Static equations for thin inextensible elastic rods, or elastica as they are sometimes called, have been studied since before the time of Euler. In this paper, we examine how to model the dynamic behavior of elastica. We present a fairly high speed, robust numerical scheme that uses (i) a space discretization that uses cubic splines, and (ii) a time discretization that preserves a discrete version of the Hamiltonian. A good choice of numerical scheme is important because these equations are very stiff; that is, most explicit numerical schemes will become unstable very quickly. The authors conducted this research anticipating describing the dynamic Kirchhoff problem, that is, the behavior of general springs that have natural curvature, and for which the equations take into account torsion of the rod.

References

References
1.
Love
,
A. E. H.
,
1927
,
A Treaties on the Mathematical Theory of Elasticity
,
4th ed.
,
Dover
,
New York
.
2.
Montgomery-Smith
,
S. J.
, and
Huang
,
W.
,
2012
, “
Dynamic Equations of Non-Helical Springs
,” (to be published).
3.
Levian
,
R.
,
2008
, “
The Elastica: A Mathematical History
,” University of California, Berkeley, Technical Report No. UCB/EECS-2008-103.
4.
Falk
,
R. S.
, and
Xu
,
J.-M.
,
1995
, “
Convergence of a Second-Order Scheme for the Non-Linear Dynamical Equations of Elastic Rods
,”
SIAM J. Numer. Anal.
,
32
(
4
), pp.
1185
1209
.10.1137/0732055
5.
Caflisch
,
R. E.
, and
Maddocks
,
J. H.
,
1984
, “
Nonlinear Dynamical Theory of the Elastica
,”
Proc. R. Soc. Edinburgh
,
99A
, pp.
1
23
.10.1017/S0308210500025920
6.
Coleman
,
B. D.
, and
Dill
,
E. H.
,
1992
, “
Flexure Waves in Elastic Rods
,”
J. Acoust. Soc. Am.
,
91
, pp.
2663
2673
.10.1121/1.402974
7.
Dichmann
,
D. J.
,
Maddocks
,
J. H.
, and
Pego
,
R. L.
,
1996
, “
Hamiltonian Dynamics of an Elastica and the Stability of Solitary Waves
,”
Arch. Rational Mech. Anal.
,
135
, pp.
357
396
.10.1007/BF02198477
8.
Ito
,
K.
,
1998
, “
Uniform Stabilization of the Dynamic Elastica by Boundary Feedback
,”
SIAM J. Control Optim.
,
37
, pp.
319
329
.10.1137/S0363012997322352
9.
Ito
,
K.
,
2009
, “
Stabilization of the Dynamic Elastica Only by Damping Torque
,”
Nonlinear Anal.: Real World Appl.
,
10
(
5
), pp.
3122
3131
.10.1016/j.nonrwa.2008.10.011
10.
Maddocks
,
J. H.
, and
Dichmann
,
D. J.
,
1994
, “
Conservation Laws in the Dynamics of Rods
,”
J. Elast.
,
34
, pp.
83
96
.10.1007/BF00042427
12.
Lin
,
Y.
, and
Pisano
,
A. P.
,
1987
, “
General Dynamic Equations of Helical Springs with Static Solution and Experimental Verification
,”
Trans. ASME
,
54
, pp.
910
917
.10.1115/1.3173138
13.
Pai
,
F. P.
,
2007
,
Highly Flexible Structures: Modeling, Computation, and Experimentation
,
AIAA
,
Reston, VA
.
14.
Pai
,
P. F.
,
2007
,
Highly Flexible Structures: Modeling, Computation and Experimentation
, AIAA, Reston, Virginia. Available at: http://web.missouri.edu/~paip/HFSs
15.
Antman
,
S. S.
,
2005
,
Nonlinear Problems in Elasticity
,
2nd ed.
,
Springer
,
New York
.
16.
Goss
,
V. G. A.
,
2003
, “
Snap Buckling, Writhing and Loop Formation in Twisted Rods
,” Ph.D. thesis, University College London, London.
17.
van der Heijden
,
G. H. M.
,
2012
, Home page of Gert van der Heijden. Available at: http://www.ucl.ac.uk/~ucesgvd
18.
Preston
,
S. C.
,
2011
, “The Geometry of Whips,” http://arxiv.org/abs/1105.1754
19.
Preston
,
S. C.
,
2011
, “
The Motion of Whips and Chains
,”
J. Differ. Equ.
,
251
(
3
), pp.
