkω is the damped natural frequency of the ��� mode which can be expressed as Eq. (4).

20 1 kkdk ςωω −= (4)

ok

kk ω

σς = (5)

In Eq. (5), kς is the damping ratio for the ��� mode and okω is the undamped natural frequency for the ��� mode. [Rk] in Eq.(2) is given as:

���� = kψ � kγ �� (6)

In Eq. (6), kψ is the mode shape of the ��� mode and kγ is the modal participation vector of the ��� mode. If the input to the system is considered as the white noise, power spectral density matrix can be taken as a constant matrix ( CjwGxx =)( ) and Eq. (1) takes the following form.

[ ] [ ] [ ] [ ] [ ]∑∑

= =

−+

−

−+

−=

m

k

m

s s

H

s

s

Ts

k

k

k

kyy j

R

j

RC

j

R

j

RG

1 1**

*

)(λωλωλωλω

ω (7)

Where superscript H represents the complex conjugate transpose. With using the expression

[ ]yyG in Eq. (1) and with some mathematical operations, output power spectrum matrix can be

written as: [2]

[ ] [ ] [ ] [ ] [ ]*

*

1*

*

)(k

k

k

km

k k

k

k

kyy j

B

j

B

j

A

j

AG

λωλωλωλωω

−−+

−−+

−+

−=∑

=

(8)

Where, [ ]kA is the ��� residue matrix of [ ]yyG matrix. This matrix is a Hermitian matrix

and can be expressed as:

[ ] [ ] [ ] [ ]∑

= −−+

−−=

m

s sk

Ts

sk

Hs

kk

RRCRA

1* λλλλ

(9)

The contribution of the residue for the ��� mode has the following expression:

[ ] [ ] [ ]k

Hkk

k

RCRA

σ2= (10)

Here kσ is the negative of the real part of the pole as kkk jωσλ +−= . If lightly damped model is considered, modal contribution matrix becomes proportional to the mode shape vector and can be written as:

[ ] [ ] [ ] { }{ } { }{ } { }{ }T

kkk

T

kk

T

kkT

kklightdamping

k dCRCRA ψψψγγψ ===→

lim (11)

Where kd is a scalar constant. At a certain frequency (ω ) only a limited number of modes will contribute significantly, typically one or two modes. These modes are indicated by )(ωSub As a result, for the lightly damped structures, output spectral density matrix is expressed as the final form in Eq. (12) [2].

[ ] ∑∈ −

+−

=)(

*

**

)(wSubk k

Hkkk

k

Tkkk

yyj

d

j

dG

λωψψ

λωψψω (12)

This final form of the matrix is decomposed to singular values and singular vectors using the singular value decomposition technique. This decomposition is performed to determine modal parameters of the system.

3 NUMERICAL SIMULATIONS

In order to assess the performance and numerical simulations are performedcalculated and these results are used as input to the FDD method for operational modal analysis. In other words, measured data are simulated is used here to create numeric200 mm and a thickness of 3 mm. Material is steel. The finite element model created and shown in Figure 1. Thenusing classical modal analysis. After thatforced response of the system is calculatedplate at many locations are “recorded”results of the OMA are then compared to those obtained via traditional modal analysis.

Figure 1: Finite element model of the plate

In this simulation, a swept sine force is applied to a point in the numerical model of the plate shown in Figure 1. For OMA purposes, thisresponses of the system are recorded at 50 coof acceleration is plotted in Figure 2.

Figure 2

NUMERICAL SIMULATIONS

performance and the applicability of FDD technique numerical simulations are performed first. In these simulations, the system response is calculated and these results are used as input to the FDD method for operational modal analysis. In other words, measured data are simulated in a numerical environment. Finite element method

ical models of the test structure which has dimensions200 mm and a thickness of 3 mm. Material is steel. The finite element model

and shown in Figure 1. Then, the natural frequencies and mode shapes are modal analysis. After that, the structure is excited using a specified force and

d response of the system is calculated. The resulting accelerations over the surface of the “recorded” for further operational modal analysis using FDD. The

results of the OMA are then compared to those obtained via traditional modal analysis.

Figure 1: Finite element model of the plate structure

, a swept sine force is applied to a point in the numerical model of the plate For OMA purposes, this excitation force is considered unknown and forced

recorded at 50 co-ordinates. A typical response spectruin Figure 2.

Figure 2: Frequency dependent acceleration spectrum

in practice, some the system response is

calculated and these results are used as input to the FDD method for operational modal analysis. Finite element method

dimensions of 450 mm x 200 mm and a thickness of 3 mm. Material is steel. The finite element model of the plate is

natural frequencies and mode shapes are calculated the structure is excited using a specified force and the

over the surface of the ysis using FDD. The

results of the OMA are then compared to those obtained via traditional modal analysis.

