SYMMETRY AND STABILITY IN NONLINEAR MECHANICS

WITH INTERDISCIPLINARY APPLICATIONS

BUDAPEST, JULY 10-14th 2000.

LOCATION: MAIN ("K") BUILDING OF THE TECHNICAL UNIVERSITY,
FACULTY CLUB (I.66)

Organizers:

Gabor Domokos     and      Tim Healey
(Budapest)                      (Cornell)

Hosts:

The goal of the workshop is to provide opportunity
for the nonlinear mechanics/applied mathematics
community to exchange ideas on the subjects given
in the title.

Financial support is provided by the US-Hungarian Science and Technology Joint  Fund and the Hungarian Scientific Research Fund.

The list of speakers is now final.

Whenever available, a link to the personal homepages is provided.

 LIST OF SPEAKERS

 Name Affiliation Title of talk Time Alan Champneys Univ.Bristol, UK, Eng. Mathematics The Indian rope trick' for a continuously flexible rod;  linear and nonlinear analysis. Mo. 1450 Bernard Coleman Rutgers, Program in Mechanics On solutions of the self-contact problem for elastic rods and applications to DNA supercoiling Wed 1600 Harry Dankowicz Virginia Tech, Eng. Sci. and Mechanics Stability analysis of passive, bipedal gait in a three-dimensional environment Mo. 0950 Eusebius Doedel Caltech, Applied Mathematics Discontinuous Piecewise Polynomial Collocation Methods for Elliptic PDEs Fri 0900 Gábor Domokos Tech. Univ. Budapest, Strength of Materials Ghost solutions in BVPs Wed 1150 Zsolt Gáspár Tech. Univ. Budapest, Structural Mechanics Special shapes of bent and twisted rods Wed  0950 Oded Gottlieb Technion, Israel, Mechanical Eng. Nonlinear dynamics and stability of spherical membranes Mo. 1100 Alain Goriely U.Arizona, Mathematics Homoclinic collapse in the statics and dynamics of biological filaments Wed 1400 George Haller Brown Univ., Applied Mathematics Coherent structures and mixing in two-dimensional turbulence Fri 1100 Tim Healey Cornell, Theoretical and Applied Mechanics Hidden Symmetry of Global Solutions in Twisted Elastic Rings Fri 1450 Mike Henderson IBM Watson Research Center Multiple Parameter Continuation Mo. 1600 Phil Holmes Princeton Univ., Applied Math. Unconstrained Euler buckling in a potential field. Mo. 1400 Giles Hunt Bath, UK, Structural Mechanics Maxwell Lower Bounds for Bifurcation at Infinite Load Wed 1100 György Károlyi, Tamás Tél TU Budapest, Struct.Mech., Eötvös University, Theor.Physics Fractality, chaos, and reactions in imperfectly mixed open hydrodynamical flows Fri 1150 Mark Levi Penn State,  Mathematics Curvature and parallel transport in averaging Tue 0950 John Maddocks EPFL, Lausanne, Applied Aanalysis Symmetry Breaking and Stability of Equilibria of DNA Mini-Circles Wed 1450 Sri Namachchivaya Illinois, Aeronautical & Astr. Eng. Nonstandard Reduction of Noisy Mechanical Systems Tue 1100 Andy Ruina Cornell, Theoretical and Applied Mechanics The symmetry of some of the simple non-holonomic examples precludes assymtotic stability. Mon 0900 Steve Shaw Michigan State, Mech. Engineering. Dynamic Stability and Performance of Systems of Nonlinear Vibration Absorbers Fri 0950 Gábor Stépán Tech. Univ. Budapest, Applied Mechanics Stability and bifurcations in dynamical systems with time delay and parametric excitation Wed 0900 Alois Steindl Tech. Univ. Vienna., Inst. Mechanics Application of Normal Form theory in an optimal controlled production model Mon 1650 David Swigon Rutgers, Program in Mechanics Elastic stability of knotted DNA plasmids Wed 1650 Tibor Tarnai Tech. Univ. Budapest,  Structural Mechanics Transition from spherical circle packing to covering Mon 1150 Michael Thompson ,      Gert Van Der Heijden University College, Center for Nonlinear Dynamics. London, UK. The chaotic instability of a slowly spinning asymmetric top Tue 0900 Hans Troger Tech. Univ. Vienna, Inst. Mechanics Stability Analysis of Relative Equilibria of Tethered Satellite Systems Tue 1150 Alex Vakakis Illinois, Mechanical and Indust. Eng. Passive Nonlinear Energy Pumping due to Symmetry Breaking in Coupled Oscillators Fri 1400

