Bifurcation and stability analysis for a non-smooth friction oscillator

Summary Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self-and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Erwin Stein on the occasion of his 65th birthday.     

Bifurcation sequences of a Coulomb friction oscillator

In some parameter ranges, the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity is reducible to the dynamics of a one-dimensional map. In numerical simulations, period-doubling bifurcations are observed for the oscillator. In this bifurcation procedure, the map arising from the Coulomb model may not have standard form. The bifurcation sequence of the Coulomb model is compared to that of the standard one-dimensional maps to see if it exhibits universal behavior. All observed components of the bifurcation sequence fit the universal sequence, although some universal events are not witnessed.     

Bifurcation of piecewise-linear nonlinear vibration system of vehicle suspension

A kinetic model of the piecewise-linear nonlinear suspension system that consists of a dominant spring and an assistant spring is established. Bifurcation of the resonance solution to a suspension system with two degrees of freedom is investigated with the singularity theory. Transition sets of the system and 40 groups of bifurcation diagrams are obtained. The local bifurcation is found, and shows the overall character- istics of bifurcation. Based on the. relationship between parameters and the topological bifurcation solutions, motion characteristics with different parameters are obtained. The results provides a theoretical basis for the optimal control of vehicle suspension system parameters.     

Bifurcation and chaos in discrete-time BVP oscillator

In this paper, we investigate the discrete-time Bohöffer-Van der Pol (BVP) oscillator obtained by Euler method. We provide the sufficient conditions of existence, asymptotic stability of the fixed points, then give theoretical analysis for local bifurcations of the fixed points, and derive the conditions under which the local bifurcations such as pitchfork, saddle-node, flip and Hopf occur at the fixed points. Furthermore, we prove that the fixed point eventually evolves into a snap-back repeller which generates chaotic behavior in the sense of Marotto's chaos when certain conditions are satisfied. Finally, several numerical simulations are provided to demonstrate the theoretical results of the previous and to show the new complex dynamical behaviors of the system.     

Non-smooth stability analysis of the parametrically excited impact oscillator

The aim of this paper is to give a Lyapunov stability analysis of a parametrically excited impact oscillator, i.e. a vertically driven pendulum which can collide with a support. The impact oscillator with parametric excitation is described by Hill's equation with a unilateral constraint. The unilaterally constrained Hill's equation is an archetype of a parametrically excited non-smooth dynamical system with state jumps. The exact stability criteria of the unilaterally constrained Hill's equation are rigorously derived using Lyapunov techniques and are expressed in the properties of the fundamental solutions of the unconstrained Hill's equation. Furthermore, an asymptotic approximation method for the critical restitution coefficient is presented based on Hill's infinite determinant and this approximation can be made arbitrarily accurate. A comparison of numerical and theoretical results is presented for the unilaterally constrained Mathieu equation.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

Filtration problems with a piecewise-linear resistance law

The article discusses elementary solutions of problems of nonlinear filtration with a piece-wise-linear resistance law, and analyzes their behavior with a relative increase in the resistance in the region of small velocities, and a transition to the law of filtration with a limiting gradient. The results obtained are applied to a determination of the dimensions of the stagnant zones in stratified strata. The law of filtration with a limiting gradient (0.1) $$\begin{gathered} w = - \frac{k}{\mu }\left( {grad p - G\frac{{grad p}}{{|grad p|}}} \right),|grad p| > G \hfill \\ w = 0,|grad p|< G \hfill \\ \end{gathered}$$ describes motion in some intermediate range of velocities w, but its satisfaction in the region of the smallest velocities, as a rule, remains unverified. It is natural to pose the problem of the degree to which a divergence between the true filtration law and its approximation (0.1) affects the accuracy of calculation of the flow fields, and the significance of a determination of the dimensions of the stagnant zones under such conditions. To answer this problem to some measure, there are considered below several simple exact (elementary) solutions obtained for a more general nonlinear filtration law (0.2) $$\begin{gathered} grad p = - (\mu /k) (w + \lambda )w/w,\lambda = kG/\mu ,w \geqslant w_0 \hfill \\ grad p = - \mu w/k\varepsilon ), w \leqslant w_0 ,\varepsilon = w_0 /w_0 + \lambda \hfill \\ \end{gathered}$$ going over into (0.1) with w₀→ 0. The solutions obtained are applied also to an evaluation of the dimensions of stagnant zones, forming in stratified strata when the effects of the limiting gradient in one of the intercalations are considerable.     

