This paper describes a very simple beam model, amenable to be used in multibody applications, for cases where the effects of torsion and shear are negligible. This is the case of slender rods connecting different parts of many space mechanisms, models useful in polymer physics, computer animation, etc. The proposed new model follows a lumped parameter method that leads to a rotation-free formulation. Axial stiffness is represented by a standard nonlinear truss model, while bending is modeled with a force potential. Several numerical experiments are carried out in order to assess accuracy, which is usually the main drawback of this type of approach. Results reveal a remarkable accuracy in nonlinear dynamical problems, suggesting that the proposed model is a valid alternative to more sophisticated approaches.

References

1.
Sadler
,
J. P.
, and
Sandor
,
G. N.
,
1973
, “
A Lumped Parameter Approach to Vibration and Stress Analysis of Elastic Linkages
,”
J. Eng. Ind.
,
95
(
2
), pp.
549
557
.
2.
Sadler
,
J. P.
,
1975
, “
On the Analytical Lumped-Mass Model of an Elastic Four-Bar Mechanism
,”
J. Eng. Ind.
,
97
(
2
), pp.
561
565
.
3.
Wang
,
Y.
, and
Huston
,
R. L.
,
1994
, “
A Lumped Parameter Method in the Nonlinear Analysis of Flexible Multibody Systems
,”
Comput. Struct.
,
50
(
3
), pp.
421
432
.
4.
Wittbrodt
,
E.
,
Adamiec-Wójcik
,
I.
, and
Wojciech
,
S.
,
2006
,
Dynamics of Flexible Multibody Systems. Rigid Finite Element Method
,
Springer-Verlag
, Berlin.
5.
Shabana
,
A. A.
,
2010
,
Computational Dynamics
,
Wiley
, Chichester, UK.
6.
Crisfield
,
M. A.
, and
Moita
,
G. F.
,
1996
, “
A Unified Co-Rotational Framework for Solids, Shells and Beams
,”
Int. J. Numer. Methods Eng.
,
33
(
20–22
), pp.
2969
2992
.
7.
Mayo
,
J.
, and
Domínguez
,
J.
,
1996
, “
Geometrically Non-Linear Formulation of Flexible Multibody Systems in Terms of Beam Elements: Geometric Stiffness
,”
Comput. Struct.
,
59
(
6
), pp.
1039
1050
.
8.
Liu
,
J. Y.
, and
Hong
,
J. Z.
,
2004
, “
Geometric Stiffening Effect on Rigid-Flexible Coupling Dynamics of an Elastic Beam
,”
J. Sound Vib.
,
278
(
4–5
), pp.
1147
1162
.
9.
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
.
10.
Simó
,
J. C.
, and
Vu-Quoc
,
L.
,
1986
, “
On the Dynamics of Flexible Beams Under Large Overall Motions—The Planar Case
,”
ASME J. Appl. Mech.
,
53
(
4
), pp.
849
863
.
11.
Simó
,
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
.
12.
Simó
,
J. C.
, and
Vu-Quoc
,
L.
,
1988
, “
On the Dynamics in Space of Rods Undergoing Large Motions—A Geometrically Exact Approach
,”
Comput. Methods Appl. Mech. Eng.
,
66
(
2
), pp.
125
161
.
13.
Han
,
S.
, and
Bauchau
,
O. A.
,
2015
, “
Nonlinear Three-Dimensional Beam Theory for Flexible Multibody Dynamics
,”
Multibody Syst. Dyn.
,
34
(
3
), pp.
211
242
.
14.
Shabana
,
A. A.
,
1996
, “
Finite Element Incremental Approach and Exact Rigid Body Inertia
,”
ASME J. Mech. Des.
,
118
(
2
), pp.
171
178
.
15.
Escalona
,
J. L.
,
Hussien
,
H. A.
, and
Shabana
,
A. A.
,
1998
, “
Application of the Absolute Nodal Coordinate Formulation to Multibody System Dynamics
,”
J. Sound Vib.
,
214
(
5
), pp.
833
851
.
16.
Omar
,
M. A.
, and
Shabana
,
A. A.
,
2001
, “
A Two Dimensional Shear Deformable Beam for Large Rotation and Deformation Problems
,”
J. Sound Vib.
,
243
(
3
), pp.
565
576
.
17.