, a swept sine force is applied to a point in the numerical model of the plate on force is considered unknown and forced

ordinates. A typical response spectrum in terms

As can be seen from Figure 2, taccelerations data on the structure are taken situation where the response levels are recorded at desired comany sensors as required. In such cases, a dimension 50 x 50, containing data taken from 50 locations using singular value decomposition of the

A plot of singular values isare seen 50 curves as accelerationfrequencies are identified from this figure considering the frequencies values. It should be noted in Figure 3 that the first singular values are much larger than the other singular values, indicating that there are no double modes. has 4 natural frequencies within 0 to

Figure 3:

The corresponding mode shape is obtained via singular value decomposition of the spectral density matrix at a natural frequency. two natural frequencies and mode shapes analysis.

As can be seen from Figure 2, this plate is analyzed up to 400 Hz. In this simulationon the structure are taken simultaneously. This simulates the measureme

situation where the response levels are recorded at desired co-ordinates simultaneously In such cases, there is no need to use a reference point.

containing power and cross spectrums is obtained using of the structure. Natural frequencies of the structure

singular value decomposition of the spectral density matrix. s is presented in Figure 3 as function of frequency. In this

accelerations are “measured” at 50 co-ordinates on the structure. dentified from this figure considering the frequencies with highest singular

It should be noted in Figure 3 that the first singular values are much larger than the other singular values, indicating that there are no double modes. As seen in Figure 3,

within 0 to 400 Hz.

Figure 3: Singular Value Plot for plate structure

corresponding mode shape is obtained via singular value decomposition of the spectral density matrix at a natural frequency. In figure 4a and 4b there is shown the comparison of first two natural frequencies and mode shapes that are obtained via classical and operational modal

a)

this simulation, all the simultaneously. This simulates the measurement

ordinates simultaneously using as reference point. A matrix with

using the acceleration Natural frequencies of the structure are obtained

In this graph, there on the structure. Natural

with highest singular It should be noted in Figure 3 that the first singular values are much larger than the

As seen in Figure 3, this plate

corresponding mode shape is obtained via singular value decomposition of the spectral In figure 4a and 4b there is shown the comparison of first

classical and operational modal

Figure 4a 4b: First two modes obtained

It is also aimed to assess the performance of the FDD method in terms of damping estimation. For this purpose, a known level ofmodel. Then, the response levels in terms of acceleration are recorded and the spectrums are analyzed using estimated damping levels are listed in Table 1.OMA analysis deviates from the actual levels by about 5%.from the actual damping level may be considered acceptable, it is still significant since OMA is simulated here under ideal conditions.

In this section, both classical and and the corresponding natural frequencies, mode shapes and damping ratios seen that the modal parameters obtained from FDD techniquefrom classical modal analysishere represents ideal conditions. In the next section, the performance of the FDD is examined in real situation using measured data.

4 EXPERIMENTAL STUDY

After validating FDD method performance and the applicabilitTwo experimental test cases are performed here. In the first case, a plate

b)

First two modes obtained via Classical Modal Analysis and OMA

also aimed to assess the performance of the FDD method in terms of damping estimation. For this purpose, a known level of (i.e. 1%) structural damping is specified in the FE

Then, the response levels in terms of acceleration are recorded and the using the half-power bandwidth method [10]. The actual and the

g levels are listed in Table 1. It is seen that the estimated damping level in this OMA analysis deviates from the actual levels by about 5%. Although this level of deviation from the actual damping level may be considered acceptable, it is still significant since

is simulated here under ideal conditions.

Table 1: Estimated damping levels

both classical and operational modal analyses are applied to natural frequencies, mode shapes and damping ratios

modal parameters obtained from FDD technique are very close to those obtained ical modal analysis. However, it should be restated here that the OMA simulation

here represents ideal conditions. In the next section, the performance of the FDD is examined in real situation using measured data.

EXPERIMENTAL STUDY

After validating FDD method using numerical simulations, this section aims to assess the performance and the applicability of this OMA method in practice using experimental data. Two experimental test cases are performed here. In the first case, a plate structure is used. The

Classical Modal Analysis and OMA

also aimed to assess the performance of the FDD method in terms of damping g is specified in the FE

Then, the response levels in terms of acceleration are recorded and the response The actual and the

It is seen that the estimated damping level in this though this level of deviation

from the actual damping level may be considered acceptable, it is still significant since the

applied to a plate structure natural frequencies, mode shapes and damping ratios are obtained. It is

are very close to those obtained the OMA simulation

here represents ideal conditions. In the next section, the performance of the FDD is examined in

aims to assess the y of this OMA method in practice using experimental data.

structure is used. The

second case, however, is using an actual rotating machinery in order to assess the capabilities and limitations of FDD method in practice.