Bifurcation scenarios of the noisy duffing-van der pol oscillator

This paper presents a numerical study of the bifurcation behavior of the noisy Duffing-van der Pol oscillator% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttLeary% qr1ngBPrgaiuGacuWF4baEgaWaaiaaiccacqWF9aqpcaaIGaGaaiik% aerbtLhBMfwzUbacgiGaa4xSdiaaiccacqGHRaWkcaaIGaGaeq4Wdm% 3ccaaIXaGcceqGxbGbaiaaliaaigdakiGacMcacqWF4baEcaaIGaGa% ci4kaiaaiccacqaHYoGycuWF4baEgaGaaiaaiccacqGHsislcaaIGa% Gae8hEaG3aaWbaaSqabeaacaaIZaaaaOGaaGiiaiabgkHiTiaaicca% cqWF4baEdaahaaWcbeqaaiaaikdaaaGccuWF4baEgaGaaiaaiccaci% GGRaGaaGiiaiabeo8aZTGaaGOmaOGabe4vayaacaGaaeOmaiaabYca% aaa!5F62!\[\ddot x = (\alpha + \sigma 1{\rm{\dot W}}1)x + \beta \dot x - x^3 - x^2 \dot x + \sigma 2{\rm{\dot W2,}}\]where , are bifurcation parameters, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceqGxbGbaiaali% aaigdakiqabEfagaGaaSGaciOmaaaa!35B4!\[{\rm{\dot W}}1{\rm{\dot W}}2\] are independent white noise processes, and ₁, ₂ are intensity parameters. A stochastic bifurcation here means (a) the qualitative change of stationary measures or (b) the change of stability of invariant measures and the occurrence of new invariant measures for the random dynamical system generated by (1). The first type of bifurcation can be observed when studying the solution of the Fokker-Planck equation, this stationary measure is a quantity corresponding to the one-point motion. More generally, if one is interested in the simultaneous motion of n points (n1) forward and backward in time, then the second type of bifurcation arises naturally, capturing all the stochastic dynamics of (1). Based on the numerical results, we propose definitions of the stochastic pitchfork and Hopf bifurcations.     

Bifurcation analysis of a double pendulum with internal resonance

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Chaotifying 2-D piecewise-linear maps via a piecewise-linear controller function

A simple method for chaotifying piecewise-linear maps of the plane using a piecewise-linear controller function is given. A domain of chaos in the resulting controlled map was determined exactly and rigorously.     

Bifurcation trees of period-3 motions to chaos in a time-delayed Duffing oscillator

Nonlinear Dynamics - In this paper, bifurcation trees of period-3 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator are investigated by a semi-analytical method....     

Periodic sticking motion in a two-degree-of-freedom impact oscillator

Periodic sticking motions can occur in vibro-impact systems for certain parameter ranges. When the coefficient of restitution is low (or zero), the range of periodic sticking motions can become large. In this work the dynamics of periodic sticking orbits with both zero and non-zero coefficient of restitution are considered. The dynamics of the periodic orbit is simulated as the forcing frequency of the system is varied. In particular, the loci of Poincaré fixed points in the sticking plane are computed as the forcing frequency of the system is varied. For zero coefficient of restitution, the size of the sticking region for a particular choice of parameters appears to be maximized. We consider this idea by computing the sticking region for zero and non-zero coefficient of restitution values. It has been shown that periodic sticking orbits can bifurcate via the rising/multi-sliding bifurcation. In the final part of this paper, we describe three types of post-bifurcation behavior which occur for the zero coefficient of restitution case. This includes two types of rising bifurcation and a border orbit crossing event.     