 LOGISTICS

**How to get from the airport to the Guesthouse or the Gellert Hotel

Cabs at the airport are by default rather expensive (~USD 20) and tend to overcharge, sometimes badly.
If you decide to take a cab, make sure that you agree on the rate with the driver in advance. They can charge
ONLY FLAT RATES, which depend on the district of Budapest where you drive.
I recommend the Airport Minibus Service (~USD 7), especially for single travellers. This service is available
immediately after you emerge from the customs area. Normally you have to wait for 15-20 minutes, the minibuses
take a couple of travellers and drop them in sequence.
In both cases (cab or minibus) a tip 10-15% to the driver is recommended.
The EXACT address of the guesthouse is in Hungarian:
BME Professzori Vendégház
Stoczek utca 5-7  (Martos kollégium)
1111 Budapest
If you print this out and show it to the receptionist of the Minibus service or the cab driver, they will know.
Both of them would know Hotel Gellert without any further specifications.

 ABSTRACTS

The Indian rope trick' for a continuously flexible rod;
linear and nonlinear analysis.

Alan R. Champneys
Applied Nonlinear Mathematics Group
Dept of Engineering Maths        email:  a.r.champneys@bristol.ac.uk
University of Bristol            phone:  (+44) (0)117 928 7510
Bristol  BS8 1TR  UK             fax:    (+44) (0)117 925 1154

Acheson and Mullin have demonstrated both experimentally and
theoretically that it is possible to balance a system of N coupled
rigid pendulums upside down by oscillating its support. In the limit as
N tends to  infinity the theoretical region of stability in frequency
vs. amplitude disappears. Nevertheless experiments of Mullin on a
continuously flexible rod (a piece of curtain wire just long enough to
fall over under its own weight) suggest it can be stabilised using the
right frequency and amplitude of excitation.

We propose a theory for this. First, a dimensionless PDE is derived
from rod theory that is fourth-order in arclength and second-order in
time. The stability of the vertical solution subject to parametric
excitation is then studied using Floquet theory, which is the
infinite-dimensional analogue of the equivalent analysis for the
Mathieu equation. The results of double-scale asymptotics and numerics
agree and lead to an upside down stability region, albeit one which is
punctured by thin Arnold tongues corresponding to higher-order
resonances.

The asymptotic analysis is extended to include geometric
nonlinearities and viscous damping. The stability of various
bifurcating states at the simplest resonances is calculated. Emphasis
is placed on the distinction between planar and circular motions which
bifurcate into different symmetry subgroups of the problem. A
qualitative agreement is found with the experimental data on the
observed stability boundaries being caused by an interaction between a
fundamental and sub-harmonic instability.
Back to List of Speakers

On solutions of the self-contact problem for elastic rods and applications
to DNA supercoiling

Bernard D. Coleman, Rutgers University, Department of Mechanics and
Materials Science, Piscataway, NJ 08854-8058, USA

A plasmid is a DNA molecule in which the two strands of the Watson-Crick
double helix (duplex) structure are intertwined closed curves.  There are
circumstances under which it is useful to model a segment of DNA as an
inextensible, homogeneous, naturally straight, elastic rod of circular
cross-section. (This can be called the idealized rod model for DNA.)
Research of David Swigon, Irwin Tobias, and the speaker has yielded exact
analytical solutions of Kirchhoff's equations of mechanical equilibrium for
knotted and unknotted plasmids, with the effects of impenetrability and
self-contact forces taken into account.  Such explicit solutions are now
available for cases in which self contact occurs both at isolated points
and along curves. To be presented in this talk are examples of
configurations and results about the stability of equilibrium
configurations. The emphasis will be laid on bifurcation diagrams in which
the bifurcation parameter is the excess link, a topological parameter that
is closely related to the Gauss linking number for the two DNA strands.
General rules will be presented concerning (i) the relation of the symmetry
of equilibrium solutions to their location in bifurcation diagrams and (ii)
the graph-theory structure of the set of points of self-contact.  A
discussion will be given of the way in which the analytical solutions may
be employed to calculate bounds on the activation energy for transitions
between distinguishable metastable equilibrium configurations with the same
topological parameters.

