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Research Papers

Experimental Study on Complex Modes of an End-Damped Continuous Beam

[+] Author and Article Information
Xing Xing

Dynamics and Vibrations Research Laboratory,
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: xingxing@msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: feeny@egr.msu.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 23, 2016; final manuscript received July 11, 2017; published online August 16, 2017. Assoc. Editor: Patrick S. Keogh.

J. Vib. Acoust 139(6), 061014 (Aug 16, 2017) (9 pages) Paper No: VIB-16-1414; doi: 10.1115/1.4037301 History: Received August 23, 2016; Revised July 11, 2017

The complex modes of an end-damped cantilevered beam are studied as an experimental example of a nonmodally damped continuous system. An eddy-current damper is applied, for its noncontact and linear properties, to the end of the beam, and is then characterized to obtain the effective damping coefficient. The state-variable modal decomposition (SVMD) is applied to extract the modes from the impact responses in the cantilevered beam experiments. Characteristics of the mode shapes and modal damping are examined for various values of the end-damper damping coefficient. The modal frequencies and mode shapes obtained from the experiments have a good consistency with the results of the finite element model. The variation of the modal damping ratio and modal nonsynchronicity with varying end-damper damping coefficient also follow the prediction of the model. Over the range of damping coefficients studied in the experiments, we observe a maximum damping ratio in the lowest underdamped mode, which correlates with the maximum modal nonsynchronicity. Complex orthogonal decomposition (COD) is applied in comparison to the modal identification results obtained from SVMD.

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Figures

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Fig. 1

The magnetic field of the eddy-current damper

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Fig. 2

The experimental setup of the cantilevered beam with an eddy-current damper

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Fig. 3

Schematic diagram showing the arrangement of the copper sheet and permanent magnet in the eddy-current damper

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Fig. 4

Real parts of SVMD complex modal state variables are shown for mode 2 to mode 5 based on N = 12,500 sample points

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Fig. 5

The FFT plot of the impact response with and without the eddy-current damper

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Fig. 6

The variation of the eigenvalues (rad/s) with increasing damping coefficient (circles indicate the experimental values and the asterisks indicate the relative values of the model)

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Fig. 7

Comparison of modes in the complex plane when the end-damper damping coefficient is c = 10 kg/s (the horizontal axis of mode 4 is stretched for a better view of the mode shape)

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Fig. 8

The real and imaginary parts of the modes when the end-damper damping coefficient is c = 10 kg/s

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Fig. 9

The modal damping ratio comparison between the model prediction and experiments at different end-damper damping coefficients (in kg/s)

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Fig. 10

The modal traveling index comparison between the model prediction and experiments at different end-damper damping coefficients (in kg/s)

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Fig. 11

The modes comparison in complex plane when the end-damper damping coefficient is c = 10 kg/s

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Fig. 12

The real and imaginary parts of the modes when the end-damper damping coefficient is c = 10 kg/s

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