Piecewise Linear Oscillator Research Articles

Исследование динамики кусочно-линейного осциллятора с двумя степенями свободы

Исследование динамики кусочно-линейного осциллятора с двумя степенями свободы

Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness

In this paper the existence condition and generation mechanism of the possible bursting phenomenon in a piecewise mechanical system with different time scales are studied. As an example of mechanical systems, a piecewise linear oscillator with parameter perturbation in stiffness and subject to external excitation is examined. The order gaps between the time scales are considered in the model, which are related to the periodic excitation and the changing rates of the variables. The focus-type periodic bursting oscillation with two time scales is presented, and the corresponding generation mechanism is revealed by using slow–fast analysis method. Furthermore, the analytical solution of piecewise linear subsystem as well as the stability condition of the fast subsystem are explored to explain the transition of bursting behaviors coming from the variation of intrinsic parameter and external excitation. The results about bursting phenomenon and its generation mechanism would provide important theoretical basis on the mechanical manufacturing and engineering practice.

A wavelet-based method for the forced vibration analysis of piecewise linear single- and multi-DOF systems with application to cracked beam dynamics

The present investigation outlines a method based on the wavelet transform to analyze the vibration response of discrete piecewise linear oscillators, representative of beams with breathing cracks. The displacement and force variables in the governing differential equation are approximated using Daubechies compactly supported wavelets. An iterative scheme is developed to arrive at the optimum transform coefficients, which are back-transformed to obtain the time-domain response. A time-integration scheme, solving a linear complementarity problem at every time step, is devised to validate the proposed wavelet-based method. Applicability of the proposed solution technique is demonstrated by considering several test cases involving a cracked cantilever beam modeled as a bilinear SDOF system subjected to a harmonic excitation. In particular, the presence of higher-order harmonics, originating from the piecewise linear behavior, is confirmed in all the test cases. Parametric study involving the variations in the crack depth, and crack location is performed to bring out their effect on the relative strengths of higher-order harmonics. Versatility of the method is demonstrated by considering the cases such as mixed-frequency excitation and an MDOF oscillator with multiple bilinear springs. In addition to purporting the wavelet-based method as a viable alternative to analyze the response of piecewise linear oscillators, the proposed method can be easily extended to solve inverse problems unlike the other direct time integration schemes.

A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System

We analyze a four-dimensional slow-fast piecewise linear system consisting of two coupled McKean caricatures of the FitzHugh--Nagumo system. Each oscillator is a continuous slow-fast piecewise linear system with three zones of linearity. The coupling is one-way, that is, one subsystem evolves independently and is forcing the other subsystem. In contrast to the original FitzHugh--Nagumo system, we consider a negative slope of the linear nullcline in both the forcing and the forced system. In the forcing system, this lets us, by just changing one parameter, pass from a system having one equilibrium and a relaxation cycle to a system with three equilibria keeping the relaxation cycle. Thus, we can easily control the changes in the oscillation frequency of the forced system. The case with three equilibria and a linear slow nullcline is a new configuration of the McKean caricature, where the existence of the relaxation cycle was not studied previously. We also consider a negative slope of the $y$-nullcline in ...

Study of a piecewise linear dynamic system with negative and positive stiffness

The present paper mainly focuses on numerical and analytical study of a piecewise linear dynamic oscillator with negative stiffness followed by positive stiffness which has not been studied to date. The dynamic system of interest stems from a previous analytical and experimental research on adaptive negative stiffness for the purpose of seismic protection. Numerical algorithms meant specifically for simulating piecewise smooth (PWS) systems like this nonlinear system are studied. An appropriate combination of negative stiffness and adequate damping can reduce the peak restoring or transmitted force with a slightly larger peak displacement. Essentially, the negative stiffness system in a dynamic system is very beneficial in reducing the amount of force transmitted. The exact solution is derived for free vibration. A modified Lindstedt–Poincaré method (modified L–P method) is adopted to derive approximate periodic solutions for the forced and damped system and its frequency-response curves are obtained through numerical simulation. The modified L–P solution obtained for the forced and damped case is found to agree well with the numerical results. In the piecewise linear dynamic system with initial negative stiffness followed by positive stiffness, it is found that the response remains bounded in a limit cycle. This system behaves similar to a van der Pol oscillator wherein negative damping is followed by positive damping. Presented herein is a special case as defined by the specified parameter ranges; thus, to make it more general future work is needed.

