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Research Papers

# A Method to Simulate Structural Intensity Fields in Plates and General Structures Induced by Spatially and Temporally Random Excitation Fields

[+] Author and Article Information
Michael J. Daley

Akustica, Inc., 2835 East Carson Street, Suite 301, Pittsburgh, PA 15203mdaley@akustica.comApplied Research Laboratory, The Pennsylvania State University, PO Box 30, State College, PA 16804mdaley@akustica.com

Stephen A. Hambric

Akustica, Inc., 2835 East Carson Street, Suite 301, Pittsburgh, PA 15203sah19@only.arl.psu.eduApplied Research Laboratory, The Pennsylvania State University, PO Box 30, State College, PA 16804sah19@only.arl.psu.edu

$m$ denotes the mode order along the length of the plate (11.2 cm) and $n$ the mode order along the width (7.5 cm), where the mode order indicates the number of antinodes, or peaks, in the displacement mode shape.

The finite differencing approach, however, may be used on complex flat plates, such as corrugated and ribbed structures, or plates with anisotropic material properties, provided equivalent homogenous isotropic properties are computed and applied to the formulas. See Ref. 29 for an example of this approach.

J. Vib. Acoust 131(1), 011006 (Jan 05, 2009) (9 pages) doi:10.1115/1.2980381 History: Received August 21, 2007; Revised July 15, 2008; Published January 05, 2009

## Abstract

The structure-borne power in bending waves is well understood, and has been studied by many investigators in ideal beam and plate structures. Most studies to date, however, have considered only the structural intensity induced by deterministic localized drives. Many structures of practical interest are excited by spatially random pressure fields, such as diffuse and turbulent boundary layer pressure fluctuations. Additionally, such studies typically employ finite differencing techniques to estimate the shear, bending, and twisting components of intensity, and are therefore only applicable to simple homogenous uniform structures such as thin plates and beams. Often, however, finite differencing techniques are not applicable to practical structures of interest. The present study introduces a new analytic method to compute the structural intensity induced by spatially random pressure fields in general structures, which does not require the use of finite differencing techniques. This method uses multiple-input multiple-output random analysis techniques, combining frequency response function matrices generated from analytic or finite element (FE) models with cross-spectral density matrices of spatially random pressure fields to compute intensities in structures. The results of this method are validated using those obtained using finite-difference-based techniques in a flat plate. Both methods show intensity patterns different from those caused by deterministic point drives. The new general method, combined with FE analysis techniques, may be applied in the future to complex nonhomogenous structures, which include discontinuities, curvature, anisotropic materials, and general three-dimensional features.

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## Figures

Figure 1

Example of notional structural intensity field on the blade of a propeller excited by a convecting turbulent pressure field, along with intensities on the shaft

Figure 2

Free body diagram of differential element of the plate

Figure 3

Diagram of the response array used for the 13-point finite differencing scheme. The SI vector is computed at the center point (point 2).

Figure 4

Schematic of the system used for simulations

Figure 5

Schematic of the test plate with input and absorbed power shown. Note the rectangular contour surrounding the damper. The power crossing the contour is compared with the input and absorbed power via Eq. 31.

Figure 6

Input and absorbed power spectra normalized to pressure field magnitude for acoustically excited plate. Mode orders (m,n) shown above resonance peaks.

Figure 7

SI for the (3,1) mode using the finite difference method. The contours highlight the lines of equal intensity, and have units of W/m2 Pa2. The vector lengths are proportional to the intensity magnitude.

Figure 8

SI for the (3,1) mode using the analytic method

Figure 9

Divergence of SI for the (3,1) mode using the finite difference method. The units are W/m3 Pa2. Note the clear region of negative divergence (convergence) of energy to damper, and the lack of any clear region of positive divergence.

Figure 10

Divergence of SI for the (3,1) mode using the analytic method

Figure 11

SI for the (2,3) mode using the finite difference method

Figure 12

SI for the (2,3) mode using the analytic method

Figure 13

Beam element intensity field

Figure 14

Plate in-plane (membrane) element intensity field

Figure 15

Solid element intensity field

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