3,330 research outputs found
Visualizing the decisions of the first boundary value problem for of the autonomous duffing equation with the use of the methods of the theory of branch
Mathematicalmodelsareusedtodescribephysicalprocessesinthetheoryofoscillations. The Duffing oscillator or an oscillator with a cubic nonlinearity is one of the most common models of the theory of oscillations. The Duffing Oscillator is the simplest one-dimensional nonlinear system. A feature of the Duffing oscillator is the possibility of obtaining chaotic dynamics
Stability analysis of a noise-induced Hopf bifurcation
We study analytically and numerically the noise-induced transition between an
absorbing and an oscillatory state in a Duffing oscillator subject to
multiplicative, Gaussian white noise. We show in a non-perturbative manner that
a stochastic bifurcation occurs when the Lyapunov exponent of the linearised
system becomes positive. We deduce from a simple formula for the Lyapunov
exponent the phase diagram of the stochastic Duffing oscillator. The behaviour
of physical observables, such as the oscillator's mean energy, is studied both
close to and far from the bifurcation.Comment: 10 pages, 8 figure
Sparse Identification of Nonlinear Duffing Oscillator From Measurement Data
In this paper we aim to apply an adaptation of the recently developed
technique of sparse identification of nonlinear dynamical systems on a Duffing
experimental setup with cubic feedback of the output. The Duffing oscillator
described by nonlinear differential equation which demonstrates chaotic
behavior and bifurcations, has received considerable attention in recent years
as it arises in many real-world engineering applications. Therefore its
identification is of interest for numerous practical problems. To adopt the
existing identification method to this application, the optimization process
which identifies the most important terms of the model has been modified. In
addition, the impact of changing the amount of regularization parameter on the
mean square error of the fit has been studied. Selection of the true model is
done via balancing complexity and accuracy using Pareto front analysis. This
study provides considerable insight into the employment of sparse
identification method on the real-world setups and the results show that the
developed algorithm is capable of finding the true nonlinear model of the
considered application including a nonlinear friction term.Comment: 6 pages, 8 figures, conference pape
On artifact solutions of semi-analytic methods in nonlinear dynamics
Nonlinear dynamics is a topic of permanent interest in mechanics since decades. The authors have recently published some results on a very classical topic, the dynamics of a softening Duffing oscillator under harmonic excitation focusing especially on low-frequency excitation (von Wagner in Arch Appl Mech 86(8):1383–1390, 2016). In this paper, it was shown that classical tools like harmonic balance and perturbation analysis may produce artificial solutions when applied without extra carefulness with respect to parameter ranges in the case of perturbation analysis or prior knowledge about the type of solution in case of harmonic balance. In the present paper these results are shortly summarized as they give the starting point for the additional investigations described herein. First, the method of slowly changing phase and amplitude is reviewed with respect to its capability of determining asymptotic stability of stationary solutions. It is shown that this method can also produce artifact results when applied without extra carefulness. As next example an extended Duffing oscillator is investigated, which shows, if harmonic balance is applied, “islands” of solutions. Using the error criterion in harmonic balance as described in von Wagner (2016) again artifact solutions can be identified
Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis
By using the Duffing oscillator as a case study, this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the Nonlinear Output Frequency Response Functions (NOFRFs) are one of the solutions obtained using the Harmonic Balance Method (HBM). A comparison of the performances of the two methods shows that the HBM can capture the well-known jump phenomenon, but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components. Although the NOFRFs cannot capture the jump phenomenon, the method has few computational restrictions. For the nonlinear damping systems, the NOFRFs can give better predictions for all the harmonic components in the system response than the HBM even when the damping system is strongly nonlinear
Evolutionary-based sparse regression for the experimental identification of duffing oscillator
In this paper, an evolutionary-based sparse regression algorithm is proposed and applied onto experimental data collected from a Duffing oscillator setup and numerical simulation data. Our purpose is to identify the Coulomb friction terms as part of the ordinary differential equation of the system. Correct identification of this nonlinear system using sparse identification is hugely dependent on selecting the correct form of nonlinearity included in the function library. Consequently, in this work, the evolutionary-based sparse identification is replacing the need for user knowledge when constructing the library in sparse identification. Constructing the library based on the data-driven evolutionary approach is an effective way to extend the space of nonlinear functions, allowing for the sparse regression to be applied on an extensive space of functions. The results show that the method provides an effective algorithm for the purpose of unveiling the physical nature of the Duffing oscillator. In addition, the robustness of the identification algorithm is investigated for various levels of noise in simulation. The proposed method has possible applications to other nonlinear dynamic systems in mechatronics, robotics, and electronics
Nonlinear Generalization of Den Hartog's Equal-Peak Method
This study addresses the mitigation of a nonlinear resonance of a mechanical
system. In view of the narrow bandwidth of the classical linear tuned vibration
absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber
(NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is
that the mathematical form of its restoring force is tailored according to the
nonlinear restoring force of the primary system. The NLTVA parameters are then
determined using a nonlinear generalization of Den Hartog's equal-peak method.
The mitigation of the resonant vibrations of a Duffing oscillator is considered
to illustrate the proposed developments
High-Intensity Discharge Lamp and Duffing Oscillator - Similarities and Differences
The processes inside the arc tube of high-intensity discharge lamps are
investigated by finite element simulations. The behavior of the gas mixture
inside the arc tube is governed by differential equations describing mass,
energy and charge conservation as well as the Helmholtz equation for the
acoustic pressure and the Navier-Stokes equation for the flow driven by the
buoyancy and the acoustic streaming force. The model is highly nonlinear and
requires a recursion procedure to account for the impact of acoustic streaming
on the temperature and other fields. The investigations reveal the presence of
a hysteresis and the corresponding jump phenomenon, quite similar to a Duffing
oscillator. The similarities and, in particular, the differences of the
nonlinear behavior of the high-intensity discharge lamp to that of a Duffing
oscillator are discussed. For large amplitudes the high-intensity discharge
lamp exhibits a stiffening effect in contrast to the Duffing oscillator.Comment: 14 pages, 8 figure
The dynamics of the pendulum suspended on the forced Duffing oscillator
We investigate the dynamics of the pendulum suspended on the forced Duffing
oscillator. The detailed bifurcation analysis in two parameter space (amplitude
and frequency of excitation) which presents both oscillating and rotating
periodic solutions of the pendulum has been performed. We identify the areas
with low number of coexisting attractors in the parameter space as the
coexistence of different attractors has a significant impact on the practical
usage of the proposed system as a tuned mass absorber.Comment: Accepte
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