Bi-linear reduced-order models of structures with friction intermittent contacts

The dynamic analysis of structures with localized nonlinearities, such as intermittent contacts of cracked structures, is a computationally demanding task because of the large size of the models involved. Thus, high-resolution finite element models are often reduced using a variety of specialized techniques which exploit spatial coherences in the dynamics. In addition, when a steady-state forced response analysis is performed, direct time integration can be replaced with multi-harmonic balance methods. Recently, a technique based on bi-linear normal modes has been successfully applied to piecewise-linear oscillators. The key idea of that approach is to represent the spatial coherences in the system dynamics with two sets of normal modes with special boundary conditions, referred to as bi-linear modes. In this paper, the bi-linear modal representation is extended to the case of intermittent contacts with friction. Furthermore, a novel reduced order modeling method is developed for the 0th order harmonic used in multi-harmonic balance methods. The forced response of a cracked structure is used to demonstrate the proposed methods.

Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator

In this paper, we make a bifurcation analysis of a mathematical model for an electric circuit formed by the four fundamental electronic elements: one memristor, one capacitor, one inductor and one resistor. The considered model is given by a discontinuous piecewise linear system of ordinary differential equations, defined on three zones in ℝ3, determined by |z| < 1 (called the central zone) and |z| > 1 (the external zones). We show that the z-axis is filled by equilibrium points of the system, and analyze the linear stability of the equilibria in each zone. Due to the existence of this line of equilibria, the phase space ℝ3 is foliated by invariant planes transversal to the z-axis and parallel to each other, in each zone. In this way, each solution is contained in a three-piece invariant set formed by part of a plane contained in the central zone, which is extended by two half planes in the external zones. We also show that the system may present nonlinear oscillations, given by the existence of infinitely many periodic orbits, each one belonging to one such invariant set and passing by two of the three zones or passing by the three zones. These orbits arise due to homoclinic and heteroclinic bifurcations, obtained varying one parameter in the studied model, and may also exist for some fixed sets of parameter values. This intricate phase space may bring some light to the understanding of these memristor properties. The analytical and numerical results obtained extend the analysis presented in [Itoh & Chua, 2009; Messias et al., 2010].

Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution ofm-PWL Variational Equations

An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on themregions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents fromm-piecewise linear variational equations and their associatedm-Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.

Algebraically computable piecewise linear nodal oscillators

A family of piecewise linear oscillators whose oscillation can be completely characterized by algebraic methods is studied. It represents up to the best of authors’s knowledge, the first planar example where all the oscillation properties can be determined for all the values of the bifurcation parameter. In fact, algebraic expressions for coordinates of representative points, period and characteristic multiplier of the corresponding periodic orbit are provided. Thus, the studied family of oscillators deserves to be considered a good benchmark for testing approximate methods of analysis in nonlinear oscillation theory.The piecewise linear oscillators studied are called nodal oscillators, since their relevant linear parts are of node type, and they are not perturbations of the harmonic oscillator. They represent real models in practice, as it is shown for an electronic circuit modeling a piecewise linear version of the classical Van der Pol oscillator.

Novel bifurcation structure generated in piecewise-linear three-[formula omitted] resonant circuit and its Lyapunov analysis

We analyse a piecewise-linear oscillator that consists of a three-LC resonant circuit with a hysteresis element. Three sets of two-dimensional linear equations, including a hysteresis function, represent the governing equations of the circuit, and all the Lyapunov exponents are calculated in a remarkably simple manner based on derived explicit solutions. Various dynamical phenomena such as two-torus, three-torus, and hyperchaos with four positive Lyapunov exponents are observed by Lyapunov analysis. We obtained a detailed bifurcation diagram in which novel bifurcation structure which we call a “two-torus Arnold tongue” is observed where two-torus generating regions exist in a three-torus generating region as if periodic states exist in a two-torus generating region.

Periodic motions and resonances of impact oscillators

Bilinear oscillators – the oscillators whose springs have different stiffnesses in compression and tension – model a wide range of phenomena. A limiting case of bilinear oscillator with infinite stiffness in compression – the impact oscillator – is studied here. We investigate a special set of impact times – the eigenset, which corresponds to the solution of the homogeneous equation, i.e. the oscillator without the driving force. We found that this set and its subsets are stable with respect to variation of initial conditions. Furthermore, amongst all periodic sets of impact times with the period commensurate with the period of driving force, the eigenset is the only one which can support resonances, in particular the multi-‘harmonic’ resonances. Other resonances should produce non-periodic sets of impact times. This funding indicates that the usual simplifying assumption [e.g., S.W. Shaw, P.J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound and Vibration 90 (1983) 129–155] that the times between impacts are commensurate with the period of the driving force does not always hold. We showed that for the first sub-‘harmonic resonance’ – the resonance achieved on a half frequency of the main resonance – the set of impact times is asymptotically close to the eigenset. The envelope of the oscillations in this resonance increases as a square root of time, opposite to the linear increase characteristic of multi-‘harmonic’ resonances.