Discontinuous Piecewise Polynomial Collocation Methods for Elliptic PDEs

Eusebius Doedel
Applied Math 217-50
California Institute of Technology
(on leave from Concordia University, Montreal)

Piecewise polynomial collocation methods, as implemented in software
such as COLSYS and AUTO, have been useful in studying the behavior
of dynamical systems represented by ODEs. For example, the behavior
and bifurcations of periodic solutions as a parameter varies can be
tracked very effectively by using collocation coupled to numerical
continuation and bifurcation techniques. Adaptive mesh selection allows
the computation of "difficult" orbits, e.g., near-homoclinic periodic
orbits. Although collocation methods have been extended to PDEs, their
use in studying bifurcation phenomena in such equations has been. In this
talk I introduce a class of high-order accurate, globally
discontinous piecewise polynomial collocation methods for solving nonlinear
elliptic PDEs, using adaptive triangulations of the problem domain.
The nested dissection technique used in solving the linearized systems
that arise in Newton's method will be briefly described. Numerical
results that illustrate the effectiveness of the methods will be given.
It is also indicated how problems with certain symmetries and invariances
can be treated.limited.
Back to List of Speakers

Ghost solutions in BVPs

G. Domokos
Technical University of Budapest
H-1521 Budapest, Hungary

It is widely known that discretization can yield
spurious solutions not only in Initial Value Problems (IVPs)
but also in Boundary Value Problems (BVPs). However,
it is still believed by many engineers that "sufficiently
smooth/continuous" solutions of discretized  models
are valid approximations of the solutions in the
continuous model.

This talk addresses the above question; the goal
is to show that there exist arbitrarily smooth/continuous
("ghost") solutions of the discretized model which are bounded away
from any of the continuous solutions. We will
illutrate this  on  Euler's elastica problem
and indicate the class of problems in which
similar phenomena can be expected. We also
discuss the physical relevance of ghosts.

Special shapes of bent and twisted rods
Zsolt Gaspar and Robert Nemeth
Budapest University of Technology and Economics
Dept. of Structural Mechanics
H-1521 Budapest Mûegyetem rkp. 3. Hungary

An inextensible, long, straight elastic rod of circular cross-
section is considered. First it is bent into a torus form, then a
relative rotation between the end cross sections will be produced. Beyond
the critical value of this relative rotation the axis of the bar will
be a spatial curve, and there will be self-contact points.
We are dealing with the special interval of the relative rotation
when the deformed rod has three symmetry axes, and there is also an
interval of the contact points. For these cases the differential
equations will be derived, the shape and the stresses of the rod will
be showed.

Homoclinic collapse in the statics and dynamics of biological filaments
by A.Goriely, Dept. of Mathematics, University of Arizona.

In this talk, I will consider models for long thin intrinsically curved
filaments. These models are used  in a variety of biological systems
ranging from bacterial flagella to climbing plants.  As tension is
decreased or intrinsic curvature is increased, these filaments bifurcate
to solutions connecting asymptotically helices with opposite handedness.
The statics of these heteroslinic structures is studied within the
framework of the Kirchhoff theory with linear and nonlinear elasticity.
A center manifold reduction and a normal form transformation for a
triple zero eigenvalue reduce the dynamics to a third order reversible
dynamical system. The analysis of this reduced system reveals that the
heteroclinic connection representing the physical solution results from
the collapse of  pairs of symmetric homoclinic orbits.
The dynamics of moving fronts and the possibilty of multi-heteroclinic orbits
will  be discussed. Applications to realistic biological systems such as the
flagellar bistability in Salmonella and the motion of spirochetes will also
be considered.

Nonlinear dynamics and stability of spherical membranes

Oded Gottlieb
Faculty of Mechanical Engineering
Technion-Israel Institute of Technology
Haifa, Israel.

This paper focuses on the nonlinear dynamic inflation process of spherical
membranes. We consider isotropic hyperelastic constitutive laws portrayed
by a classical Mooney-Rivlin material and by a family of compressible
Blatz-Ko materials where the effect of material compressibility is
characterized by the material Poisson function.

