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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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References

Semperlotti, F. , and Conlon, S. C. , 2010, “Structural Damage Identification in Plates Via Nonlinear Structural Intensity Maps,” J. Acoust. Soc. Am., 127(2), pp. EL48–EL53. [CrossRef] [PubMed]
Schmidt, W. T. , 2009, “Open-Crack Damage Assessments of Aluminum Panels Using Structural Intensity-Based Techniques,” M.Sc. thesis, The Pennsylvania State University, State College, PA. https://etda.libraries.psu.edu/files/final_submissions/6538
Arruda, J. R. F. , and Mas, P. , 1998, “Localizing Energy Sources and Sinks in Plates Using Power Flow Maps Computed From Laser Vibrometer Measurements,” Shock Vib., 5(4), pp. 235–253. [CrossRef]
Vuye, C. , 2011, “Measurement and Modeling of Sound and Vibration Fields Using a Scanning Laser Doppler Vibrometer,” Vrije Universiteit Brussel, Brussel, Belgium.
Cho, D. , Choi, T. , Kim, J. , and Vladimir, N. , 2016, “Structural Intensity Analysis of Stepped Thickness Rectangular Plates Utilizing the Finite Element Method,” Thin-Walled Struct., 109, pp. 1–12. [CrossRef]
Roozen, N. B. , Guyader, J. L. , and Glorieux, C. , 2015, “Measurement-Based Determination of the irrotational Part of the Structural Intensity by Means of Test Functional Series Expansion,” J. Sound Vib., 356, pp. 168–180. [CrossRef]
Lamberti, A. , and Semperlotti, F. , 2013, “Detecting Closing Delaminations in Laminated Composite Plates Using Nonlinear Structural Intensity and Time Reversal Mirrors,” Smart Mater. Struct., 22(12), p. 125006. [CrossRef]
Romano, A. J. , Williams, E. G. , Abraham, B. , and Williams, E. G. , 1990, “A Poynting Vector Formulation for Thin Shells and Plates, and Its Application to Structural Intensity Analysis and Source Localization—Part I: Theory,” J. Acoust. Soc. Am., 87(3), pp. 1166–1175. [CrossRef]
Gravic, L. , and Pavic, G. , 1993, “Finite Element Method for Computation of Structural Intensity by the Normal Mode Approach,” J. Sound Vib., 164(1), pp. 29–43. [CrossRef]
Zhang, Y. , 1993, “An Experimental Method for Structural Intensity and Source Location,” Ph.D. thesis, Iowa State University, Ames, IA.
Romano, A. J. , Williams, E. G. , Russo, K. L. , and Schuette, L. C. , 1992, “On the Visualization and Analysis of Fluid-Structure Interaction From the Perspective of Instantaneous Intensity,” J. Phys. IV France, pp. C1-597–C1-600.
Cho, D. S. , Kim, K. S. , and Kim, B. H. , 2010, “Structural Intensity Analysis of a Large Container Carrier Under Harmonic Excitations of Propulsion System,” Int. J. Nav. Architecture Ocean Eng., 2(2), pp. 87–95. [CrossRef]
Rothberg, S. J. , Allen, M. S. , Castellini, P. , Di Maio, D. , Dirckx, J. J. J. , Ewins, D. J. , Halkon, B. J. , Muyshondt, P. , Paone, N. , Ryan, T. , Steger, H. , Tomasini, E. P. , Vanlanduit, S. , and Vignola, J. F. , 2017, “An International Review of Laser Doppler Vibrometry: Making Light Work of Vibration Measurement,” Opt. Lasers Eng., 99, pp. 11–22. [CrossRef]
Schreier, H. , Orteu, J.-J. , and Sutton, M. A. , 2009, Image Correlation for Shape, Motion and Deformation Measurements, Springer, NY.
Pascal, J.-C. , Carniel, X. , Chalvidan, V. , and Smigielski, P. , 1996, “Determination of Phase and Magnitude of Vibration for Energy Flow Measurements in a Plate Using Holographic Interferometry,” Opt. Lasers Eng., 25(4–5), pp. 343–360. [CrossRef]
Van der Jeught, S. , and Dirckx, J. J. J. , 2015, “Real-Time Structured Light Profilometry: A Review,” Opt. Lasers Eng., 87, pp. 18–31. [CrossRef]
Ventsel, E. , and Krauthammer, T. , 2001, Thin Plates and Shells: Theory: Analysis and Applications, CRC Press, Boca Raton, FL.
Williams, E. G. , 1991, “Structural Intensity in Thin Cylindrical Shells,” J. Acoust. Soc. Am., 89(4), pp. 1615–1622. [CrossRef]
Saijyou, K. , and Yoshikawa, S. , 1999, “Structural Intensity Measurement Technique and Energy Flow of Cylindrical Influences of Shell Based on NAH a Rib on the Acoustic,” J. Acoust. Soc. Jpn., 20(2), pp. 125–136. [CrossRef]
Junger, M. C. , and Feit, D. , 1986, Sound, Structures, and Their Interaction, MIT Press, Cambridge, MA.
Lisle, J. , and Robinson, M. , 1995, “The Mohr Circle for Curvature and Its Application to Fold Description,” J. Struct. Geology, 17(5), pp. 739–750. [CrossRef]
Morse, P. M. , and Feshbach, H. , 1946, Methods of Theoretical Physics, Technology Press, McGraw-Hill, New York.
Kraus, H. , 1967, Thin Elastic Shells, Wiley, New York.
Soedel, W. , 2004, Vibrations of Shells and Plates, CRC Press, Boca Raton, FL.
Novozhilov, V. V. , 1959, Thin Shell Theory, P. Noordhoff, Groningen, The Netherlands.
Boas, M. L. , 2006, Mathematical Methods in the Physical Sciences, Wiley, Noida, India.
Hubbard, J. H. , and Hubbard, B. B. , 2015, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Matrix Editions, New York.
Lopes, H. M. R. , Guedes, R. M. , and Vaz, M. P. , 2006, “Techniques in Numerical Differentiation of Experimentally Noisy Data,” Fifth International Conference of Mechanical & Materials in Design, Porto, Portugal, July 24–26, pp. 27–28. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1001.1299&rep=rep1&type=pdf
Morikawa, R. , Ueha, S. , and Nakamura, K. , 1996, “Error Evaluation of the Structural Intensity Measured With a Scanning Laser Doppler Vibrometer and a k-Space Signal Processing,” J. Acoust. Soc. Am., 99(5), pp. 2913–2921. [CrossRef]
Pascal, J.-C. , Li, J.-F. , and Carniel, X. , 2002, “Wavenumber Processing Techniques to Determine Structural Intensity and Its Divergence From Optical Measurements Without Leakage Effects,” Shock Vib., 9(1–2), pp. 57–66. [CrossRef]
Eck, T. , and Walsh, S. J. , 2012, “Measurement of Vibrational Energy Flow in a Plate With High Energy Flow Boundary Crossing Using Electronic Speckle Pattern Interferometry,” Appl. Acoust., 73(9), pp. 936–951. [CrossRef]

Figures

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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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