Research Papers

Structural Intensity Analysis on Irregular Shells

[+] Author and Article Information
F. Pires

Department of Physics,
University of Antwerp,
Groenenborgerlaan 171,
Antwerp 2020, Belgium;
Department of Mechanical Engineering,
Vrije Universiteit Brussel,
Pleinlaan 2,
Brussel 1050, Belgium
e-mail: felipe.pires@uantwerpen.be

S. Vanlanduit

Department of Electromechanical Engineering,
University of Antwerp,
Groenenborgerlaan 171,
Antwerp 2020, Belgium;
Department of Mechanical Engineering,
Vrije Universiteit Brussel,
Pleinlaan 2,
Brussel 1050, Belgium
e-mail: steve.vanlanduit@uantwerpen.be

J. J. J. Dirckx

Department of Physics,
University of Antwerp,
Groenenborgerlaan 171,
Antwerp 2020, Belgium
e-mail: joris.dirckx@uantwerpen.be

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 7, 2017; final manuscript received July 11, 2018; published online August 13, 2018. Assoc. Editor: Miao Yu.

J. Vib. Acoust 141(1), 011011 (Aug 13, 2018) (12 pages) Paper No: VIB-17-1534; doi: 10.1115/1.4040926 History: Received December 07, 2017; Revised July 11, 2018

A method is presented to assess the transmission path of vibration energy and to localize sources or sinks on shells with arbitrary shape, constant thickness, and isotropic material properties. The derived equations of the structural intensity (SI) are based on the Kirchhoff–Love postulates and are formulated in terms of displacements, Lamé parameters, principal curvatures, and their partial derivatives with respect to the principal curvilinear coordinates (PCC). To test the accuracy of the method, two numerical models of thin shells with nonuniform curvatures were developed. The coordinates, displacements, and principal curvature directions (PCDs) at the shell's outer surface were used to estimate the SI vector fields and the energy density at the shell's middle surface. The power estimated from the surface integral of the divergence of the SI over the source was compared to the actual power injected in the shell. The absolute error in both models did not exceed 1%, showing that, in theory, the method is able to handle the high-order spatial derivatives of the displacement and geometry data. The qualitative effect of varying the internal damping in the models on the energy transmission was also investigated.

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Fig. 1

Geometric representation of the shell models from which the SI are determined. Both the rectangular shell (a) and circular shell (b) have a nonuniform curvature as can be noticed from the mean curvature field ((K1+K2)/2) on their surfaces. The tangent arrows on both geometries show the PCDs e1 and e2.

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Fig. 2

Actual location of the energy sources and sinks on both geometries. The sources and sinks are highlighted on the upper and lower surfaces from both geometries

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Fig. 3

Viewing perspective from which the Cartesian coordinates x,y,z and displacement fields U¯,V¯,W¯ at the outer surface were extracted from the rectangular (a) and the circular shell (b)

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Fig. 4

Schematic overview of the processing explained in Sec. 4.1. The fields inside the slashed contours are auxiliary fields and are used as input for the displacement processing Sec. 4.2.

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Fig. 5

Schematic overview of the processing explained in Sec.4.2

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Fig. 6

Representation of the PDs on the circular shell. Figure (a) represents the tangent vectors î and ĵ. Figure (b) shows the normal vector k̂.

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Fig. 7

Representation of the PCCs a and β as an orthogonal grid on the surfaces of the shells. If the directions of these grids are compared with the PCDs (Fig. 1), it can be noticed that both are aligned with each other.

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Fig. 8

Processing of the spatial derivatives of the fields x, ρ1 and K1h/2 of the circular shell. The first (a)–(i) and the second column (b)–(j) show the mentioned fields represented on the xy and the αβ space, respectively. Afterward, their spatial derivatives with respect to α and β can be assessed. As an example, the derivatives with respect to α are shown (c)–(k). Finally, the processed fields are transformed back to the original xy space (d)–(l).

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Fig. 9

SI vector field (Eq. (34)) and its respective DSI field (Eq. (37)) for the rectangular (a) and circular shell (b) model

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Fig. 10

DSI field of the circular shell on the xy space (a) and on the αβ space (b). To determine the injected power, Eq. (38) was used to integrate the field defined in Eq. (37) over the source.

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Fig. 11

SI vector field (Eq. (34)) and its corresponding DSI field (Eq. (37)) for the rectangular (a) and circular shell (b) model when an internal damping loss factor of 0.005% was introduced.



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