Research Papers

Extended Smooth Orthogonal Decomposition for Modal Analysis

[+] Author and Article Information
Zhi-Xiang Hu

Department of Civil Engineering,
Hefei University of Technology,
Hefei 23009, Anhui Province, China
e-mail: huzhixiang@hfut.edu.cn

Xiao Huang

The 38th Research Institute of CETC,
Hefei 230088, Anhui Province, China

Yixian Wang, Feiyu Wang

Department of Civil Engineering,
Hefei University of Technology,
Hefei 23009, Anhui Province, China

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 25, 2017; final manuscript received January 5, 2018; published online February 23, 2018. Assoc. Editor: Stefano Gonella.

J. Vib. Acoust 140(4), 041008 (Feb 23, 2018) (12 pages) Paper No: VIB-17-1428; doi: 10.1115/1.4039240 History: Received September 25, 2017; Revised January 05, 2018

The smooth orthogonal decomposition (SOD) is an output-only modal analysis method, which has simple structure and gives good results for undamped or lightly damped vibration systems. In the present study, the SOD method is extended to incorporate various measurements that contain the displacement, the velocity, the acceleration, and even the jerk (derivation of the acceleration). Several generalized eigenvalue problems (EVPs) are put forward considering different measurement combinations, and it is proved that all these EVPs can reduce to the eigenvalue problems of the undamped vibration system. These different methods are called extended smooth orthogonal decomposition (ESOD) methods in this paper. For the damped vibration system, the frequencies obtained by different ESOD methods are different from each other. Thus, a cost function is defined and a search algorithm is proposed to find the optimal frequency and damping ratio that can explain these differences. Although the search algorithm is derived for the single-degree-of-freedom (SDOF) vibration systems, it is effective for the multi-degrees-of-freedom (MDOF) vibration system after assuming that the smooth orthogonal coordinates (SOCs) computed by the ESOD methods are approximate to the modal coordinate responses. In order to verify the ESOD methods and the search algorithm, simulations are carried out and the results indicate that all ESOD methods reach correct results for undamped vibration systems and the search algorithm can give accurate frequency and damping ratio for damped systems. In addition, the effects of measurement noises are considered and the results show that the proposed method has anti-noise property to some extent.