504
550
.10.1016/j.jde.2011.05.005
20.
Thess
,
A.
,
Zikanov
,
O.
, and
Nepomnyashchy
,
A.
,
1999
, “
Finite-Time Singularity in the Vortex Dynamics of a String
,”
Phys. Rev. E
,
59
, pp.
3637
3640
.10.1103/PhysRevE.59.3637
21.
Bauchau
,
O. A.
, and
Bottasso
,
C. L.
,
1999
, “
On the Design of Energy Preserving and Decaying Schemes, for Flexible, Nonlinear Multi-Body Systems
,”
Comput. Methods Appl. Mech. Eng.
,
169
, pp.
61
79
.10.1016/S0045-7825(98)00176-5
22.
Celledoni
,
E.
,
Grimm
,
V.
,
McLachlan
,
R. I.
,
McLaren
,
D. I.
,
ONeale
,
D.
,
Owren
,
B.
, and
Quispel
,
G. R. W.
,
2012
, “
Preserving Energy Resp. Dissipation in Numerical PDEs Using the ‘Average Vector Field’ Method
,”
J. Comput. Phys.
,
231
, pp.
6770
6789
.10.1016/j.jcp.2012.06.022
23.
Futterer
,
T.
,
Klar
,
A.
, and
Wegener
,
R.
,
2012
, “
An Energy Conserving Numerical Scheme for the Dynamics of Hyperelastic Rods
,”
Int. J. Differ. Equ.
,
2012
, Article No.
718308
.10.1155/2012/718308
24.
Leimkuhler
,
B.
, and
Reich
,
S.
,
2004
,
Simulating Hamiltonian Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
25.
Pace
,
B.
,
Diele
,
F.
, and
Marangi
,
C.
,
2012
, “
Energy Preservation in Separable Hamiltonian Systems by Splitting Schemes
,”
AIP Conf. Proc.
,
1479
, pp.
1204
1207
.10.1063/1.4756367
26.
Sanz-Serna
,
J. M.
, and
Calvo
,
M. P.
,
1994
,
Numerical Hamiltonian Problems
,
Chapman and Hall
,
London, UK
.
27.
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
.10.1007/BF00251673
28.
Simo
,
J. C.
, and
Wong
,
K. K.
,
1991
, “
Unconditionally Stable Algorithms for Rigid Body Dynamics that Exactly Preserve Energy and Momentum
,”
Int. J. Numer. Methods Eng.
,
31
(
1
), pp.
19
52
.10.1002/nme.1620310103
29.
Simo
,
J. C.
, and
Tarnow
,
N.
,
1992
, “
The Discrete Energy-Momentum Method. Conserving Algorithms for Nonlinear Elastodynamics
,”
Z. Math. Phys. (ZAMP)
,
43
(
5
), pp.
757
792
.10.1007/BF00913408
30.
Simo
,
J. C.
,
Tarnow
,
N.
, and
Wong
,
K. K.
,
1992
, “
Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
100
(
1
), pp.
63
116
.10.1016/0045-7825(92)90115-Z
31.
Simo
,
J. C.
,
Tarnow
,
N.
, and
Doblare
,
M.
,
1995
, “
Non-Linear Dynamics of Three-Dimensional Rods: Exact Energy and Momentum Conserving Algorithms
,”
Int. J. Numer. Methods Eng.
,
38
(
9
), pp.
1431
1473
.10.1002/nme.1620380903
32.
Asmar
,
N. H.
,
2004
,
Partial Differential Equations with Fourier Series and Boundary Value Problems
,
2nd ed.
,
Prentice Hall
,
Englewood Cliffs, NJ
.
33.
Dunford
,
N.
, and
Schwartz
,
J. T.
,
1963
,
Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space
,
John Wiley & Sons, Inc.
,
New York
.
34.
Burden
,
R. L.
, and
Faires
,
J. D.
,
2005
,
Numerical Methods
,
8th ed.
,
Thomson Brooks/Cole Publishing Co
,
Pacific Grove, CA
.
35.
Hairer
,
E.
, and
Wanner
,
G.
,
2002
,
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics
,
2nd ed.
,
Springer
,
New York
.
36.
Hall
,
C. A.
, and
Meyer
,
W. W.
,
1976
, “
Bounds for Cubic Spline Interpolation
,”
J. Approx. Theory
,
16
, pp.
105
122
.10.1016/0021-9045(76)90040-X
You do not currently have access to this content.