Research Papers

A Higher-Order Shear Deformation Model of a Periodically Sectioned Plate

[+] Author and Article Information
Andrew J. Hull

Undersea Warfare Weapons, Vehicles,
and Defensive Systems Department,
Naval Undersea Warfare Center Division,
Newport, RI 02841
e-mail: andrew.hull@navy.mil

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 2, 2016; final manuscript received April 20, 2016; published online June 2, 2016. Assoc. Editor: Matthew Brake.This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Vib. Acoust 138(5), 051010 (Jun 02, 2016) (9 pages) Paper No: VIB-16-1063; doi: 10.1115/1.4033495 History: Received February 02, 2016; Revised April 20, 2016

This paper develops a higher-order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton's principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the system. This new model is validated with finite-element theory and Kirchhoff plate theory for a thin plate simulation and verified with comparison to experimental results for a 0.0191 m thick sectional plate.

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Graf, K. F. , 1975, Wave Motion in Elastic Solids, Dover Publications, New York.
Junger, M. C. , and Feit, D. , 1986, Sound, Structures, and Their Interaction, The MIT Press, Cambridge, MA.
Soedel, W. , 2004, Vibrations of Shells and Plates, Marcel Dekker, New York.
Timoshenko, S. P. , 1922, “ On the Transverse Vibrations of Bars of Uniform Cross-Section,” Philos. Mag., 43(253), pp. 125–131. [CrossRef]
Bickford, W. B. , 1982, “ A Consistent Higher Order Beam Theory,” 11th Southeastern Conference on Theoretical and Applied Mechanics, Huntsville, AL, Apr. 8–9, Vol. 11, pp. 137–150.
Karama, M. , Afaq, K. S. , and Mistou, S. , 2003, “ Mechanical Behavior of Laminated Composite Beam by New Multi-Layered Laminated Composite Structures Model With Transverse Shear Stress Continuity,” Int. J. Solids Struct., 40(6), pp. 1525–1546. [CrossRef]
Mindlin, R. D. , 1951, “ Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic Elastic Plates,” ASME J. Appl. Mech., 18(1), pp. 31–38.
Ambartsumyan, S. A. , 1958, “ On the Theory of Bending Plates,” Proc. Acad. Sci. USSR, 5, pp. 69–77.
Reddy, J. N. , and Phan, N. D. , 1985, “ Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher-Order Shear Deformation Theory,” J. Sound Vib., 98(2), pp. 157–170. [CrossRef]
Touratier, M. , 1991, “ An Efficient Standard Plate Theory,” Int. J. Eng. Sci., 29(8), pp. 901–916. [CrossRef]
Soldatos, K. P. , 1992, “ A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates,” Acta Mech., 94(3), pp. 195–220. [CrossRef]
Akavci, S. S. , 2007, “ Buckling and Free Vibration Analysis of Symmetric and Antisymmetric Laminated Composite Plates on an Elastic Foundation,” J. Reinf. Plast. Compos., 26(18), pp. 1907–1919. [CrossRef]
Mead, D. J. , and Pujara, K. K. , 1971, “ Space-Harmonic Analysis of Periodically Supported Beams: Response to Convected Random Loading,” J. Sound Vib., 14(4), pp. 525–541. [CrossRef]
Mace, B. R. , 1980, “ Periodically Stiffened Fluid-Loaded Plates—I: Response to Convected Harmonic Pressure and Free Wave Propagation,” J. Sound Vib., 73(4), pp. 473–486. [CrossRef]
Mace, B. R. , 1980, “ Periodically Stiffened Fluid-Loaded Plates—II: Response to Line and Point Forces,” J. Sound Vib., 73(4), pp. 487–504. [CrossRef]
Lin, G. G. , and Hayek, S. I. , 1977, “ Acoustic Radiation From Point Excited Rib-Reinforced Plate,” J. Acoust. Soc. Am., 62(1), pp. 72–83. [CrossRef]
Sylvia, J. E. , and Hull, A. J. , 2013, “ A Dynamic Model of a Reinforced Thin Plate With Ribs of Finite Width,” Int. J. Acoust. Vib., 18(2), pp. 86–90.
Cray, B. A. , 2015, “ Experimental Verification of Acoustic Trace Wavelength Enhancement,” J. Acoust. Soc. Am., 138(6), pp. 3765–3772. [CrossRef] [PubMed]
Floquet, G. , 1883, “ On Linear Differential Equations With Periodic Coefficients,” Sci. Ann. Éc. Norm. Supér., 12, pp. 47–88.
Bloch, F. , 1928, “ About the Quantum Mechanics of Electrons in Crystal Lattices,” Mag. Phys., 52, pp. 555–600.
Hussein, M. I. , Leamy, M. J. , and Ruzzene, M. , 2014, “ Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlooks,” ASME Appl. Mech. Rev., 66(4), p. 040802. [CrossRef]
Gei, M. , Movchan, A. B. , and Bigoni, D. , 2009, “ Band-Gap and Defect-Induced Annihilation in Prestressed Elastic Structures,” J. Appl. Phys., 105(6), p. 063507. [CrossRef]
Brun, M. , Giaccu, G. G. , Movchan, A. B. , and Movchan, N. V. , 2012, “ Asymptotics of Eigenfrequencies in the Dynamic Response of Elongated Structures,” Proc. R. Soc. A, 468(2138), pp. 378–394. [CrossRef]
Brun, M. , Movchan, A. B. , and Jones, I . S. , 2013, “ Phononic Band Gap Systems in Structural Mechanics: Finite Slender Elastic Structures and Infinite Periodic Waveguides,” ASME J. Vib. Acoust., 135(4), p. 041013. [CrossRef]
Carta, G. , and Brun, M. , 2015, “ Bloch–Floquet Waves in Flexural Systems With Continuous and Discrete Elements,” Mech. Mater., 87, pp. 11–26. [CrossRef]
Carta, G. , Brun, M. , and Movchan, A. B. , 2014, “ Dynamic Response and Localization in Strongly Damaged Waveguides,” Proc. R. Soc. A, 470(2167), p. 20140136. [CrossRef]
Carta, G. , Brun, M. , Movchan, A. B. , and Boiko, T. , 2016, “ Transmission and Localisation in Ordered and Randomly-Perturbed Structural Flexural Systems,” Int. J. Eng. Sci., 98, pp. 126–152. [CrossRef]
Reddy, J. N. , 1984, Energy and Variational Methods in Applied Mechanics, Wiley, New York.
Cheng, Z. , and Shi, Z. , 2013, “ Influence of Parameter Mismatch on the Convergence of the Band Structures by Using the Fourier Expansion Method,” Compos. Struct., 106, pp. 510–519. [CrossRef]
Lamb, M. , and Rouillard, V. , 2009, “ Some Issues When Using Fourier Analysis for the Extraction of Model Parameters,” J. Phys.: Conf. Ser., 181, p. 012007. [CrossRef]


