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.

References

References
1.
Love
,
A. E. H.
,
1944
,
A Treatise on the Mathematical Theory of Elasticity
,
Courier Dover Publications
,
New York
.
2.
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
.
3.
Antman
,
S. S.
,
1995
,
Nonlinear Problems in Elasticity
,
Springer-Verlag
,
New York
.
4.
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
.
5.
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
.
6.
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.
7.
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
.
8.
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
.
9.
von Dombrowski
,
S.
,
2002
, “
Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates
,”
Multibody Syst. Dyn.
,
8
(
4
), pp.
409
432
.
10.
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
.
11.
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
.
12.
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
.
13.
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
.
14.
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
.
15.
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
.
16.
Bauchau
,
O. A.
,
2010
,
Flexible Multibody Dynamics
,
Springer
,
Dordrecht, Heidelberg, London/New York
.
17.
Shabana
,
A. A.
,
2005
,
Dynamics of Multibody Systems
,
Cambridge University Press
,
Cambridge, UK
.
18.
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
.
19.
Ken
,
S.
,
1985
, “
Animating Rotation With Quaternion Curves
,”
SIGGRAPH Comput. Graphics
,
19
(
3
), pp.
245
254
.
20.
Davis
,
P. J.
, and
Rabinowitz
,
P.
,
2007
,
Methods of Numerical Integration
,
Courier Dover Publications
,
New York
.
21.
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
.
22.
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
.
23.
Timoshenko
,
S. P.
, and
Gere
,
J. M.
,
1961
,
Theory of Elastic Stability
,
McGraw-Hill
,
New York
.
24.
Blevins
,
R. D.
,
1979
,
Formulas for Natural Frequency and Mode Shape
,
Krieger Publishing Company
,
Malabar, FL
.
25.
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
.
26.
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
.
27.
Arnold
,
M.
, and
Brüls
,
O.
,
2007
, “
Convergence of the Generalized-α Scheme for Constrained Mechanical Systems
,”
Multibody Syst. Dyn.
,
18
(
2
), pp.
185
202
.
28.
Shabana
,
A. A.
,
2011
,
Computational Continuum Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
You do not currently have access to this content.