Legged vehicle control and vibration reduction

This paper presents a design method of controller reducing the vibration of the upper body of a legged vehicle during locomotion. To design locomotion pattern generator, instead of using non-linear dynamics models and solving complex equations, the control strategy is simply based on the use of piecewise linear oscillators and a few parameters that can be easily obtained and tuned. In this research framework, we present tuning algorithms of both the rolling profile and the feedback controllers gains such that the mechanical vibration in the legged vehicle during locomotion is attenuated. As a result, stable and smooth locomotion is obtained. The effectiveness of the proposed method is demonstrated by experimental results.

Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.

Synchronization regions of two pulse-coupled electronic piecewise linear oscillators

Stable synchronous states of different order were analytically, numerically and experimentally characterized in pulse-coupled light-controlled oscillators (LCOs). The Master-Slave (MS) configuration was studied in conditions where different time-scale parameters were tuned under varying coupling strength. Arnold tongues calculated analytically - based on the piecewise two-time-scale model for LCOs - and obtained numerically were consistent with experimental results. The analysis of the stability pattern and tongue shape for (1 : n) synchronization was based on the construction of return maps representing the Slave LCO evolution induced by the action of the Master LCO. The analysis of these maps showed that both tongue shape and stability pattern remained invariant. Considering the wide variation range of LCO parameters, the obtained results could have further applications on ethological models.

A Design Procedure for Wideband Micropower Generators

We developed a design procedure for wideband electromagnetic micropower generators (WMPGs) based on piecewise-linear oscillators. We find that the dominant factors in the performance of this class of WMPGs are the stiffness ratio of the oscillator and the velocity of the moving structure at the point of impact with the stopper. We also find that designing these WMPGs requires additional steps beyond those required in the design of regular MPGs. The additional steps match the output power and bandwidth of the WMPG to the probability density function of environmental vibrations. While these steps add complexity to the design of WMPGs, they are shown to significantly increase harvested energy.

Quasi-periodic solutions of forced isochronous oscillators at resonance

We deal with the existence of quasi-periodic solutions of forced isochronous oscillators with a repulsive singularity, the nonlinearity is a bounded perturbation. Using a variant of Moser's twist theorem of invariant curves, due to Ortega [R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999) 381–413], we show that there are many quasi-periodic solutions and the boundedness of all solutions.

FOLLOWING A SADDLE-NODE OF PERIODIC ORBITS' BIFURCATION CURVE IN CHUA'S CIRCUIT

Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

Dynamics of the archetypal piecewise linear oscillator close to grazing

A linear oscillator undergoing impact with a secondary support is studied numerically. Smooth as well as nonsmooth bifurcations are observed. The bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. The obtained results are verified by comparison with previous experimental studies. Experimental stability analysis is also presented, justifying the normal form maps used to show the response close to grazing for systems with and without prestress of the secondary spring. The high degree of correspondence lends support to the modelling approach, as does the close match between experiments and simulations.

BISTABILITY AND HYSTERESIS IN SYMMETRIC 3D PIECEWISE LINEAR OSCILLATORS WITH THREE ZONES

A degeneration in a two-parameter setting for the focus-center-limit cycle bifurcation of symmetric 3D piecewise linear systems with three zones is studied. The analysis gives a rigorous mathematical explanation for the appearance in piecewise linear oscillators of up to two limit cycles along with a related hysteresis phenomenon.A piecewise linear oscillator, which is a member of the generic Chua's circuit family, is shown to exhibit bistability, that is, coexistence of stable oscillations with a stable steady state. Also, the corresponding hysteresis in the appearance of periodic oscillations is explained. The analytical results included in the paper predict accurately the observed behavior.

A wideband vibration-based energy harvester

We present a new architecture for wideband vibration-based micro-power generators (MPGs). It replaces a linear oscillator with a piecewise-linear oscillator as the energy harvesting element of the MPG. A prototype of an electromagnetic MPG designed accordingly is analyzed analytically, numerically and experimentally. We find that the new architecture increases the bandwidth of the MPG during a frequency up-sweep, while maintaining the same bandwidth in a down-sweep. Closed-form expressions for the response of the new MPG as well as the up-sweep bandwidth are presented and validated experimentally. Simulations show that under random-frequency excitations, the new MPG collects more energy than the traditional MPG.