A bifurcation analyses of the integrable problem reveals conditions
controlling local stability of periodic solutions and existence of
limiting homoclinic solutions. Perturbation of the integrable structure
with (weak) dissipation and a time dependent inflation process renders the
problem nonintegrable. Approximate conditions for aperiodic dynamics are
determined using Melnikov's method complemented by numerical simulations.

The results shed light on the nonlinear (initial) inflation dynamics of
spherical membranes and reveal the influence of several controlling
parameters governing nonlinear compressible and incompressible
viscoelastic materials utilized in large (e.g. space) and small (e.g.
medical applications) structures and biological tissue materials.

Coherent structures and mixing in two-dimensional turbulence

G. Haller, Division of Applied Mathematics, Brown University

It is well known that in two-dimensional turbulent fluid flows
coherent structures tend to emerge. These structures have primarily
been described based on instantenous Eulerian quantities, such as
energy, vorticity, or enstrophy, all giving different answers for
the exact location and properties of coherent structures. At the same time,
in many industrial and geophysical flow problems the exact location of
the boundaries of these structures, as well as their impact on mixing, is
of fundamental importance. In this talk we describe new dynamical
systems-based
methods that do detect exact coherent structure boundaries in the Lagrangian
sense.
We show how such boundaries can be extracted from numerical simulations and
experimental data.

Multiple Parameter Continuation

Michael E. Henderson
IBM Research
P.O. Box 218, Yorktown Heights,   NY 10598

A new continuation method for computing implicitly defined manifolds of
arbitrary dimension will be described.
The method is based on a representation of the manifold as the projection
of a set of overlapping spherical balls,
with new points being added by choosing a point on the boundary of the
collection of balls. This will be shown
to be equivalent to  computing the vertices of a certain set of convex
polyhedra, which form a polyhedral decomposition
of the manifold, and which are a weighted Voronoi diagram for the centers
of the balls. This decomposition
can be used to generate a triangulation of the manifold similar to the
Delaunay triangulation.
Several applications will be described,  including the computation of a
manifold of periodic motions of a pair
of coupled pendula. The representation may also be used for manifolds
defined in other ways, provided that a method
can be found to project from the tangent space onto the manifold. Methods
for invariant manifolds of mappings and flows
will be presented.
Back to List of Speakers

Hidden Symmetry of Global Solutions in Twisted Elastic Rings
Tim Healey, Gabor Domokos
Cornell University
Department of Theoretical and Applied Mechanics
Ithaca, NY 14853-1503 USA

We investigate global equilibria of twisted, isotropic
elastic rings. The high degree  of symmetry in the problem
leads to non-isolated solutions. We prove that
{\em all} solutions are flip-symmetric, and from this
we can globally isolate all solution branches. We
derive  possible other choices for boundary conditions
systematically and show that other conditions necessarily
lead to {\em spurious} solutions. We show global computations
performed by the Parallel Simplex Algorithm.
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Unconstrained Euler buckling in a potential field.
P. Holmes, G. Domokos and G. Hek

We consider elastic buckling of an inextensible rod with free ends,
confined to the plane, and in the presence of distributed body forces
derived from a potential. We formulate the geometrically nonlinear
(Euler) problem with non-zero preferred curvature, and show that it may
be written as a three-degree-of-freedom Hamiltonian system. We focus on
the special case of an initially straight rod subject to body forces
derived from a quadratic potential uniform in one direction; in this
case the system reduces to two degrees of freedom. We find two classes
of trivial (straight) solutions and study the primary non-trivial
branches bifurcating from one of these classes, as a load parameter, or
the rod's length, increases. We show that the primary branches may be
followed to large loads (lengths) and that segments derived from
primary solutions may be concatenated to create secondary solutions,
including closed loops, implying the existence of disconnected
branches. At large loads all finite energy solutions approach
homoclinic and heteroclinic orbits to the other class of straight
states, and we prove the existence of an infinite set of such
spatially chaotic' solutions, corresponding to arbitrary
concatenations of simple' homoclinic and heteroclinic orbits. We
illustrate our results with numerically computed equilibria and global
bifurcation diagrams.
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Maxwell Lower Bounds for Bifurcation at Infinite Load
G.W. Hunt
Department of Mechanical Engineering and Centre for Nonlinear Mechanics
University of Bath, UK

