Piecewise Linear Oscillator Research Articles

Experimental Study of a Symmetrical Piecewise Base-Excited Oscillator

This paper presents an experimental study on a base-excited piecewise linear oscillator with symmetrical flexible constrains of high stiffness ratio (above 20). The details of the adopted design of the oscillator, the experimental setup, and calibration procedure are briefly discussed. The regions of chaotic motion predicted theoretically were confirmed by the experimental results arranged into bifurcation diagrams. Clearance, stiffness ratio, amplitude, and frequency of the external force were used as branching parameters. The discussion of the system dynamics is based on bifurcation diagrams and Lissajous curves. The investigated system tends to be periodic for large clearances and chaotic for small ones. This picture is reversed for the amplitude of the forcing changes, where periodic motion occurred for small values and chaos dominated for larger forcing. The same behavior is observed for increasing frequency ratio where, for values below the natural frequency, the most interesting dynamics occurs. For the investigated parameter values, the stiffness ratio variation produces only periodic motion.

Canonical Forms of Periodic Solution for Piecewise Linear Oscillators. Proposition of Feedback Superposition Method.

A new method for getting periodic solutions for Piecewise Linear Oscillator is proposed. From the steady state responses, periodic solution for the oscillators can be derived by using pseudo feedback approach. It is shown that for some piecewise linear systems, periodic solution can be globaly represented by the form of superposition of steady response of linear system and periodic pseudo feedback responses. By comparing the results through the method with those of numerical simulations, the validity of the proposed method is confirmed.

Global Representation of General Solution and Canonical Forms for piecewise Linear Oscillators. Formulation for Impact Oscillator.

A new global representation of a general solution and Poincare mapping from local exact solutions for impact oscillators is proposed. A general solution can be globally represented by the form of superposition of linear system responses and pseudo-feedback responses. According to this formulation, Poincare mapping can be expressed in a global analytical form depending on impact time parameters. By comparing the results through this method with those obtained by numerical simulations, the validity of the proposed method is confirmed.

Measurement of chaotic vibration in a symmetrically piecewise linear oscillator

The paper describes a measurement of chaotic oscillations occurring in a base excited symmetrically bilinear oscillator. The details of the current experimental rig and possible modifications to improve the measurement accuracy are explained. The regions of chaotic motion predicted theoretically were confirmed by the experimental results arranged into bifurcation diagrams, which were obtained by numerical and experimental means. Clearance and modulus of the external force were used as branching parameters. The discussion of the system dynamics is based on bifurcation diagrams, Lissajous curves and power spectra. The investigated system tended to be periodic for large clearances and chaotic for small ones. This picture is repeated for the modulus variation, where periodic motion occurred for its small values and chaos dominated for larger forcing.

Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos

A method of detecting parameter changes using analytical and experimental nonlinear dynamics and chaos is applied to a piecewise-linear oscillator. Experimental data show the chaotic nature of the system through phase portraits, Poincare´ maps, frequency spectra and bifurcation diagrams. Unstable periodic orbits were extracted from each chaotic time series obtained from the system with six different parameter values. Movement of the unstable periodic orbits in phase space is used to detect parameter changes in the system.

Two-parameter experimental investigation of co-existing oscillation modes in torus breakdown

The onset of to chaos in a quasiperiodic piecewise-linear two-frequency oscillator is studied by experiments and numerical simulation. The fine structure of the multistable behavior of the phaselocking regions of Arnold's tongues near the Hopf bifurcation line is plotted in a two-parameter bifurcation diagram.

DETECTION OF GRAZING ORBITS AND INCIDENT BIFURCATIONS OF A FORCED CONTINUOUS, PIECEWISE-LINEAR OSCILLATOR

A critical phenomenon in a continuous, piecewise-linear oscillator subjected to periodic excitation is considered, in which a periodic orbit happens to graze a switching plane between two linear regions of the oscillator at zero velocity. The analysis shows that if the piecewise linearity of the oscillator changes dramatically on the switching plane, the phenomenon turns the stability trend of the periodic orbit so abruptly that it may be hard to predict an incident bifurcation, although it is of a local type, according to the concepts of smooth dynamical systems. To detect the critical phenomenon induced by the non-smooth nature of the oscillator, a numerical scheme of searching for all periodic grazing orbits is presented, as well as a test function for grazing phenomena in tracing a branch of a periodic orbit. There follows a numerical simulation based on the scheme and the test function, which reveals an abundance of grazing phenomena and incident bifurcations of the oscillator.

Simulation complexities in the dynamics of a continuously piecewise-linear oscillator

The vector field of a continuously piecewise-linear oscillator under periodic excitation is by nature nonsmooth. However, the nonsmoothness in studying the oscillator dynamics has drawn little attention. The paper presents the nonsmoothness effect on the differentiability of the Poincaré mapping, and then gives the dynamics comparison between the oscillator and a corresponding smoothed model. The numerical evidence, with the assistance from geometric concepts of dynamical systems, suggests that great care should be taken during locating a periodic orbit and tracing a branch of the periodic orbit when the orbit approaches a bifurcation of saddle-node type and relevant degenerated types or touches a switching plane at very low velocity. In these critical cases, the oscillator may behave quite different from an oscillator having smooth vector field.