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Luz, E. , 1987, “ Experimental Modal Analysis of Large-Scale Structures,” International Conference on Mechanical Dynamics, Shenyang, China, pp. 257–262.
Brincher, R. , Zhang, L. , and Andersen, P. , 2000, “ Output-Only Modal Analysis by Frequency Domain Decomposition,” 25th International Seminar on Modal Analysis (ISMA), Leuven, Belgium, pp. 717–723.
Yu, D. J. , and Ren, W. X. , 2005, “ EMD-Based Stochastic Subspace Identification of Structures From Operational Vibration Measurement,” Eng. Struct., 27(12), pp. 1741–1751. [CrossRef]
Ibrahim, S. R. , and Mikulcik, E. C. , 1977, “ A Method for the Direct Identification of Vibration Parameters From the Free Response,” Shock Vib. Bull., 47(4), pp. 183–198.
Wang, B. T. , and Cheng, D. K. , 2008, “ Modal Analysis of MDOF System by Using Free Vibration Response Data Only,” J. Sound Vib., 311(3–5), pp. 737–755. [CrossRef]
Wang, B. T. , and Cheng, D. K. , 2011, “ Modal Analysis by Free Vibration Response Only for Discrete and Continuous Systems,” J. Sound Vib., 330(16), pp. 3913–3929. [CrossRef]
Rezaee, M. , Yam, G. F. , and Fathi, R. , 2015, “ Development of ‘Modal Analysis Free Vibration Response Only’ Method for Randomly Excited Systems,” Acta Mech., 226(12), pp. 4031–4042. [CrossRef]
Feeny, B. F. , and Kappagantu, R. , 1998, “ On the Physical Interpretation of Proper Orthogonal Modes in Vibration,” J. Sound Vib., 211(4), pp. 607–616. [CrossRef]
Kerschen, G. , and Golinval, J. C. , 2002, “ Physical Interpretation of the Proper Orthogonal Modes Using the Singular Value Decomposition,” J. Sound Vib., 249(5), pp. 849–865. [CrossRef]
Feeny, B. F. , and Liang, Y. , 2003, “ Interpreting Proper Orthogonal Modes in Randomly Excited Vibration Systems,” J. Sound Vib., 265(5), pp. 953–966. [CrossRef]
Yi, J. H. , and Yun, C. B. , 2004, “ Comparative Study on Modal Identification Methods Using Output-Only Information,” Struct. Eng. Mech., 17(3–4), pp. 445–466. [CrossRef]
Bellizzi, S. , and Sampaio, R. , 2006, “ POMs Analysis of Randomly Vibrating Systems Obtained From Karhunen-Loève Expansion,” J. Sound Vib., 297(3–5), pp. 774–793. [CrossRef]
Caldwell, R. A. , and Feeny, B. F. , 2016, “ Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition,” ASME J. Vib. Acoust., 138(4), p. 041007. [CrossRef]
Cao, S. , and Ouyang, H. , 2017, “ Output-Only Damage Identification Using Enhanced Structural Characteristic Deflection Shapes and Adaptive Gapped Smoothing Method,” ASME J. Vib. Acoust., 140(1), p. 011005. [CrossRef]
Park, J. Y. , Wakin, M. B. , and Gilbert, A. C. , 2014, “ Modal Analysis With Compressive Measurements,” IEEE Trans. Signal Process., 62(7), pp. 1655–1670. [CrossRef]
Chelidze, D. , and Zhou, W. , 2006, “ Smooth Orthogonal Decomposition-Based Vibration Mode Identification,” J. Sound Vib., 292(3–5), pp. 461–473. [CrossRef]
Zhou, W. , and Chelidze, D. , 2007, “ Blind Source Separation Based Vibration Mode Identification,” Mech. Syst. Signal Process., 21(8), pp. 3072–3087. [CrossRef]
Zhou, W. , and Chelidze, D. , 2008, “ Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification,” ASME J. Vib. Acoust., 130(1), p. 011001. [CrossRef]
Bellizze, S. , and Sampaio, R. , 2009, “ Smooth Karhunen-Loève Decomposition to Analyze Randomly Vibrating Systems,” J. Sound Vib., 325(3), pp. 491–498. [CrossRef]
Farooq, U. , and Feeny, B. F. , 2008, “ Smooth Orthogonal Decomposition for Modal Analysis of Randomly Excited Systems,” J. Sound Vib., 316(1–5), pp. 137–146. [CrossRef]
Rezaee, M. , and Yam, G. F. , 2014, “ Improving the Accuracy of SOD for Modal Parameters Estimation of Damped Systems,” Acta Mech., 226(6), pp. 1673–1687. [CrossRef]
Rezaee, M. , Shaterian-Alghalandis, V. , and Banan-Nojavani, A. , 2013, “ Development of the Smooth Orthogonal Decomposition Method to Derive the Modal Parameters of Vehicle Suspension System,” J. Sound Vib., 332(7), pp. 1829–1842. [CrossRef]
Allemang, R. J. , and Brown, D. , 1982, “ A Correlation Coefficient for Modal Vector Analysis,” International Modal Analysis Conference, Orlando, FL, Nov. 8–10, pp. 110–116.
Souza, K. , and Epureanu, B. , 2011, “ Noise Rejection for Two Time-Based Multi-Output Modal Analysis Techniques,” J. Sound Vib., 330(6), pp. 1045–1051. [CrossRef]


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Fig. 1

Flowchart of the search algorithm for the optimal ω and ξ corresponding to the minimum of the cost function

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Fig. 2

Comparison between frequencies estimated by the ESOD methods and these computed by theoretical equation

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Fig. 3

Distribution of the value of log10(I1,3)

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Fig. 4

The optimal frequencies and damping ratios of 100 simulations for cost functions I1,3, I1,4 and I3,4 with the real damping ratios set to be 0.2

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Fig. 5

Optimal ω and ξ estimated with different noise level for the SDOF system

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Fig. 6

The 8DOF vibration system

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Fig. 7

Modal shapes for the MDOF system with α = β = 1 × 10−5

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Fig. 8

Modal shapes for the damped MDOF system with α = β = 0.2

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Fig. 9

Percentage error of identified optimal ω and ξ of the each mode estimated with different noise level for the damped MDOF system with α = β = 0.2




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