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Research Papers

Output-Only Modal Identification of a Nonuniform Beam by Using Decomposition Methods

[+] Author and Article Information
Rickey A. Caldwell

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI
e-mail: caldwe20@msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI
e-mail: feeny@egr.msu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 25, 2013; final manuscript received March 18, 2014; published online May 19, 2014. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(4), 041010 (May 19, 2014) (10 pages) Paper No: VIB-13-1259; doi: 10.1115/1.4027243 History: Received July 25, 2013; Revised March 18, 2014

Reduced-order mass weighted proper orthogonal decomposition (RMPOD), smooth orthogonal decomposition (SOD), and state variable modal decomposition (SVMD) are used to extract modal parameters from a nonuniform experimental beam. The beam was sensed by accelerometers. Accelerometer signals were integrated and passed through a high-pass filter to obtain velocities and displacements, all of which were used to build the necessary ensembles for the decomposition matrices. Each of these decomposition methods was used to extract mode shapes and modal coordinates. RMPOD can directly quantify modal energy, while SOD and SVMD directly produce estimates of modal frequencies. The extracted mode shapes and modal frequencies were compared to an analytical approximation of these quantities, and to frequencies estimated by applying the fast Fourier transform to accelerometer data. SVMD is also applied to estimate modal damping, which was compared to the estimate by logarithmic decrement applied to modal coordinate signals, with varying degrees of success. This paper shows that these decomposition methods are capable of extracting lower modal parameters of an actual experimental beam.

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Figures

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Fig. 2

Top: second mode shape extracted by RMPOD (○) plotted with the analytical approximation's discretized mode shape. Middle: second mode, modal acceleration coordinate from RMPOD (not to scale). Bottom: FFT of the modal coordinate acceleration.

Grahic Jump Location
Fig. 3

Top: third mode shape extracted by RMPOD (○) plotted with the analytical approximation's discretized mode shape. Middle: third mode, modal acceleration coordinate from the RMPOD. Bottom: FFT of the modal coordinate acceleration.

Grahic Jump Location
Fig. 4

Top: fourth mode shape extracted by RMPOD (○) plotted with the analytical approximation's discretized mode shape. Middle: fourth mode, modal acceleration coordinate from RMPOD. Bottom: FFT of modal the acceleration coordinates.

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Fig. 5

Seventh mode shape extracted by RMPOD of instructive purpose as an example of poor extraction

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Fig. 6

Top: second mode shape extracted by SOD (○) plotted with the analytical approximation's discretized mode shape. Middle: second mode, modal acceleration coordinate from SOD. Bottom: FFTs of modal coordinates.

Grahic Jump Location
Fig. 7

Top: third mode shape extracted by SOD (○) plotted with the analytical approximation's discretized mode shape. Middle: third mode, modal acceleration coordinate from SOD. Bottom: FFTs of modal coordinates.

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Fig. 8

Top: fourth mode shape extracted by SOD (○) plotted with the analytical approximation's discretized mode shape. Middle: fourth mode, modal acceleration coordinate from SOD. Bottom: FFTs of modal coordinates.

Grahic Jump Location
Fig. 9

Top: second mode shape extracted by SVMD (○) plotted with the analytical approximation's discretized mode shape (-). Middle: second modal coordinate displacement from SVMD. Bottom: FFT of modal coordinate displacement.

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Fig. 10

Top: third mode shape extracted by SVMD (○) plotted with the analytical approximation's discretized mode shape (-). Middle: third modal coordinate displacement from SVMD. Bottom: fast Fourier transform of modal coordinate displacement.

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Fig. 11

Top: fourth mode shape extracted by SVMD (○) plotted with the analytical approximation's discretized mode shape (-). Middle: fourth modal coordinate displacement from SVMD. Bottom: FFT of modal coordinate displacement.

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Fig. 12

SVMD modal coordinates plotted with maximum peaks (○), and boundaries for log decrement calculations (•←). Time segment 1.1142 ≤ t ≤ 2.2000. Top: second mode; middle: third mode; bottom: fourth mode.

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Fig. 13

Top: SVMD modal coordinates for the second mode 1.1142 ≤ t ≤ 2.2000 (solid curve), standard SVMD damping(◻), logarithmic damping (○) for peak 3 to peak 20, least square solution to 2ωiζi=α+βΛ damping (*). Bottom: SVMD modal coordinates for the second mode 1.1142 ≤ t ≤ 1.5330 (solid curve), standard SVMD damping (◻), logarithmic damping (○) for peak 3 to peak 14, least square solution to 2ωiζi=αI+βΛ damping (*).

Grahic Jump Location
Fig. 14

Top: SVMD modal coordinates for the third mode 1.1142 ≤ t ≤ 2.2000 (solid curve), standard SVMD damping (◻), logarithmic damping (○) for peak 3 to peak 60, least square solution to 2ωiζi=αI+βΛ damping (*). Bottom: SVMD modal coordinates for the third mode 1.1142 ≤ t ≤ 1.5330 (solid curve), standard SVMD damping (◻), logarithmic damping (○) for peak 3 to peak 40, least square solution to 2ωiζi=αI+βΛ damping (*).

Grahic Jump Location
Fig. 15

Top: SVMD modal coordinates for the fourth mode 1.1142 ≤ t ≤ 2.2000 (solid curve), standard SVMD damping (◻), logarithmic damping (○) for peak 3 to peak 60, least square solution to 2ωiζi=αI+βΛ damping (*). Bottom: SVMD modal coordinates for the fourth mode 1.1142 ≤ t ≤ 1.5330 (solid curve), standard SVMD damping (◻), logarithmic damping (○) for peak 3 to peak 40, least square solution to 2ωiζi=αI+βΛ damping (*).

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