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Research Papers

Mixed-Coordinate ANCF Rectangular Plate Finite Element

[+] Author and Article Information
Zuqing Yu

Department of Mechanic and
Electronic Engineering,
Harbin Institute of Technology,
Harbin, Heilongjiang 150001, China

Ahmed A. Shabana

Department of Mechanical and
Industrial Engineering,
University of Illinois at Chicago,
842, West Taylor Street,
Chicago, IL 60607

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 31, 2014; final manuscript received July 21, 2014; published online April 9, 2015. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 10(6), 061003 (Nov 01, 2015) (14 pages) Paper No: CND-14-1086; doi: 10.1115/1.4028085 History: Received March 31, 2014; Revised July 21, 2014; Online April 09, 2015

Higher order finite elements (FEs) based on the absolute nodal coordinate formulation (ANCF) may require the use of curvature vectors as nodal coordinates. The curvature vectors, however, can be difficult to define at the reference configuration, making such higher order ANCF FEs less attractive to use. It is the objective of this investigation to use the concept of the mixed-coordinate ANCF FEs to ensure that the gradient vectors are the highest spatial derivatives in the element nodal coordinate vector regardless of the order of the interpolating polynomials used. This concept is used to convert the curvature vectors to nodes, called position nodes, which have only position coordinates. These new position nodes can be defined at a preprocessing stage, leading to two different sets of nodes: one set of nodes has position and gradient coordinates, while the second set of nodes has position coordinates only. The new position nodes can be used to obtain better distribution of the forces, including contact forces. Higher degree of continuity, including curvature continuity, can still be achieved at the element interface by using, at a preprocessing stage, linear algebraic equations that can reduce significantly the model dimension and ensure higher degree of smoothness. The procedure proposed in this investigation also allows for the formulation of mechanical joints at arbitrary points and nodes using linear algebraic constraint equations. The difficulties that arise when formulating these joint constraints using B-spline and NURBS (Nonuniform Rational B-Splines) representations are discussed. In order to explain the concepts introduced in this paper, low and high order ANCF thin plate elements are used. For the high order thin plate element, the curvature vectors at the interface nodes are converted to internal nodes with position coordinates only, leading to a mixed-coordinate ANCF thin plate element. This element preserves the desirable ANCF features including a constant mass matrix and zero Coriolis and centrifugal forces. Kirchhoff plate theory is used to formulate the element elastic forces. The equations of motion of the structure are formulated in terms of an independent set of structure coordinates. The resulting mass matrix associated with the independent coordinates remains constant. Numerical examples are presented in order to demonstrate the use of the mixed-coordinate ANCF thin plate element when the continuity constraints are imposed.

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References

Figures

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Fig. 1

Two mixed-coordinate thin plate element connected by two spherical joints at arbitrary points

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Fig. 2

Nodal coordinates of the mixed-coordinate thin plate element

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Fig. 4

A simply supported plate subjected to a constant force at its center

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Fig. 5

Boundary conditions of two connected mixed-coordinate thin plate elements

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Fig. 6

Track modeled as one ANCF plate element mesh

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Fig. 7

Control points of cubic B-spline surface

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Fig. 10

Configurations of the pendulum using the 4×4 element mesh

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Fig. 11

Configurations of the pendulum using the 8×8 element mesh

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Fig. 8

Two ANCF plate elements connected by two spherical joints

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Fig. 9

Configurations of the pendulum using the 2×2 element mesh

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Fig. 12

z coordinate of the tip of the plate (2×2, 4×4, 8×8)

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Fig. 13

Four external forces are imposed at a plate

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Fig. 14

Change of the external force magnitude with respect to time

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Fig. 15

Displacement of the center point (——— one mixed-coordinate thin plate element, – – – – one 36 degrees of freedom thin plate element)

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Fig. 16

Displacement of the center point (——— 3×3 nine mixed-coordinate thin plate element mesh, – – – – one mixed-coordinate thin plate element)

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Fig. 17

Displacement of the center point (——— 3×3 mixed-coordinate thin plate element mesh, – – – – 3×3 36 degrees of freedom thin plate element mesh)

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Fig. 18

Displacement of the center point (——— one mixed-coordinate thin plate element, – – – – 3×3 36 degrees of freedom thin plate element mesh)

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Fig. 19

Simply supported plate before imposing continuity conditions

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Fig. 20

x,y, and z coordinates of rx at the two element interface (x coordinates, y coordinates, z coordinates, ——— element 1, – – – – element 2)

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Fig. 21

x,y, and z coordinates of rxy at two element interface (x coordinates, y coordinates, z coordinates, ——— element 1, – – – – element 2)

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Fig. 22

Simply supported plate after imposing continuity conditions

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Fig. 23

x,y, and z coordinates of rx at the two element interface (x coordinates, y coordinates, z coordinates, ——— element 1, – – – – element 2)

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Fig. 24

x,y, and z coordinates of rxy at the two element interface (x coordinates, y coordinates, z coordinates, ——— element 1, – – – – element 2)

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