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Research Papers

Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media

[+] Author and Article Information
B. F. Feeny

Dynamics and Vibrations Research Laboratory,
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
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 October 26, 2010; final manuscript received September 3, 2012; published online April 5, 2013. Assoc. Editor: Bogdan Epureanu.

J. Vib. Acoust 135(3), 031010 (Apr 05, 2013) (8 pages) Paper No: VIB-10-1260; doi: 10.1115/1.4023047 History: Received October 26, 2010; Revised September 03, 2012

A method of complex orthogonal decomposition is summarized for the time-domain, and then formulated and justified for application in the frequency-domain. The method is then applied to the extraction of modes from simulation data of sampled multimodal traveling waves for estimating wave parameters in one-dimensional continua. The decomposition is first performed on a transient nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The decomposition is finally applied to a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler–Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of modal components (i.e., mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. Modal signal energies are also quantified. The results are robust to random noise.

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Figures

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Fig. 1

(a) The single-humped waveform in the t=0 position. This waveform propagates intact until it moves off of the right end of the measurement domain at x=25. (b) The square root of the complex orthogonal values (+ symbols) of the transient waveform, representing the root mean squared amplitudes of the modal coordinates, and the amplitudes of the Fourier coefficients (° symbols) of a periodic pulse of the same shape.

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Fig. 2

The superposition of snapshots of the noise-free transient wave in the beam, with higher spatial resolution to clearly show the waveform. In the electronic version, colored lines are used only to help distinguish the snapshots.

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Fig. 3

The superposition of snapshots of (a) the first-mode-only reconstructed transient wave in the beam, (b) the second-mode-only reconstructed transient wave in the beam, and (c) the fourth-mode-only reconstructed transient wave in the beam. The first mode is of low frequency and speed, and remains in the measurement domain throughout the sampling record. The fourth mode motion starts with large amplitude, and then becomes small after the mode has mostly traveled off of the measurement domain.

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Fig. 4

The first mode plotted (a) as the real part (solid line) and imaginary part (dashed line) versus x, and (b) as the complex values in the complex plane. The fourth mode plotted (c) as the real part versus x, and (d) as the complex values in the complex plane.

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Fig. 5

The first modal coordinate time history plotted (a) as the real part versus time, and (b) as the complex values in the complex plane. The fourth modal coordinate time history plotted (c) as the real part versus time, and (d) as the complex values in the complex plane.

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Fig. 6

(a) The frequency versus wave number for the extracted modes (° symbols) compared to the theoretical curve (solid line), from data with added uniform random noise in the range of (-εymax,εymax), where ε=2-6. (b) Group velocity versus wave number for the extracted modes calculated by centered differences spanning two intervals (° symbols) and spanning four intervals (dots), compared to the theoretical curve (solid line).

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Fig. 7

The root mean squared normalized error, between the estimated frequency and the theoretical Euler–Bernoulli beam frequency at each of the estimated wave numbers, for various levels of noise as defined by ε

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Fig. 8

The COVs (in mm2) versus modal frequency

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