There are a number of  problems of structural stability, one-way buckling of
(heavy) railway tracks and pipelines from impenetrable beds for example, or
the well-known failure mechanism for fibrous and layered media known as
kink-banding, which carry quite pathological stability characteristics.
A realistic modelling process leads in each case to a symmetrical
(pre-buckling) equilibrium state that is apparently a local energy minimum
for all values of applied compressive load. Not only does this suggest,
erroneously, that the system can maintain stability up to infinite load, but
the usual practices of linearization can now play no useful purpose. Such
systems are clearly sensitive to external disturbances and imperfections, but
assessing the level of this sensitivity is difficult.

Instabilities of this kind are often associated with a form of localization in
which one part of a structure unloads elastically into a localizing region.
One measure of sensitivity then lies in the ratio of the lengths of the
energy per unit length, averaged over the structure as a whole, necessary to
trigger the instability. Such considerations lead naturally to lower bound
descriptions for critical loads or critical applied displacements in terms
of the Maxwell criterion, where stability rests only with global, as opposed
to local, energy minima. The paper will review the accuracy of  the criterion
for simple one and two degree of freedom systems, modelled as dynamical systems
and including effects of stochastic noise.
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Fractality, chaos, and reactions in imperfectly mixed
open hydrodynamical flows

György Károlyi, Tamás Tél
TU Budapest   , Eötvös University

Recent developments in the field of chaotic advection in
hydrodynamical or environmental flows encourage us to revisit the
chemical and biological dynamics of active tracers in open aquatic
systems.
Passively advected tracers in open hydrodynamical flows
are known to trace out permanent fractal patterns.
If these advected tracers undergo certain chemical or biological
interactions, the activity takes place on a fractal set.
This observation leads to a novel form of a reaction
equation containing essential chaos parameters.
In biological respects,
while in homogeneous, well mixed environments only
one species could survive a competition for a common limiting resource,
coexistence of competitors is typical in our hydrodynamical system.

Curvature and parallel transport in averaging

Mark Levi

Stabilization of the inverted pendulum by vibration of its
suspension point is a well known and surprising phenomenon. A deeper study
of this effect leads to a new observation: there is some hidden geometry
geometry behind various mechanical phenomena associated with high-frequency
vibrations. One of these is a relationship of normal form (averaging theory)
and parallel transport.

Symmetry Breaking and Stability in DNA mini-circles

Abstract

DNA mini-circles are formed when the double helix of a short DNA
molecule bonds with its own tail to form a closed loop. Mini-circles of
200 or so base-pairs are a very convenient experimental motif for
chemists. In the first part of this talk I will describe some old work
in which my collaborators and I used computation of the bifurcation
diagram for an idealized, highly symmetric, elastic rod model, combined
with symmetry breaking techniques, and stability analysis to model
various experimental data for mini-circles. The computations and
analysis make extensive use of the appropriate Hamiltonian form of the
equilibrium conditions and a new theory of isoperimetric conjugate
points for constrained variational principles. In the second part of the
talk I will describe more recent work that addresses the issue of
identifying perturbations for which the symmetry breaking is not
generic. This analysis explains recent Monte Carlo simulations of
Katritch and Vologodski in which multi-peaked Link distributions have
been observed, and offers the promise of designing particular base-pair
sequences that should exhibit atypical behaviour in mini-circle
experiments.

Nonstandard Reduction of Noisy Mechanical Systems

N. Sri Namachchivaya and Richard Sowers
Department of Aeronautical and Astronautical Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA

The purpose of this paper is to formulate and develop
methods to analyze certain behaviors of {\em mechanical systems}.
Small random excitations affect many mechanical systems, and
the presence of random vibrations can lead to a host of undesirable
effects: reduction of the dynamic strength via fatigue crack growth
or other modes of failure, stochastic instabilities, etc.
An understanding of their dynamics necessitates a study of the
complex interactions between {\em noise},
{\em symmetries}, and {\em nonlinearities}.
Our approach will consist of the application of some recent
abstract theories of stochastic dimensional reduction to some relevant
mechanical models.

