0
Research Papers

A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters

[+] Author and Article Information
W. Fan

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

H. Ren

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250;
MSC Software Corporation,
201 Depot Street, Suite 100,
Ann Arbor, MI 48105

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 14, 2015; final manuscript received September 15, 2015; published online January 4, 2016. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 11(4), 041013 (Jan 04, 2016) (13 pages) Paper No: CND-15-1217; doi: 10.1115/1.4031769 History: Received July 14, 2015; Revised September 15, 2015

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Love, A. E. H. , 1944, A Treatise on the Mathematical Theory of Elasticity, Courier Dover Publications, New York.
Goyal, S. , Perkins, N. C. , and Lee, C. L. , 2005, “ Nonlinear Dynamics and Loop Formation in Kirchhoff Rods With Implications to the Mechanics of DNA and Cables,” J. Comput. Phys., 209(1), pp. 371–389. [CrossRef]
Antman, S. S. , 1995, Nonlinear Problems in Elasticity, Springer-Verlag, New York.
Simo, J. C. , 1985, “ A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem. Part I,” Comput. Methods Appl. Mech. Eng., 49(1), pp. 55–70. [CrossRef]
Simo, J. C. , and Vu-Quoc, L. , 1986, “ A Three-Dimensional Finite-Strain Rod Model. Part II: Computational Aspects,” Comput. Methods Appl. Mech. Eng., 58(1), pp. 79–116. [CrossRef]
Shabana, A. A. , 1996, “ An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies,” University of Illinois at Chicago, Technical Report No. MBS96-1-UIC.
Shabana, A. A. , and Yakoub, R. Y. , 2001, “ Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory,” ASME J. Mech. Des., 123(4), pp. 606–613. [CrossRef]
Yakoub, R. Y. , and Shabana, A. A. , 2001, “ Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications,” ASME J. Mech. Des., 123(4), pp. 614–621. [CrossRef]
von Dombrowski, S. , 2002, “ Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates,” Multibody Syst. Dyn., 8(4), pp. 409–432. [CrossRef]
Zhao, Z. H. , and Ren, G. X. , 2012, “ A Quaternion-Based Formulation of Euler–Bernoulli Beam Without Singularity,” Nonlinear Dyn., 67(3), pp. 1825–1835. [CrossRef]
Zhu, W. D. , Ren, H. , and Xiao, C. , 2011, “ A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables,” ASME J. Appl. Mech., 78(4), p. 041017. [CrossRef]
Huang, J. L. , and Zhu, W. D. , 2014, “ Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler—Bernoulli Beam Under the Gravity Load,” ASME J. Appl. Mech., 81(10), p. 101007. [CrossRef]
Cardona, A. , and Geradin, M. , 1988, “ A Beam Finite Element Non-Linear Theory With Finite Rotations,” Int. J. Numer. Methods Eng., 26(11), pp. 2403–2438. [CrossRef]
Zupan, E. , Saje, M. , and Zupan, D. , 2009, “ The Quaternion-Based Three-Dimensional Beam Theory,” Comput. Methods Appl. Mech. Eng., 198(49), pp. 3944–3956. [CrossRef]
Geradin, M. , and Cardona, A. , 1988, “ Kinematics and Dynamics of Rigid and Flexible Mechanisms Using Finite Elements and Quaternion Algebra,” Comput. Mech., 4(2), pp. 115–135. [CrossRef]
Bauchau, O. A. , 2010, Flexible Multibody Dynamics, Springer, Dordrecht, Heidelberg, London/New York.
Shabana, A. A. , 2005, Dynamics of Multibody Systems, Cambridge University Press, Cambridge, UK.
Sugiyama, H. , Gerstmayr, J. , and Shabana, A. A. , 2006, “ Deformation Modes in the Finite Element Absolute Nodal Coordinate Formulation,” J. Sound Vib., 298(4), pp. 1129–1149. [CrossRef]
Ken, S. , 1985, “ Animating Rotation With Quaternion Curves,” SIGGRAPH Comput. Graphics, 19(3), pp. 245–254. [CrossRef]
Davis, P. J. , and Rabinowitz, P. , 2007, Methods of Numerical Integration, Courier Dover Publications, New York.
Campanelli, M. , Berzeri, M. , and Shabana, A. A. , 2000, “ Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems,” ASME J. Mech. Des., 122(4), pp. 498–507. [CrossRef]
Sugiyama, H. , and Suda, Y. , 2007, “ A Curved Beam Element in the Analysis of Flexible Multi-Body Systems Using the Absolute Nodal Coordinates,” Proc. Inst. Mech. Eng., Part K, 221(2), pp. 219–231. [CrossRef]
Timoshenko, S. P. , and Gere, J. M. , 1961, Theory of Elastic Stability, McGraw-Hill, New York.
Blevins, R. D. , 1979, Formulas for Natural Frequency and Mode Shape, Krieger Publishing Company, Malabar, FL.
Hindmarsh, A. C. , Brown, P. N. , Grant, K. E. , Lee, S. L. , Serban, R. , Shumaker, D. E. , and Woodward, C. S. , 2005, “ Sundials: Suite of Nonlinear and Differential/Algebraic Equation Solvers,” ACM Trans. Math. Software, 31(3), pp. 363–396. [CrossRef]
Ren, H. , 2015, “ A Simple Absolute Nodal Coordinate Formulation for Thin Beams With Large Deformations and Large Rotations,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061005. [CrossRef]
Arnold, M. , and Brüls, O. , 2007, “ Convergence of the Generalized-α Scheme for Constrained Mechanical Systems,” Multibody Syst. Dyn., 18(2), pp. 185–202. [CrossRef]
Shabana, A. A. , 2011, Computational Continuum Mechanics, Cambridge University Press, Cambridge, UK.

Figures

Grahic Jump Location
Fig. 1

Geometrical description of a three-dimensional Euler–Bernoulli beam

Grahic Jump Location
Fig. 2

Static equilibria of a rolling cantilever beam calculated with two elements in the current formulation; circles indicate nodes of the beam

Grahic Jump Location
Fig. 3

Static equilibria of a cantilever beam under different compressive forces that are over the critical load, calculated with 20 elements in the current formulation; θ represents the rotation angle of the free end of the beam

Grahic Jump Location
Fig. 4

Relative errors of rotations of the free end of the beam under different compressive loads: (a) P=1.015 Pcr and (b) P=9.116 Pcr

Grahic Jump Location
Fig. 5

The first four (a) in-plane and (b) out-of-plane mode shapes of an elastic ring; dashed lines correspond to the undeformed ring

Grahic Jump Location
Fig. 6

Nonlinear dynamic responses at the free end of a cantilever beam under end moments Mx=50 sin(πt) and Mz=100 sin(πt/2) from the current formulation and ADAMS

Grahic Jump Location
Fig. 7

(a) Angular velocity of a falling pinned-free beam under gravity and (b) its transverse response at its free end

Grahic Jump Location
Fig. 8

Evolution of various energies of the beam during its motions

Grahic Jump Location
Fig. 9

Violation of the normalization constraint of Euler parameters at the free end of the beam

Grahic Jump Location
Fig. 10

Schematic of a double elastic pendulum falling under gravity

Grahic Jump Location
Fig. 11

The (a) X, (b) Y, and (c) Z displacements at the free end of the beam B2 from the current formulation and recurdyn

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In