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.

References

References
1.
Olshevskiy
,
A.
,
Dmitrochenko
,
O.
,
Dai
,
M. D.
, and
Kim
,
C. W.
,
2015
, “
The Simplest 3-, 6- and 8-Noded Fully-Parameterized ANCF Plate Elements Using Only Transverse Slopes
,”
J. Multibody Syst. Dyn.
,
34
(
1
), pp.
23
51
.
2.
Dmitrochenko
,
O. N.
, and
Pogorelov
,
D. Y.
,
2003
, “
Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation
,”
J. Multibody Syst. Dyn.
,
10
(
1
), pp.
17
43
.
3.
Bathe
,
J. K.
,
2007
,
Finite Element Procedures
,
Prentice Hall
,
Upper Saddle River, NJ
.
4.
Chang
,
H.
,
Liu
,
C.
,
Tian
,
Q.
,
Hu
,
H.
, and
Mikkola
,
A.
,
2015
, “
Three New Triangular Shell Elements of ANCF Represented by Bezier Triangles
,”
J. Multibody Syst. Dyn.
,
35
(
4
), pp.
321
351
.
5.
Dufva
,
K.
,
Kerkkanen
,
K.
,
Maqueda
,
L. G.
, and
Shabana
,
A. A.
,
2007
, “
Nonlinear Dynamics of Three-Dimensional Belt Drives Using the Finite-Element Method
,”
J. Nonlinear Dyn.
,
48
(
4
), pp.
449
466
.
6.
Hamed
,
A. M.
,
Jayakumar
,
P.
,
Shabana
,
A. A.
, and
Letherwood
,
M. D.
,
2011
, “
Nonstructural Geometric Discontinuities in Finite Element/Multibody System Analysis
,”
J. Nonlinear Dyn.
,
66
(
4
), pp.
809
824
.
7.
Hamed
,
A. M.
,
Jayakumar
,
P.
,
Letherwood
,
M. D.
,
Gorsich
,
D. J.
,
Recuero
,
A. M.
, and
Shabana
,
A. A.
,
2015
, “
Ideal Compliant Joints and Integration of Computer Aided Design and Analysis
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021015
.
8.
Hu
,
W.
,
Tian
,
Q.
, and
Hu
,
H. Y.
,
2014
, “
Dynamics Simulation of the Liquid-Filled Flexible Multibody System Via the Absolute Nodal Coordinate Formulation and SPH Method
,”
Nonlinear Dyn.
,
75
(
4
), pp.
653
671
.
9.
Kerkkanen
,
K. S.
,
Garcia-Vallejo
,
D.
, and
Mikkola
,
A. M.
,
2006
, “
Modeling of Belt-Drives Using a Large Deformation Finite Element Formulation
,”
J. Nonlinear Dyn.
,
43
(
3
), pp.
239
256
.
10.
Liu
,
C.
,
Tian
,
Q.
, and
Hu
,
H.
,
2012
, “
New Spatial Curved Beam and Cylindrical Shell Elements of Gradient-Deficient Absolute Nodal Coordinate Formulation
,”
J. Nonlinear Dyn.
,
70
(
3
), pp.
1903
1918
.
11.
Li
,
P.
,
Liu
,
C.
,
Tian
,
Q.
,
Hu
,
H.
, and
Song
,
Y.
,
2015
, “
Dynamics of a Deployable Mesh Reflector of Satellite Antenna: Form Finding and Modal Analysis
,”
ASME J. Comput. Nonlinear Dyn.
(submitted).
12.
Nachbagauer
,
K.
,
2013
, “
Development of Shear and Cross Section Deformable Beam Finite Elements Applied to Large Deformation and Dynamics Problems
,” Ph.D. dissertation, Johannes Kepler University, Linz, Austria.
13.
Pappalardo
,
C. M.
,
Patel
,
M. D.
,
Tinsley
,
B.
, and
Shabana
,
A. A.
,
2015
, “
Contact Force Control in Multibody Pantograph/Catenary Systems
,”
Proc. Inst. Mech. Eng., Part K
(in press).
14.
Patel
,
M. D.
,
Orzechowski
,
G.
,
Tian
,
Q.
, and
Shabana
,
A. A.
,
2015
, “
A New Multibody System Approach for Tire Modeling Using ANCF Finite Elements
,”
Proc. Inst. Mech. Eng., Part K
, pp.
1
16
(in press).
15.
Seo
,
J. H.
,
Sugiyama
,
H.
, and
Shabana
,
A. A.
,
2005
, “
Three-Dimensional Large Deformation Analysis of the Multibody Pantograph/Catenary Systems
,”
J. Nonlinear Dyn.
,
42
(
2
), pp.
199
215
.
16.
Shabana
,
A. A.
,
2012
,
Computational Continuum Mechanics
,
2nd ed.
,
Cambridge University Press
,
Cambridge, UK
.
17.
Shabana
,
A. A.
,
2013
,
Dynamics of Multibody Systems
,
4th ed.
,
Cambridge University Press
,
Cambridge, UK
.
18.
Yan
,
D.
,
Liu
,
C.
,
Tian
,
Q.
,
Zhang
,
K.
