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A review of operational modal analysis techniques for in-service modal identification

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Abstract

Vibrations are the root cause of many mechanical and civil structure failures. Dynamic characteristics of a structure must be extracted to better understand structural vibrational problems. Modal analysis is used to determine the dynamic characteristics of a system like natural frequencies, damping ratios and mode shapes. Some of the applications of modal analysis include damage detection, design of a structure/machine for dynamic loading conditions and structural health monitoring. The techniques used for modal analysis are experimental modal analysis (EMA), operational modal analysis (OMA) and a less known technique called impact synchronous modal analysis (ISMA), which is a new development. EMA is performed in simulated controlled environment, while OMA and ISMA are performed when the system is in operation. Although EMA is the oldest modal analysis technique, there is an increasing interest in operational modal analysis techniques in recent years. In this paper, operational modal analysis techniques OMA and ISMA are reviewed with their development over the years and their pros and cons discussed.

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Abbreviations

\( \omega \) :

Angular frequency

\( \left[ {\varvec{G}_{{\varvec{xx}}} (\varvec{j\omega })} \right] \) :

Power spectral density (PSD) matrix of the input

\( \left[ {\varvec{G}_{{\varvec{yy}}} (\varvec{j\omega })} \right] \) :

PSD matrix of the output

\( \left[ {\varvec{H}(\varvec{j\omega })} \right] \) :

Frequency response function (FRF) matrix

\( \left[ \varvec{U} \right]_{\varvec{i}} \) :

Eigenvectors of \( \left[ {G_{yy} (j\omega )} \right] \)

\( \left[ \varvec{S} \right]_{\varvec{i}} \) :

Eigenvalues of \( \left[ {G_{yy} (j\omega )} \right] \)

\( \varvec{y}\left( \varvec{t} \right) \) :

System time response/output displacement

\( \varvec{\varphi } \) :

Mode shape

\( c_{i} \) :

\( i \)th modal contribution factor

\( \varvec{Y}_{\varvec{i}} \) :

Mode-isolated output acceleration time history

\( \varvec{E}_{\varvec{i}} \) :

Cross-correlation matrix

\( \varvec{U} \) :

Singular vector matrix of \( Y_{i} \)

\( {\varvec{\Omega}} \) :

Singular value matrix of \( Y_{i} \)

\( f(t) \) :

Input/excitation force

\( x_{ik} \left( t \right) \) :

System response at coordinate I due to force at coordinate k

\( \zeta \) :

Damping ratio

\( m_{r} \) :

Modal mass of mode r

\( E \) :

Expectation operator

\( \varvec{M} \) :

Mass matrix

\( \varvec{C} \) :

Damping matrix

\( \varvec{K} \) :

Stiffness matrix

\( \varvec{y}_{\varvec{k}} \) :

System response/output vector

\( \varvec{e}_{\varvec{k}} \) :

White noise/input vector

\( \varvec{A}_{\varvec{i}} \) :

Auto-regressive (AR) coefficient matrix

\( \varvec{C}_{\varvec{i}} \) :

Moving average (MA) coefficient matrix

\( \varvec{w}_{\varvec{k}} \) :

Process noise

\( \varvec{v}_{\varvec{k}} \) :

Measurement noise

A :

State transition matrix

\( {\varvec{\Psi}} \) :

Eigenvector matrix of A

\( {\varvec{\Lambda}} \) :

Eigenvalue matrix of A

\( R_{i} \) :

Output covariance

\( \varvec{T}_{\varvec{i}} \) :

Output covariance matrix

\( \varvec{O}_{\varvec{i}} \) :

Observability matrix

\( {\varvec{\Gamma}}_{\varvec{i}} \) :

Controllability matrix

\( \lambda_{k } \) :

Eigenvalue of discrete-time system

\( \lambda_{\text{ck}} \) :

Eigenvalue in continuous time system

s r :

Pole of mode r

β :

Phase angle

A :

Amplitude of the impact signature

x(t):

Desired response signal due to impact

e(t):

Undesired signal of random ambient noises

σ r :

Decay rate

R 2 :

Amplitude of the undesired signal

β 1 :

Phase of the desired response signal

β 2 :

Phase of undesired signal of the periodic response of cyclic load

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Acknowledgements

The authors wish to acknowledge the financial support and advice given by University of Malaya Faculty Research Grant (GPF001A-2018), Impact-Oriented Interdisciplinary Research Grant (IIRG007B-2019), Advanced Shock and Vibration Research (ASVR) Group of University of Malaya and other project collaborators.

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  1. Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia

    Fahad Bin Zahid, Zhi Chao Ong & Shin Yee Khoo

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  1. Fahad Bin Zahid
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  2. Zhi Chao Ong
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  3. Shin Yee Khoo
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Correspondence to Zhi Chao Ong.

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Zahid, F.B., Ong, Z.C. & Khoo, S.Y. A review of operational modal analysis techniques for in-service modal identification. J Braz. Soc. Mech. Sci. Eng. 42, 398 (2020). https://doi.org/10.1007/s40430-020-02470-8

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