First of all, classical modal analyses are performed on the experimental structures. For thispurpose, Frequency Response Functions (FRFs) are measured using either hammer or shaker testing. Then measured FRFs are analyzed using classical modal analysis approach so as to have reference modal parameters to compare with OMA results.

Shaker or modal hammer is also used for operational modal analyses in this study. However, the forces applied to the structure are not measured for OMA purposes. Instead, only the resulting vibrations are recorded.

The first experimental stud450 mm x 200 mm x 3 mm. Figure 5a. 3 accelerometers are used simultaneously and a total ofpurpose of classical modal ana

Then, a random white noise type of excitation is applied to in Figure 5b. However, this force is not measured or recorded only the resulting vibrations are recorded.during data collection for OMA and vibration data areHowever, the location of one of the accelerometers is kept fixed during these measuresponse measured by this accelerometer being used as a reference for phase determination.

Figure 5a 5b: Plate

Figure 6:

second case, however, is using an actual rotating machinery in order to assess the capabilities and limitations of FDD method in practice.

First of all, classical modal analyses are performed on the experimental structures. For thispurpose, Frequency Response Functions (FRFs) are measured using either hammer or shaker testing. Then measured FRFs are analyzed using classical modal analysis approach so as to have reference modal parameters to compare with OMA results.

is also used for operational modal analyses in this study. However, the forces applied to the structure are not measured for OMA purposes. Instead, only the resulting vibrations are recorded.

The first experimental study is carried out on a steel plate structure which has450 mm x 200 mm x 3 mm. As stated, FRFs are measured using modal hammer

3 accelerometers are used simultaneously and a total of 50 FRFs are measured for modal analysis. A typical measured FRF is presented in Figure 6.

a random white noise type of excitation is applied to this plate using aHowever, this force is not measured or recorded for the purpose of OMA. Instead,

resulting vibrations are recorded. Again 3 accelerometers are used during data collection for OMA and vibration data are collected at 50 coHowever, the location of one of the accelerometers is kept fixed during these measuresponse measured by this accelerometer being used as a reference for phase determination.

a) b)

Plate structure excited using a modal hammer and a shaker

Figure 6: A Measured FRF on a plate structure

second case, however, is using an actual rotating machinery in order to assess the capabilities

First of all, classical modal analyses are performed on the experimental structures. For this purpose, Frequency Response Functions (FRFs) are measured using either hammer or shaker testing. Then measured FRFs are analyzed using classical modal analysis approach so as to

is also used for operational modal analyses in this study. However, the forces applied to the structure are not measured for OMA purposes. Instead, only the

structure which has dimensions of modal hammer as shown in

50 FRFs are measured for the A typical measured FRF is presented in Figure 6.

a shaker as shown or the purpose of OMA. Instead,

ometers are used simultaneously collected at 50 co-ordinates again.

However, the location of one of the accelerometers is kept fixed during these measurements, the response measured by this accelerometer being used as a reference for phase determination.

shaker

Power and cross spectrums plots of a typical cross spectrum are givenspectrum is obtained by using the response measurements at natural frequencies of the system are identified via singular value decomposition ofdensity matrix as shown in Figure 8. Each peak in this graph the corresponding mode shape density matrix at a natural frequency.

Figure 7a 7b: Plate structure:

Power and cross spectrums are calculated using 4 x 4 matrices. The magnitude and phase a typical cross spectrum are given in Figure 7a and 7b, respectively. is obtained by using the response measurements at two points on the plate.

natural frequencies of the system are identified via singular value decomposition ofshown in Figure 8. Each peak in this graph identifies a natural

mode shape is obtained via singular value decomposition of natural frequency.

a)

b)

Plate structure: Measured Cross Spectrum– Magnitude & Phase

magnitude and phase , respectively. This cross

two points on the plate. Then, the natural frequencies of the system are identified via singular value decomposition of the spectral

natural frequency and singular value decomposition of the spectral

Magnitude & Phase

Figure 8:

In this study the plate structuremodes are identified up to 800 Hz. operational modal analyses are listed and compared in Table 2. mode shapes are compared in Figure 9. two methods are fairly close to each other. Also, the mode shapes are found to be quite compatible. The damping level for the to compare the damping levels

Figure 8: Plate structure: Singular Value Graph

structure is analyzed up to 800 Hz with a 0.5Hz frequency resolution

up to 800 Hz. The natural frequencies determined via classical and operational modal analyses are listed and compared in Table 2. Some of tmode shapes are compared in Figure 9. It is seen that the natural frequencies identified using

irly close to each other. Also, the mode shapes are found to be quite The damping level for the bare steel plate is very low, hence it was not appropriate

to compare the damping levels.