Stability analysis of the periodic solution of a piecewise-linear non-linear dynamic system

The stability of the periodic solution under harmonic excitation of a non-linear dynamic system with “linear hysteretic damping” is examined proceeding from first principles. The method can be extended to the case of multi-degree of freedom systems unlike regular perturbation procedure.     

Bifurcation analysis of a nonlinear viscoelastic panel

The dynamic behavior of a nonlinear viscoelastic panel subjected to a simple harmonic excitation is studied. Using the Galerkin principle, the double mode model is presented in this paper. The bifurcation behavior of the panel is examined in detail in the case of internal response. The method of averaging is used to derive a set of autonomous equations. The averaged differential equations are then examined to determine their bifurcation behavior. Finally, the results of theoretical analysis are numericaly verified.     

Bifurcation analysis for sand samples with a non-linear constitutive equation

Summary A thoroughly non-linear constitutive equation connecting stress-rates and strain-rates is shown to yield realistic predictions for the conditions at which bifurcation may appear in the course of axisymmetric and plane homogeneous deformations of sand samples. The features of the non-homogeneous deformation after the bifurcation are also realistically predicted, a fact which is of importance in understanding and describing rupture phenomena (formation of shear bands) in geomechanics.

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Übersicht In dieser Arbeit wird gezeigt, daß eine nichtlineare Beziehung zwischen der Spannungsrate und der Deformationsgeschwindigkeit realistische Voraussagen für Verzweigungen liefert, die sich im Verlauf von achsensymmetrischer und ebener Verformung von Sandproben einstellen können. Die Geometrie der inhomogenen Verformung, die sich nach der Verzweigung einstellt, kann ebenfalls realistisch vorausgesagt werden. Diese Tatsache ist von Bedeutung für das Verständnis und die Beschreibung von Brucherscheinungen (Bildung von Scherfugen u. ä.) in der Geomechanik.

     

Bifurcation analysis for a modified Jeffcott rotor with bearing clearances

A HB (Harmonic Balance)/AFT (Alternating Frequency/Time) technique is developed to obtain synchronous and subsynchronous whirling motions of a horizontal Jeffcott rotor with bearing clearances. The method utilizes an explicit Jacobian form for the iterative process which guarantees convergence at all parameter values. The method is shown to constitute a robust and accurate numerical scheme for the analysis of two dimensional nonlinear rotor problems. The stability analysis of the steady-state motions is obtained using perturbed equations about the periodic motions. The Floquet multipliers of the associated Monodromy matrix are determined using a new discrete HB/AFT method. Flip bifurcation boundaries were obtained which facilitated detection of possible rotor chaotic (irregular) motion as parameters of the system are changed. Quasi-periodic motion is also shown to occur as a result of a secondary Hopf bifurcation due to increase of the destabilizing cross-coupling stiffness coefficients in the rotor model.     

Versatile mass excited impact oscillator

Nonlinear Dynamics - This paper presents the design and the initial experimental results of a novel impact oscillator rig developed by the Centre of Applied Dynamic Research at the University of...     

Bifurcation of water column oscillator behavior analyzed by two-dimensional graphical method

A piecewise linear model which consists of the set of two linear ordinary differential equations with four parameters was derived to investigate the behavior of the water column oscillator simulating a safety system of an advanced reactor. The model shows various bifurcation. The behavior of the model can be discussed using the two-dimensional mapping function. When the system is governed by the two (or more)-dimensional mapping function, it is difficult to draw the map because the map has four dimensions, from two dimensions to two dimensions. In this study, therefore, a “two-dimensional graphical method” is proposed and the bifurcation of the two-dimensional system is visualized.