116 research outputs found
Modelling elastic structures with strong nonlinearities with application to stick-slip friction
An exact transformation method is introduced that reduces the governing
equations of a continuum structure coupled to strong nonlinearities to a low
dimensional equation with memory. The method is general and well suited to
problems with point discontinuities such as friction and impact at point
contact. It is assumed that the structure is composed of two parts: a continuum
but linear structure and finitely many discrete but strong nonlinearites acting
at various contact points of the elastic structure. The localised
nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite
the discontinuities in the model, we demonstrate that contact forces are
Lipschitz continuous in time at the onset of sticking for certain classes of
structures. The general formalism is illustrated for a continuum elastic body
coupled to a Coulomb-like friction model
Non-deterministic dynamics of a mechanical system
A mechanical system is presented exhibiting a non-deterministic singularity,
that is, a point in an otherwise deterministic system where forward time
trajectories become non-unique. A Coulomb friction force applies linear and
angular forces to a wheel mounted on a turntable. In certain configurations the
friction force is not uniquely determined. When the dynamics evolves past the
singularity and the mechanism slips, the future state becomes uncertain up to a
set of possible values. For certain parameters the system repeatedly returns to
the singularity, giving recurrent yet unpredictable behaviour that constitutes
non-deterministic chaotic dynamics. The robustness of the phenomenon is such
that we expect it to persist with more sophisticated friction models,
manifesting as extreme sensitivity to initial conditions, and complex global
dynamics attributable to a local loss of determinism in the limit of
discontinuous friction.Comment: 22 pages, 8 figure
Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the
smoothest invariant manifolds tangent to linear modal subspaces of an
equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the
classic backbone curves sought in experimental nonlinear model identification.
We develop here a methodology to compute analytically both the shape of SSMs
and their corresponding backbone curves from a data-assimilating model fitted
to experimental vibration signals. Using examples of both synthetic and real
experimental data, we demonstrate that this approach reproduces backbone curves
with high accuracy.Comment: 32 pages, 4 figure
Invariant spectral foliations with applications to model order reduction and synthesis
The paper introduces a technique that decomposes the dynamics of a nonlinear
system about an equilibrium into low order components, which then can be used
to reconstruct the full dynamics. This is a nonlinear analogue of linear modal
analysis. The dynamics is decomposed using Invariant Spectral Foliations (ISF),
which is defined as the smoothest invariant foliation about an equilibrium and
hence unique under general conditions. The conjugate dynamics of an ISF can be
used as a reduced order model. An ISF can be fitted to vibration data without
carrying out a model identification first. The theory is illustrated on a
analytic example and on free-vibration data of a clamped-clamped beamComment: New revision in response to reviewers' comment
Characteristic matrices for linear periodic delay differential equations
Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction
for characteristic matrices for systems of linear delay-differential equations
with periodic coefficients. First, we show that matrices constructed in this
way can have a discrete set of poles in the complex plane, which may possibly
obstruct their use when determining the stability of the linear system. Then we
modify and generalize the original construction such that the poles get pushed
into a small neighborhood of the origin of the complex plane.Comment: 17 pages, 1 figur
Data-driven reduced order models using invariant foliations, manifolds and autoencoders
This paper explores how to identify a reduced order model (ROM) from a
physical system. A ROM captures an invariant subset of the observed dynamics.
We find that there are four ways a physical system can be related to a
mathematical model: invariant foliations, invariant manifolds, autoencoders and
equation-free models. Identification of invariant manifolds and equation-free
models require closed-loop manipulation of the system. Invariant foliations and
autoencoders can also use off-line data. Only invariant foliations and
invariant manifolds can identify ROMs, the rest identify complete models.
Therefore, the common case of identifying a ROM from existing data can only be
achieved using invariant foliations.
Finding an invariant foliation requires approximating high-dimensional
functions. For function approximation, we use polynomials with compressed
tensor coefficients, whose complexity increases linearly with increasing
dimensions. An invariant manifold can also be found as the fixed leaf of a
foliation. This only requires us to resolve the foliation in a small
neighbourhood of the invariant manifold, which greatly simplifies the process.
Combining an invariant foliation with the corresponding invariant manifold
provides an accurate ROM. We analyse the ROM in case of a focus type
equilibrium, typical in mechanical systems. The nonlinear coordinate system
defined by the invariant foliation or the invariant manifold distorts
instantaneous frequencies and damping ratios, which we correct. Through
examples we illustrate the calculation of invariant foliations and manifolds,
and at the same time show that Koopman eigenfunctions and autoencoders fail to
capture accurate ROMs under the same conditions.Comment: 48 pages, 16 figures. Update: some bugs were fixed in the numerical
calculation
Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces
In this paper a model reduction technique is introduced for piecewise-smooth
(PWS) vector fields, whose trajectories fall into a Banach space, but the
domain of definition of the vector fields is a non-dense subset of the Banach
space. The vector fields depend on a parameter that can assume different
discrete values in two parts of the phase space and a continuous family of
values on the boundary that separates the two parts of the phase space. In
essence the parameter parametrizes the possible vector fields on the boundary.
The problem is to find one or more values of the parameter so that the solution
of the PWS system on the boundary satisfies certain requirements. In this paper
we require continuous solutions. Motivated by the properties of applications,
we assume that when the parameter is forced to switch between the two discrete
values, trajectories become discontinuous. Discontinuous trajectories exist in
systems whose domain of definition is non-dense. It is shown that under our
assumptions the trajectories of such PWS systems have unique forward-time
continuation when the parameter of the system switches. A finite-dimensional
reduced order model is constructed, which accounts for the discontinuous
trajectories. It is shown that this model retains uniqueness of solutions and
other properties of the original PWS system. The model reduction technique is
illustrated on a nonlinear bowed string model.Comment: 11 figures, 55 pages. Accepted for publication in Journal of
Nonlinear Scienc
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