Research Papers

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

[+] Author and Article Information
Rickey A. Caldwell, Jr.

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: caldwe20@msu.edu

Brian F. Feeny

Fellow ASME
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 11, 2014; final manuscript received March 19, 2016; published online May 23, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 138(4), 041007 (May 23, 2016) (7 pages) Paper No: VIB-14-1436; doi: 10.1115/1.4033268 History: Received November 11, 2014; Revised March 19, 2016

Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.

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Fig. 1

Schematic diagram of the experimental setup: top view

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Fig. 2

Acceleration data versus position along the beam

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Fig. 3

Accelerations from sensors 1, 16, and 31

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Fig. 4

Displacements from sensors 1, 16, and 31

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Fig. 5

FFT of sensors: (a) sensor 1, (b) sensor 16, and (c) sensor 31

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Fig. 6

FFT of the modal impact hammer signal

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Fig. 7

Selected COMs. Top left: second modal COM. Top right: sixth COM which is less circular than second. This trends continues and is illustrated in the lower left: seventh COM. Lower right: tenth COM.

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Fig. 8

Selected COCs. Top left: second modal COC. Top right: sixth COC which shows attenuation after some initial oscillations. This trends continues and is illustrated in the lower left: seventh COC. Lower right: tenth COC.

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Fig. 9

Real part of complex orthogonal coordinate number 4

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Fig. 10

Real part of complex orthogonal coordinate number 8

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Fig. 11

Experimental results. Theoretical dispersion relationship for the Euler–Bernoulli beam (solid line). COD extracted data points (○).

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Fig. 14

COD extracted modal amplitude versus frequency (○) compared to theory




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