Our main analytical tool is a certain method of {\em dimensional reduction}
of nonlinear systems with symmetries and small noise.
As the noise becomes asymptotically small, one can exploit symmetries and a
separation of scales to use well-known methods (viz. stochastic averaging) to
find an appropriate lower-dimensional description of the system.
The interest of this paper is when classical methods fail because the
original Hamiltonian system has bifurcations, which give rise
to singularities in the lower-dimensional description.  Such singularities
complicate the analysis of standard design-related statistical measures of
response and stability (e.g., mean exit times and stationary measures); our
methods are uniquely adapted to study exactly such problems.

The focus of this work is the development
of general techniques of stochastic averaging of randomly-perturbed
four-dimensional integrable Hamiltonian systems.
The integrable system here has certain nontrivial (yet generic) types
of fixed points.
Stochastic averaging makes use of these integrable structures to
identify a reduced diffusive model on a space which
encodes the structure of the fixed points and can have dimensional
singularities.
At these singularities, {\em glueing conditions} will be derived,
these glueing conditions completing the specification of the
dynamics of the reduced model.
Qualitative dependence of statistical measures of the reduced system upon
various coefficients can be studied by extensions of known techniques
such as stochastic bifurcation theory.

The symmetry of some of the simple non-holonomic examples precludes assymtotic stability.

Andy Ruina
Cornell University
Department of Theoretical and Applied Mechanics
ruina@cornell.edu

Conservative non-holonomic mechanical systems are not Hamiltonian
in general and thus are not constrained to obey the theorems
applicable to Hamiltonion systems.  In particular Hamiltonian
systems cannot have assymptotic stability and non-holonomic
systems can.

But this fact has been often missed by the non-experts.  Perhaps
this is because many of the common examples of non-holonomic
systems have symmetries that preclude stability.  In particular,
a system that has a reflection symmetry that correponds to
moving backwards cannot be so stable.  Such systems include
a rolling hoop, ball, or symmetric top while rolling on
a flat plane, a surface of revolution or a prismatic channel.
The Chapligin sleigh is also an example of such a system if
the center of mass is on the line normal to the skate.

Non-holonomic systems that violate this symmetry, and thus
better reveal the stability possibilities of non-holonomic
systems, include a general Chaplygin Sleigh, Monte-Hubbards
skate-board, and a bicycle.

KEY WORDS: non-holonomic, symmetry, stability
Back to List of Speakers

Dynamic Stability and Performance of Systems of Nonlinear Vibration Absorbers

Steven W. Shaw
Department of Mechanical Engineering
Michigan State University
East Lansing, MI  48824-1226 USA

Centrifugal pendulum vibration absorbers are devices that are used to
attenuate torsional vibrations in rotating machinery.  They consist of a
set of masses that are mechanically suspended from a rotating shaft and are
free to move relative to it in a plane perpendicular to the axis of
rotation.  The paths along which these masses move are designed so that
their motions dynamically cancel the effects of external fluctuating
torques applied to the shaft.  Some key features of these passive absorbers
include the following: they are linearly tuned to match the frequency of
the applied torque over all operating speeds; they operate most effectively
when lightly damped; one can design the paths of the absorber masses to
achieve desired performance; the absorbers are typically identical and
coupled to one another indirectly through the (relatively large) inertia of
the rotor.  It is shown that the equations of motion of such a system
possess SN symmetry, and the fully symmetric response is desired in
applications.  Scaling of the system parameters allows one to express the
equations of motion in a form that is amenable to asymptotic analysis.
Application of the method of averaging shows that as system parameters are
varied, the desired response can exhibit bifurcations that maintain the
full symmetry as well as those that result in a lower order of symmetry.
It is also shown that imperfections among the absorbers can lead to
localized responses in which a small subset of absorbers undergo large
amplitude motions.  The effects of system and excitation parameters on
these undesirable behaviors are examined, with the goal of minimizing
torsional vibration levels of the rotor.  It is shown that by proper
selection of the absorber paths, both the instabilities and the
localization can be avoided, thereby providing good performance over a wide
range of operating conditions.
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Stability and bifurcations in dynamical systems
with time delay and parametric excitation

G. Stépán and T. Insperger

Department of Applied Mechanics
Budapest University of Technology and Economics
Budapest  H-1521  Hungary
e-mail: stepan@mm.bme.hu, inspi@m.bme.hu

The stability analysis of linear autonomous ordinary differential equations can based on their characteristic roots. The Routh–Hurwitz criterion (back to 1875) supports to check whether all the characteristic roots are located in the left half of the complex plane. In case of single degree of freedom mechanical systems, the necessary and sufficient condition of asymptotic stability is the presence of positive stiffness and damping.

