Complex and chaotic response of a non-linear oscillator with an isothermal gas spring

In this paper we study the dynamics of a non-linear one-degree-of-freedom system subjected to an external harmonic excitation, representing a simplified model for the synchronous hydraulic oscillations that can occur in the draft tube of Francis turbines at partial loads. The application of different typical numerical techniques has shown the existence of multiple coexisting periodic solutions, and the non-periodic bounded solutions which exhibit deterministic chaotic behaviour. The relevant strange attractor has been defined and the loss of memory associated with an exponential divergence in time of close initial conditions resulting in chaotic dynamics have been found and measured. A partial classification of qualitatively different dynamical behaviours for the system has been outlined in the control parameter space.

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Sommario In questo articolo viene studiata la dinamica di un sistema non-lineare ad un singolo grado di liberta' soggetto ad una forzante armonica esterna, rappresentante un modello semplificato per le oscillazioni idrauliche sincrone che hanno luogo nei diffusori delle turbine tipo Francis a carico parziale. Applicando differenti tecniche numeriche, viene mostrata l'esistenza di soluzioni periodiche multiple, oltre che soluzioni non-periodiche limitate con tipico comportamento caotico deterministico. L'attrattore strano corrispondente e' stato definito e caratterizzato: la perdita di memoria associata alla divergenza esponenziale di orbite inizialmente vicine, tipica della dinamica caotica, e' stata individuata e calcolata numericamente. Una prima parziale classificazione dei vari comportamenti dinamici per il sistema viene evidenziata attraverso la rappresentazione nello spazio parametrico.

     

Bifurcations in a parametrically excited non-linear oscillator

A non-linear parametrically excited oscillator, that includes van der Pol as well as Duffing type non-linearities, is studied for its small non-linear motions using the method of averaging. The averaged equations, which form a dynamical system on the plane and depend on the linear damping and the detuning, are analyzed for their constant and periodic solutions. Bendixon's criterion is used to deduce the existence and the non-existence of limit cycle solutions for various values of the parameters. Then, using local bifurcation theory for “saddle-node”, pitchfork and “Hopf” bifurcations and some results from one and two parameter unfoldings of degenerate singularities, a partial bifurcation set is constructed. Since constant and periodic solutions of the averaged system correspond, respectively, to the periodic solutions and almost periodic or amplitude modulated motions of the original oscillator, the bifurcation set indicates some ways in which periodic solutions can become “entrained” or can break the entrainment for almost periodic oscillations.     

Optimal control of a non-linear noisy sine wave oscillator

This paper deals with the optimal control of a random non-linear sine wave oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a generalized type of a Poisson process. Sufficient conditions on the optimal controls are derived. These conditions are given by a set of two coupled non-linear partial integro-differential equations. A numerical procedure for the solution of these equations is suggested and its efficiency and applicability are demonstrated with examples.     

General perturbational solution of a harmonically forced non-linear oscillator equation

An extension of a general perturbational method in the theory of harmonically forced non-linear oscillations has been presented, in which the interplay of two system parameters appears. The method clearly exhibits the entrainment of subharmonic, super-harmonic and other harmonic responses for various interplays of the system parameters.The mathematical insufficiency of the method to predict the behavior of the system in a limiting case of parameter interplay, which is usually attributed to the perturbational method in the Poincaré sense, has been recognized and the method for its removal suggested.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Analysis of a weakly non-linear autonomous oscillator by means of the field method

In this paper the field method is extended to the study of oscillatory systems with two degrees of freedom and weak quadratic non-linearity. The basic field method concept is combined with the technique of multiple time scales and the solution for both non-resonant case and the case out of the first resonance are found. The qualitative analysis of behavior in the resonant area is done by determining the values of the “adelphic” integral.     

Behaviour of an oscillator with even non-linear damping

In this short note we prove two theorems on the behaviour of a single degree of freedom oscillator with linear stiffness and even non-linear damping terms.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Generalized lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator

Generalized Lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator (GHRNO) are considered. The constraint which the variable-coefficient functions must satisfy for the GHRNO to have infinite-dimensional symmetry algebras is derived. The integral of motion for the GHRNO under this condition can be read off from the symmetry vector fields. The structure of the symmetry algebras is also presented.     

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

The use of non-linear energy sink to passively control vibrations of a non-linear main structure under the effect of bi-frequency harmonic excitation is addressed here. The excitation is assumed to induce both 1:1 and 1:3 resonance, and the response of the system is studied after using the Multiple Scale/Harmonic Balance Method, applied to obtain amplitude modulation equations in the slow time scale. The efficiency of the non-linear energy sink to reduce or suppress vibrations of the main structure is finally discussed.     

Characterization of the non-linear response of asphalt mixtures using a torsional rheometer

There is sparse documentation of the large normal stresses that develop in asphalt mixtures subjected to shear, a typical characteristic of non-linear materials. In this study, we continue our initial investigations of using a torsional rheometer for measuring the normal stresses developed in asphalt mixtures when subjected to torsion. Samples of sand-asphalt mixture are subjected to rotation rates as low as one revolution in ninety minutes. This study provides further clear evidence of significant development of normal stresses due to shearing and emphasizes the need for the development of models that can describe such a phenomenon.     

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

Harmonic non-linear response of beck's column to a lateral excitation

The non-linear response of a column with a follower force (Beck's column) subjected to a distributed periodic lateral excitation, or to a support excitation, is determined. An analytical solution for the response amplitude in terms of the loading and system parameters is obtained by a perturbation analysis of the differential equations of motion. Non-linear inertia and non-linear curvature terms are taken into account in the formulation of the differential equations.     

Response statistic of strongly non-linear oscillator to combined deterministic and random excitation

The principal resonance of a van der Pol-Duffing oscillator to the combined excitation of a deterministic harmonic component and a random component has been investigated. By introducing a new expansion parameter , the method of multiple scales is adapted for the strongly non-linear system. Then the method of multiple scales is used to determine the equations of modulation of response amplitude and phase. The behavior and the stability of steady-state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady-state solutions. Random jump may be observed under some conditions. The results obtained in the paper are adapted for a strongly non-linear oscillator that complement previous results in the literature for the weakly non-linear case.     

On the resonance response of an asymmetric Duffing oscillator

The primary resonance response of an asymmetric Duffing oscillator with no linear stiffness term and with hardening characteristic is investigated in this paper. An approximate solution corresponding to the steady-state response is sought by applying the harmonic balance method. Its stability is also studied. It is found that different shapes of frequency-response curves can exist. Multiple-valued solutions, indicating the occurrence of jump phenomena, are observed analytically and confirmed numerically. The influence of the system parameters on the primary resonance response is also examined.     

On the response of a preloaded mechanical oscillator with a clearance: Period-doubling and chaos

The dynamic behavior of a harmonically excited, preloaded mechanical oscillator with dead-zone nonlinearity is described quantiatively. The governing strongly nonlinear differential equation is solved numerically. Damping coefficient-force ratio maps for two different values of the excitation frequency have been formed and the boundaries of the regions of different motion types are determined. The results have been compared with the results of the forced Duffing's equation available in the literature in order to identify the differences between cubic and dead-zone nonlinearities. Period-doubling bifurcations, which take place with a change of any of the system parameters, have been found to be the most common route to chaos. Such bifurcations follow the scaling rule of Feigenbaum. b half length of the clearance.     

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.