Table 2: Plate Natural Frequencies

with a 0.5Hz frequency resolution. 9 The natural frequencies determined via classical and

Some of the corresponding It is seen that the natural frequencies identified using

irly close to each other. Also, the mode shapes are found to be quite very low, hence it was not appropriate

Figure 9a 9b: First two mode

FDD method’s practical applicability machine under actual operating compared to those obtained using data are acquired from one of theof singular value decomposition of the spectr

a)

b)

First two mode shapes obtained via Classical Modal Analysis and OMA

FDD method’s practical applicability is also tested here. The dynamic properties ofactual operating conditions are determined via OMA and

to those obtained using traditional modal analysis method. In this case, the vof the side panels of the rotating machine. Figure 10 shows the result

of singular value decomposition of the spectral density matrix.

Classical Modal Analysis and OMA

he dynamic properties of a rotating and the results are

In this case, the vibration Figure 10 shows the result

Figure 10:

In this study, the particular Hz, so in frequency spectrums peaks will appear at rotation frequency 11.5 Hz and harmonics. This situation is one of the major disadvantages of operational modal analysis, leads to the need for distinguishing virtual modes. If the rotationfrequencies of the structure, frequencies. In this study, the rotational frequency of the mTherefore, it was possible to identifyfrequencies via OMA. The identified natural frequencies are listed and compared to those estimated from classical approach in Table 3.compatible, are shown in Figure 11.

Table 3: Natural Frequencies

Figure 10: Singular Value Plot for a Rotating Machine

the particular rotating machine is running at 690 rpm which corresponds to 11.5 Hz, so in frequency spectrums peaks will appear at rotation frequency 11.5 Hz and harmonics. This situation is one of the major disadvantages of operational modal analysis, leads to the need for distinguishing between the real physical modes of the structurevirtual modes. If the rotational frequency or its harmonics are very close to the natural

it will be very difficult to determine modal parameters at In this study, the rotational frequency of the machine under test was measured.

Therefore, it was possible to identify the physical modes and the corresponding natural frequencies via OMA. The identified natural frequencies are listed and compared to those

rom classical approach in Table 3. The corresponding mode shapes, which are quite Figure 11.

Natural Frequencies of the side panel of a rotating machine.

rotating machine is running at 690 rpm which corresponds to 11.5 Hz, so in frequency spectrums peaks will appear at rotation frequency 11.5 Hz and its harmonics. This situation is one of the major disadvantages of operational modal analysis, as it

the real physical modes of the structure and the re very close to the natural

will be very difficult to determine modal parameters at those achine under test was measured.

the physical modes and the corresponding natural frequencies via OMA. The identified natural frequencies are listed and compared to those

The corresponding mode shapes, which are quite

of the side panel of a rotating machine.

Figure 11a 11b: 4th and 7

5 CONCLUDING REMARKS

The accuracy and the applicability of experimental case studies. The results show that OMA can yield satisfactory underlying assumptions behind this approach, the most significant one being that the excitation is stationary white noise, are satare not scaled as the excitationstarts to decline when a structure is subjected to complicated excitations as in rotating machinery. In operating conditions, close to the one or more natural frequencies of the structure, determine modal parameters at

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[1] G.H.James III, T.G.Carne and J.P.Laufer,parameter extraction from operating structures”, Experimental Modal Analysis 10(4)

a)

b)

and 7th modes obtained using Classical Modal Analysis and OMA

CONCLUDING REMARKS

The accuracy and the applicability of the FDD method is examined using both numerical and The results show that OMA can yield satisfactory

underlying assumptions behind this approach, the most significant one being that the excitation is stationary white noise, are satisfied. Mode shapes obtained from operational modal analysis

the excitation forces are not known. It is found that the accuracy of the method starts to decline when a structure is subjected to complicated excitations as in rotating

In operating conditions, if the main excitation frequency and its harmonicsnatural frequencies of the structure, it appears that it will be

determine modal parameters at those frequencies.

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Classical Modal Analysis and OMA

examined using both numerical and The results show that OMA can yield satisfactory answers if the

underlying assumptions behind this approach, the most significant one being that the excitation Mode shapes obtained from operational modal analysis

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