Grahic Jump Location
Fig. 1

Periodically sectioned plate with a Cartesian coordinate system

Grahic Jump Location
Fig. 2

Schematic of the (a) validation problem and the magnitude of the displacement versus x-direction position at (b) 100 Hz, (c) 300 Hz, and (d) 500 Hz for the sectioned plate with a thickness of 0.001 m. The solid lines are the higher-order shear deformation plate solution, the circular markers are the Kirchhoff plate solutions, and the square markers are finite-element results. The vertical dashed line is the location where the periodic sections abut one another.

Grahic Jump Location
Fig. 3

Magnitude of the wave propagation coefficients Wm versus m index at (a) 100 Hz, (b) 300 Hz, and (c) 500 Hz for the sectioned plate with a thickness of 0.001 m. The horizontal dashed line is the location that is −90 dB down from the maximum value in each plot.

Grahic Jump Location
Fig. 7

Model (a) and experimental results (b) of plate normal displacement in the decibel scale versus x- and y-wavenumbers at 1200 Hz

Grahic Jump Location
Fig. 6

Model (a) and experimental results (b) of plate normal displacement in the decibel scale versus x- and y-wavenumbers at 800 Hz

Grahic Jump Location
Fig. 5

Model (a) and experimental results (b) of plate normal displacement in the decibel scale versus x- and y-wavenumbers at 400 Hz

Grahic Jump Location
Fig. 4

The experimental setup showing the mechanical shaker attached to the plate. The spacing of the adjacent individual vertical strips is 0.0572 m (2.25 in.).



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