,
Liu
,
X. N.
, and
Hu
,
G. K.
,
2013
, “
A New Curved Gradient Deficient Shell Element of Absolute Nodal Coordinate Formulation for Modeling Thin Shell Structures
,”
J. Nonlinear Dyn.
,
74
, pp.
153
164
.
19.
Wang
,
L.
,
Octavio
,
J. R. J.
,
Wei
,
C.
, and
Shabana
,
A. A.
,
2015
A, “
Low Order Continuum-Based Liquid Sloshing Formulation for Vehicle System Dynamics
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021022
.
20.
Wang
,
L.
,
Wang
,
Y.
,
Recuero
,
A. M.
, and
Shabana
,
A. A.
,
2015
, “
ANCF Analysis of Textile Systems
,”
ASME J. Comput. Nonlinear Dynam
11
(
3
), p.
031005
.
21.
Wei
,
C.
,
Wang
,
L.
, and
Shabana
,
A. A.
,
2015
, “
A Total Lagrangian ANCF Liquid Sloshing Approach for Multibody System Applications
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
051014
.
22.
Shabana
,
A. A.
,
1998
, “
Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics
,”
J. Nonlinear Dyn.
,
16
(
3
), pp.
293
306
.
23.
Shabana
,
A. A.
, and
Yakoub
,
R. Y.
,
2000
, “
Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory
,”
ASME J. Mech. Des.
,
123
(
4
), pp.
606
613
.
24.
Shabana
,
A. A.
,
1998
, “
Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics
,”
J. Nonlinear Dyn.
,
16
(3), pp.
293
306
.
25.
Yakoub
,
R. Y.
, and
Shabana
,
A. A.
,
2000
, “
Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications
,”
ASME J. Mech. Des.
,
123
(
4
), pp.
614
621
.
26.
Lan
,
P.
, and
Shabana
,
A. A.
,
2010
, “
Integration of B-Spline Geometry and ANCF Finite Element Analysis
,”
J. Nonlinear Dyn.
,
61
, pp.
193
206
.
27.
Lan
,
P.
, and
Shabana
,
A. A.
,
2010
, “
Rational Finite Elements and Flexible Body Dynamics
,”
ASME J. Vib. Acoust.
,
132
(
4
), p.
041007
.
28.
Sanborn
,
G. G.
, and
Shabana
,
A. A.
,
2009
, “
A Rational Finite Element Method Based on the Absolute Nodal Coordinate Formulation
,”
J. Nonlinear Dyn.
,
58
(
3
), pp.
565
572
.
29.
Piegl
,
L. A.
, and
Tiller
,
W.
,
1997
,
The NURBS Book
,
Springer
,
New York
.
30.
Mortenson
,
M.
,
2006
,
Geometric Modeling
,
Industrial Press
,
New York
.
31.
Yu
,
Z.
,
Lan
,
P.
, and
Lu
,
N.
,
2014
, “
A Piecewise Beam Element Based on Absolute Nodal Coordinate Formulation
,”
J. Nonlinear Dyn.
,
77
(
1–2
), pp.
1
15
.
32.
Mikkola
,
A.
,
Shabana
,
A. A.
,
Sanchez-Rebollo
,
C.
, and
Jimenez-Octavio
,
J. R.
,
2013
, “
Comparison Between ANCF and B-Spline Surfaces
,”
J. Multibody Syst. Dyn.
,
30
(
2
), pp.
119
138
.
33.
Shabana
,
A. A.
,
2015
, “
Definition of ANCF Finite Elements
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
054506
.
34.
Yu
,
Z.
, and
Shabana
,
A. A.
,
2015
, “
Mixed-Coordinate ANCF Rectangular Plate Finite Element
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061003
.
35.
Dufva
,
K.
, and
Shabana
,
A. A.
,
2005
, “
Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation
,”
Proc. Inst. Mech. Eng., Part K
,
219
(
4
), pp.
345
355
.
36.
Hyldahl
,
P.
,
2013
, “
Large Displacement Analysis of Shell Structures Using the Absolute Nodal Coordinate Formulation
,” Department of Engineering, Aarhus University, Aarhus, Denmark, Technical Report No. ME-TR-6.
37.
Shabana
,
A. A.
,
2015
, “
ANCF Reference Node for Multibody System Analysis
,”
Proc. Inst. Mech. Eng., Part K
,
229
(
1
), pp.
109
112
.
38.
Shabana
,
A. A.
,
2015
, “
ANCF Tire Assembly Model for Multibody System Applications
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
024504
.
39.
Cook
,
R. D.
,
Malkus
,
D. S.
, and
Plesha
,
M. E.
,
1989
,
Concepts and Applications of Finite Element Analysis
,
3rd ed.
,
Wiley
,
Hoboken, NJ
.
40.
Hussein
,
B. A.
, and
Shabana
,
A. A.
,
2011
, “
Sparse Matrix Implicit Numerical Integration of the Stiff Differential/Algebraic Equations: Implementation
,”
J. Nonlinear Dyn.
,
65
(
4
), pp.
369
382
.
41.
Aboubakr
,
A. K.
, and
Shabana
,
A. A.
,
2015
, “
Efficient and Robust Implementation of the TLISMNI Method
,”
J. Sound Vib.
,
353
, pp.
220
242
.
You do not currently have access to this content.