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Research Papers

High Frequency Analysis of a Point-Coupled Parallel Plate System

[+] Author and Article Information
Dean R. Culver

Aeroelasticity Laboratory,
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27705
e-mail: dean.culver@duke.edu

Earl H. Dowell

Aeroelasticity Laboratory,
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27705
e-mail: earl.dowell@duke.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 January 11, 2017; final manuscript received February 17, 2017; published online May 26, 2017. Assoc. Editor: Alper Erturk.

J. Vib. Acoust 139(5), 051002 (May 26, 2017) (11 pages) Paper No: VIB-17-1013; doi: 10.1115/1.4036212 History: Received January 11, 2017; Revised February 17, 2017

The root-mean-square (RMS) response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, asymptotic modal analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely, the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system considered is: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to classical modal analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including statistical energy analysis (SEA).

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Figures

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Fig. 1

Schematic of the point-coupled parallel plate system

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Fig. 2

Response of plate 1 in the point-coupled plate system under a broadband transverse point excitation with a 600 Hz band centered at 1 kHz. One hundred and seventy-one modes are excited in plate 1. The excitation and coupling points are illustrated by vertical dashed lines. The spatial intensification zones and their mirrors are clearly visible, creating an almost tic-tac-toe-like pattern on the plate.

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Fig. 3

Response of plate 2 in the point-coupled plate system under a broadband transverse point excitation with a 600 Hz band centered at 1 kHz. Two hundred and nine modes are excited in plate 2. Spatial intensification is less clear. In this case, there appears to be a hot line and a mirror image in the x-direction, but why it exists without a corresponding hot line in the y-direction is likely a consequence of the coupling force power spectrum being neither sparse nor definitively broadband.

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Fig. 4

Ratio of response predictions of a free point (a point not experiencing any spatial intensification) on plate 1 from dCMA to CMA-FD for various bandwidths and center frequencies ranging from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 5

Ratio of response predictions of a free point (a point not experiencing any spatial intensification) on plate 2 from dCMA to CMA-FD for various bandwidths and center frequencies ranging from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 6

The ratio of ζM after modal averaging to ζM before modal averaging. The frequency axis is normalized by the location within the excitation band. More specifically, 0 refers to ωmin and 1 refers to ωmax.

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Fig. 7

Illustration of the definition of γi, where a given ωM is illustrated by a solid vertical line and the adjacent component natural frequencies ωmi are illustrated by the dashed vertical lines. The horizontal axis of this one-dimensional plot is frequency.

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Fig. 8

Relative distribution of plates 1 (solid) and 2 (dashed) natural frequencies assuming a constant modal density. The horizontal axis of this one-dimensional plot is frequency. Recall that exactly one coupled natural frequency occurs between two component natural frequencies of any component, or rather, only one coupled natural frequency exists between any two adjacent dashed lines in this plot. It can be seen that there are few cases where one might place a coupled natural frequency such that γ1 and γ2 are simultaneously close to 1 or simultaneously close to 0.

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Fig. 9

Ratio of predictions of AMA to CMA-FD for the plate 1 excitation point response experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 10

Ratio of predictions of AMA to CMA-FD for a sample of the plate 1 hot line response experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 11

Ratio of predictions of AMA to CMA-FD for the response of a nonintensified point on plate 1 experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 1. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 12

Ratio of predictions of AMA to CMA-FD for the response of a point on plate 2 experiencing excitation from various center frequencies and bandwidths, exciting different numbers of modes in plate 2. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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Fig. 13

Ratio of computer runtimes for the two-plate system experiencing excitation from various center frequencies and bandwidths, exciting different numbers of total system modes. Center frequencies used range from 1 kHz to 100 kHz exciting prescribed numbers of modes. The bandwidths used to capture the array of modes illustrated on the x-axis are 30 Hz, 50 Hz, 100 Hz, 300 Hz, 500 Hz, and 1000 Hz.

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