When the parameters of the ordinary differential equation depend on the time periodically, that is in case of parametric excitation, the linearization of the system still describes the stability of the trivial solution. However, the so-called Floquet theory (back to 1883) does not provide closed form results even for the simplest single degree of freedom mechanical system with time periodic stiffness and zero damping. The stability chart of the corresponding Mathieu equation was constructed via power series by Incze and Strutt in the last century.

When time delay appears in the autonomous systems, the differential equation is not ordinary anymore. It belongs to the theory of functional differential equations which involves differential-difference (in other words delay-differential) equations. The linear stability criterion is based on characteristic roots again. This time, infinitely many characteristic roots are to be checked whether they have negative real parts. In some cases, closed form results can be given. For example, the undamped, single degree of freedom mechanical system with delay in its stiffness was fully analyzed by Hsu and Bhatt in 1966.

In this paper, the combination of the two cases above, that is, the stability investigation of the time-periodic delay-differential equation

is presented. As a mechanical application, the stability and bifurcation analysis of the milling process is considered.

Transition from spherical circle packing to covering

Tibor Tarnai
Budapest University of Technology and Economics
Department of Structural Mechanics
Budapest, Muegyetem rkp. 3., H-1521 Hungary
email: tarnai@ep-mech.me.bme.hu

How must n equal circles (spherical caps) of given angular
radius r be arranged on the surface of a sphere so that the
area covered by the circles will be as large as possible?
In this paper, conjectural solutions of this  problem for
n=5,7,8,9,10,11,13 are given when r varies from the maximum
that this variation of r provides a transition from the
maximum circle packing to the minimum circle covering on
a sphere. To a circle arrangement, there is associated a contact
polyhedron whose vertices are the centres of the spherical
circles and whose edges appear between the centres of
two circles having at least one point in common. During this
configuration transition, the symmetry and the number of edges of
the contact polyhedra change. The change is in general continuous,
but sometimes at certain points, discontinuous as in the case
n=7,11,13. For n=2,3,4,6,12 we obtain the regular polyhedra
as solutions for any r, that means that the configurations
are constant during the packing to covering transition. The case
n=13 introduces several features not present for lower numbers
of circles: large numbers  of symmetry and edge transitions in contact
polyhedra, crowding of transitions within small ranges of r and
three separate intervals where the position of one circle has
a limited freedom to "rattle" without changing the fractional
covering of the sphere. The relation of the results of this
problem to the extremal configurations of n mutually repulsive equal
point charges on the surface of a sphere, and to chemical isomerization
processes will be discussed. The role of symmetry will be also analyzed.

Application of Normal Form theory in an optimal controlled production model
Alois Steindl, Gustav Feichtinger
Institute of Mechanics II and Institute of Econometrics, Operations Research
and System Theory, Vienna University of Technology

We investigate the creation of periodic solutions in a continuous production
model with non-convex production cost and constant demand.
For vanishing interest rate a Hamiltonian Hopf bifurcation'' occurs,
state. Normal Form Theory introduces a mathematical phase shift
the periodic solutions. If the interest rate is assumed small, but
different from zero, almost all periodic
solutions vanish; only an hyperbolic unique branch of periodic solutions is
obtained.
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Elastic stability of knotted DNA plasmids

David Swigon, Rutgers University,  Piscataway, New Jersey 08854-8058 USA

In research of Bernard Coleman, Irwin Tobias, and the speaker on the theory
of the idealized elastic rod model for DNA exact analytical representations
have been obtained for equilibrium configurations of DNA segments showing
both isolated points and intervals of self-contact.  Criteria have been
derived for determining whether an equilibrium configuration is stable in
the sense that it gives a strict local minimum to elastic energy.  Here
this theory will be applied to knotted DNA plasmids with emphasis placed on
the dependence on excess link of minimum energy configurations of DNA
trefoil knots.  Among the results to be presented is the following: the
minimum bending energy configuration a (2,q) torus knot shows self-contact
along a circle. This result remains valid even when the integer q is even,
i.e., when the "knot" reduces to a link of 2 unknots.

