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Research Papers

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

[+] Author and Article Information
K. Wu

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
Baltimore, MD 21250
e-mail: wukai1@umbc.edu

W. D. Zhu

Professor
Fellow ASME
Department of Mechanical Engineering,
University of Maryland,
Baltimore County, Baltimore, MD 21250
e-mail: wzhu@umbc.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 February 10, 2017; final manuscript received June 22, 2017; published online August 17, 2017. Assoc. Editor: Stefano Gonella.

J. Vib. Acoust 140(1), 011002 (Aug 17, 2017) (18 pages) Paper No: VIB-17-1054; doi: 10.1115/1.4037176 History: Received February 10, 2017; Revised June 22, 2017

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

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Figures

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Fig. 1

Example of a two-dimensional continuous system defined in D with piecewise ℂ1-continuous boundary curves Sr and corners Cr (r=1,2,…,6)

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Fig. 2

Logarithms of condition numbers of mass matrices with different numbers of truncated terms and M = N, where “◯” and “△” correspond to the AMM and the NGSDM, respectively

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Fig. 3

Computational times with different numbers of truncated terms and M = N, where “◯” and “△” correspond to the AMM and the NGSDM, respectively

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Fig. 4

(a) Displacements w and (b) velocities wt at the location (x,y)=(0.5,0.25) on the plate in the free vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), and the FDM (dashed–dotted lines)

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Fig. 5

(a) Rotational angles wx about the x axis and (b) rotational angles wy about the y axis at the location (x,y)=(0.5,0.25) on the plate in the free vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), and the FDM (dashed–dotted lines)

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Fig. 6

(a) Bending moments Mx and (b) transverse shearing forces Qx at the location (x,y)=(0.5,0.25) on the plate in the free vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), and the FDM (dashed–dotted lines)

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Fig. 7

(a) Displacements w and (b) velocities wt at the location (x,y)=(1,0.75) on the edge beam in the free vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), and the FDM (dashed–dotted lines)

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Fig. 8

(a) Rotational angles wx about the x axis and (b) rotational angles wy about the y axis at the location (x,y)=(1,0.75) on the edge beam in the free vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), and the FDM (dashed–dotted lines)

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Fig. 9

(a) Bending moments Mx and (b) transverse shearing forces Qx at the location (x,y)=(1,0.75) on the edge beam in the free vibration case, calculated from the AMM (solid lines); (c) bending moments Mx and (d) transverse shearing forces Qx at the location (x,y)=(1,0.75) on the edge beam in the free vibration case, calculated from the NGSDM (dashed lines) and the FDM (dashed–dotted lines)

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Fig. 10

(a) Displacements w and (b) velocities wt at the location (x,y)=(1,0.75) on the edge beam in the forced vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), the FDM (dashed–dotted lines), and the FEM (dotted lines)

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Fig. 11

(a) Rotational angles wx about the x axis and (b) rotational angles wy about the y axis at the location (x,y)=(1,0.75) on the edge beam in the forced vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), the FDM (dashed–dotted lines), and the FEM (dotted lines)

Grahic Jump Location
Fig. 12

(a) Bending moments Mx and (b) transverse shearing forces Qx at the location (x,y)=(1,0.75) on the edge beam in the forced vibration case, calculated from the AMM (solid lines), the NGSDM (dashed lines), the FDM (dashed–dotted lines), and the FEM (dotted lines)

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