The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators

A simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro-impacting systems. The presence of impacts makes the system discontinuous and non-linear and to study it effectively the theory of dynamical systems is used. By a combination of analytical and numerical techniques, many phenomena are identified which are characteristic of non-linear dynamical systems as the system parameters are varied. In particular, these include chaotic regimes and multiply coexisting stable states. The domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states. In addition, phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems; chatter, when infinitely many impacts occur in a finite time, and grazing, when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters.

A note on stochastic phase lockings in a relaxation oscillator forced by a sine input with additive noise

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. Using the probability density functions of the first-passage time, the operator which governs a transition of an input phase density after one cycle of the oscillator is defined. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

Vibration of Harmonically Excited Oscillators With Asymmetric Constraints

investigation is carried out for a class of piecewise linear oscillators with asymmetric characteristics. The damping and restoring forces are general trilinear functions of the system velocity and displacement, respectively, while the excitation is harmonic in time. First, an analysis is presented which determines harmonic and subharmonic steady-state response. Then, a special formulation is employed in examining the stability of located periodic motions. Finally, numerical results are presented for several representative sets of the system parameters. Effects of asymmetries in the response due to unequal gaps as well as unequal stiffness and damping coefficients are analyzed in detail. Asymmetric response of a system with symmetric technical characteristics is also investigated. The behavior of the systems examined resembles response of similar nonlinear systems with continuous characteristics, like the response of the Duffing and van der Pol oscillators. Complicated nonperiodic response is also encountered and analyzed.

Local and global stability of a piecewise linear oscillator

Machines and mechanisms with moving parts, subjected to periodic excitation, often show unexpected dynamic behaviour, and impacts due to their connection clearances may occur. The most simple mathematical model is a one degree-of-freedom nonlinear oscillator governed by a piecewise linear symmetric function to describe the restoring force. The system’s response, which can be quite rich and complicated, is described in detail. Modern methods for a combined analytical and numerical analysis are used to study local and global bifurcation conditions, coexisting solutions and their associated domains of attraction.

Response and stability analysis of piecewise-linear oscillators under multi-forcing frequencies

Steady-state solutions of a piecewise-linear oscillator under multi-forcing frequencies are obtained using the fixed point algorithm (FPA). Stability analysis is also performed using the same technique. For the periodic solutions of a piecewise-linear oscillator with single forcing frequency, the harmonic balance method (HBM) is also used along with the FPA. Although both FPA and HBM generate accurate solutions, it is observed that the HBM failed to converge to solutions in the superharmonic range of the forcing frequency. The fixed point algorithm was also applied to the oscillator under multifrequency excitation. The algorithm proved to be very effective in obtaining torus solutions and in locating corresponding bifurcation thresholds. A piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior. A second order Poincare mapping technique reveals the hidden fractal-like nature of the resulting chaotic response. A parametric study is performed for the response of the ALP.

Quantitative and qualitative analysis of a piecewise-linear symmetrical oscillator

A single degree of freedom oscillator is considered. The nonlinearity is in the restoring force which is piecewise-linear and symmetrical. A stability criterion for harmonic solutions is discussed. Through digital simulations the existence of harmonic, subharmonic and irregular motions is established

Global Representation of General Solution and Canonical forms for Piecewise Linear Oscillators. The Case of a Discontinuous Nonlinearity.

In this paper, we describe a construction method of global representaion of a general solution and Poincare mapping (or phase return mapping) from local exact solutions (or half return maps) for piecewise linear oscillators with discontinuous nonlinearity by using a pseudo-feedback approach. It is shown that for some piecewise linear systems, a general solution can be global represented by the form of superposition of linear system response and pseudo-feedback responses. Based on these global representations, we propose canonical forms of general solutions, steady states and poincare mapping. According to this formulation, phase return mapping can be expressed in a global analytical form depending on return time parameters. Finally, numerical simulations are performed, and the validity of the presented approach can be confirmed.

Energy levels of a piecewise linear oscillator with a barrier

We study the spectrum of a piecewise linear one-dimensional oscillator with a δ-function barrier. The spectrum outside the quasiclassical region is highly influenced by the nature of the ratio ω 1 ω 2 , ω i being the frequencies in the linear domains of t oscillator.

Subharmonic solution of a piecewise linear oscillator with tow degrees of freedom

In the present paper subharmonic resonance solution of a piecewise linear oscillator with two degrees of freedom is studied. It is shown that in this system there exist a series of subharmonic resonance solutions, among them there are 1/2, 1/3, 1/4, 1/5, 1/6 ... ... subharmonic resonance solutions. The calculated results by the analogy computer and the field experiments in the factory partly verify this theory. Under certain circumstances, the generation of chaotic states of the oscillation is observed in analogy computer solutions.

A periodically forced piecewise linear oscillator

A single-degree of freedom non-linear oscillator is considered. The non-linearity is in the restoring force and is piecewise linear with a single change in slope. Such oscillators provide models for mechanical systems in which components make intermittent contact. A limiting case in which one slope approaches infinity, an impact oscillator, is also considered. Harmonic, subharmonic, and chaotic motions are found to exist and the bifurcations leading to them are analyzed.