The chaotic instability of a slowly spinning asymmetric top

J.M.T. Thompson and
Gert Van der Heijden

Centre for Nonlinear Dynamics,
University College London

We consider a top that is spinning in an upright state at an angular velocity
that is less than the critical value predicted by classical bifurcation
theory. The top is then in a state of unstable relative equilibrium, and
given an infinitesimal perturbation it will fall away from the upright
configuration. In the absence of dissipation, it will however return, again
and again, very close to the upright configuration. In technical terms, its
motion will lie very close to a homoclinic orbit that, as time passes from
minus infinity to plus infinity, departs from the upright state and returns
to it. For a top spinning about an axis of symmetry, the repeated returns
will be regular and predictable. However, for a non-symmetric top there are
an infinite number of different homoclinic orbits, implying that the repeated
returns are chaotic and unpredictable. We use homoclinic path-following
techniques to explore the structure of these homoclinic orbits, and
illustrate the physical motions involved.

Stability Analysis of Relative Equilibria of Tethered Satellite Systems
Martin Krupa, Martin Schagerl, Alois Steindl, and
Hans Troger,
Vienna University of Technology, A-1040 Vienna, Austria

After shortly explaining the concept of tethered satellite
systems and giving some practical applications of this concept, we present two
different mathematical models of such systems.
The first is a finite dimensional model consisting of a rigid
rod and two point masses. The second more complicated model is
infinite dimensional and
consists of two  masses, which are connected by a massive perfectly
flexible and extensible string.

For both systems relative equilibrium positions for a circular motion around
the Earth exist,  due to the $O(3)$-symmetry of the system. Their stability
will be investigated by means of the reduced energy momentum method
since by symmetry, besides energy, also the angular momentum
is conserved.

For the continuous system we distinguish between two different
types of relative equilibria.
One are the so-called trivial relative equilibria, for example,
The second are  nontrivial (wavy) tether configurations which require active
control of the motion of at least one of the satellites.
For the solution of the nontrivial
relative equilibrium problem we formulate a bifurcation problem, where
we vary the position of one of the satellites relative to the other.

Numerical methods have to be applied, first, in the calculation of
the nontrivial relative equilibria by Finite Differences and,
further, in the evaluation
of the stability condition following from the reduced energy
momentum method.
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Passive Nonlinear Energy Pumping due to Symmetry Breaking in Coupled
Oscillators

Alexander F. Vakakis
Mechanical and Industrial Engineering
Theoretical and Applied Mechanics
University of Illinois

Oleg Gendelman
Institute of Chemical Physics

Igor Rozhkov
Theoretical and Applied Mechanics
University of Illinois

We present numerical evidence of nonlinear energy pumping from a linear system
of coupled oscillators to a symmetry-breaking essentially nonlinear element. By
energy pumping we mean the one-way, irreversible, transient \221channeling\222 of
externally induced energy from the linear to the nonlinear part of the system.
This phenomenon occurs only when the energy is above a critical level. To
analytically study  energy pumping we first restrict our attention to a two
degree-of-freedom system, transform the equations of motion into action-angle
form, and apply two-frequency averaging. We show that energy pumping is due to
resonance capture on a 1:1 resonance manifold of the system, and perform a
perturbation analysis in an O() neighborhood of this manifold to study the
time-varying attracting region responsible for this phenomenon. An alternative
analytical approach based on the assumption of 1:1 internal resonance in the
fast dynamics of the system, complexifies and averages the fast dynamics. This
leads to satisfactory analytical approximations of the nonlinear transient
responses of the system in the energy pumping regime.

In the next step of our step we consider energy pumping in a periodic
semi-infinite linear periodic system of coupled oscillators, attached to an
essentially nonlinear oscillator. We study in detail the interaction of the
linear chain with the essentially nonlinear element in order to identify the
harmonic components of the interaction that are responsible for the energy
pumping phenomenon. An analytical technique is then employed to analytically
study the transient response of the system in the energy pumping regime. The
practical applications of the nonlinear energy pumping phenomenon to the
active/passive control of large-scale mechanical systems are discussed.
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