Romero
,
I.
,
2008
, “
A Comparison of Finite Elements for Nonlinear Beams: The Absolute Nodal Coordinate and Geometrically Exact Formulations
,”
Multibody Syst. Dyn.
,
20
(
1
), pp.
51
68
.
18.
Bauchau
,
O. A.
,
Han
,
S.
,
Mikkola
,
A.
, and
Matikainen
,
M. K.
,
2014
, “
Comparison of the Absolute Nodal Coordinate and Geometrically Exact Formulations for Beams
,”
Multibody Syst. Dyn.
,
32
(
1
), pp.
67
85
.
19.
Mata
,
P. L.
,
Barbat
,
A. H.
,
Oller
,
S.
, and
Boroschek
,
R.
,
2008
,
Inelastic Analysis of Geometrically Exact Rods
(Monograph Series in Earthquake Engineering),
CIMNE
, Barcelona, Spain.
20.
Romero
,
I.
,
Urrecha
,
M.
, and
Cyron
,
C. J.
,
2014
, “
A Torsion-Free Non-Linear Beam Model
,”
Int. J. Non Linear Mech.
,
58
(
0
), pp.
1
10
.
21.
Duan
,
Y.
,
Li
,
D.
, and
Frank Pai
,
P.
,
2013
, “
Geometrically Exact Physics-Based Modeling and Computer Animation of Highly Flexible 1D Mechanical Systems
,”
Graphical Models
,
75
(
2
), pp.
56
68
.
22.
Phaal
,
R.
, and
Calladine
,
C. R.
,
1992
, “
A Simple Class of Finite Elements for Plate and Shell Problems—I: Elements for Beams and Thin Flat Plates
,”
Int. J. Numer. Methods Eng.
,
35
(
5
), pp.
955
977
.
23.
Flores
,
F. G.
, and
Oñate
,
E.
,
2006
, “
Rotation-Free Finite Element for the Non-Linear Analysis of Beams and Axisymmetric Shells
,”
Comput. Methods Appl. Mech. Eng.
,
195
(41–43), pp.
5297
5315
.
24.
Battini
,
J.-M.
,
2008
, “
A Rotation-Free Corotational Plane Beam Element for Non-Linear Analyses
,”
Int. J. Numer. Methods Eng.
,
75
(
6
), pp.
672
689
.
25.
Zhou
,
Y. X.
, and
Sze
,
K. Y.
,
2013
, “
A Rotation-Free Beam Element for Beam and Cable Analyses
,”
Finite Elem. Anal. Des.
,
64
, pp.
79
89
.
26.
Bauchau
,
O. A.
,
Betsch
,
P.
,
Cardona
,
A.
,
Gerstmayr
,
J.
,
Jonker
,
B.
,
Masarati
,
P.
, and
Sonneville
,
V.
,
2016
, “
Validation of Flexible Multibody Dynamics Beam Formulations Using Benchmark Problems
,”
Multibody Syst. Dyn.
,
37
(
1
), pp.
29
48
.
27.
Bonet
,
J.
, and
Wood
,
R. D.
,
2008
,
Nonlinear Continuum Mechanics for Finite Element Analysis
, 2nd ed.,
Cambridge University Press
,
Cambridge, UK
.
28.
Mattiasson
,
K.
,
1981
, “
Numerical Results From Large Deflection Beam and Frame Problems Analysed by Means of Elliptic Integrals
,”
Int. J. Numer. Methods Eng.
,
17
(
1
), pp.
145
153
.
29.
Mayo
,
J. M.
,
García-Vallejo
,
D.
, and
Domínguez
,
J.
,
2004
, “
Study of the Geometric Stiffening Effect: Comparison of Different Formulations
,”
Multibody Syst. Dyn.
,
11
(
4
), pp.
321
341
.
30.
Simeon
,
B.
,
2013
,
Computational Flexible Multibody Dynamics. A Differential-Algebraic Approach
,
Springer-Verlag
, Berlin.
31.
Tian
,
Q.
,
Zhang
,
Y.
,
Chen
,
L.
, and
Flores
,
P.
,
2009
, “
Dynamics of Spatial Flexible Multibody Systems With Clearance and Lubricated Spherical Joints
,”
Comput. Struct.
,
87
(
13–14
), pp.
913
929
.
You do not currently have access to this content.