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Research Papers

Rational ANCF Thin Plate Finite Element

[+] Author and Article Information
Carmine M. Pappalardo

Department of Industrial Engineering,
University of Salerno,
Fisciano (Salerno) 84084, Italy

Zuqing Yu

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

Xiaoshun Zhang

School of Science,
Nanjing University of Science and Technology,
Nanjing 210094, China

Ahmed A. Shabana

Department of Mechanical and
Industrial Engineering,
University of Illinois at Chicago,
Chicago, IL 60607

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 22, 2015; final manuscript received December 12, 2015; published online February 3, 2016. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 11(5), 051009 (Feb 03, 2016) (15 pages) Paper No: CND-15-1252; doi: 10.1115/1.4032385 History: Received August 22, 2015; Revised December 12, 2015

In this paper, a rational absolute nodal coordinate formulation (RANCF) thin plate element is developed and its use in the analysis of curved geometry is demonstrated. RANCF finite elements are the rational counterpart of the nonrational absolute nodal coordinate formulation (ANCF) finite elements which employ rational polynomials as basis or blending functions. RANCF finite elements can be used in the accurate geometric modeling and analysis of flexible continuum bodies with complex geometrical shapes that cannot be correctly described using nonrational finite elements. In this investigation, the weights, which enter into the formulation of the RANCF finite element and form an additional set of geometric parameters, are assumed to be nonzero constants in order to accurately represent the initial geometry and at the same time preserve the desirable ANCF features, including a constant mass matrix and zero centrifugal and Coriolis generalized inertia forces. A procedure for defining the control points and weights of a Bezier surface defined in a parametric form is used in order to be able to efficiently create RANCF/ANCF FE meshes in a straightforward manner. This procedure leads to a set of linear algebraic equations whose solution defines the RANCF coordinates and weights without the need for an iterative procedure. In order to be able to correctly describe the ANCF and RANCF gradient deficient FE geometry, a square matrix of position vector gradients is formulated and used to calculate the FE elastic forces. As discussed in this paper, the proposed finite element allows for describing exactly circular and conic sections and can be effectively used in the geometry and analysis modeling of multibody system (MBS) components including tires. The proposed RANCF finite element is compared with other nonrational ANCF plate elements. Several numerical examples are presented in order to demonstrate the use of the proposed RANCF thin plate element. In particular, the FE models of a set of rational surfaces, which include conic sections and tires, are developed.

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Figures

Grahic Jump Location
Fig. 1

RANCF thin plate element

Grahic Jump Location
Fig. 2

Cylinder meshed using eight rectangular RANCF thin plate elements

Grahic Jump Location
Fig. 8

Tip vertical position of the ellipse section ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

Grahic Jump Location
Fig. 11

Cross section of the ANCF tire mesh

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Fig. 9

Tip vertical position of the hyperbola section ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

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Fig. 20

Normal pressure distribution in the contact patch when the tire is accelerated. (a) Nonrational 36-degrees-of-freedom ANCF thin plate element, (b) nonrational 48-degrees-of-freedom ANCF thin plate element, and (c) RANCF thin plate element

Grahic Jump Location
Fig. 3

Conic sections meshed using eight RANCF thin plate elements: (a) Parabola, (b) circle, (c) ellipse, and (d) hyperbola

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Fig. 4

Straight rectangular plate pendulum

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Fig. 5

Convergence of RANCF thin plate element (2×2 elements, 4×4 elements, 8×8 elements)

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Fig. 6

Tip vertical position of the parabola section ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

Grahic Jump Location
Fig. 7

Tip vertical position of the circle section ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

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Fig. 12

Function C(v) of Eq. (12)

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Fig. 13

Function R(v) of Eq. (12)

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Fig. 14

Torque applied to the rim node

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Fig. 15

Longitudinal displacement of the rim node ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

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Fig. 16

x-Component of the rz vector of the rim node ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

Grahic Jump Location
Fig. 17

Normal pressure distribution on contact patch. (a) Nonrational 36-degrees-of-freedom ANCF thin plate element, (b) nonrational 48-degrees-of-freedom ANCF thin plate element, and (c) RANCF thin plate element.

Grahic Jump Location
Fig. 18

Normal pressure distribution in the longitudinal direction ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

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Fig. 19

Normal pressure distribution in the lateral direction ( nonrational 36-degrees-of-freedom ANCF thin plate element, nonrational 48-degrees-of-freedom ANCF thin plate element, and RANCF thin plate element)

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