In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

References

References
1.
Love
,
A. E.
,
1944
,
A Treatise on the Mathematical Theory of Elasticity
,
4th ed.
,
Dover Publications
,
New York
.
2.
Reissner
,
E.
,
1972
, “
On One-Dimensional Finite-Strain Beam Theory: The Plane Problem
,”
J. Appl. Math. Phys. (ZAMP)
,
23
(
5
), pp.
795
804
.
3.
Reissner
,
E.
,
1973
, “
On One-Dimensional Large-Displacement Finite-Strain Beam Theory
,”
Stud. Appl. Math.
,
52
(
2
), pp.
87
95
.
4.
Reissner
,
E.
,
1981
, “
On Finite Deformation of Space-Curved Beams
,”
J. Appl. Math. Phys. (ZAMP)
,
32
(
6
), pp.
734
744
.
5.
Simonds
,
J. G.
, and
Danielson
,
D. A.
,
1972
, “
Nonlinear Shell Theory With Finite Rotation and Stress-Function Vectors
,”
ASME J. Appl. Mech.
,
39
(
4
), pp.
1085
1090
.
6.
Antman
,
S. S.
,
1972
, “
The Theory of Rods
,”
Handbuch der Physik
, Vol.
2
,
Springer
,
Berlin
, pp.
641
703
.
7.
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
.
8.
Simo
,
J. C.
, and
Vu-Quoc
,
L.
,
1986
, “
A Three-Dimensional Finite Strain Rod Model—Part II: Computational Aspect
,”
Comput. Methods Appl. Mech. Eng.
,
58
(
1
), pp.
79
116
.
9.
Marsden
,
J. E.
, and
Hughes
,
T. J. R.
,
1983
,
Mathematical Foundations of Elasticity
,
Prentice Hall
,
Englewood Cliffs, NJ
.
10.
Cartan
,
É.
,
1928
,
Leçons sur la Géométrie des Espaces de Riemann
,
Gauthiers-Villars
, Paris,
France
.
11.
Cartan
,
É.
,
1986
,
On Manifolds With an Affine Connection and the Theory of General Relativity
,
A.
Magnon
, and
A.
Ashtekar
, eds.,
Bibiliopolis
,
Napoli, Italy
.
12.
Flanders
,
H.
,
1963
,
Differential Forms With Applications to the Physical Science
,
Academic Press
,
New York
.
13.
Chern
,
S. S.
,
Chen
,
W. H.
, and
Lam
,
K. S.
,
1999
,
Lectures on Differential Geometry
,
World Scientific
,
Singapore
.
14.
Frankel
,
T.
,
2012
,
The Geometry of Physics: An Introduction
,
3rd ed.
,
Cambridge University Press
,
New York
.
15.
Hunter
,
R. D.
,
Moss
,
V. A.
, and
Elder
,
H. Y.
,
1983
, “
Image Analysis of the Burrowing Mechanism of Polyphysia crassa (Annelida: Polychaeta) and Priapulus caudatus (Priapulida)
,”
J. Zool.
,
199
(3), pp.
305
323
.
16.
Murakami
,
H.
, and
Yamakawa
,
J.
,
2000
, “
Development of One-Dimensional Models for Elastic Waves in Heterogeneous Beams
,”
ASME J. Appl. Mech.
,
67
(
4
), pp.
671
684
.
17.
Murakami
,
H.
,
2013
, “
A Moving Frame Method for Multi-Body Dynamics
,”
ASME
Paper No. IMECE2013-62833.
18.
Murakami
,
H.
,
2015
, “
A Moving Frame Method for Multi-Body Dynamics Using SE(3)
,”
ASME
Paper No. IMECE2015-51192.
19.
Murakami
,
H.
,
Rios
,
O.
, and
Impelluso
,
T. J.
,
2016
, “
A Theoretical and Numerical Study of the Dzhanibekov and the Tennis Racket Phenomena
,”
ASME J. Appl. Mech.
,
83
(
11
), p.
111006
.
20.
O'Neill
,
B.
,
1997
,
Elementary Differential Geometry
,
2nd ed.
,
Academic Press
,
New York
.
21.
Do Carmo
,
M. P.
,
1976
,
Differential Geometry of Curves and Surfaces
,
Prentice Hall
,
Englewood Cliffs, NJ
.
22.
Truesdell
,
C.
, and
Toupin
,
R. A.
,
1960
, “
The Classical Field Theories
,”
Handbuck der Physik
, Vol.
III-1
, Springer-Verlag, Berlin, pp.
226
790
.
23.
Simo
,
J. C.
,
Marsden
,
J. E.
, and
Krishnaprasad
,
P. S.
,
1988
, “
The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representations of Solids, Rods, and Plates
,”
Arch. Ration. Mech. Anal.
,
104
(
2
), pp.
125
183
.
24.
Murakami
,
H.
,
2016
, “
Development of an Active Curved Beam Model Using a Moving Frame Method
,”
ASME
Paper No. IMECE2016-65294.
25.
Smith
,
D. R.
,
1974
,
Variational Methods in Optimization
,
Prentice Hall
,
Englewood Cliffs, NJ
.
26.
Holm
,
D.
,
2008
,
Geometric Mechanics, Part II: Rotating, Translating and Rolling
,
World Scientific
,
London
.
27.
Noble
,
B.
, and
Daniel
,
J. W.
,
1977
,
Applied Linear Algebra
,
Prentice Hall
,
Englewood Cliffs, NJ
.
28.
Wittenburg
,
J.
,
2008
,
Dynamics of Multibody Systems
,
2nd ed.
,
Springer
,
Berlin
.
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