140,886 research outputs found
Harmonic analysis of oscillators through standard numerical continuation tools
In this paper, we describe a numerical continuation method that enables
harmonic analysis of nonlinear periodic oscillators. This method is formulated
as a boundary value problem that can be readily implemented by resorting to a
standard continuation package - without modification - such as AUTO, which we
used. Our technique works for any kind of oscillator, including electronic,
mechanical and biochemical systems. We provide two case studies. The first
study concerns itself with the autonomous electronic oscillator known as the
Colpitts oscillator, and the second one with a nonlinear damped oscillator, a
non-autonomous mechanical oscillator. As shown in the case studies, the
proposed technique can aid both the analysis and the design of the oscillators,
by following curves for which a certain constraint, related to harmonic
analysis, is fulfilled.Comment: 20 pages, 4 figure
Nonlinear normal modes, modal interactions and isolated resonance curves
The objective of the present study is to explore the connection between the
nonlinear normal modes of an undamped and unforced nonlinear system and the
isolated resonance curves that may appear in the damped response of the forced
system. To this end, an energy balancing technique is used to predict the
amplitude of the harmonic forcing that is necessary to excite a specific
nonlinear normal mode. A cantilever beam with a nonlinear spring at its tip
serves to illustrate the developments. The practical implications of isolated
resonance curves are also discussed by computing the beam response to sine
sweep excitations of increasing amplitudes.Comment: Journal pape
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Identification of Nonlinear Normal Modes of Engineering Structures under Broadband Forcing
The objective of the present paper is to develop a two-step methodology
integrating system identification and numerical continuation for the
experimental extraction of nonlinear normal modes (NNMs) under broadband
forcing. The first step processes acquired input and output data to derive an
experimental state-space model of the structure. The second step converts this
state-space model into a model in modal space from which NNMs are computed
using shooting and pseudo-arclength continuation. The method is demonstrated
using noisy synthetic data simulated on a cantilever beam with a
hardening-softening nonlinearity at its free end.Comment: Journal pape
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
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